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The voluntary contribution mechanism with complementarity Fenig, Guidon 2016

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The Voluntary Contribution Mechanismwith ComplementaritybyGuidon FenigB.A., Instituto Teconolo´gico Auto´nomo de Me´xico, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Guidon Fenig 2016AbstractThis dissertation investigates the effect of complementarity in the private provision of public goods.The inclusion of complementarity in contributions transforms the classic voluntary contributionmechanism into a coordination game, in which we are able to study how different behavioral typesinteract.The first chapter generalizes the linear voluntary contribution mechanism case by allowingagents’ contributions to be complements in production. When complementarity is sufficiently high,an additional full-contribution equilibrium emerges. We experimentally investigate subjects’ be-havior using a between-subject design that varies complementarity. When two equilibria exist, sub-jects tend to coordinate on contributions close to the efficient equilibrium. When complementarityis sizable but only a zero-contribution selfish-equilibrium exists, subjects persistently contributeabove it. Observed choices and other nonchoice data indicate heterogeneity among subjects andtwo distinct types. Homo pecuniarius maximizes profits by best responding to beliefs, while Homobehavioralis identifies this strategy but chooses to deviate from it, sacrificing pecuniary rewards tosupport altruism or competitiveness.The second chapter studies the effect of introducing thresholds on equilibrium selection in thevoluntary contribution mechanism with complementarity (VCMC). The introduction of thresholdsexpands the basin of attraction of the socially inefficient equilibrium by raising the minimum groupoutput necessary for a positive return from cooperation. If contributions are not sufficient to gener-ate the minimum output, the social return is zero. The data suggest that the introduction of thresh-olds does not alter the equilibrium selection or the pattern of contribution dynamics. Individualsare still able to coordinate on the socially preferable outcome.The third chapter examines experimentally the effect of displaying feedback about the incomeof other group members on the evolution of contributions in the VCMC. The results show thatthe degree of coordination responds to the way information about outcomes is made available tosubjects. Emphasis on income differences can be detrimental from a social perspective and mayresult in the unraveling of cooperation.iiPrefaceAll chapters of this thesis are coauthored with Giovanni Gallipoli, University of British Columbia,and Yoram Halevy, University of British Columbia. All coauthors contributed to all aspects ofthe project equally. This work was approved by the Behavioural Research Ethics Board of theUniversity of British Columbia under the project title “Strategic Complementarity in ExperimentalPublic Good Games” (certificate number: ID H14-02460).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Complementarity in the Private Provision of Public Goods by Homo Pecuniarius andHomo Behavioralis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Voluntary Contribution Mechanism with Complementarity . . . . . . . . . . 31.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 How Do Players Choose Their Contributions? . . . . . . . . . . . . . . . . . . . 131.6 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Coordination and Contribution Thresholds in the Presence of Complementarity . . 332.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 VCMC With a Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37ivTable of Contents2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Understanding Deviations From the Monetary-Profit-Maximizing Strategy . . . . 442.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 The Race to the Bottom in the Voluntary Contribution Mechanism With Complemen-tarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Environment and Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 563.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 How Do Subjects Make Choices? . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73AppendicesA Appendix for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.1 Derivation of the BR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2 Best-Response Range and Contributions . . . . . . . . . . . . . . . . . . . . . . . 80A.3 Deviations from the Profit-Maximizing Strategies, by Type . . . . . . . . . . . . . 84A.4 Computer Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.5 Control Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.6 Contributions by Gender and Field of Study . . . . . . . . . . . . . . . . . . . . . 89A.7 Myopic Best Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A.8 Persistence of Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A.9 Intensity and Processing Speed on Calculator Usage . . . . . . . . . . . . . . . . 93A.10 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100B.1 Derivation of the Best-Response Function . . . . . . . . . . . . . . . . . . . . . 100B.2 Additional equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.3 Best-Response Range and Contributions . . . . . . . . . . . . . . . . . . . . . . . 108B.4 Computer Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.5 Control Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114vTable of ContentsC Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118C.1 History-Dependence of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 118C.2 Best-Response Range and Contributions . . . . . . . . . . . . . . . . . . . . . . . 119C.3 Main Computer Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126C.4 Computer Interface (Feedback) . . . . . . . . . . . . . . . . . . . . . . . . . . . 126viList of Tables1.1 Experimental Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Average Conjecture About Others’ Contributions . . . . . . . . . . . . . . . . . . 101.3 Response of Subjects’ Conjectures to Others’ Contributions . . . . . . . . . . . . . 161.4 Mechanical Use of the Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 Warm-Glow Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 Competitive-Motive Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7 Response Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Average Hypothetical Contribution and Conjecture About Others’ Contributions . . 412.2 Response of Subjects’ Conjectures to Others’ Contributions . . . . . . . . . . . . . 462.3 Mechanical Use of the Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4 Average Conjecture About Others’ Contributions . . . . . . . . . . . . . . . . . . 522.5 Average Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1 Experimental Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Average Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Average Proportional Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4 Competitiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Mechanical Use of the Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.6 Average Response Time and Feedback Time . . . . . . . . . . . . . . . . . . . . 703.7 Payoff-Relevant Use of the Calculator . . . . . . . . . . . . . . . . . . . . . . . . 71A.1 Contributions by Gender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.2 Contributions by Field of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A.3 Mechanical Use of the Calculator (Myopic BR) . . . . . . . . . . . . . . . . . . . 91A.4 Persistence of Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92viiList of TablesC.1 Response of Subjects’ Conjectures to Others’ Contributions . . . . . . . . . . . . . 119viiiList of Figures1.1 Best-response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Average contribution over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Cumulative distribution functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Dispersion loss index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Use of the calculator over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Calculated deviation from BR versus actual deviation from BR . . . . . . . . . . . 231.7 Cumulative distribution of the individual minimum loss . . . . . . . . . . . . . . . 251.8 Cost of deviating from the money-profit-maximizing strategy . . . . . . . . . . . . 271.9 Response-time frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Best-response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Average contribution over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Public good output below the threshold . . . . . . . . . . . . . . . . . . . . . . . 412.4 Cumulative distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5 Dispersion loss index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 Histogram of group account output . . . . . . . . . . . . . . . . . . . . . . . . . . 452.7 Monetary cost of zero contribution and full contribution . . . . . . . . . . . . . . . 452.8 Average contribution over rounds, by types . . . . . . . . . . . . . . . . . . . . . 482.9 Calculated deviation from BR versus actual deviation from BR . . . . . . . . . . . 493.1 Best-response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Average contribution over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Cumulative distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 Dispersion loss index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5 Average contribution over time by type . . . . . . . . . . . . . . . . . . . . . . . . 65ixList of FiguresA.1 Session 1 (LVCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.2 Session 2 (LVCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.3 Session 3 (ρ = 0.54) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.4 Session 4 (ρ = 0.54) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.5 Session 5 (ρ = 0.65) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.6 Session 6 (ρ = 0.65) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.7 Session (ρ = 0.70) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.8 Session 8 (ρ = 0.58) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.9 Contributions versus best response . . . . . . . . . . . . . . . . . . . . . . . . . . 84A.10 Main computer interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.11 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.12 Control question 1/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.13 Control question 2/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.14 Control question 3/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.15 Control question 4/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.16 Control question 5/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.17 Control question 6/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.18 Control question 7/7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.19 New conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93B.1 Best-response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104B.2 Session 1 (T = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.3 Session 2 (T = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.4 Session 3 (T = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.5 Session 4 (T = 30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.6 Session 5 (T = 30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110B.7 Session 6 (T = 30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110B.8 Session 7 (T = 60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.9 Session 8 (T = 60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.10 Session 9 (T = 60) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112B.11 Main computer interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.12 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.13 Control question 1/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114xList of FiguresB.14 Control question 2/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114B.15 Control question 3/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.16 Control question 4/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.17 Control question 5/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116B.18 Control question 6/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116B.19 Control question 7/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117B.20 Control question 8/8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C.1 Average conjecture about others’ contributions . . . . . . . . . . . . . . . . . . . 118C.2 Session 1 (Show Contr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119C.3 Session 2 (Show Contr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120C.4 Session 3 (Show Contr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120C.5 Session 4 (Show Payoffs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.6 Session 5 (Show Payoffs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.7 Session 6 (Show Ranking) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122C.8 Session 7 (Show Ranking) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122C.9 Session 8 (No Link) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123C.10 Session 9 (No Link) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123C.11 Session 10 (Show Contr Linear) . . . . . . . . . . . . . . . . . . . . . . . . . . . 124C.12 Session 11 (Show Contr Linear) . . . . . . . . . . . . . . . . . . . . . . . . . . . 124C.13 Session 12 (Show Payoffs Linear) . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.14 Session 13 (Show Payoffs Linear) . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.15 Main computer interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126C.16 Feedback Show Contr treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 126C.17 Feedback Show Payoffs treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.18 Feedback Show Ranking treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.19 Feedback No Link treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128xiAcknowledgementsI would like to express my sincere gratitude to several people who have made this project possible.I am particularly indebted to my supervisor and coauthor, Yoram Halevy. I would like to thankhim for the continuous support, of my PhD study and related research. Yoram is an excellent super-visor, he is the kind of advisor most doctoral candidates wish they had. He constantly pushed meto reach my research potential, helped me improve my work, mentored me, and helped me preparefor the job market. I thank him for the knowledge and wisdom he shared with me.I would like to express my most sincere gratitude to Giovanni Gallipoli for his guidance andmentorship. His co-authorship in this project was invaluable. He constantly encouraged me andshowed me how rewarding hard work is. He is passionate about research and his enthusiasm iscontagious. His support during this past two years was instrumental to my success.I also would like to thank Li, Hao, who was generous with his time in his role of commit-tee member. He made insightful comments, and raised relevant questions which motivated me towiden my research to different perspectives and improve it in many aspects.I would like to acknowledge the financial support and funding of the experiments from SSHRC.Ruzhi Zhu provided valuable research assistance in this project.I benefited from comments from Evan Calford, Anujit Chakraborty, John Duffy, Mukesh Eswaran,Vitor Farinha Luz, Patrick Francois, Ara Norenzayan, Jesse Perla, Franck Portier, Thorsten Rogall,Sergei Severinov, Michal Szkup, Yaniv Yedid-Levi, Lanny Zrill, as well as numerous seminar par-ticipants.xiiAcknowledgementsI also would like to thank Luba Petersen for the opportunity of running my first experiment.I would like to thank my family, especially my parents, Bernardo and Esther Fenig, they who,throughout the years, have taught me the importance of education and have always supported mein my decisions. I also thank Hilel and Liora Fenig, Jack and Karen Milstein, Carlos Herna´ndez,Dave and Teresa Schwartz and the VanBand for all their support.Finally, I am immensely grateful to my wife, Bella Fenig. I am sure that this journey would nothave been possible without her constant love and support.xiiiDedicationDedicado a Bella y a mis papa´sxivChapter 1Complementarity in the PrivateProvision of Public Goods by HomoPecuniarius and Homo Behavioralis1.1 IntroductionPublic good games highlight the interaction between individual motives and group outcomes. Intheir simplest form they emphasize the tension between private incentives and social efficiency,allowing researchers to investigate how individual actions shape social consequences.The linear voluntary contribution mechanism (LVCM) has been the most common experimentaldesign employed in studies of public good games. It assumes a production technology of the publicgood which is linear and additively separable in agents’ contributions. Under this key assumptionthe dominant strategy for agents with self-regarding preferences is to contribute nothing at all (i.e.,to free ride) rather than make a positive contribution that results in a private cost and a socialbenefit.1 The robust experimental finding is that contributions are significantly higher than zero inearly rounds but diminish over time.2 Positive contributions have been interpreted (among otherexplanations) as reflecting confusion, altruism, or willingness to cooperate if others do.3Identifying why subjects may want to coordinate in voluntary contribution contexts is essential1Assuming that the marginal per capita return (MPCR) is lower than one.2Henrich et al. (2004) show that there are some differences between the levels of contributions of university studentsand non-students. They show evidence of LVCM experiments with non-students from different ethnic groups. Studentsusually contribute between 40 and 60 percent of their endowment in one-shot games. The Machiguenga (Peru), forexample, contribute on average 22 percent of their endowment, whereas the Ache (Ecuador) contribute 65 percent oftheir endowment.3The experimental literature is much too vast and thoughtful to be covered fairly here. An interested reader is referredto Ledyard (1995) for an older but very helpful survey, and a more recent survey by Vesterlund (forthcoming). Typically,changes in the environment have been shown to increase cooperation, for example by allowing communications betweenparticipants, increasing the group size, setting a higher MPCR on total contributions, and introducing the ability toadminister punishment.11.1. Introductionto understanding empirical observations. In this study we generalize the linear mechanism usedin most public good experiments by letting agents’ contributions be complements in production.This provision technology captures two essential features. First, an increase in one’s contributionraises the marginal return on others’ contributions, and second, the provision is more efficient whenagents’ contributions are relatively homogeneous.Complementarity is fundamental when the provision is performed through effort. Throughoutevolution, Homo sapiens has learned to coordinate efforts in order to hunt and guard. A familymay be viewed as an environment in which public goods are provided through effort and in whichcomplementarity is instrumental. Similarly, many modern charities that provide for public goodsrely on efforts by stakeholders (mainly board members) in order to raise funds and produce theirpublic good of choice. Crucially, in several joint endeavors such as school funding activities, neigh-borhood improvement initiatives and even scientific research projects, the return on a participant’seffort depends on the level of effort that all other participants choose to exert.For low levels of complementarity the unique Nash equilibrium (assuming agents are selfish)remains the zero-contribution equilibrium, although in the nonlinear case, it is not in dominantstrategies. When complementarity is sufficiently high, a new, second full-contribution equilibriumemerges, transforming the selection of equilibrium into a coordination problem.Our experimental design varies the degree of complementarity, encompassing the special lin-ear case. For the linear (no-complementarity) benchmark, we replicate the usual result of positivebut diminishing contributions. When we introduce complementarity, subjects visibly respond to it.With strong complementarity, subjects are able to coordinate on a high-contribution level. Whencomplementarity is sizable but insufficient to support a new selfish-equilibrium, subjects persis-tently contribute above the unique zero-contribution selfish equilibrium and we observe little or noconvergence towards this equilibrium.To understand what motivates subjects to make these choices, we investigate the decision-making processes underlying their choices. This analysis relies on a wealth of unique nonchoicedata, including accurate information about calculations made by each subject before submitting achoice and how long it took to submit a choice. We document a variety of facts about the waysubjects form conjectures about other players’ contributions, whether subjects are able to identifyprofit-maximizing responses to their conjectures, and how these calculations relate to their choices.The examination of choice and nonchoice data allows us to reduce the rich heterogeneity in ob-served contributions to two modus operandi, which we associate to two different types of agents,denoted as Homo pecuniarius and Homo behavioralis. Homo pecuniarius maximizes money-21.2. The Voluntary Contribution Mechanism with Complementarityprofits by best responding to his or her beliefs, which are shaped by recent history. Homo be-havioralis, on the other hand, is able to identify the profit-maximizing choice but chooses to sys-tematically deviate from it. We find no strong evidence of confusion: Homo behavioralis subjectsappear willing to sacrifice some pecuniary rewards to pursue other goals. When complementar-ity is low, some agents may have altruistic motives and they contribute above their monetary bestresponse. When complementarity is high, altruistic behavior is indistinguishable from profit max-imization, but a new motive surfaces: by lowering their contribution below the pecuniary bestresponse, some subjects are able to make relatively higher profits than other participants. Thus,this behavior is rational when subjects are competitive. In Chapter 3 we are able to manipulate thedegree of competitiveness by modifying the feedback information we show subjects.4When quantifying the magnitude of these behavioral motives we show that they are relativelymodest but lead to significant and systematic deviations from the pecuniary best response.These two types of agents coexist and are able to best respond to each other in equilibrium.Over time their interaction shapes aggregate dynamics and provides a way to interpret the patternsobserved under different degrees of complementarity.This chapter is organized as follows. Section 1.2 presents an overview of the theoretical modeland selfish-equilibrium predictions. The experimental design and laboratory procedures are de-scribed in Section 1.3. In Section 1.4 we report results from aggregate data and show that contri-bution behavior converges towards equilibrium values, with one conspicuous exception which weexamine in detail. Section 1.5 explores individual-level behavior. The combined use of choice andnonchoice data is instrumental in explaining deviations from the profit-maximizing strategies. Wethen classify subjects into two types, Homo pecuniarius and Homo behavioralis, and we estimatethe magnitude of altruistic and competitive motives. Section 1.6 provides a summary of relatedresearch, and Section 1.7 concludes the chapter.1.2 The Voluntary Contribution Mechanism with ComplementarityConsider a set of n individuals, indexed by i ∈ {1, ...,n}, each endowed with ω > 0, who mustdecide whether—and how much—to invest in a public project that maps private contributions intoan output that is equally shared among all group members. Let gi denote individual i’s contributionto the public good. The remainder of the endowment not allocated to the public good (ω−gi) is4In the low-complementarity treatment, competition is indistinguishable from profit-maximizing behavior.31.2. The Voluntary Contribution Mechanism with Complementarityconsumed privately by player i. Individual investments in the public good are aggregated through aconstant elasticity of substitution production function that exhibits constant returns to scale. Playeri’s preferences are additively separable between the private and public goods:pii = ω−gi +β(n∑i=1gρi)1/ρ, (1.1)where ρ ≤ 1 denotes the degree of complementarity and β > 0 is a constant. The voluntary con-tribution mechanism with complementarity (VCMC) encompasses, as a special case when ρ = 1,the standard LVCM. The individual return from an investment in the public good depends on thecontributions of all n players and on the degree of complementarity between their investments.1.2.1 Best-response functionIn the well-studied special case of LVCM (ρ = 1), the unique dominant strategy is to contribute zerowhenever β is below one or to allocate the entire endowment to the public good when β is greaterthan one. In the general VCMC environment, the best response (BR) of agent i, denoted as g∗i (g−i),is a linear function of the generalized ρ-mean of his or her conjecture about the contributionsof other group members, denoted by the vector g−i ∈ ℜn−1+ . The generalized ρ-mean of g−i isMρ(g−i)≡(∑n−1i=1 gρ−in−1)1/ρ.5 To see this, consider the first order condition with respect to gi:6∂pii∂gi= β((g∗i )ρ +∑gρ−i) 1−ρρ((g∗i )ρ−1)−1 = 0. (1.2)Rearranging terms, we obtain g∗i (g−i):g∗i (g−i) =kMρ(g−i) if kMρ(g−i)≤ ωω otherwise, (1.3)where k ≡(n−1βρρ−1−1) 1ρis a constant that depends on the model’s parameters. If k > 0, the contri-butions are complementary; moreover, as the degree of complementarity diminishes (ρ increases),5The arithmetic mean is a special case of the generalized mean when ρ = 1. The arithmetic and the generalizedmeans are identical when all contributions are equal, that is when g−i = g1n−1.6Details on the derivation of the BR can be found in Appendix A.1.41.3. Experimental Designk decreases as well. In the limit, when ρ approaches one, k goes to zero and the BR of player i is toinvest zero in the public good regardless of other players’ actions. Because agent i’s BR depends onthe generalized mean of g−i, it depends also on the dispersion of other players’ contributions: for agiven arithmetic mean, player i’s optimal contribution decreases as the dispersion of other players’contributions increases. Put simply, there is an additional benefit from coordination. Figure 1.1summarizes the BR g∗i (g−i) for different values of the complementarity parameter ρ (each used inthe experiments that follow). The generalized ρ-mean of other group members’ contributions ismeasured on the horizontal axis, and player i’s contribution is shown on the vertical axis. The solidlines represent the BR of player i.Imposing the symmetry condition gi + G−i = ngi in Equation (1.2)7 and solving for gi, wecharacterize the symmetric equilibria:geqi =0 if k < 1{0,ω} if k > 1. (1.4)Thus, for given β and n and with sufficiently high complementarity, there exist two equilibria.8When k = 1, any symmetric strategy profile is a Nash equilibrium.9,101.3 Experimental DesignThe baseline parameters are chosen so that the linear treatment (ρ = 1) is easily comparable tosimilarly parameterized LVCM experiments.11 Specifically, we assign the following values: (a)number of players in a group, n = 4; (b) initial token endowment, ω = 20; and (c) β = 0.4. Thelatter is a commonly assumed value of the MPCR in the linear case. In the nonlinear case, however,the MPCR also depends on the curvature parameter ρ and on contributions of other players.Our treatments consist of variations in the degree of complementarity, ρ . Table 1.1 presents anoverview of the experimental design, highlighting key aspects for each value of ρ . The equilibrium7Where G−i = ∑ j 6=i g j.8Alternatively, k T 1 if and only if ρ S ln(n)ln(n/β) .9It is straightforward to verify that only symmetric equilibria exist. Suppose that there exists a nonsymmetric equilib-rium g∗ and denote by g∗min = min{g∗}< max{g∗}= g∗max . For the case of k ≤ 1, it follows that kMρ(g∗−max)< g∗max,which is a contradiction. Similarly, if k ≥ 1, it follows that kMρ(g∗−min)> g∗min, which is a contradiction.10There are no Nash equilibria in mixed strategies. The proof can be found in Appendix A.1.1.11See, among others, Fehr and Ga¨chter (2000), Kosfeld et al. (2009), and Fischbacher and Ga¨chter (2010).51.3. Experimental Design0 2 4 6 8 10 12 14 16 18 200246810121416182045◦ρ = 0.54 ρ = 0.58ρ = 0.65ρ = 0.70LVCMMρ(g−i)giFigure 1.1. Best-response functions. In this figure the x-axisshows the generalized mean of others’ contributions; the y-axis displays player i’s contributions. The figure shows theBR as a function of others’ contributions, g∗i (g−i). The solidlines represent g∗i (g−i) of player i.contribution is displayed in the third column. For sufficiently large values of ρ there exists aunique equilibrium of zero contribution. There also exists a threshold value of ρ below whichthe equilibrium contribution is either zero or the whole endowment ω (given baseline parameters,this happens when ρ < 0.602). Finally, the fourth column reports the exchange rate used in eachtreatment, adjusted so that expected payoffs were similar across treatments.Table 1.1Experimental TreatmentsDegree of Number of Equilibrium Exchange RateComplementarity Sessions Contribution (tokens per CAD)ρ = 1 2 {0} 1ρ = 0.70 1 {0} 2ρ = 0.65 2 {0} 2ρ = 0.58 1 {0,20} 2.5ρ = 0.54 2 {0,20} 361.3. Experimental Design1.3.1 Experimental proceduresIn each experimental session we recruited 16 subjects with no prior experience in any treatment ofour experiment. Subjects were recruited from the broad undergraduate population of the Universityof British Columbia using the online recruitment system ORSEE (Greiner, 2015). The subject poolincludes students with many different majors.Each session was developed in the following way: upon arriving at the lab, subjects wereseated at individual computer stations and given a set of written instructions; at the same time theinstructions were displayed on their computer screens.12 After reading the instructions, subjectswere required to answer a set of control questions. The goal of the control questions was to verifyand measure subjects’ basic understanding of how to use the tools in their computer interfaces andhow to interpret information displayed on the screens. Subjects received cash for answering controlquestions correctly.13 The experiment did not proceed further until all participants had answeredall control questions correctly.At the beginning of each round of the experiment, subjects were matched with three otherparticipants. They then played the static game described in Section 1.2. This process was repeated20 times.To avoid reputation effects we used an extreme version of the stranger matching protocol.14The group composition was predetermined and unknown to the participants. We preselected thegroups so that the subjects were matched with a given participant in only four rounds. Each timesomeone was matched with a participant he or she had encountered before, all other group memberswere different. This meant that any given grouping of four players never occurred more than once.All eight sessions were computerized using the software z-Tree (Fischbacher, 2007). Given thedifficulty of computing potential earnings using the nonlinear payoff function, we provided subjectswith a computer interface which eliminated the need to make calculations. Through this interfacesubjects were able to enter as many hypothetical choices and conjectures of other group members’12The instructions can be found in Appendix A.10.13The questions’ goal was to facilitate subjects’ learning of the main features of the VCMC. Relevant features included(a) decreasing marginal productivity in the group account given a fixed level of others’ contributions, (b) efficiency gainsdue to coordination, and (c) absence of a dominant strategy (for treatments in which ρ < 1). Subjects were credited $0.20,$0.15 or $0.10 for each question answered correctly in, respectively, the first, second, and third attempt. There were 19control questions, which can be found in Appendix A.5.14There is a vast experimental literature on the comparison between groups whose members stay in the same groupover repeated rounds and groups whose members switch groups every round. There are mixed results with respect todifferences in levels of provision of public goods for these matching protocols.71.4. Resultscontributions as they wanted, visualizing the potential payoff associated with each combination.15In each round, subjects had 95 seconds to submit their chosen contribution. At the end of eachround, they were informed about their own earnings and the contribution choices of other groupmembers.16At the end of the experiment, subjects were paid the payoff they obtained in a singlerandomly selected round.The sessions were conducted at the Experimental Lab of the Vancouver School of Economics(ELVSE) at the University of British Columbia, in January 2015. The experiments lasted 90 min-utes. Subjects were paid in Canadian dollars (CAD). On average, participants earned $30.60. Thisamount includes a $5 show-up fee and the cash received for the control questions.In the following sections we examine how changes in the degree of complementarity in differenttreatments are reflected in both the level and the evolution of individual contributions. Next, usinga combination of choice and nonchoice data, we document various interesting aspects of the choiceprocess: we examine the scope of history dependence in subjects’ decision making and documenthow past contributions of other partners in previous rounds shape the subject’s current choice.This history dependence allows us to define a notion of BR to past contributions and assess towhat extent subjects’ choices can be rationalized as profit-maximizing behavior, both in the crosssection and over time. Nonchoice data also reveal differences in calculator usage and responsetime, showing how subjects process information and make choices in different ways.1.4 ResultsManipulating the degree of complementarity induces stark changes in subjects’ behavior. This isreflected in the average contribution chosen by subjects, as well as in the heterogeneity of contri-butions. In this section we study how changes in complementarity affect the level and evolution ofindividual contributions.1.4.1 Average contributionsEach solid line in Figure 1.2 represents the evolution of the average contribution over the 20rounds of an individual treatment (dotted lines identify 95% confidence intervals). Figure 1.215Figure A.10 in Appendix A.4 displays a screenshot of the main interface.16Figure A.11 in Appendix A.4 shows the screenshot of the feedback given to subjects at the end of each round.Subjects were shown their overall income, as well as the breakdown between their private account income and groupaccount income. Given that group income was the same for each group member, subjects could easily infer the earningsof each of the other group members by looking at their contributions, reported in the same screen.81.4. Results1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2002468101214161820ρ = 0.54ρ = 0.58ρ = 0.65ρ = 0.70LVCMRoundAvg.ContributionFigure 1.2. Average contribution over time. This figureshows the evolution of the average contribution in each treat-ment (solid lines).clearly shows that average contributions increase with complementarity.17 With the exception ofρ = 0.65, average contributions converge towards the socially inefficient equilibrium when the de-gree of complementarity supports only one zero-contribution selfish-equilibrium. In contrast, theyconverge towards the socially efficient equilibrium when complementarity introduces an additionalfull-contribution equilibrium.Initial contributionsThe average contribution in the first round is significantly higher than zero in all treatments. Thisis not surprising given existing evidence about LVCM experiments. Some variation exists in first-round contributions across treatments: subjects in high-complementarity (HC) treatments, with ρequal to 0.54 or 0.58, contribute 4.3 tokens more, on average, than subjects in low-complementarity(LC) ones, with ρ equal to 0.65 or 0.70. The difference in contributions across treatments issubstantial, even in the first round when subjects have yet to receive any feedback. This may beattributed to the training subjects receive before deciding on contributions: their understandingof the rules of the game is reflected in their initial beliefs about others’ contributions, and thesebeliefs are likely to be treatment specific. To verify the role of training we compare the initial17Average contributions when ρ = 0.58 look marginally higher than average contributions when ρ = 0.54. However,this difference is not statistically significant.91.4. Resultsconjectures on others’ contributions across different treatments. Table 1.2 shows the average ofthe generalized mean of the conjectures in each treatment.18 Column 2 reports conjectures madeduring the practice period, before the experiment started; unsurprisingly no significant differenceacross treatments is apparent at this stage, as subjects are still learning about the payoff space andmay experiment with any conjectures that come to mind. However, starting from round 1 (column3) we observe significant differences across treatments. When a subject chooses to best respondto beliefs, his or her contributions will decrease as the degree of complementarity diminishes (ρincreases).Table 1.2Average Conjecture About Others’ ContributionsTreatment Practice Round 1 Round 2 Round 5 Round ≥ 10LVCM9.1 6.6 5.2 3.7 3(5.5) (6.3) (5) (4.6) (4.6)ρ = 0.70 8.7 7.6 5.3 2.3 4.6(5.3) (5) (4.5) (2.5) (6.3)ρ = 0.65 9.4 10 10.1 6.9 6.8(5.6) (5.2) (4.4) (4.1) (5)ρ = 0.58 9.5 11.7 12.1 17.7 15.4(5.9) (5.3) (4.5) (2.8) (5.5)ρ = 0.54 8.9 10.3 12.3 14.5 13.8(5.7) (5.8) (6.1) (4.3) (6.1)No. of conjectures 3,885 263 150 139 537Note: Each cell reports the average value for the generalized mean of the conjectures ofothers’ contributions (standard deviations are reported in parentheses).Treatment-specific dynamicsIn the LVCM environment we observe a pattern consistent with many previous experiments. Ini-tially the average contribution is significantly larger than zero; as rounds progress, there is anincremental decline in contributions.In the linear case the dominant (selfish) strategy is to contribute zero.19 The treatment with18We did not elicit beliefs. Rather, we collected data on the inputs subjects entered in the payoff calculator. Thisincludes conjectures about other group members’ contributions, which are a proxy of beliefs about others’ contributions.In Section 1.5 we describe these data extensively.19The linear treatment is also useful to benchmark our experimental design. While differences exist in instructions and101.4. Resultsρ = 0.70 introduces very slight complementarity in contributions. Yet the pattern remains similarto that of the LVCM treatment despite the fact that a zero contribution is no longer the dominantstrategy.By contrast, when complementarity is sufficiently strong to generate an additional full-contributionequilibrium (ρ = 0.58 and ρ = 0.54), the evolution of the average contribution exhibits the oppo-site pattern, as contributions tend to grow. In other words, intense complementarity changes theway agents interact, as they increase their contributions moving towards the efficient equilibrium.Finally, a unique pattern emerges when ρ = 0.65, a value which supports only a unique selfish-equilibrium of zero contribution but is closer to the threshold at which a full-contribution equilib-rium emerges. Experimental results show little or no evidence of variation in the average contri-bution as rounds elapse; it is apparent that contributions remain range-bound even as players gainexperience in advanced rounds. It is important to note that, at this level of complementarity, theunique selfish-equilibrium of zero contribution has a full basin of attraction. Even if an agent be-lieves that all other group members will fully contribute, his or her BR is to contribute only half ofhis or her endowment. The dynamics observed in this treatment are possible only if many subjectscontribute significantly above their pecuniary BR.1.4.2 Distribution and dispersion of contributionsThe confidence intervals reported in Figure 1.2 suggest that there is substantial heterogeneity incontributions. In this section we start by examining how the distribution of contributions variesover time and across treatments, and we conclude the analysis by investigating the patterns ofcontribution dispersion. Specifically we study the effect of complementarity on coordination.Distribution of contributionsFigure 1.3 displays the cumulative distribution of contributions by treatment (i.e., by complemen-tarity). Individual contributions fall into one of two categories: dashed lines show the cumulativedistribution for rounds 1 to 10, and solid lines show the cumulative distribution for rounds 11 to20. The plots confirm the finding of the previous subsection: the distributions in the LVCM andρ = 0.70 treatments look similar; the same is true for HC treatments, with not much differenceexperimental interface, aggregate results appear remarkably similar to those from standard LVCM experiments, albeitwith slightly lower contributions (probably due to the extensive control questions that minimized confusion).111.4. Resultsbetween the distributions under ρ = 0.58 and ρ = 0.54. Contributions concentrate at the extremesas sessions progress towards the end.By contrast, when ρ is set to 0.65, the mass distribution is more heavily concentrated in theinterior of the strategy space. Subjects choose to contribute nontrivial amounts even after 10 rounds.For example, in rounds 11 to 20, more than half of all contributions are larger than 5 tokens.Contributions are range-bound and show little tendency towards convergence. In Section 1.5 weexamine these patterns in detail.0 2 4 6 8 10 12 14 16 18 2000.250.50.751ρ = 0.54ρ = 0.58ρ = 0.65ρ = 0.70LVCMgiCDFRounds 1-10 Rounds 11-20Figure 1.3. Cumulative distribution functions. The dashedlines display the cumulative distribution function for the in-dividual contributions from rounds 1 to 10. The solid linesshow the cumulative distribution function for the individualcontributions from rounds 11 to 20..Coordination and complementarityA key feature of the VCMC production technology is that individuals not only benefit from others’contributions but also enjoy incremental gains as coordination improves. The cost of less-than-perfect coordination depends on the degree of complementarity; in the linear case there is no ad-ditional loss due to lack of coordination. As complementarity increases, the impact of dispersiongrows and it becomes more costly to forego coordination; on the other hand, when complementarityis high, a potential obstacle to coordination is the multiplicity of equilibria.We measure coordination in each treatment by capturing the loss due to dispersion. We define121.5. How Do Players Choose Their Contributions?the dispersion loss index (DLI) for group k in round t asDLIk,t =14 ∑4i=1 gi,t −(14 ∑4i=1 gρi,t)1/ρ10−(2021/ρ) .The numerator of the DLIk,t identifies the dispersion loss, as it measures the difference between ac-tual group account output and hypothetical output under perfect coordination. The denominator isjust a normalization factor making the index comparable across treatments. When the contributionsof the four group members are identical (zero dispersion) the arithmetic mean and the generalizedmean are identical for any ρ , and DLIk,t = 0; when dispersion is highest, DLIk,t = 1.20 This in-dex may be sensitive to outliers because there are only four groups in each session. To accountfor this sensitivity, in each round/session we take the 16 actual contributions and average over allpossible combinations of contributions that can be made by groups of four players; for any suchcombination we compute DLIk,t and, finally, we record the median DLIk,t for that round.21Figure 1.4 reports median DLI by treatment, averaged over five-round intervals,22 and its 95%confidence interval.23 This analysis illustrates that in HC treatments, despite the multiplicity ofequilibria, dispersion decreases over time. This is reflected in significantly lower DLI, after multi-ple rounds, than in LC treatments and lends support to the evidence in Figure 1.3. Subjects in HCtreatments manage to better coordinate their actions.1.5 How Do Players Choose Their Contributions?So far the analysis has highlighted three main findings: (a) when complementarity is sufficientlystrong, subjects are able to better coordinate close to the socially efficient equilibrium; (b) similarly,when complementarity is sufficiently weak contributions diminish, approaching the unique zero-contribution selfish-equilibrium; and (c) in the intermediate case of ρ = 0.65, there appears to beno visible convergence to equilibrium over 20 rounds, as some subjects persistently deviate fromtheir money-maximizing strategies.20This is achieved at the vector of contributions (0,0,20,20) in which the discrepancy between the arithmetic and thegeneralized mean is maximized.21The total number of possible combinations is 16!12!×4! = 1,820.22We pool together LC treatments (ρ = 0.70 and ρ = 0.65) and HC ones (ρ = 0.58 and ρ = 0.54).23Confidence intervals are calculated using a binomial-based method. We also compute confidence intervals by ran-domly selecting 500 samples with replacement of the 1,820 combinations in each round/session. We obtain very similarresults.131.5. How Do Players Choose Their Contributions?Rounds 1-5 Rounds 6-10 Rounds 11-15 Rounds 16-2000.020.040.060.080.10.12MedianDLI kLC HC 95% CIFigure 1.4. Dispersion loss index. This figure reports themedian DLI for HC, and LC treatments, averaged over five-round intervals. The dotted lines display the 95% confidenceinterval.In addition, we find recurrent overcontribution in LC treatments and undercontribution in HCtreatments. While observed choices provide some support for the complementarity hypothesis, thisis not sufficient evidence to ascribe individual actions to profit-seeking motives. This is especiallytrue in the case of ρ = 0.65.It is challenging to interpret individual choices through the examination of choice data alone.Therefore, we complement the analysis by resorting to nonchoice data. Throughout each sessionparticipants were given access to a payoff calculator. By using the calculator subjects could seethe monetary payoff associated with as many hypothetical contributions as they wished, includingdifferent hypothetical values of their own choice. We recorded every trial that subjects entered inthe calculator during both the practice period and the experiment.This nonchoice data is different from information collected using “mouse lab”24 or “eyetrack-ing”25 techniques. When employing these techniques participants are aware that experimenters aregathering data and this may influence their choices. Moreover, finding the optimal strategy in theVCMC makes the use of the calculator often necessary, as payoff functions are nonlinear and in-dividual gains are affected by the dispersion of players’ contributions. For these reasons, subjects24See, among others, Camerer et al. (1993), Costa-Gomes et al. (2001), Johnson et al. (2002), Costa-Gomes andCrawford (2006), and Brocas et al. (2014).25See, among others, Knoepfle et al. (2009), Wang et al. (2010), Reutskaja et al. (2011), and Arieli et al. (2011).141.5. How Do Players Choose Their Contributions?depend on the calculator to evaluate different strategies and to make informed choices, and theinput they enter into the calculator can be considered a proxy for their beliefs about the contribu-tion of others. Our method is a novel technique to collect high-quality nonchoice data which hasnot been exploited in the experimental literature.26 An additional advantage of our method is thatcollection of data is very simple because there is no need for any special technology or equipment;thus, it can be applied easily to other individual or group decision environments.Combining choice and nonchoice data makes it possible to ask questions such as these: Areconjectures influenced by the history of other players’ contributions? How do subjects adjust theirbehavior from one round to the next? Do they use history-dependent BR strategies? If so, howis this reflected in the use of the calculator? Do subjects in specific treatments use the calculatormore or less intensively? How do they experiment with hypothetical contributions? Are they ableto find the profit-maximizing strategy given their conjectures? How does this relate to their actualcontribution? And can we classify subjects according to the way they use the calculator?1.5.1 Classifying subjects into typesLarge differences exist in subjects’ behavior within each treatment. Some contribute consistentlymore than others; many change their choices repeatedly, while others do not. Also, as we documentbelow, the calculator is used with different intensity. This suggests that not all agents conductthemselves in the same way when it comes to choosing a contribution. To facilitate the analysis,we classify subjects into two broad groups, or types, based on the discrepancy between the payoffassociated with the history-dependent BR and the payoff from the actual contribution.27 A largerdiscrepancy indicates larger foregone earnings. We then examine whether there are differences inthe calculator usage of different subject types.Our grouping criterion considers the payoff associated with the history-dependent BR. Thus,we begin by providing evidence of history dependence of subjects’ beliefs about others, buttressingthe choice of history-dependent BR as our benchmark. Next, we assess the length of the subjects’memory span; to do this we regress the conjectures about others’ contributions on the actual con-tributions by group partners in the previous five rounds. Table 1.3 reports the results, showing thatsubjects’ conjectures respond to other members’ contributions in the previous two rounds.2826Cherry et al. (2015) also provide subjects a payoff calculator and analyze nonchoice data. A key difference withrespect to our design is that, in their case, subjects have to enter a conjecture first, and then the experimenters display atable with the payoffs associated with each hypothetical choice given that conjecture.27We classify subjects into two different groups only to simplify the analysis.28To confirm the results of Table 1.3, we consider all conjectures from round 2 onwards and find that roughly 11%151.5. How Do Players Choose Their Contributions?Table 1.3Response of Subjects’ Conjectures to Others’ Contributions( 1n−1 ∑gρ−i)1/ρ 1n−1 ∑g−iF(g−i,t−1)0.564∗∗∗ 0.570∗∗∗(0.07) (0.08)F(g−i,t−2)0.138∗∗ 0.170∗∗(0.07) (0.07)F(g−i,t−3)0.044 0.035(0.06) (0.06)F(g−i,t−4)0.016 0.003(0.08) (0.08)F(g−i,t−5)0.088∗ 0.076(0.08) (0.07)Constant 1.653∗∗∗ 1.609∗∗∗(0.52) (0.51)Observations 936 936Note: We estimate the following least-squares specification: F ( ˆgi,t) = C +∑5L=1 ALF (g−i,t−L)+ ui,t , where gˆiis a vector of player i’s conjectures about other group members’ contributions, g−i,t−L contains the vector ofcontributions made by other members in round t−L, C is a common constant, and ui,t is an idiosyncratic error. Welet the function F(·) be either the arithmetic or the generalized mean of degree ρ . The standard errors (reported inparentheses) are clustered by individuals and obtained by bootstrap estimations with 1,000 replications. *p < 0.1,**p < 0.05, ***p < 0.01. As a robustness check, we also estimate this specification including dummy variablesto control for different treatments. Results look very similar.How should one use information about contributions in the previous two rounds to define aBR? Restricting subjects to respond to the specific contributions observed in a given round seemsunreasonable because subjects are well aware that they will not be matched with the same set ofindividuals in subsequent rounds. Instead, we ask if a subject’s contribution can be rationalizedbased on recent history. We posit that subjects may respond to any possible combination thatcan be obtained by combining group members’ contributions in rounds t − 1 and t − 2. Then,for each subject/round and for every combination of the partners’ contributions, we compute thecoincide exactly with previous-round contributions by other group members. In 28% of the cases the conjecture matchesexactly with one of the 10 possible combinations that can be formed from the group members’ contributions in the priorround. Finally, in 38% of the cases the conjecture matches exactly one of the 56 possible combinations that can beformed from group members’ contributions in the two previous rounds. These relative frequencies are extremely highwhen compared to the three most recurring individual conjectures, namely (0,0,0), (20,20,20), and (10,10,10), whichwere considered in only 9%, 7%, and 3% of the cases, respectively. Agents clearly appear to make conjectures based onpast experiences.161.5. How Do Players Choose Their Contributions?difference between the profit associated with the BR, piBRi,t , and the profit associated with the ac-tual choice, piACTi,t . We keep only the lowest such difference per subject/round and denote it byMin Lossi,t = min{piBRi,t −piACTi,t}.29 Next, we define the proportional loss as Min Lossi,tpiBRi,t. This is amoney-metric index that measures how close actual contributions are to the money-maximizingcontributions (conditional on conjectures). If the lowest proportional loss is zero, then the choicecan be rationalized through the lens of pecuniary-profit-seeking behavior. The final step is to com-pute the average proportional loss of each subject. Then for each treatment group—LVCM (ρ = 1),LC (ρ ∈ {0.65,0.70}), and HC (ρ ∈ {0.54,0.58})—we obtain a median proportional loss by se-lecting the median value among all the individual averages in that treatment group. Subjects aredenoted as Type 1 if their individual proportional loss is not higher than the median value for theirgroup; otherwise they are denoted as Type 2. It is worth stressing again that this grouping criterionrequires the joint use of choice and nonchoice data.There are alternative ways to group subjects. One natural way to classify them is by usingdemographic information such as gender or field of study. In Appendix A.6 we show that there areno statistical differences in contributions levels by male or female participants within treatments. Interms of field of study, interestingly economics and business students contribute more than studentsfrom other fields in HC treatments. It is likely that many of this participants are classified as Type1 subjects. Although it is worthwhile to examine these findings in a deeper way, the demographicinformation might not be precise given that the variables are self-reported.1.5.2 Patterns of individual contributionsValuable information about individual decision making can be elicited from the evolution of in-dividual contributions. Crucially one can measure how close contributions are to the notion ofhistory-dependent pecuniary BR, as defined in Section 1.5.1. In HC treatments, despite much het-erogeneity, a remarkable two thirds of all contributions are consistent with BR behavior. Moreover,subjects commit a full 20 tokens in over half of the cases in which a full contribution is within therange classified as BR. Even when a deviation exists, it is often small. Most deviations are due toundercontributions; in HC treatments subjects undercontribute in 30% of the cases, but overcon-tribute in only 5% of them.In LC treatments just 42% of contributions are consistent with BR and, when deviations occur,29We sort the piBRi,t values from highest to lowest. We then remove the two lowest and highest values. We do this toavoid bias due to outlying contributions, whether unusually high or unusually low.171.5. How Do Players Choose Their Contributions?they mostly result in overcontributions; in over half of all cases subjects overcontribute, whileundercontributions occur in only 3% of cases.In Appendix A.2 we present plots of the complete sequence of contributions made by eachsubject. Contributions are juxtaposed to the rationalizable set (gray)—an area consisting of theset of BRs computed using the steps described in Section 1.5.1. This allows one to visualizewhether a subject’s contribution can be rationalized by pecuniary-profit-maximizing motives, andto appreciate how contributions drift into and out of the BR range. In these same figures wesuperimpose a red line representing the myopic BR; that is, a function of the contributions bymembers of the group to which the subject belonged in the previous round. This provides a moredirect counterpart to assess the path dependence of actual contributions. In Appendix A.3 weinclude a graphical representation of the patterns of deviation for each type of subject.1.5.3 Linking types to behavioral categoriesWhat drives Type 2 subjects to deviate from profit-maximizing strategies? One possibility is thatovercontribution in LC treatments reflects motives that are beyond simple profit seeking. For exam-ple, when optimal contributions become smaller, some agents may find joy in the act of contributingto a group account. Such joy of giving would be harder to experience when complementarity ishigh and profit-seeking behavior dictates high contributions.30On the other hand, undercontribution in HC treatments might be due to competitive motives;when other subjects contribute relatively high amounts, marginally reducing one’s own contribu-tions may guarantee the highest payoff in the group. Competitiveness can be magnified whensubjects are aware of the payoffs of others and infer the negative correlation between contributionsand payoffs (we test this hypothesis in Chapter 3). This motive would be indistinguishable frompecuniary-profit-maximizing when complementarity is low, as both usually lead to lower contribu-tions relative to other group members.It is conceivable that subjects—even profit-seeking ones—may deviate from the profit-maximizingstrategy because they do not understand the rules of the game. Given their conjectures, they mayfail to calculate the profit-maximizing choices. To discriminate between confusion and alternativebehavioral motives we examine both the mechanical use of the calculator (the number of roundsthe calculator is activated and the number of hypothetical contributions and conjectures about other30Another way to explain overcontribution is by assuming that subjects make mistakes due to confusion. We rule outthis possibility in the analysis below.181.5. How Do Players Choose Their Contributions?players) and what we call payoff-relevant use of it. We exploit this information to identify whethersubjects are able to compute the BR to their conjectures using the calculator and whether theysystematically play a BR strategy after they identify it.31We adopt two measures of payoff-relevant use: (a) the difference between hypothetical con-tributions and the BR to conjectures about other players’ choices (gˆi− g∗i (gˆ−i)), denoted as Cal-culated Deviation from BR32,33 and (b) the difference between actual contributions and the BR toconjectures (gi−g∗i (gˆ−i)), called Actual Deviation from BR.Mechanical use of the calculatorWe begin by reporting in Table 1.4 the summary statistics of the mechanical variables for differenttypes and treatments.34Number of rounds. Table 1.4 confirms that the LVCM is arguably the easiest environmentfor Type 1 subjects: they end up using the calculator very little (in only 4.4 rounds).35 In contrast,Type 2 agents use the calculator in the LVCM as much as in other LC treatments. This suggeststhat Type 1 may use the calculator to identify the BR and then mechanically play it to maximizepecuniary rewards.The degree of complementarity noticeably affects calculator usage: subjects in LC treatmentsuse the calculator in twice as many rounds as subjects in HC sessions (roughly, in 10 versus 5rounds). This supports the view that subjects find it easier to calculate BR strategies in HC treat-ments.36 For example, when ρ = 0.54, the BR is to invest the whole endowment in the group31To analyze the mechanical use of the calculator we examine the following variables: (a) CalcRound, number ofrounds the calculator was used by a subject, (b) Hyp, number of own hypothetical contributions entered in the calculator,(c) Conj, number of conjectures about other players’ contributions that were entered into the calculator, and (d) Hyp perConj, number of own hypothetical contributions entered, given a conjecture about other players’ contributions.32We consider all conjectures and hypothetical contributions starting from the practice session.33For cases in which an individual entered multiple hypothetical contributions for the same conjecture, we set a ruleto match a hypothetical contribution with a conjecture: namely, we select the current or past hypothetical contributionthat maximizes the monetary payoff given the conjecture. We consider past hypothetical contributions (in additionto current ones) because we find evidence that subjects select a given conjecture and then adjust their hypotheticalcontributions over several rounds (more details can be found in Appendix A.8). Finally, to simplify the analysis wegroup conjectures within different bins based on their generalized ρ-mean. The bins, B, are defined as follows: ifMρ ≤ 0.5 then Mρ ∈ {B = 1}; if Mρ ≥ 19.5 then Mρ ∈ {B = 21}; if j− 1.5 < Mρ ≤ j− 0.5 then Mρ ∈ {B = j} forj = 2, . . . ,20. When ρ = 0.54 (ρ = 0.58) we group in the same bin all conjectures for which Mρ ≥ 10(Mρ ≥ 15).34As a robustness check for the results in Table 1.4, Appendix A.7 contains a table displaying results for a groupingof subjects based on the assumption that subjects respond only to other group members’ contributions in the previousround.35Three Type 1 participants did not even activate the calculator after the practice round.36Six subjects in the HC treatment did not use the calculator after the practice period.191.5. How Do Players Choose Their Contributions?Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2001020304050607080%ofSubjectsWhoUsedtheCalculatorLVCM LC HCFigure 1.5. Use of the calculator over time. This figure re-ports the percentage of subjects that activated the calculator,for the LVCM, LC, and HC treatments, averaged over four-round intervals.account if other group members invest at least half of their endowment; this means that, after a fewrounds, agents may effectively adopt something close to a high-investment strategy, which requiresno further refinement through the use of the calculator. In LC treatments, instead, choosing a strat-egy that maximizes payoff requires more fine tuning. For example, when ρ = 0.70, a subject wouldoptimally choose to invest one quarter of the average contribution made by others to maximize hisor her payoff, assuming all other players contribute the same amount. Hence, it may be harder toidentify a BR strategy in LC treatments. For all treatments, calculator usage declined over time.This can be observed in Figure 1.5.Conjectures and hypothetical choices. Looking at conjectures, and at the number of own hy-pothetical choices per conjecture, there is no significant difference across types. However, subjectsin LC and HC treatments enter more hypothetical choices than in LVCM. A Type 1 subject enterson average slightly more hypothetical contributions per conjecture than does a Type 2 subject in theLC and the HC sessions. One may expect this behavior from an individual who is very concernedabout maximizing his or her money earnings. The difference, however, is not significant, possi-bly because a Type 2 subject may be able to approximate the monetary BR; the next subsectionprovides evidence to this effect.201.5. How Do Players Choose Their Contributions?Table 1.4Differences in Mechanical Use of the Calculator, by Subject Type Within Complemen-tarity LevelLVCM LC HCType 1 Type 2 t-test Type 1 Type 2 t-test Type 1 Type 2 t-test(p-value) (p-value) (p-value)CalcRound 4.4 9.5 0.0 11.1 9.5 0.3 4.9 5.5 0.7(1.2) (1.9) (1.1) (0.9) (0.7) (1.2)Hyp 17.1 21.9 0.2 31.1 29.5 0.7 27.6 24.9 0.6(2.6) (3.1) (2.8) (2.6) (3.8) (3.6)Conj 13.9 14.1 0.9 15.0 15.8 0.5 9.9 10.1 0.9(1.1) (1.1) (0.8) (0.7) (0.6) (0.6)Hyp Per Conj 3.7 4.3 0.4 7.2 6.3 0.2 8.0 7.3 0.6(0.4) (0.5) (0.5) (0.4) (1.1) (1.0)Observations 16 16 24 24 24 24Note: Each cell reports the average value for the respective category (standard errors are reported in parentheses).The t-tests of the means are reported in the third column of each treatment. CalcRound, number of rounds in whichsubjects used the calculator; Hyp, number of hypothetical own contributions; Conj, number of conjectures aboutothers; Hyp per Conj, number of own hypothetical contributions entered, given a conjecture about other players’contributions. We include the practice rounds.Homo pecuniarius versus Homo behavioralisNext, we examine how the payoff-relevant measures Calculated Deviation from BR and ActualDeviation from BR are distributed among participants. When a subject identifies the pecuniary-profit-maximizing strategy using the calculator, the discrepancy between his or her hypotheticalown contribution and BR to his or her conjectures (Calculated Deviation from BR) is close tozero. Similarly, a value of Actual Deviation from BR close to zero indicates that a participant haspursued the pecuniary-profit-maximizing strategy for a given conjecture. Figure 1.6 displays ascatter plot of the average value of Calculated Deviation from BR and Actual Deviation from BRfor each subject. Blue circles and red squares refer to Type 1 and Type 2 subjects, respectively.The plot confirms that both types are usually capable of finding the profit-maximizing contributionusing the calculator (Calculated Deviation from BR is never very far from zero). This means thatconfusion cannot account for most of the observed choices.37 Considering actual choices (ActualDeviation from BR), significant differences become apparent: Type 1 subjects (Homo pecuniarius)clearly pursue the pecuniary-profit-maximizing strategy, whereas Type 2 individuals (Homo behav-ioralis) often choose to deviate from it. Type 2 subjects seem to exhibit an altruistic behavior in LC37This is also confirmed when looking at the performance on the control questions. There is a negligible differencebetween the payoffs each type obtained from answering the control questions correctly. Type 1 subjects earned $3.73,whereas Type 2 received $3.63.211.5. How Do Players Choose Their Contributions?treatments, while in HC environments Type 2 subjects act as if they have a competitive motive.38Crucially, variation in the degree of complementarity and the magnitude of optimal contributionsmay play a role in the occurrence of different behavioral motives. When BR choices are very low(LC treatments) some agents may enhance their payoff through altruistic overcontributions. Suchjoy of giving could be tainted, or less salient, in an environment where a high payoff is associatedwith a high contribution. When the optimal contribution is high, a competitive motive may becomemore appealing as agents recognize that small reductions in contribution are both costly to otherplayers and useful to boosting relative performance within a group. This competitive motive isindistinguishable from pecuniary-profit-maximizing in LC environments.Behavioral motives may operate side by side with profit-seeking behavior as agents consider allthese aspects in their decision making. This observation motivates the analysis in the next section.1.5.4 Quantifying altruistic and competitive motivesGiven that deviations from profit-maximizing strategies cannot be simply attributed to confusion,Type 2 subjects appear to pursue a combination of monetary and nonmonetary rewards. In whatfollows we attempt to quantify the magnitude of nonpecuniary motives by estimating how muchmoney these subjects are willing to forego in the process of making gifts (in LC treatments) or toobtain a relatively higher payoff within their group (in HC treatments).Measuring nonpecuniary motivesAndreoni et al. (2008) define the warm-glow of giving as “the utility one gets simply from the actof giving” (p. 1). Therefore an individual’s utility function can be defined as Ui = pi(gi,g−i,ρ)+γ ,where γ captures the joy-of-giving motive. We use observed choices by Homo behavioralis (Type2) to estimate γ for each treatment. By definition, γ is the difference between the pecuniary-profit-maximizing contribution and the pecuniary profits from the actual contribution of Type 2 subjects.pi(ρ,g∗i (g¯), g¯)−pi(ρ, g¯Type2, g¯) = γ, (1.5)38Since this is a between-subject study, we make no claim as to the identity of types across treatments. That is, anagent may appear as Homo pecuniarius in LC treatments (since competitive behavior coincides with profit maximizing)while undercontributing in HC treatments—like a Homo behavioralis. The opposite pattern may emerge as well.221.5. How Do Players Choose Their Contributions?−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(a) LVCM−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(b) LC−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(c) HCFigure 1.6. Calculated Deviation from BR versus Actual Deviation from BR. The blue circles display the average Cal-culated Deviation from BR and Actual Deviation from BR for each Type 1 subject. The red squares display the averageCalculated Deviation from BR and Actual Deviation from BR for each Type 2 subject.231.5. How Do Players Choose Their Contributions?where g¯ is the average contribution observed among all players and g¯Type2 is the observed averagecontribution of Type 2 subjects.39 Equation (1.5) describes the choice of a Homo behavioralissubject: when other subjects contribute g¯, he or she prefers to contribute g¯Type2 tokens rather thang∗i (g¯). We assume that the warm-glow compensates the subject for the pecuniary loss. Table 1.5reports the estimated average magnitude of γ within each treatment; the estimates are roughlysimilar when comparing across treatments (between 0.75 and 0.85 tokens).40Table 1.5Warm-Glow Estimatesρ g¯ g∗i (g¯) g¯Type2 γ1 0.7 0 1.4 0.840.70 1.9 0.5 3.7 0.880.65 7.2 3.7 9.2 0.76Note: The first column displays the degree of complementarity, g¯ is the overall average contribution,g∗i (g¯) is the BR given the average contribution, g¯Type2 is the average contribution of Type 2, and γcaptures the warm-glow. We only consider the last 10 rounds.Using similar reasoning, one can quantify the intensity of competitive motives in HC treat-ments; that is, the pecuniary payoff a subject is willing to sacrifice in exchange for a higher incomerank within a group. We define the individual utility function as Ui = pi(gi,g−i,ρ)+κ , where κmeasures the joy of winning. Table 1.6 reports estimates for κ . The competitive motive is esti-mated to be higher for ρ = 0.54 than for ρ = 0.58. This is consistent with two observations: (a)Type 2 deviations are marginally larger in ρ = 0.54, and (b) for any given κ , the cost of deviating isnontrivially higher when complementarity is stronger. In the next subsection we discuss the latterpoint in some detail.39We assume that g−i = g¯. To account for possible early learning of the game and the environment, we concentrateon the last 10 rounds.40These amounts are even lower if one converts the tokens to CAD based on the exchange rates in Table 1.1.241.5. How Do Players Choose Their Contributions?Table 1.6Competitive-Motive Estimatesρ g¯ g∗i (g¯) g¯Type2 κ0.58 17.6 20 13.9 0.660.54 16.2 20 13.4 1.59Note: The first column displays the degree of complementarity, g¯ is the overall average contribution,g∗i (g¯) is the BR given the average contribution, g¯Type2 is the average contribution of Type 2, and κcaptures the competitive motives. We only consider the last 10 rounds.Finally, we examine the distribution of estimated γ and κ for Type 2 subjects. To do this, weuse the Min Loss (defined in Section 1.5.1). Figure 1.7 displays the cumulative distribution ofthe individual Min Loss values for rounds 11 to 20 in the LC, LVCM, and HC treatments. Weconsistently find that for more than 80% of Type 2 subjects the nonpecuniary motive is at most 1.5tokens, which is fairly low given the monetary stakes in the game.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1500.10.20.30.40.50.60.70.80.91Minimum Loss (tokens)CDFLVCM ρ = 0.70 ρ = 0.65(a) LVCM and LC0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1500.10.20.30.40.50.60.70.80.91Minimum Loss (tokens)CDFρ = 0.58 ρ = 0.54(b) HCFigure 1.7. Cumulative distribution of the individual minimum loss. Each line of the left panel displays the cumulativedistribution of the per-round Min Loss of Type 2 subjects for the LC and LVCM treatments, whereas each line of theright panel displays cumulative distribution of the per-round Min Loss of Type 2 subjects for the HC treatments. Weconsider the minimum loss per subjects for rounds 11 to 20.Complementarity and cost of deviations from pecuniary best responseIn the VCMC, the cost of a constant deviation from the money-maximizing strategy changes withρ . As ρ decreases, the payoff function becomes flatter and any marginal change in strategy has251.5. How Do Players Choose Their Contributions?a smaller effect on the final reward. This implies that rationalizing similar deviations from profit-maximizing behavior requires a higher warm-glow value (γ) as ρ increases. This observation helpsexplain the contributions of Type 2 subjects when ρ = 0.65 as opposed to when ρ = 0.70.To illustrate this point we assume that subject i makes a contribution equal to the average con-tribution of Type 2 subjects when ρ = 0.65.41 Then we calculate the difference between the moneyearnings that subject i would make following this strategy and that obtained when best responding(monetarily) to group members’ contributions.42 This difference measures the monetary cost ofdeviating from the profit-maximizing strategy, which is plotted in the left panel of Figure 1.8. Thex-axis displays the contributions of others (g¯−i), and the y-axis reports the cost for each treatment.When g¯−i = 0, the cost is the same irrespective of complementarity; as the investment by otherplayers grows, overcontributing becomes generally less costly. Comparing between treatmentsin panel (a), as complementarity increases, the cost of overcontributing is reduced. This impliesthat in treatments with higher complementarity it is less expensive to behave altruistically, whichaccounts for the different behavior of players in the ρ = 0.65 and ρ = 0.70 treatments. For theLVCM the cost is constant and it is higher than in LC treatments. In other words, an identicalvalue of the joy-of-giving motive is translated into a higher overcontribution as complementarityincreases (from LVCM to ρ = 0.7 to ρ = 0.65).The right panel of Figure 1.8 displays the cost of deviating in HC treatments. Here we assumethat player i makes a contribution equal to the average contribution of Type 2 subjects in HCtreatments, which is 13.3 tokens. In HC the cost function does not monotonically decrease in otherplayers’ contributions, and losses start mounting if one does not best respond to high contributionsby others. In these cases, if ρ decreases (that is, complementarity increases), the competitive motiveκ must become stronger to justify similar deviations below pecuniary BR.41Type 2 subjects contribute an average of 9.2 tokens when ρ = 0.65 (last 10 rounds).42To facilitate the analysis we assume that other members’ contributions are equal.261.5. How Do Players Choose Their Contributions?0 2 4 6 8 10 12 14 16 18 20012345678910g¯−iCostofDeviating(tokens)ρ = 0.65 ρ = 0.70 LVCM(a) LVCM and LC0 2 4 6 8 10 12 14 16 18 20012345678910g¯−iCostofDeviating(tokens)ρ = 0.54 ρ = 0.58(b) HCFigure 1.8. Cost of deviating from the money-profit-maximizing strategy. Each line of the left panel displays the cost ofdeviating from the profit-maximizing strategy (in tokens) for the LC and LVCM treatments when subject i contributes9.3 tokens (the observed average contribution of Type 2 when ρ = 0.65). Each line of the right panel displays the cost ofdeviating from the profit-maximizing strategy (in tokens) for the HC treatments when subject i contributes 13.3 tokens(the observed average contribution of Type 2 in HC). The cost is equal to pi(ρ,g∗i , g¯−i)−pi(ρ,gi, g¯−i).1.5.5 Evidence from response timesPrecise measures of subjects’ response times are available for all treatments. This informationprovides an alternative way to peek at the mechanics of individual decision making. Analyzingdecision times in public good games has become increasingly popular following a study by Randet al. (2012). In a one-shot LVCM experiment, they show how shorter response times positivelycorrelate with higher contributions, and they interpret this as evidence that humans are instinc-tively generous. This interpretation has been challenged by, among others, Recalde et al. (2015),who point out that in the LVCM the only possible deviation is to overcontribute, making it hard todistinguish between subjects who instinctively overcontribute and those who make genuine mis-takes.43 We combine qualitative nonchoice data and response-time information to illustrate thatsome of the conclusions drawn by Rand et al. (2012) with respect to instinctive generosity are notconsistent with our findings. More generally we argue that valuable information can be extracted43Recalde et al. (2015) design a voluntary contribution mechanism experiment in which the dominant strategy is inthe interior of the strategy space, and they replicate the finding of Rand et al. (2012) when the equilibrium contributionis below the midpoint of the choice space. However, when the equilibrium is located above the midpoint, they find anegative correlation between response time and contributions.271.5. How Do Players Choose Their Contributions?from differences in the length of time it takes subjects to enter their contributions and from theintensity of their calculator usage over that interval.Response time in the first roundFirst, we replicate the analysis of Rand et al. (2012). For comparability we consider only the first-round contributions in the LVCM treatment. The results confirm the findings of Rand et al. (2012):subjects who contribute zero wait 33 seconds, on average, before logging their choice. In contrastit takes only an average of 25 seconds to select a positive contribution. Our experimental designallows us to go beyond the one-shot game, and in the rest of this section we report evidence aboutresponse times after the first round. The analysis of sequential rounds makes it feasible to assesshow response times differ when observed contributions are closer, or farther, from hypotheticalBRs.Differences across treatmentsBefore proceeding, we categorize observed contributions into those that can and those that can-not be rationalized using the procedure described in Section 1.5.1. This distinction unveils someremarkable differences in both the quantity and quality of time use. As shown in Figure 1.9 andTable 1.7, the response times of subjects in the HC and LVCM treatments appear quite similar andare significantly shorter than those of their counterparts in the LC treatments.We can look separately at Type 1 and Type 2 subjects. In the HC and LVCM treatments Type 1subjects respond faster. This highlights a new and interesting discrepancy: in one set of treatmentsthe fastest subjects are the ones who contribute little or nothing, while in another set the quickestsubjects are those who get closer to a full contribution. These results suggest that both responsetimes and the direction of deviations from BR depend on the specific environment and that speedychoices do not necessarily imply overcontribution.44In contrast, in LC treatments Type 1 subjects take more time before submitting their choices,possibly because calculating the (pecuniary) optimal level of contribution with precision is moredifficult when complementarity is low. This interpretation is borne out by additional measurements;as we show in Appendix A.9, agents who play close to pecuniary BR in the LC treatments not only44Rubinstein (2007) obtains similar results. He finds that it takes more time to make decisions that require cognitivereasoning than it does to make instinctive choices.281.5. How Do Players Choose Their Contributions?take longer to log a choice but also use the calculator more intensively and consider a higher numberof potential combinations.45Table 1.7Response TimeType 1 Type 2 OverallAvg.(SD) obs.Avg.(SD) obs.Avg.(SD) obs.seconds seconds secondsLVCM 9.4 (13.1) 304 13.8 (14.5) 304 11.6 (14.0) 608LC 25.9 (24.6) 456 20.7 (21.2) 456 23.3 (23.1) 912HC 8.9 (13.0) 456 13.2 (15.9) 456 11.1 (14.7) 9120 10 20 30 40 50 60 70 80 900102030405060708090100SecondsCDF(%)Type 1 LVCMType 2 LVCMType 1 LCType 2 LCType 1 HCType 2 HCFigure 1.9. Response-time frequencies. Each solid line rep-resents the cumulative distribution function for Type 1 sub-jects for each of the treatments. Each dashed line representsthe cumulative distribution function for Type 2 subjects foreach of the treatments. The y-axis is displayed in percentageterms..45The response times of Type 1 and Type 2 subjects in the LC treatments are consistent with the typology describedin Rubinstein (fothcoming). He divides subjects into two types according to their response time, arguing that subjectswho make quick decisions are more instinctive while those who are slower often make strategic considerations.291.6. Related Literature1.6 Related LiteratureThe experimental literature has focused on coordination failures in games with strategic comple-mentarities in players’ decisions. The classic example is the two-by-two stag hunt game in whichthere are two Nash equilibria in pure strategies, one payoff dominant and the other risk dominant(see Cooper et al., 1992). In this type of coordination game, the Pareto superior (payoff-dominant)outcome is not always chosen; the equilibrium selection depends on the basin of attraction and theoptimization premium (see Battalio et al., 2001; Van Huyck, 2008). The current study introducescoordination considerations in a public good game. Our experimental result of no convergence tothe unique Nash equilibrium in the case of ρ = 0.65 is in sharp contrast to experimental resultsin binary-action games, and testifies that a richer strategy space may induce different behavioraldynamics.Another example of a coordination game is the weakest-link game in which n agents mustchoose an integer from the set 1 to k. The agents’ payoff depends on the minimum of all the chosennumbers. This is the extreme case of strategic complementarities. The seminal paper by Van Huycket al. (1990) shows that subjects fail to coordinate on the efficient outcome when groups are large.In terms of complementarity in public goods provision, there are experiments based on theweakest-link mechanism of Hirshleifer (1983). In this framework, public goods provision dependson the minimum contribution. Moreover, there are multiple Pareto-ranked equilibria because everyset of symmetric choices is an equilibrium. Harrison and Hirshleifer (1989) were the first to im-plement this in the lab. They compare simultaneous and sequential two-player contribution gamesin which the provision of the public good depends on the sum of contributions, on the minimumcontribution (weakest link), or on the maximum contribution (best shot). They find that underthe weakest-link mechanism, subjects’ contributions are very close to the Pareto-dominant equi-librium. Croson et al. (2005) report results from a voluntary contribution experiment in which theprovision of the public good depends on the lowest contributor (weakest link). They contrast thistreatment with the LVCM and find that in most periods subjects are unable to coordinate on anyof the equilibria. As in the linear case, the average contribution decreases over time. The authorssuggest that imitation of the lowest contributors may explain this pattern.With respect to the payoff function, there are alternative ways to introduce nonlinear payofffunctions in voluntary contribution mechanism environments. In Keser (1996) individuals’ privateaccount exhibits diminishing marginal returns. Therefore the NE is at the interior of the strategyspace. However, individuals’ BR does not depend on the contributions of others as in the standard301.7. ConclusionsLVCM. In Andreoni (1993) the NE is at the interior of the strategy space as well. To achieve this heelicited a Cobb-Douglas utility function in which the arguments are the private and public accounts.In another related paper, Steiger and Zultan (2014) compare the linear case and a case in whichthe marginal return from the public good increases as the number of contributors increases (throughincreasing returns to scale, or IRS). Subjects have a binary choice: either to contribute or not. Inthe IRS treatment there are two equilibria: zero contribution and full contribution. The authorsimplement a partner-matching protocol and find that only groups that cooperate in early rounds areable to converge to the full-contribution equilibrium. Overall, they find that contributions decreaseover time and that the average contribution is not significantly different than what is observed inthe linear case.Finally, Potters and Suetens (2009) design an experiment in which there is a unique equilibriumat the interior of the choice space. They find that subjects converge faster to the equilibrium understrategic complementarity than under strategic substitutability.46In terms of competitiveness our environment enables within-group competition. However, theliterature on public goods includes experiments that elicit between-group competition. Gunnthors-dottir and Rapoport (2006), Tan and Bolle (2007), Puurtinen and Mappes (2009), and Burton-Chellew et al. (2010) find evidence that suggests that inter-group competition increases coopera-tion. However, these results may rely in the fact that subjects are rewarded by their group perfor-mance, therefore the cost of cooperation is lower than in the standard LVCM.As mentioned earlier, several studies have used mouse lab and eyetracking technology to collectnonchoice data that shed light on the decision process of players. Cherry et al. (2015) implementan output-sharing game with negative externalities. They use subjects’ conjectures to analyze devi-ations from the theoretical predictions and conclude that deviations are consistent with preferencesfor altruism and conformity.1.7 ConclusionsIn this chapter we investigate how the introduction of complementarity between private contribu-tions to a public good affects choices to contribute. Consistent with theoretical predictions we finda positive relationship between aggregate contributions and the degree of complementarity. In HC46The experiments implement a static game over 31 successive rounds. Between rounds there is no change in groupcomposition.311.7. Conclusionsenvironments subjects learn to coordinate, moving towards the socially preferable equilibrium.47By contrast, when complementarity is very low, choices converge to the unique zero-contributionequilibrium. Subjects also seem to respond to complementarity when its intensity is sizable but notsufficiently high to introduce a second full-contribution equilibrium; in this case they persistentlyovercontribute and show little or no tendency towards the unique zero contribution.Manipulating the intensity of complementarity allows us to look at the decision making pro-cess and identify alternative motives underlying observed choices. We find that deviations from theprofit-maximizing strategy cannot be attributed to confusion, but rather originate from nonpecu-niary motives. Moreover, different motives are present under different degrees of complementarity.Not all subjects are equally sensitive to nonpecuniary motives. We find evidence that whilesome individuals (Homo pecuniarius) can be clearly described as profit seekers who follow pe-cuniary BR strategies, others (Homo behavioralis) are able to calculate the payoff-maximizingstrategy but deliberately deviate from it. The interaction of different types of participants is key tounderstanding how groups behave and why we observe different aggregate patterns under differentlevels of complementarity. The fact that Homo behavioralis subjects are willing to sacrifice somepecuniary rewards to deviate from BR strategies may lead to imperfect convergence to equilibrium.The presence of Homo behavioralis increases social welfare when complementarity is low, as it re-strains group contributions from collapsing to zero, but it reduces welfare when complementarityis high and full contributions would be optimal. We also find strong evidence that Homo pecu-niarius and Homo behavioralis subjects respond to the presence of each other by adjusting theircontributions.47In a related study (available upon request) we investigate how convergence to the payoff-dominant (full-contribution) equilibrium is affected when its basin of attraction is reduced. We do this by requiring a minimal level ofpublic good to be produced; if this level is not attained, then all contributions to the public good are lost. We show that insuch environments (and even when the threshold is high) players tend to coordinate on the full-contribution equilibrium.32Chapter 2Coordination and ContributionThresholds in the Presence ofComplementarity2.1 IntroductionOne key challenge for mechanisms promoting private provision of public goods is how to inducecoordination of individual contributions on efficient outcomes. In these environments there ex-ists an inherent tension between self-interest and socially preferable outcomes. Chapter 1 of thisdissertation develops and implements a mechanism (the voluntary contribution mechanism withcomplementarity, or VCMC) that enables coordination in the Pareto optimal equilibrium by allow-ing the inputs of the public good production technology to be complementary. In the VCMC whenthe degree of complementarity is high enough, there exist two equilibria (a zero-contribution and afull-contribution equilibrium). In the experimental settings with high complementarity in Chapter1 there is no evidence of convergence to the zero-contribution equilibrium. However, the zero-equilibrium is not a stable one. Only if individuals believe that none of their partners will makepositive contributions will they find it optimal to contribute zero. Could the lack of convergence tothe inefficient equilibrium be due to its instability? To test the robustness of the findings in Chapter1 we expand the basin of attraction of the socially inefficient equilibrium of the VCMC by varyingthe minimum group output required to obtain a positive return from cooperation. There are numer-ous papers examining the effect of adding thresholds in linear-technology public good games (seeLedyard, 1995).48 In the LVCM, the introduction of thresholds creates new equilibria. Any vectorof contributions whose sum times the MPCR is equal to the threshold is an equilibrium. Resultsfrom existing experiments suggest that, as thresholds are set higher, contributions increase but the48The experimental design of Isaac et al. (1989) is the one that more closely resembles our experiment.332.1. Introductionprobability of provision decreases.For the two equilibria in the VCMC, the introduction of thresholds does not alter the equilib-rium set. We analyze differences in contributions for three different treatments: (a) no threshold(benchmark), (b) low threshold, and (c) high threshold. We find that the introduction of thresholdsdoes not alter the equilibrium selection or the pattern of contribution dynamics relative to the base-line. For the high-threshold case, contributions are initially higher but in later rounds they convergeto the average value of the benchmark. In this treatment there are two focal points: (a) zero con-tribution and (b) full contribution. However, subjects might be aware that they have veto power interms of provision (when contributing zero), and for this reason nonpositive contributions are in-frequent. Low thresholds are more disruptive; here, contributions are lower than in the benchmarktreatment as subjects find it more challenging to coordinate. Unlike the high-threshold case, in thisframework provision is guaranteed even if there is a low contributor in the group. The experimentalresults show little evidence of variation in the average contribution as rounds elapse.To understand what motivates subjects to make decisions, we collect and analyze nonchoicedata. Following the procedure of Chapter 1, we divide participants into two types: Homo behav-ioralis and Homo pecuniarius. We do not find significant differences in the behavior of Homopecuniarius subjects across the different treatments. They are able to calculate the pecuniary BRand this is consistent with their choices. In contrast, Homo behavioralis deviates from the pecuniaryBR due to competitiveness. One feature of the environment is that the introduction of thresholdincreases the monetary cost of deviations relative to the benchmark. This is consistent with ourfindings that show that the deviations of Homo behavioralis are less pronounced in sessions withhigh threshold than in the benchmark treatment. For the case of the low-threshold sessions, Homobehavioralis’ deviations from the monetary-profit-maximizing strategy are due to (a) less intensiveuse of the calculator and (b) competitive motives; they attempt to contribute the minimum amountthat guarantees that thresholds are met, given their conjectures about their partners.This chapter is organized as follows. Section 2.2 describes the theoretical model based onChapter 1. Section 2.3 presents our experimental design and laboratory procedures. Section 2.4reports the results from aggregate data. The data suggest that there is no convergence to the zero-equilibrium when thresholds are introduced. Section 2.5 explores individual-level behavior, ana-lyzing both choice and nonchoice data. It considers some explanations for the observed behavior,focusing specifically on the differences between Homo pecuniarius and Homo behavioralis. Sec-tion 2.6 concludes.342.2. VCMC With a Threshold2.2 VCMC With a ThresholdThe VCMC, as defined in Chapter 1, refers to a public good provision mechanism in which thepublic good exhibits nonlinear returns due to complementarity. In this setting n individuals, indexedby i ∈ {1, ...,n}, are given an endowment (ω) and decide whether—and how much—to invest ina group account which maps private contributions into an aggregate output. This output is equallyshared among group members. Crucially, individual contributions are complementary inputs in theproduction of output.Let gi denote individual i’s contribution to the public good; the remainder of i’s endowment isdeposited in a risk-free private account that yields no interest. Individual investments in the groupaccount are aggregated through a constant elasticity of substitution production function. In thischapter we modify the environment of Chapter 1 by introducing a minimum output threshold. Thisimplies that agents obtain a positive return in the public good only if output is greater than somethreshold. Player i’s payoff is given bypii =ω−gi +β(∑ni=1 gρi)1/ρ if β (∑ni=1 gρi )1/ρ > Tω−gi if β(∑ni=1 gρi)1/ρ ≤ T , (2.1)where ρ ≤ 1 denotes the degree of complementarity, β is a constant, and T is a threshold.49 Equa-tion 2.1 implies that agents obtain a positive return from the group account if the public good’soutput is greater than T .2.2.1 Best-response functionTo obtain the best-response (BR) function of agent i, denoted as g∗i (g−i), the first step is to derivethe first order condition with respect to gi:∂pii∂gi= β((g∗i )ρ +∑gρ−i) 1−ρρ((g∗i )ρ−1)−1 = 0. (2.2)g∗i (g−i) = kMρ(g−i),49In the experimental literature there is a distinction between threshold and provision point. The former is used insetups in which the return from the public is fixed when the threshold is met (see Bagnoli and McKee 1991, Suleiman andRapoport, 1992, and Cadsby and Maynes, 1998). The latter is consistent with the payoff function defined in Equation2.1 (see Isaac et al., 1989, and Bougherara et al., 2011).352.2. VCMC With a Thresholdwhere Mρ(g−i) ≡(∑n−1i=1 gρ−in−1)1/ρis the generalized ρ-mean of agent i’s conjecture about the con-tributions of other group members, denoted by the vector g−i ∈ ℜn−1+ , and k ≡(n−1βρρ−1−1) 1ρis aconstant that depends on the model’s parameters. The next step is to check whether the threshold ismet given g∗i (g−i) and g−i. In other words, one has to confirm that β((g∗i (g−i))ρ +∑gρ−i)1/ρ> T .The latter condition can be restated in terms of Mρ(g−i): Mρ(g−i) ≥(β ρ/(1−ρ)/k)T . If this isnot the case, it would be optimal for agent i to increase his or her contribution to the point wherethe public good output is marginally above the threshold, gi =((T/β )ρ −∑gρ−i)1/ρ+ ε ,50 as longas the following two conditions hold: (a) gi ≤ ω and (b) pii(gi,g−i) ≥ pii(0,g−i). If either of theconditions does not hold, then the optimal contribution is zero.51 The BR function is summarizedbelow:g∗i (g−i) =ω if Mρ (g−i)≥( 1k)ωkMρ (g−i) if(β ρ/(1−ρ)k)T < Mρ (g−i)<( 1k)ω((Tβ)ρ −∑gρ−i)1/ρ + ε if ( 1n−1 [( Tβ )ρ −min(ωρ ,Tρ )])1/ρ + ε ≤Mρ (g−i)≤ ( β ρ/(1−ρ)k )T0 otherwise.(2.3)Imposing the symmetry condition gi +G−i = ngi in Equation (2.2)52 and solving for gi, we char-acterize the equilibria: geqi = {0,ω}.53,54Figure 2.1 plots the BR function for the no-threshold case (T = 0) and for two positive thresh-old values used in our experiments (T = 30 and T = 60). The generalized ρ-mean of other groupmembers’ contributions is measured on the horizontal axis, and player i’s contribution is shownon the vertical axis. The solid lines represent the BR of player i. For T = 0, the zero-contributionequilibrium is not stable; the BR is to contribute zero only if other group members contribute zero.This is not the case for treatments in which the threshold is positive. For example, if the generalizedmean of others’ contributions is smaller than 3 tokens for T = 30, and smaller than 10 tokens forT = 60, agent i’s BR is to contribute zero.50Where ε is a small constant.51If T > ω , only the first condition is relevant. If ω > T , only the second is relevant. Details on the derivation of theBR can be found in Appendix B.1.52Where G−i = ∑ j 6=i g j.53This assumes that k > 1, which is the case of our treatments.54It is shown in footnote 9 that for Mρ (g−i)>(β ρ/(1−ρ)k)T , only a symmetric equilibrium exists. In Appendix B.2 itis proven that for the case of Mρ (g−i)≤(β ρ/(1−ρ)k)T , nonsymmetric equilibria do not exist.362.3. Experimental Design0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789101112131415161718192045◦NoThresholdThreshold=30Threshold=60Mρ(g−i)giFigure 2.1. Best-response functions. The x-axis shows thegeneralized mean of others’ contributions; the y-axis dis-plays player i’s contributions. The figure plots the BR as afunction of others’ contributions, g∗i (g−i). The solid linesrepresent g∗i (g−i) of player i.2.3 Experimental DesignWe study the case of ρ = 0.54. This value corresponds to one of the HC treatments (two equilibria)considered in Chapter 1 (n = 4, β = 0.4 and ρ = 0.54). Under this parameterization there are twoequilibria: zero contribution and full contribution. We further distinguish treatments by varying theoutput threshold. In the baseline treatment the threshold is zero (T = 0). We also implement twotreatments with a positive threshold (T = 30 and T = 60). We ran three sessions per treatment.55In all treatments, at the end of the sessions subjects received 1 CAD for every 3 tokens.One key feature of the high-threshold treatment, T = 60, is that a decision from a single membermay determine whether the public good is produced or not. Specifically, if an individual contributeszero, the public good output will be below the threshold irrespective of other group members’contributions. Thus, contributing the entire endowment may be very costly relative to the othertreatments.5655For T = 0, two sessions belong to the ρ = 0.54 treatment in Chapter 1.56This aspect is analyzed in more detail in Section 2.4.372.3. Experimental Design2.3.1 Experimental proceduresIn each experimental session we recruited 16 subjects with no prior experience in any treatment ofour experiment. Subjects were recruited from the broad undergraduate population of the Universityof British Columbia using the online recruitment system ORSEE (Greiner, 2015). The subject poolincludes students with many different majors.Each session was administered in the following way: upon arriving at the lab, subjects wereseated at individual computer stations and given a set of written instructions; at the same time theinstructions were displayed on their computer screens.57 After reading the instructions subjectswere required to answer a set of control questions. The goal of the control questions was to ver-ify and measure each subject’s basic understanding of how to use the tools in his or her computerinterface and how to interpret information displayed on screens. Subjects received cash for answer-ing control questions correctly.58 The experiment did not proceed further until all participants hadanswered all control questions correctly.At the beginning of each round of the experiment, subjects were matched with three otherparticipants. They then played the static game described in Section 2.2. This game was repeatedover 20 rounds.To avoid reputation effects, we used an extreme version of the stranger matching protocol.The group composition was predetermined and unknown to the participants. We preselected thegroups so that the subjects were matched with a given participant in only four rounds and eachtime someone was matched with a participant he or she had encountered before, all other groupmembers were different. As a result, any given grouping of four players never occurred more thanonce.All nine sessions were computerized using the software z-Tree (Fischbacher, 2007). Giventhe difficulty of computing potential earnings using the nonlinear payoff function, we providedsubjects with a computer interface which eliminated the need to make hard calculations. Throughthis interface subjects were able to enter as many hypothetical choices and conjectures of othergroup members’ contributions as they wanted, visualizing the potential payoff associated with each57The instructions are in Appendix A.10.58The goal of these questions was to facilitate learning of the main features of the VCMC. Relevant features included(a) decreasing marginal returns in group production, given a fixed level of others’ contributions; (b) efficiency gains dueto coordination; (c) absence of a dominant strategy; and (d) understanding how the threshold operates. Subjects werecredited $0.20, $0.15, or $0.10 for each question answered correctly in, respectively, the first, second, and third attempt.There were 19 control questions in the baseline sessions and 22 when thresholds were present; they can be found inAppendix B.5.382.4. Resultscombination.59 In each round, subjects had 95 seconds to submit their chosen contribution. At theend of each round they were informed about their own earnings and the contribution choices ofother group members.60 At the end of the experiment, subjects received the payoff from a singlerandomly selected round.The sessions were conducted at the Experimental Lab of the Vancouver School of Economics(ELVSE) at the University of British Columbia in January and November 2015. The experimentslasted 90 minutes. Subjects were paid in CAD. On average, participants earned $34.21. Thisamount includes a $5 show-up fee and the additional cash received for the control questions.2.4 ResultsIn this section we examine the dynamic behavior of players’ contributions across treatments withdifferent thresholds. This evidence is helpful to draw inference about equilibrium selection.2.4.1 Average contributionsInitial contributionsEach solid line in Figure 2.2 represents the evolution of the average contribution over the 20 roundsof each individual treatment (dotted lines identify 95% confidence intervals). Initial contributionsin all treatments are above 10 tokens (half of the endowment). However, the average contribution inthe first round for the T = 60 treatment is higher relative to the other treatments, and this differenceis statistically significant when comparing T = 60 to T = 30 (Mann-Whitney test, p = 0.003).Given that subjects receive some training before the experiment starts, their understanding ofthe game should be reflected in their initial hypothetical contributions and beliefs about others’ con-tributions, which are likely to be treatment specific, keeping in mind that as the threshold increases,contributions should increase to reach it; added pressure to “make the threshold” may cause highcontributions to be a focal point. Table 2.1 confirms this hypothesis: the data from the payoff calcu-lator (nonchoice data) suggest that the generalized mean of the conjectures of others’ contributionsin T = 60 is highest both for the practice sessions and during the first round (that is, before subjects59Figure B.11 of Appendix B.4 displays a screenshot of the main interface.60Figure B.12 of Appendix B.4 shows the screenshot of the feedback given to subjects at the end of each round.Subjects were shown their overall income, as well as the breakdown between their private account income and groupaccount income. Given that group income is the same for each group member, subjects could easily infer the earningsof each of the other group members by looking at their contributions, reported in the same screen.392.4. Results2 4 6 8 10 12 14 16 18 2068101214161820T = 0T = 30T = 60RoundAvg.ContributionFigure 2.2. Average contribution over time. Evolution ofthe average contribution in each treatment (solid lines). Thedotted lines identify confidence intervals at the 95% confi-dence level.receive any feedback about group members’ contributions). With respect to the hypothetical owncontributions, they are lower on average in T = 30 than in the rest of the treatments.To verify the presumption that the threshold levels operate as focal points for subjects, in Figure2.3 we show histograms of the hypothetical public good output that subjects computed using thepayoff calculator. The left panel displays the hypothetical group payoffs that are below 30 tokens,while the right panel shows the hypothetical group payoffs that are below 60 tokens. The differ-ence between T = 0 and T > 0 illustrates that the introduction of a threshold has an effect on thehypothetical scenarios that subjects consider, pushing them to pay more attention to hypotheticalsin which the threshold is reached.Dynamics of the contributionsIn all treatments the average contribution increases over the rounds. Hence we observe a pattern ofconvergence towards the full-contribution equilibrium and there is no evidence of convergence tothe zero-contribution equilibrium. In the last 10 rounds, average contribution for T = 0 and T = 60does not differ much. For the case of T = 30, contributions are lower.To gain a more general picture of the dynamics of the contributions, we also analyze the dis-tribution of contributions. Figure 2.4 displays the cumulative distribution of contributions by treat-ment. The dashed lines show the cumulative distribution for rounds 1 to 10, and solid lines show402.4. ResultsTable 2.1Average Hypothetical Contribution and Conjecture AboutOthers’ ContributionsTreatment Variable Practice First RoundMean Std. Dev. Mean Std. Dev.T = 0gˆi 9.3 5.6 12.7 5.6Mρ(gˆ−i) 8.9 5.7 11.0 5.9T = 30 gˆi 9.7 5.4 11.2 5.4Mρ(gˆ−i) 9.0 4.8 8.8 4.6T = 60gˆi 11.0 5.7 12.1 5.2Mρ(gˆ−i) 11.6 4.3 12.4 4.4Note: The cells in the third and fifth columns report the average valuefor the hypothetical contributions and the generalized mean of the con-jectures of others’ contributions for the practice session and the firstround, respectively. The cells in the fourth and sixth columns reportthe standard deviation for the hypothetical contributions and general-ized mean of the conjectures of others’ contributions for the practicesession and the first round, respectively.Practice Rounds 1 to 10 Rounds 11 to 200510152025303540PercentageofOverallTrialsThreshold=0 Threshold=30 Threshold=60(a) Below T = 30Practice Rounds 1 to 10 Rounds 11 to 200102030405060708090100PercentageofOverallTrialsThreshold=0 Threshold=30 Threshold=60(b) Below T = 60Figure 2.3. Hypothetical public good output below the threshold. Each bar of the left (right) panel displays the numberof times that the hypothetical group payoff is below 30 (60) tokens as a percentage of overall trials for the T = 0 (inblue), T = 30 (in red), and T = 60 (in green) treatments.412.4. Resultsthe cumulative distribution for rounds 11 to 20. The plots show that contributions tend to con-centrate at one of the two extremes towards the end of the sessions. For T = 0 and T = 30, thedistribution shifts to the right for the last 10 rounds, whereas for T = 60 there is not much change.The Kolmogorov-Smirnov test shows that there is no significant difference in the T = 0 and T = 60distributions for the late rounds (p = 0.11). The median individual contribution is greater than 15tokens for rounds 11 to 20 in all treatments, while less than 4% of the contributions are equal to orless than 1 token in these late rounds. This is consistent with evidence shown in Figure 2.2; thereis no evidence of convergence to the zero-contribution equilibrium in any of the treatments.0 2 4 6 8 10 12 14 16 18 2000.250.50.751T = 0T = 30T = 60giCDFRounds 1-10 Rounds 11-20Figure 2.4. Cumulative distribution function. The dashedlines display the cumulative distribution function for the in-dividual contributions from rounds 1 to 10. The solid linesshow the cumulative distribution function for the individualcontributions from rounds 11 to 20.2.4.2 CoordinationThe fact that the confidence intervals for the average contributions are considerably wide (dottedlines in Figure 2.2) suggests that there is significant dispersion in contributions. This is more clearlythe case of T = 30 as shown in Figure 2.4. This motivates the analysis of Section 2.5, in whichwe analyze the behavior of different types of subjects in order to understand the decision-makingprocess of each group of participants.The lack of coordination may also reduce the monetary welfare due to two features of the422.4. Resultsframework: (a) for the same aggregate contribution, provision of the public good is more efficientif contributions are more uniform, and (b) it is more likely that the threshold is not met.Dispersion loss indexTo analyze the first feature, we follow the procedure of Chapter 1 and compare the DLIs for thedifferent treatments. The DLI is defined as follows:DLI =14 ∑4i=1 gi−(14 ∑4i=1 giρ)1/ρ10−(2021/ρ) .Figure 2.5 reports median DLI for each treatment. The value is averaged over four-round inter-vals to highlight the general tendency. The plots show that groups in T = 60 manage to coordinatebetter than subjects in other treatments. The difference in coordination is significant with respect tothe other treatments even in the first rounds; thus, coordination is not the result of a long learningprocess. This result is especially interesting for this high-threshold case: when T = 60 the mon-etary BR of subjects is to contribute zero if the average contribution of others is less than 10 andto contribute 20 otherwise. This could potentially increase dispersion within groups because thereare two different focal points. However, it appears that two offsetting motive are at play: (a) agentsinternalize the higher threshold and jointly move towards higher contributions, and (b) agents areaware of their veto power in terms of the provision of the public goods and thus they avoid con-tributing zero. In contrast, for the T = 30 treatment, the hurdle does not appear to be sufficientlyhigh to induce the same push towards higher contributions.Public good outputTo analyze the second feature we study provision of the public good. In principle, it would notbe surprising to frequently observe groups failing to reach the threshold, especially in the T = 60treatment. However, this turns out not to be the case. As shown in Figure 2.6, the public goodoutput was below threshold in only 7.5% of the rounds for the T = 30 treatment. When T = 60 thishappens in only 13% of the rounds. Such low occurrence is perhaps striking, but it can be explainedby noticing that the cost of undercontributing when other group members make high contributionsincreases with threshold size.The left panel of Figure 2.7 shows how much money is left on the table when an individualcontributes zero for different combinations of others’ contributions. Assuming that the average432.5. Understanding Deviations From the Monetary-Profit-Maximizing StrategyRounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2000.020.040.060.080.10.12MedianDLI kT=0 T=30 T=60 95% CIFigure 2.5. Dispersion loss index. This figure reports themedian DLI for T = 0, T = 30, and T = 60, averaged overfour-round intervals. The dotted lines display the 95% con-fidence interval.contribution of others is greater than 10 tokens, the cost of contributing zero relative to the monetaryBR increases monotonically with others’ contributions. At the extreme, if all other members fullycontribute, the cost of contributing zero is as high as 80 tokens.On the other hand, confirming the findings about dispersion using the DLI, groups achievedalmost the maximum level of provision (80 tokens or higher) in 44% of the rounds when T = 60.This happened in only 22% of the cases when T = 30, and in 33% of the cases when T = 0. Highcontributions are themselves quite surprising, as for a given player there are high potential lossesassociated with contributing high amounts. The right panel of Figure 2.7 shows the monetary lossof contributing 20 tokens (relative to the pecuniary BR). For T > 0, individuals who fully contributemay end up losing the entire endowment, which is costly relative to the T = 0 treatment.2.5 Understanding Deviations From theMonetary-Profit-Maximizing StrategyIn what follows we examine the information that subjects processed in the calculator in makingtheir choices. To do this we analyze the inputs subjects entered in the payoff calculator in eachround. Specifically, we focus on different hypothetical own contributions and on conjectures about442.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategy0 (0, 10] (10, 20] (20, 30] (30, 40] (40, 50] (50, 60] (60, 70] (70, 80] (80, 90](90, 100]0510152025303540Group AccountPercentageThreshold=0 Threshold=30 Threshold=60Figure 2.6. Histogram of group account output. This fig-ure shows relative frequency (in percentage) of the groupaccount output by treatment.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2001020304050607080Mρ(g−i)TokensNo Threshold Threshold=30 Threshold=60(a) Zero contribution0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2005101520Mρ(g−i)TokensNo Threshold Threshold=30 Threshold=60(b) Full contributionFigure 2.7. Monetary cost of zero contribution and full contribution. Each line of the left panel displays agent i’scost in tokens of contributing zero relative to the profit-maximizing strategy. The cost is equal to pi(ρ,g∗i ,Mρ (g−i))−pi(ρ,0,Mρ (g−i)). Each line of the right panel displays agent i’s cost in tokens of contributing ω relative to the profit-maximizing strategy. The cost is equal to pi(ρ,g∗i ,Mρ (g−i))−pi(ρ,ω,Mρ (g−i)).452.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategyother players’ contributions.Given the large heterogeneity in contributions, our first goal is to classify subjects in groupsaccording to how close their contributions are to the monetary BR. Then, we assess whether thereare systematic differences across groups in the use of the calculator. To compute the monetaryBR we have to characterize beliefs about others’ contributions and establish to what extent pastcontributions influence current conjectures. Table 2.2 displays the outcome of a regression of theaverage conjectures on the average of past contributions of other group members.Table 2.2Response of Subjects’ Conjectures to Others’ Contributions( 1n−1 ∑gρ−i)1/ρ 1n−1 ∑g−iF(g−i,t−1)0.444∗∗∗ 0.492∗∗∗(0.09) (0.09)F(g−i,t−2)0.262∗∗∗ 0.260∗∗∗(0.09) (0.10)F(g−i,t−3)-0.009 -0.028(0.10) (0.10)F(g−i,t−4)-0.098 -0.119(0.07) (0.08)F(g−i,t−5)-0.027 -0.053(0.10) (0.10)Constant 5.598∗∗∗ 6.039∗∗∗(1.95) (2.14)Observations 609 609Note: We estimate the following least-squares specification: F ( ˆgi,t) = C +∑5L=1 ALF (g−i,t−L)+ ui,t , where gˆiis a vector of player i’s conjectures about other group members’ contributions, g−i,t−L contains the vector ofcontributions made by other members in round t−L, C is a common constant, and ui,t is an idiosyncratic error. Welet the function F(·) be either the arithmetic or the generalized mean of degree ρ . The standard errors (reported inparentheses) are clustered by individuals and obtained by bootstrap estimations with 1,000 replications. *p < 0.1,**p < 0.05, ***p < 0.01. As a robustness check, we also estimate this specification including dummy variablesto control for different treatments. Results look very similar.The results show that conjectures are a function of other group members’ past contributionsand this dependence goes back for up to two rounds. This evidence is consistent with the resultsin Chapter 1. In what follows, we adopt the procedure used in Chapter 1 and classify subjectsinto two groups based on each individual’s proportional loss.61 A subject is denoted as Type 1 if61Appendix B.3 shows the observed contributions and the BR range per individual.462.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategythat individual’s proportional loss is not larger than the median value for his or her group. Theremaining subjects are denoted as Type 2.2.5.1 Average contribution by typeThe evolution of the contributions for different types is quite informative. Figure 2.8 plots theaverage contribution over rounds, by type.Type 1For Type 1 subjects, the contributions tend to increase over time in all treatments. Notice that forthe last rounds, there is no significant difference in terms of contributions. The average contributionin the last five rounds is 19.3, 18.6, and 19.2 tokens for T = 0, T = 30, and T = 60, respectively.Type 2In Chapter 1 we argue that subjects who undercontribute in the presence of high complementaritymight do so because of a competitive motive.62 However, as shown in Figure 2.8, our experimentalresults highlight interesting differences in Type 2 behavior across treatments. For the case of T =60, contributions are initially higher (relative to other treatments) and the average contributiondoes not change much over time. For T = 0, Type 2 subjects increase their contributions over time.Considering the last 10 rounds, the average contribution is very similar to the T = 60 treatment. Incontrast, for the case of T = 30, average contribution remains constant and its level is significantlylower than in the other two treatments.In the next subsection, we use calculator data to explore the motives for the deviations from theprofit-maximizing strategy. This analysis reveals some interesting heuristic rules followed by Type2 subjects in their decision-making process.2.5.2 Evaluating deviations from the pecuniary BRAs discussed in Chapter 1, deviations from the pecuniary BR may be due to simple confusion orto other motives. To discriminate among different drivers for deviations, we construct two distinctvariables for each individual:62In Chapter 3 we support the hypothesis that associates undercontributions with competitive motives.472.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategy1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2068101214161820RoundAvg.ContributionType 1 Type 2 T = 0 T = 30 T = 60Figure 2.8. Average contribution over rounds, by types.This figure shows the evolution of the average contribu-tion of Type 1 subjects in each treatment (solid lines). Thedashed lines identify the average contribution of Type 2 sub-jects.• Calculated Deviation from BR: gˆi− g∗(gˆ−i). This is the difference between hypotheticalchoices and the monetary BR, given the conjectures. It measures the accuracy of the mone-tary BR computed by subjects with the calculator. Values close to zero imply higher precisionin calculating the BR.• Actual Deviation from BR: gi−g∗(gˆ−i). This is the difference between observed choices andmonetary BR, given the conjectures. If this is not zero, then subjects either overcontribute orundercontribute.Figure 2.9 shows a series of scatter plots displaying the average value of Calculated Deviation fromBR and Actual Deviation from BR. We differentiate Type 1 (blue circles) and Type 2 (red squares)subjects. The plots show that Type 1 subjects do not deviate much from zero in the two dimensions;they are able to compute the pecuniary BR with accuracy and maximize monetary gains. There are482.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategy−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(a) T = 0−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(b) T = 30−20 −15 −10 −5 0 5 10 15 20−20−15−10−505101520Calculated Deviation from BRActualDeviationfromBRType 1 Type 2 45◦ line(c) T = 60Figure 2.9. Calculated Deviation from BR versus Actual Deviation from BR. The blue circles display the average Cal-culated Deviation from BR and Actual Deviation from BR for each Type 1 subject. The red squares display the averageCalculated Deviation from BR and Actual Deviation from BR for each Type 2 subject.492.5. Understanding Deviations From the Monetary-Profit-Maximizing Strategyno significant differences in the mechanical use of the calculator of Type 1 subjects in T = 0 andin T = 30 (top panel of Table 2.3), which suggests that the inclusion of a threshold does not havea great impact on Type 1 subjects’ behavior. For the case of T = 60, Type 1 subjects use thecalculator significantly less. This may simply be a byproduct of the fact that in this case the BRis either to contribute 0 or to contribute 20; this makes the decision easier for subjects who areattempting only to maximize their monetary payoffs and who have little or no competitive motive.For Type 2 subjects there is some heterogeneity across treatments. Subjects in the T = 0 andT = 60 treatments are very precise in calculating the monetary BR (median Calculated Deviationfrom BR are -0.5 and -0.2, respectively). However these subjects systematically undercontribute.Interestingly, deviations from BR are more pronounced when T = 0 (median Actual Deviation fromBR is -7 for T = 0 and -5.3 for T = 60). This suggests that Type 2 subjects are aware that deviationsfrom the pecuniary BR may be very costly when T = 60 (as confirmed in Figure 2.7). Indulgingtheir competitive motives comes with a stiff cost, and this may prompt them to contribute morethan in T = 0.For the case of Type 2 subjects in T = 30, the median of both Calculated Deviation from BRand Actual Deviation from BR is negative (-4.2 and -7.5). This indicates that median deviationsare consistent with the data entered in the calculator. In other words, Type 2 subjects are not veryprecise in calculating the pecuniary BR using the calculator and this is reflected in their observedchoices.Why are these subjects less precise in the low-threshold treatment? The lack of accuracy is inpart due to a less intensive use of the calculator (relative to the other treatments), as shown in thebottom panel of Table 2.3. Although Type 2 subjects activate the calculator in the same number ofrounds on average across treatments, for T = 30 they enter significantly fewer combinations (trials),hypothetical own contributions, and conjectures of others’ contributions than they do in the rest ofthe treatments. Another way to view these results is to consider the evidence of Chapter 1, whichsuggests that Homo behavioralis subjects exhibit competitive motives in the HC environments.When thresholds are low, competitive behaviors are relatively cheaper. Hence, some subjects seekto obtain the highest payoff in their group (competitive motive), but at the same time they wantto make sure that, given their conjecture about others’ contributions, their own contribution ishigh enough to guarantee that the threshold is met. Thus, these subjects may follow a two-stepprocedure: (a) enter a conjecture about others’ contributions and then (b) find the hypothetical owncontribution that corresponds to a public good output just above the threshold. In Table 2.4, wereport evidence corroborating this kind of behavior: Type 2 subjects in T = 30 use the calculator502.5. Understanding Deviations From the Monetary-Profit-Maximizing StrategyTable 2.3Differences in Mechanical Use of the Calculator, by Treatment WithinSubject TypeT = 0 T = 30 T = 60t-test(p-value)T = 0 vs T = 0 vs T = 30 vsT = 30 T = 60 T = 60Type1CalcRound 4.5 6.3 3.6 .11 .42 .05(0.57) (1.01) (.85)Trials 103.3 87 58.5 .48 .03 .09(18.14) (13.96) (9.06)Hyp 28.2 22.1 19.4 .28 .08 .56(4.07) (3.76) (2.76)Conj 8.9 8.5 4.5 .59 0 0(.46) (.62) (.5)Hyp Per Conj 11.4 11.5 14.6 .99 .27 .36(1.37) (2.31) (2.52)Type2CalcRound 4.3 4.2 7 .96 .13 .1(1.12) (1.04) (1.29)Trials 56.9 39.1 76.8 .08 .29 .04(7.91) (5.95) (16.72)Hyp 17.2 11.9 20.3 .06 .41 .02(2.14) (1.74) (3.06)Conj 8.2 6.7 5.4 .06 0 .14(.44) (.62) (.65)Hyp Per Conj 7.9 5.6 14.5 .05 0 0(.9) (.7) (2.01)Note: Each cell reports the average value for the respective category (standard er-rors are reported in parentheses). The t-tests of the means are reported in columns5, 6, and 7. CalcRound, number of rounds subjects used the calculator; Trials, num-ber of combinations subject entered in the calculator (each combination includes ahypothetical own contribution and a conjecture about others); Hyp, number of hy-pothetical own contributions; Conj, number of conjectures about others; Hyp perConj, number of own hypothetical contributions entered, given a conjecture aboutother players’ contributions. We include the practice rounds.twice as much to calculate the contribution that guarantees that the threshold is met than to computethe pecuniary BR. Table 2.4 also suggests that this type of reasoning is not very common for Type1 subjects, who use the calculator primarily to compute the pecuniary BR.512.5. Understanding Deviations From the Monetary-Profit-Maximizing StrategyTable 2.4Average Conjecture About Others’ ContributionsTreatment Subject Pecuniary- Contributing JustType BR Above ThresholdT = 0 1 21.8 NA2 18.8 NAT = 30 1 16.7 10.92 11 20.1T = 60 1 19.6 13.22 16.1 14.1Note: The cells in the third column report the number oftimes subjects’ hypothetical choice was equal to or less than1 token away from the pecuniary BR, given the conjectures,as a percentage of total trials in the calculator. The fourthcolumn reports the number of times subjects’ hypotheticalcontribution was equal to or less than 1 token away from thestrategy that guarantees that, given the conjectures, the groupaccount output is above the threshold. This is reported as apercentage of total trials in the calculator.2.5.3 Cost of deviations by type and treatmentAs shown in Figure 2.7, deviations from the pecuniary BR can be very costly in the presence ofthresholds. Table 2.5 confirms this by displaying the average payoff for different groups of subjectsand across treatments. For T = 0, the difference in payoffs between Type 1 and Type 2 is less than2 tokens: such a small difference means that Type 2 subjects may indulge their competitive motiveswithout much of a monetary setback. However, in the case of T > 0, the difference in earnings fordifferent types is significantly higher. Low contributions by Type 2 subjects in T = 30 not only areharmful for the subjects themselves, but they also reduce payoffs for Type 1 subjects.Table 2.5Average Payoff per Round (in tokens)T = 0 T = 30 T = 60Type 1 78.1 71.5 77(15.9) (18.7) (27.1)Type 2 76.3 66 71.8(14.1) (22.6) (27.8)Note: Each cell reports the average prof-its by type and for each treatment. Stan-dard deviations are displayed in paren-theses.522.6. Conclusions2.6 ConclusionsIn this chapter we investigate how subjects coordinate in a multiple equilibrium environment inwhich we vary the basin of attraction of the inefficient equilibrium. We do not find evidence ofconvergence to zero-contribution equilibrium in any of the treatments. When the threshold is set toa high level, subjects manage to coordinate on the sociably preferable equilibrium, and contributionlevels do differ much from the case in which there is no threshold.However, convergence to the full-contribution equilibrium is not perfect due to heterogeneityin contributions. As in Chapter 1, we identify two types of participants: Homo pecuniarius andHomo behavioralis. The former subjects behave in a similar manner irrespective of the thresholdlevel; they maximize their monetary payoffs by choosing contributions that are close to the BRto the contributions of other Homo pecuniarius subjects and to the contributions of the Homobehavioralis subjects.Homo behavioralis subjects undercontribute relative to the pecuniary BR. Deviations are lesspronounced when the threshold is set to a high level. This is consistent with the fact that themonetary cost of deviations increases with the threshold. For the low-threshold case, some Homobehavioralis subjects exhibit competitiveness by attempting to make the minimum contributionthat guarantees that the threshold is met given their beliefs about others’ contributions.53Chapter 3The Race to the Bottom in the VoluntaryContribution Mechanism WithComplementarity3.1 IntroductionEnvironments in which actions exhibit sufficient complementarity often lead to situations in whichmultiple equilibria arise. Stag hunt games are the classic example of such environments. In theirsimplest structure there are two equilibria, one payoff dominant and the other risk dominant. De-spite the payoff-dominant equilibrium being socially desirable, it is not always the prevailing out-come.63 In this chapter we explore whether making minimal modifications to the ex-post feedbackabout contributions and payoffs of other agents alters equilibrium outcomes. The analysis is per-formed in the context of the VCMC, introduced in Chapter 1. In the VCMC, as well as in most othervoluntary contribution frameworks, there is a tension between contributions and payoffs: when asubject reduces his or her contribution, he or she increases the probability of a higher relative pay-off within the group. Chapter 1 shows that this tension is reflected in undercontribution by Homobehavioralis subjects due to competitive motives. However, the VCMC has at least two featuresthat distinguish it from other voluntary contribution mechanisms (including the linear-technologycase, known as the linear voluntary contribution mechanism or LVCM): (a) allowing sufficientcomplementarity in players’ contributions creates a Pareto superior equilibrium in which all indi-viduals contribute their entire endowment (in addition to the no-contribution equilibrium), and (b)depending on the contributions of others, when a subject reduces his or her contribution, he or shemay also lower his or her overall payoff.To test for differential effects of variations in the type of feedback about individual outcomes63Equilibrium selection depends mainly on the basin of attraction of the equilibria (Van Huyck, 2008) and on theoptimization premium (Battalio et al., 2001).543.1. Introduction(contributions and payoffs), we implement four different treatments. All the treatments clearlyshow to subjects the individual contributions of all group members in the previous round, and differonly in the additional information that we choose to make salient. In the first treatment (baseline)we add no additional information; in the second treatment we explicitly display information aboutindividual payoffs from the previous round; in the third we let subjects look at the within-groupranking in terms of payoffs; and finally, in the fourth treatment we show subjects’ individual payoffsbut we do not explicitly link them to individual members’ contributions. The last treatment allowsus to test the importance of making salient the negative correlation between contributions andpayoffs, which is key in environments in which agents have to interact through costly coordination.It is important to emphasize that in all these treatments we vary only the saliency of the information.Even in the baseline, in which we display data only on individual contributions, it is fairly easy torecover what the individual payoffs are—if one has the inclination to do so. Payoffs are equal tothe difference between the endowment and contributions, plus the constant group earnings, whichare the same for all players within a group. The way the payoff is determined is clearly explainedbefore any interaction takes place, and subjects’ comprehension is tested before the start of theexperiment through a set of control questions.The results suggest that making information more salient does matter. When subjects observeonly individual contributions in the previous round, we find that contributions increase over timeand the average contribution converges to the socially desirable outcome. In contrast, when subjectsare given additional information about individual payoffs or individual rankings (in addition to thecontributions), average contributions tend to become lower as rounds proceed and do not convergeto any of the equilibria. Interestingly, when subjects observe both individual contributions andpayoffs but the correspondence between them is omitted, contributions also decrease over time, butfar less than in the cases in which outcomes are directly linked to individual contributions.Acknowledging that when subjects reduce their contributions it is not possible to distinguishbetween their intention of obtaining the highest absolute or relative payoff, we implement an LVCMin which the dominant strategy is full contribution. In this case the monetary-payoff-maximizingstrategy does not depend on others’ contributions. We find that when no additional information isshown, contributions converge towards the equilibrium, whereas when we display the payoffs ofothers, contributions exhibit a slight decrease over time.Using a wealth of nonchoice data we also examine the differences in the decision-makingprocesses of subjects in different treatments. We divide subjects into two groups following theprocedure in Chapter 1. The evidence suggests that both groups of subjects are able to calculate553.2. Environment and Experimental Designthe best response (BR) to their conjectures and that deviations from the pecuniary BR are due tocompetitive motives and not to confusion or imitation of the best performers.The chapter is organized as follows. Section 3.2 describes the framework based on Chapter1, introduces the experimental design, characterizes the different treatments, and discusses thelaboratory procedures. In Section 3.3 we report results from aggregate data. We find that disclosinginformation about others’ choices and outcomes deteriorates cooperation significantly. Section 3.4analyzes individual-level behavior. We explore different hypotheses that explain deviations fromthe pecuniary BR using choice and nonchoice data. Section 3.5 concludes.3.2 Environment and Experimental DesignThe experimental design is based on the VCMC on Chapter 1. In this framework there are n indi-viduals, indexed by i ∈ {1, ...,n}, who are given an endowment ω and must decide whether—andhow much—to invest in a group account which maps private contributions into an output equallyshared among group members. Let gi denote individual i’s contribution to the group account. Theremainder of the endowment is deposited in a risk-free private account which yields no interest.Individual investments in the group account are aggregated through a constant elasticity of substi-tution production function, which we refer to as VCMC. Player i’s payoff is given bypii = ω−gi +β(n∑i=1gρi)1/ρ, (3.1)where ρ denotes the degree of complementarity and β is a constant. By setting the degree ofcomplementarity sufficiently high, ρ < ln(n)ln(n/β) , two Nash equilibria arise: zero contribution and fullcontribution.64,653.2.1 TreatmentsThe treatment design features two sets of parameters: (a) in the VCMC treatments, ρ = 0.54and β = 0.4 , and (b) in the LVCM treatments, ρ = 1 and β = 1.25. For each set, the onlytreatment variation is the extent of the feedback subjects get at the end of each round. Table 3.164Given the baseline parameters, this happens when ρ < 0.602.65See Chapter 1 for details on the derivation of the equilibrium.563.2. Environment and Experimental Designsummarizes the treatment design.66 The Show Contr treatment is the baseline treatment:67 atthe end of each round subjects observe their own contribution, the contributions of other groupmembers, and their own payoff. All treatments provide this basic information. In addition, in theShow Payoffs treatment, subjects are provided with an explicit display of the payoffs of other groupmembers; in the Show Ranking treatment, subjects can look at the ranking in terms of payoffs of allgroup members. Finally, in the No Link treatment, they are given the list of individuals’ payoffs;however, these payoffs are not directly linked to the contributions. Appendix C.4 shows screenshotsof the feedback given to subjects at the end of each round for the different treatments.Table 3.1Experimental TreatmentsTreatmentNumber ofβ ρAdditional Exchange RateSessions Feedback (tokens per CAD)Show Contr 30.4 0.54- 3Show Payoffs 2 Payoffs of others 3Show Ranking 2 Ranking of others 3No Link 2Payoffs of others3no link to contributionsShow Contr Linear 21.25 1- 4Show Payoffs Linear 2 Payoffs of others 4For the VCMC the BR depends on the generalized mean of degree ρ of others’ contributionsand on the parameters of the model (ρ,β , and n). Figure 3.1 summarizes the BR, g∗(g−i). Thegeneralized mean contribution of other group members is measured on the x-axis, and player i′scontribution is reported on the y-axis. The solid line represents the BR of player i. For ρ = 0.54 andβ = 0.4, the two equilibria are located at the intersection of the black solid line and the 45-degreeline (gi = 0 and gi = ω). The red line displays the BR for the linear case (ρ = 1 and β = 1.25), inwhich the dominant strategy (and unique equilibrium) is full contribution.66We adjusted the exchange rate so the maximum possible payoff is equivalent (in CAD) in both the VCMC andLVCM treatments.67Two of the sessions of this treatment correspond to the HC (ρ = 0.54) treatment in Chapter 1.573.2. Environment and Experimental Design0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2001234567891011121314151617181920ρ = 1, β = 1.25ρ = 0.54, β = 0.445◦Mρ(g−i)giFigure 3.1. Best-response functions. In this figure the x-axisshows the generalized mean of others’ contributions; the y-axis displays player i’s contributions. The figure shows theBR as a function of others’ contributions, g∗i (g−i). The solidlines represent g∗i (g−i) of player i.3.2.2 Experimental proceduresIn each experimental session we recruited 16 subjects with no prior experience in any treatment ofour experiment. Subjects were recruited from the broad undergraduate population of the Universityof British Columbia using the online recruitment system ORSEE (Greiner, 2015). The subject poolincludes students with many different majors.Each session developed in the following way: upon arriving at the lab, subjects were seated atindividual computer stations and given a set of written instructions; at the same time the instructionswere displayed on their computer screens.68 After reading the instructions, subjects were requiredto answer a set of control questions. The goal of the control questions was to prove that subjects hada basic understanding about how to use the tools in their computer interfaces and how to interpretthe information displayed on their screens. Subjects received cash for answering control questionscorrectly.69 The experiment did not proceed further until all participants had answered all controlquestions correctly.At the beginning of each round of the experiment, subjects were matched with three other68The instructions can be found in Appendix A.10.69They were credited $0.20, $0.15, and $0.10 for each question answered correctly in, respectively, the first, second,and third attempt. There were 19 control questions.583.3. Resultsparticipants. Subjects then played the static game described above with payoffs determined as inEquation 3.1. This game was repeated over 20 rounds. To avoid reputation effects, we used astranger matching protocol. The group composition was predetermined and unknown to the partic-ipants. We preselected the groups so that the participants were matched with any other participantin only four rounds and each time a participant was matched with a participant he or she hadencountered before, all other group members were different.All 13 sessions were computerized, using the software z-Tree (Fischbacher, 2007). Giventhe difficulty of computing potential earnings using the nonlinear payoff function, we providedsubjects with a computer interface which eliminated the need to make calculations. Through thisinterface subjects were able to enter as many hypothetical choices and conjectures as they wanted,visualizing the potential payoff associated to their own, and the other group members’, choices.70During each round subjects had 95 seconds to submit their chosen contribution. At the end ofeach round they were informed about their own earnings and the contribution made by other groupmembers. This information is sufficient to compute other players’ payoffs, even if they were notexplicitly displayed. Subjects also saw additional information conditional on the specific treatmentthey were in (Table 3.1). At the end of the experiment, subjects were paid based on the payoff of asingle randomly selected round.The sessions were conducted at the Experimental Lab of the Vancouver School of Economics(ELVSE) at the University of British Columbia in January and November 2015. The experimentslasted 90 minutes. Subjects were paid in CAD (the exchange rate is specified in Table 3.1). Onaverage, participants earned $30.20. This amount includes the $5 show-up fee and the additionalcash received for the control questions.3.3 Results3.3.1 Contributions over timeFigure 3.2.a shows the evolution of the average contribution in each of the VCMC treatments. Theaverage contribution increases over time for the Show Contr treatment, but it appears to decreaseover time for the rest of the treatments. There is no statistical difference in the first round’s con-tributions across treatments (Mann-Whitney tests, p > 0.2). This is not surprising considering thattreatments differ only in the feedback provided to the subjects after a round is played, and at the70Figure C.15 of Appendix C.3 shows a screenshot of the main interface.593.3. Resultstime subjects submit their first choice they have not seen any screen feedback. However, in thelast period the difference between average contribution in the Show Contr and any of the othertreatments is significant (Mann-Whitney tests, p < 0.004).On the other hand, contributions in the No Link treatment are significantly higher than in theShow Payoffs treatment, despite the fact that the only difference between these treatments is thatin the former we list the individual payoffs but do not associate them with individual contributions.This suggests that the negative correlation between contributions and payoffs becomes more salientto subjects when there are connected. Likewise, revealing the rankings (instead of the payoffs) hasa stronger impact on behavior.Figure 3.2.b displays the average contribution over time for the linear treatments. In this caseinitial contributions are initially high in both treatments (Mann-Whitney tests, p = 0.2). As roundselapse, average contributions slightly increase for the Show Contr Linear treatment and slightlydecrease for Show Payoff Linear treatment; focusing on the last five rounds, the average con-tribution is 16.3 tokens and 14.6 tokens for the latter. This difference is statistically significant(Mann-Whitney tests p = 0.05).711 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2002468101214161820Show ContrShow PayoffsShow RankingNo-LinkRoundAvg.Contribution(a) VCMC1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2002468101214161820Show Contr LinShow Payoffs LinRoundAvg.Contribution(b) LVCMFigure 3.2. Average contribution over time. These figures show the evolution of the average contribution in eachtreatment.A similar pattern can be observed for the whole distribution of contributions (Figure 3.3). For71Saijo and Nakamura (1995) were the first to run an LVCM experiment with β > 1. They did not find perfectconvergence to the full-contribution equilibrium.603.3. Resultsthe VCMC, even in the first 10 rounds, subjects contributed less in the Show Payoffs, Show Rank-ing, and No Link treatments than in Show Contr. The median contributions are 10, 8, 15, and 17tokens, respectively. For the last 10 rounds, the distribution for the Show Contr treatment shiftsoutwards, implying an increase in contributions. For the rest of the treatments the distributionsshift inwards, which means that there is a reduction in contributions. There is a similar pattern forthe linear treatments: for Show Contr Linear the distribution shifts inwards, whereas for the ShowPayoffs Linear the distribution shifts to the left.0 2 4 6 8 10 12 14 16 18 2000.250.50.751Show ContrShow PayoffsShow RankingNo LinkgiCDFRounds 1-10 Rounds 11-20(a) VCMC0 2 4 6 8 10 12 14 16 18 2000.250.50.751Show Contr LinShow Payoffs LingiCDFRounds 1-10 Rounds 11-20(b) LVCMFigure 3.3. Cumulative distribution function. The dashed lines display the cumulative distribution function for theindividual contributions from rounds 1 to 10. The solid lines show the cumulative distribution function for the individualcontributions from rounds 11 to 20.3.3.2 CoordinationGiven that in our VCMC setup there are multiple equilibria, coordination becomes an importantissue. In principle the degree of coordination should not vary in the different treatments. Toevaluate this hypothesis we follow the procedure in Chapter 1 and calculate the DLI. The DLI isdefined asDLI =14 ∑4i=1 gi−(14 ∑4i=1 giρ)1/ρ10−(2021/ρ) .613.3. ResultsRounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2000.050.10.150.20.250.3MedianDLI kShow Contr Show Payoffs Show Ranking No Link 95% CIFigure 3.4. Dispersion loss index. This figure reports themedian DLI for the different treatments, averaged over four-round intervals. The dotted lines display the 95% confidenceinterval.The numerator of the DLI captures the dispersion in contributions and the denominator nor-malizes the index, so the minimum is 0 and the maximum is 1. Figure 3.4 reports the four-periodmean of the median DLI by treatment, and it is also superimposes the 95% confidence interval.72The figure illustrates that in the Show Contr sessions, dispersion decreases over time: subjectswere able to better coordinate on the efficient outcome towards the end of the experiment. Thisis not the case for the three treatments in which dispersion increases over time, suggesting thatheterogeneity in contributions becomes more prevalent. The No Link treatment is the one with thehighest DLI, which implies that not disclosing the association between actions and outcomes mayintroduce some disagreement on the dimension on which subjects plan to coordinate.The lack of coordination and the lower level of the contributions have a significant effect onthe earnings of subjects in the VCMC treatments. This can be seen in Table 3.2. Subjects in the NoLink treatment earn 13 tokens less on average relative to the baseline, while dispersion of payoffsis the highest. Subjects in the Show Ranking treatment earn 33 tokens less on average relative tothe baseline. However, earning dispersion is the lowest, which confirms the evidence of Figure 3.4,which suggests that subjects manage to coordinate, although in a low contribution level.72The confidence interval was calculated using a binomial-based method. We also computed the confidence intervalby randomly selecting 500 samples with replacement for the 1,820 combinations by round/session. We get very similarresults.623.4. How Do Subjects Make Choices?Table 3.2Average Payoff per Round (in tokens)Treatment Mean StandardDeviationShow Contr 77.2 15Show Payoffs 52 14.9Show Ranking 44.5 12.3No Link 64.2 17.7Show Contr Linear 84.2 13.6Show Payoffs Linear 85.5 13.4Note: Each cell of the second column reports theaverage per-round payoff by treatment (in tokens).Each cell of the third column reports the standarddeviation of the payoffs.In the case of the linear treatments, payoffs are not significantly different in the two treatments.3.4 How Do Subjects Make Choices?The experimental design gives subjects access to profit calculators. This reduces the difficulty ofmapping own hypothetical contributions and conjectures about others’ contributions to potentialpayoffs. Using the calculator, subjects can try out as many conjectures and hypothetical contribu-tions as they want. Whenever a new combination is entered, the potential payoff is displayed onthe computer screen.73 We recorded all calculator inputs entered by subjects. This allowed us togather information about the reasoning of subjects before they submitted their choices and abouthow they learned throughout the experiment. In particular, we can explore what types of conjec-tures they make about other players, whether these conjectures are affected by past experiences andhistory of contributions, and how subjects adjust their hypothetical choices given some conjectures.Moreover, this analysis allows us to rule out the possibility that different behaviors in the distincttreatments are due to a differences in calculator usage.3.4.1 TypesGiven the heterogeneity in contributions in the different treatments, we follow the procedure inChapter 1 and divide the participants into two types according to how close their contributions are73In this type of experiment, subjects are usually provided with a payoff table. For this experiment a payoff table doesnot cover the entire payoff space, unless conjectures are symmetric.633.4. How Do Subjects Make Choices?to the pecuniary BR. Our data support the findings in Chapter 1 that the pecuniary BR is a functionof the contributions of other group members in the past two rounds.74 Calculating the pecuniaryBR allows us to compute the average proportional loss by individual, which we use as a criterionto classify subjects into two groups. An individual is classified as Type 1 if his or her proportionalloss is lower than the median value of the treatment. Otherwise he or she is classified as Type 2.75Contribution by typesWe analyze whether different types change their behavior as the experiment progresses. Figures3.5.a and 3.5.b show the average contribution for each subject type for the VCMC treatments. Forthe Show Contr treatment, Type 1 and Type 2 subjects slightly increase their contribution over time.Interestingly, in the rest of the treatments, average contribution decreases over time for both types.For the LVCM treatments, the average contribution of Type 1 subjects is close to 20 and doesnot vary much over time in either treatment. Type 2 subjects moderately increase their contributionsover time in Show Contr Linear and considerably decrease their contributions in Show PayoffLinear.3.4.2 Why do subjects reduce their contributions?In this subsection we analyze the possible motives for decreases in contributions in the sessions inwhich information about others’ payoffs or ranking is disclosed.Money-profit-maximizing and competitivenessThe evidence presented in Chapter 1 suggests that Type 1 subjects (Homo pecuniarius) and Type2 subjects (Homo behavioralis) have different arguments in their utility function. The former aremoney-profit maximizers, whereas the latter value their relative payoff in addition to the pecu-niary rewards. In Chapter 1, Homo pecuniarius’ contributions converge to the full-contributionequilibrium, whereas Homo behavioralis subjects undercontribute systematically. One feature ofthe VCMC is that when other group members reduce their contribution, the set of rationalizablecontributions (BR range) expands, which is not the case in the LVCM. In Appendix C.2 we plotthe actual contributions and the BR range per subject and over time. When comparing the area ofthe BR range for the different treatments, it is clear that on average the BR ranges of the baseline74See Table C.1 in Appendix C.1.75See Chapter 1 for details on the calculation of the proportional loss.643.4. How Do Subjects Make Choices?Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2002468101214161820Avg.ContributionShow Contr Show Payoffs Show Ranking No Link(a) Type 1 (VCMC)Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2002468101214161820Avg.ContributionShow Contr Show Payoffs Show Ranking No Link(b) Type 2 (VCMC)Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2002468101214161820Avg.ContributionShow Contr Linear Show Payoffs Linear(c) Type 1 (LVCM)Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16 Rounds 17-2002468101214161820Avg.ContributionShow Contr Linear Show Payoffs Linear(d) Type 2 (LVCM)Figure 3.5. Average contribution over time by type. These figures show the evolution of the average contribution in eachtreatment (solid lines). The left panel displays the average for Type 1 subjects; the right panel shows the average forType 2 subjects.653.4. How Do Subjects Make Choices?treatment are smaller.76 This may justify lowering contributions even for subjects whose objectiveis to maximize the pecuniary payoff. The evidence of Table 3.3 supports this. It can be seen thatthe proportional loss does not vary much across treatments within the types and it is close to zerofor Type 1, which suggests that even Homo pecuniarius subjects who reduce their contributionsmay be driven by money-profit-maximizing motives.In the linear treatments, the contributions of other group members in past rounds do not affectthe BR range. Thus, in this case we would not expect any change in the behavior of Homo pecu-niarius. This is confirmed by Figure 3.5.c, in which it can be seen that the average contribution isvery close to 20 for both treatments.Table 3.3Average Proportional LossTreatment Type 1 Type 2Show Contr 0.07 1.99Show Payoffs 0.24 1.98Show Ranking 0.12 1.91No Link 0.07 2.70Show Contr Linear 0.05 2.33Show Payoffs Linear 0.03 2.10Note: Each cell reports the average proportional loss per treatment andtype .In terms of competitive motives, a relevant question is whether competitiveness of Homo be-havioralis subjects is magnified when the payoffs or ranking of other players is displayed. This iskey to explaining the differences across treatments in the aggregate data. To study this, we lookat the frequency in which subjects’ contributions in round t are less than or equal to the lowestcontributions in their group in rounds t−1 and t−2. Table 3.4 suggests that showing the payoffsor ranking increases competition among Homo behavioralis, whereas Homo pecuniarius subjectsrarely make use of this strategy, even when the payoffs or ranking of others are visible.7776The BR range includes the set of contributions that are rationalizable given the contributions of the other groupmembers in t−1 and t−2.77Retaliation can explain this behavior as well. However, this is hard to justify since groups are shuffled in each round.A subject who reduces his or her contribution in period t will not be able to “punish” the subjects that undercontributedin t−1.663.4. How Do Subjects Make Choices?Table 3.4CompetitivenessTreatment Type 1 Type 2Show Contr 0.7 26.3Show Payoffs 7.6 29.3Show Ranking 3.3 41.8No Link 0.3 32.2Show Contr Linear 1.0 30.3Show Payoffs Linear 1.0 32.9Note: Each cell displays the percentage of rounds for which the invest-ment of a player in round t was equal or lower than the minimum invest-ment of the other group members in rounds t−1 and t−2 .Imitation of best performersSome of the experimental literature has focused on learning by imitating the strategy of other play-ers. This has been tested extensively in experimental oligopoly economies. Huck et al. (1999) andAltavilla et al. (2006) find evidence that suggests that agents imitate the best performers. Withrespect to voluntary contribution experiments, Bigoni and Suetens (2012) study the effects of feed-back on the contribution levels in public good experiments with a linear technology. They findthat displaying information about others’ earnings, rather than showing only others’ contributions,lowers contributions. They claim that this is explained in part by the fact that when subjects seeothers’ payoffs, they tend to imitate the best performers.One should be cautious when interpreting our results as a consequence of subjects’ imitating thebest performers. There is a clear distinction between the LVCM with a dominant strategy of zerocontribution and our framework. In the former, by imitating the best performer, a subject increasesboth absolute and relative monetary profit. Thus, disclosing information about the payoffs of othersmay accelerate the convergence to the equilibrium, as subjects have an additional way to learn howto maximize their monetary profits.This is not the case in the VCMC or the LVCM with a dominant strategy of full contribution.By imitating the best performers, subjects increase their relative payoff but they do not maximizetheir monetary payoff. On the other hand, if subjects do not understand the rules of the game, theymay not make a distinction between maximizing relative and absolute payoff. Thus, imitating thebest performers makes sense. However, one feature of our experimental design is that subjects areable to learn how to maximize their profit even before the experiment starts.In the next subsection we show that most of the subjects are able to calculate the pecuniary-673.4. How Do Subjects Make Choices?profit-maximizing strategies given their conjecture about others’ contribution. Therefore, we ruleout the possibility that undercontributions are driven by imitation.3.4.3 Use of the calculatorMechanical use of the calculatorWe start by studying differences in the use of the calculator across the different treatments. Ta-ble 3.5 shows (a) the number of rounds in which the calculator was activated, (b) the number ofdifferent hypothetical own contributions, and (c) the number of conjectures about other agents’contributions by treatment and type. The four rightmost columns report whether the average vari-able is significantly different from baseline. The outcome of the tests suggests that for the VCMCtreatments, subjects in the Show Ranking treatment use the calculator more intensively than thosein the baseline. Analogously, subjects in the Show Payoffs treatment use the calculator less thanthose in the baseline.78 For the linear treatments, there are no significant differences across treat-ments.A feature of the Show Payoffs treatment is the fact that the payoffs of the other groups membersare explicitly displayed as part of the feedback information; this eliminates the need to computethe payoffs of others with the calculator and can explain in part the difference in calculator usagebetween the Show Contr or Show Ranking treatments and the Show Payoffs treatment. Subjectsused the calculator to compute the payoffs of other group members in the previous round in 103trials (out of 1,226) and in 77 (out of 529) in the Show Ranking and Show Contr treatments,respectively. This is significantly higher than in the Show Payoffs treatment, in which subjects didthis in only 3 out of 338 trials.79These patterns are also confirmed when studying the response time discussed below.78The difference in calculator usage is not statistically significant.79This is also the case in the linear treatments. In Show Contr Linear, subjects computed the payoffs of others in 62out of 169 trials, whereas in Show Payoffs Linear they did so in 32 out of 182 trials.683.4. How Do Subjects Make Choices?Table 3.5Differences in Mechanical Use of the Calculator, by Treatment Within Subject Typet-testShow Show Show No Show Show (p-value)Contr Payoffs Ranking Link Contr Payoffs (1) (1) (1) (5)(Linear) (Linear) vs vs vs vs(1) (2) (3) (4) (5) (6) (2) (3) (4) (6)Type1CalcRound 4.5 5.3 9.7 4.6 2.3 1.1 .57 0 .89 .34(.57) (1.47) (1.67) (1.2) (1.24) (.37)Hyp 28.2 16.7 31.6 28.5 11.3 13.7 .04 .59 .97 .4(4.07) (2.22) (4.34) (5.83) (1.91) (2.16)Conj 8.9 8.6 10.8 9.3 6.3 6.3 .7 .01 .54 .95(.46) (.73) (.46) (.55) (.71) (.76)Type2CalcRound 4.3 2.3 7.9 3.4 4.4 5.9 .19 .07 .58 .5(1.12) (.69) (1.73) (1.07) (1.5) (1.69)Hyp 17.2 19.9 22.8 20.9 15.1 17.6 .5 .14 .38 .63(2.14) (3.82) (3.24) (4.11) (3.56) (3.82)Conj 8.2 7.5 9.4 8.8 6.5 6.7 .45 .11 .46 .85(.44) (.85) (.61) (.82) (.71) (.72)Note: Each cell reports the average value for the respective category (standard errors are reportedin parentheses). The t-tests of the means are reported in columns 8 to 11. CalcRound, number ofrounds subjects used the calculator; Hyp, number of hypothetical own contributions; Conj, number ofconjectures about others. We include the practice rounds.Response timeThe time taken by a subject to provide a response can also convey information about the waychoices are made. To the extent that subjects learn and adjust their behavior based on the infor-mation received at the end of each round, there should be observable differences across treatmentsin the time participants spend looking at the information. In treatments in which subjects observeinformation about others’ payoffs, they may spend more time making sense of this feedback. Someof them may try to confirm the mapping between contributions and payoffs. Others might find thatthe extra information confounds their decision making as it highlights more clearly the conflictingaspects of the game (i.e., the negative correlation between contributions and payoffs). The exper-imental design records both the time a subject spent submitting a contribution and the time he or693.4. How Do Subjects Make Choices?she spent looking at the feedback information on the screen.As mentioned before, we recorded the time subjects spent before making their contributionchoices and the time they spent looking at the feedback information. The second column of Table3.6 shows the average response time across treatments and the third column shows how long, onaverage, subjects visualized the feedback on their screen. Some clear differences are apparent.Subjects in the Show Ranking treatment spent more time before logging in their choices, but thereis no significant difference among the other treatments. In terms of feedback, subjects in the ShowContr treatment spent the least amount of time looking at the feedback information, whereas par-ticipants in the Show Ranking treatment spent the most. This is consistent with the hypothesis thatsubjects may attempt to explore more carefully the link between contributions and payoffs, andsuggests that subjects do not disregard the additional information.80Table 3.6Average Response Time and Feedback TimeDecision FeedbackTreatment Time Time(seconds) (seconds)Show Contr 11.5 6.3Show Payoffs 12.4 9.0Show Ranking 22.1 10.9No Link 12.2 7.6Show Contr Linear 8.6 3.8Show Payoffs Linear 10.1 6.2Note: Each cell in the second column reports the numbers of seconds onaverage subjects spent before submitting their choices. Each cell in thethird column reports the number of seconds on average subjects looked atthe feedback information.Are subjects able to calculate the pecuniary BR to their conjectures?In Chapter 1 we analyze the payoff-relevant use of the calculator, specifically whether the hypothet-ical contributions are consistent with the pecuniary BR to the conjectures about others’ contribu-tions (Calculated Deviation from BR) and whether actual contributions correspond to the pecuniaryBR to the conjectures (Actual Deviation from BR). Table 3.7 reports the median value for these twomeasures by treatment and by type. For the VCMC treatments, we confirm the findings in Chapter1: both Type 1 and Type 2 subjects are able to find the pecuniary BR to their conjectures using the80This is similar for the linear treatments; subjects in the Show Contr Linear spent less time submitting the choicesand looking at the feedback than those in Show Payoffs Linear.703.5. Conclusionscalculator. The median value for the Calculated Deviation from BR is close to zero and there is notmuch variation across treatments within types.Subjects in the linear treatments are less precise calculating the BR. Here, the use of the calcu-lator is not essential and it is reflected in less intensive use, as 40% of the subjects did not use thecalculator after the practice period. This is considerably higher than in the VCMC, in which only14% of the subjects did not activate the calculator at all.Table 3.7Payoff-Relevant Use of the CalculatorType 1 Type2Treatment Calculated Actual Calculated ActualDeviation Deviation Deviation Deviationfrom BR from BR from BR from BRShow Contr -0.1 0 -1.1 -6.4Show Payoffs -0.7 -1.5 -1.7 -6.3Show Ranking -0.7 -0.3 -0.6 -8.8No Link -0.2 -0.1 -1.9 -5.2Show Contr Linear -3.6 0 -7.5 -9.2Show Payoffs Linear -0.7 0 -5.9 -5.6Note: Columns 2 and 4 display the median Calculated Deviation from BR for Type 1 subjectsand Type 2 subjects, respectively. Columns 3 and 5 display the median Actual Deviation fromBR for Type 1 subjects and Type 2 subjects, respectively.3.5 ConclusionsIn this chapter we study the effect of information about other players’ actions and outcomes onprivate provision of public goods. We implement experiments using two different frameworks: (a)VCMC with two Nash equilibria (zero contribution and full contribution) and (b) LVCM with aunique Nash equilibrium of full contribution. A key feature of these environments is that strategiesthat maximize the relative payoff within a group do not necessarily maximize the monetary payoff.Thus, changes in the information provided to the agents may have significant repercussions on thedynamics of provision since they may affect convergence to the equilibrium and may impact theequilibrium selection (in the multiple equilibria case). This is not the case in the standard LVCMwith a unique Nash equilibrium of zero contribution, in which changes in information alter onlythe speed of convergence to the equilibrium (see Bigoni and Suetens, 2012).We find in our experiment that disclosing information about others’ earnings in the previousround tends to decrease contributions. This is essential to the support of the findings of Chapter 1,713.5. Conclusionswhich suggest that Homo behavioralis subjects undercontribute due to competitiveness. We findthat revealing subjects’ rankings has an even a bigger impact on contributions, perhaps becausefree-riding behavior becomes more obvious and nobody wants to be the “sucker” of the group.In this treatment, subjects coordinate in low levels of contributions and the dispersion of payoffsacross participants is the lowest. 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Finally, defining k ≡(n−1βρρ−1−1) 1ρyieldsg∗i (g−i) = k(∑gρ−in−1)1/ρ.The second order condition78A.1. Derivation of the BR∂ 2pii∂g2i= (1−ρ)β (gρi +∑gρ−i) 1−ρρ −1 g2(ρ−1)i +(ρ−1)β (gρi +∑gρ−i) 1−ρρ gρ−2i= (ρ−1)β (gρi +∑gρ−i) 1−ρρ gρ−2i(1− gρigρi +∑gρ−i)< 0implies concavity of pii.A.1.1 Absence of Nash equilibrium in mixed strategiesA symmetric Nash equilibrium in mixed strategies is a joint distribution µn−1 over g−i such thati is indifferent between all gi ∈ supp(µ). In other words, for any two strategies, g′i and g′′i , in thesupport of µ , it must be thatω−g′i +βˆsupp(µn−1)(g′ρi +∑gρ−i)1/ρdµn−1 (g−i) = ω−g′′i +βˆsupp(µn−1)(g′′i +∑gρ−i)1/ρdµn−1 (g−i) .We will show that g∗i , the BR of player i to µn−1 , is a singleton, and therefore there is nosymmetric Nash equilibrium in mixed strategies. The first order condition is∂pii(gi,µn−1 (g−i))∂gi=−1+βˆsupp(µn−1)(gρi +∑gρ−i) 1−ρρ gρ−1i dµn−1 (g−i) = 0.The second derivative of player’s i payoff is∂ 2pii(gi,µn−1 (g−i))∂g2i== βˆsupp(µn−1)((1−ρρ)(gρi +∑gρ−i) 1ρ −2 ρgρ−1i gρ−1i +(gρi +∑gρ−i) 1−ρρ (ρ−1)gρ−2i)dµn−1 (g−i)= βˆsupp(µn−1)((1−ρ) gρi(gρi +∑gρ−i) (gρi +∑gρ−i) 1ρ −1 gρ−2i + (gρi +∑gρ−i) 1ρ −1 (ρ−1)gρ−2i)dµn−1 (g−i)= βˆsupp(µn−1)((1−ρ)(gρi +∑gρ−i) 1ρ −1 gρ−2i(gρi(gρi +∑gρ−i) −1))dµn−1 (g−i)< 0.That is, pii(gi,µn−1 (g−i))is globally strictly concave and g∗i is a singleton. It follows that thereis no symmetric Nash equilibrium in mixed strategies.79A.2. Best-Response Range and ContributionsA.2 Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 Contribution Use calculator Do not use calculatorTokensRoundFigure A.1. Session 1 (LVCM)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 Contribution Use calculator Do not use calculatorTokensRoundFigure A.2. Session 2 (LVCM)80A.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.3. Session 3 (ρ = 0.54)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.4. Session 4 (ρ = 0.54)81A.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.5. Session 5 (ρ = 0.65)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.6. Session 6 (ρ = 0.65)82A.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.7. Session (ρ = 0.70)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure A.8. Session 8 (ρ = 0.58)83A.3. Deviations from the Profit-Maximizing Strategies, by TypeA.3 Deviations from the Profit-Maximizing Strategies, by TypeTo highlight the stark differences in behavior across types, Figure A.9 plots a scatter of actualcontributions (y-axis) versus the BR associated with the lowest proportional monetary loss (x-axis).A wider circle denotes a higher frequency. Panel (a) shows this relationship in the LC group; panel(b) shows the same plot for the HC group. In both panels, Type 1 subjects are shown in black circles,while Type 2 are in gray ones. As one would expect, the average Type 1 subject makes choices thatare much closer to the BR. One can simply compare the area of the circles close to the diagonal(contributions close to the BR) and the area of circles off the diagonal (contributions away from theBR). Type 2 subjects overcontribute in LC sessions and undercontribute in HC ones.0011223344556677889910101111121213131414151516161717181819192020g∗ (g−i)giType 1Type 2(a) Low complementarity0011223344556677889910101111121213131414151516161717181819192020g∗ (g−i)giType 1Type 2(b) High complementarityFigure A.9. Contributions versus best response (based on previous two rounds’ contributions by other players). The plotsdisplay actual contributions on the y-axis versus the BR associated with the lowest proportional monetary loss (x-axis).A wider circle denotes a higher frequency. Type 1 subjects are shown in black circles; Type 2 are in gray ones.84A.4. Computer InterfaceA.4 Computer InterfaceFigure A.10. Main computer interfaceFigure A.11. Feedback85A.5. Control QuestionsA.5 Control QuestionsFigure A.12. Control question 1/7Figure A.13. Control question 2/786A.5. Control QuestionsFigure A.14. Control question 3/7Figure A.15. Control question 4/787A.5. Control QuestionsFigure A.16. Control question 5/7Figure A.17. Control question 6/788A.6. Contributions by Gender and Field of StudyFigure A.18. Control question 7/7A.6 Contributions by Gender and Field of StudyIn this appendix we present evidence of difference in contributions based on alternative groupingcriteria such as gender or field of study of the participants. Table A.1 suggests that there are nostatistical differences in contributions by gender within complementarity levels. Whereas Table A.2suggests that the filed of study matters in HC treatments where subjects in economics and businessrelated fields contribute more than students that belong to other fields.Table A.1Differences in Contributions by Gender Within Complementarity LevelFemale MaleAverage Standard Number of Average Standard Number of t-testDeviation Observations Deviation Observations (p-value)LVCM 1.3 2.7 360 1.5 2.8 280 0.27LC 6.1 4.9 560 6.2 5.4 400 0.61HC 15.8 5.9 480 16.1 6.3 480 0.49Note: Each cell reports the average contribution, standard deviation of contributions and number of observationsfor the respective category. The t-tests of the means are reported in the eighth column.89A.7. Myopic Best ResponseTable A.2Differences in Contributions by Field of Study Within Complementarity LevelEconomics and OtherBusinessAverage Standard Number of Average Standard Number of t-testDeviation Observations Deviation Observations (p-value)LVCM 1.2 2.2 280 1.5 3.2 360 0.17LC 6.4 4.9 500 5.8 5.3 460 .05HC 17.6 4.6 300 15.2 6.4 660 0Note: Each cell reports the average contribution, standard deviation of contributions and number of observations forthe respective category. Economics and Business includes majors in economics, business, and mathematics and eco-nomics. Other includes majors in Asian studies, biochemistry, biology, biophysics, computer science, creative writing,earth and ocean sciences, education, engineering, film studies, forestry, geography, history, human kinetics, inter-disciplinary studies, international relations, land and food systems, mathematics, medical laboratory, pharmacology,philosophy, psychology, science, sociology, and statistics. The t-tests of the means are reported in the eighth column.A.7 Myopic Best ResponseAs a robustness check we classify subjects based on how close their contributions are to the BRgiven the other group members’ contribution in the previous round. The procedure is analogous tothe one used in Section 1.5.1, but in this case we consider the contributions made by other groupmembers only in the previous round.There are no significant differences with respect to the classification used in Section 1.5.1. Onlysix subjects would be reclassified as Type 1 and another six subjects would be reclassified as Type2 (with respect to Section 1.5.1).Table A.3 is analogous to Table 1.4.90A.8. Persistence of ConjecturesTable A.3Differences in Mechanical Use of the Calculator, by Subject Type Within Complemen-tarity Level (assuming myopic BR)LVCM LC HCType 1 Type 2 t-test Type 1 Type 2 t-test Type 1 Type 2 t-test(p-value) (p-value) (p-valueCalcRound 4.4 9.5 0.0 10.7 9.9 0.6 5.5 4.8 0.6(1.2) (1.9) (1.1) (1.0) (0.8) (1.2))Hyp 17.1 21.9 0.2 32.5 28.1 0.3 29.4 23.1 0.2(2.6) (3.1) (2.8) (2.5) (4.3) (2.8))Conj 13.9 14.1 0.9 14.8 16.0 0.3 10.2 9.8 0.7(1.1) (1.1) (0.8) (0.7) (0.7) (0.6))Hyp Per Conj 3.7 4.3 0.4 7.4 6.1 0.1 8.9 6.4 0.1(0.4) (0.5) (0.5) (0.4) (1.2) (0.8))Observations 16 16 24 24 24 24Note: Each cell reports the average value for the respective category (standard errors are reported in parentheses).The t-tests of the means are reported in the third column of each treatment. CalcRound, number of rounds subjectsused the calculator; Hyp, number of hypothetical own contributions; Conj, number of conjectures about others; Hypper Conj, number of own hypothetical contributions entered, given a conjecture about other players’ contributions.We include the practice rounds.A.8 Persistence of ConjecturesTo better understand how persistent are the conjectures about others’ contributions, in Table A.4we display the number of new conjectures in each round and across treatments. We also showthe number of overall conjectures per round. Figure A.19 shows the five-round moving averagefor the new conjectures as a percentage of the overall conjectures. Note that there is a significantdecrease in the percentage of innovations over time, especially in HC. This supports the hypothesisof persistence in subjects’ conjectures, and suggests that some subjects form conjectures early inthe experiment that do not change much. Some of them adjust only their hypothetical contributions.91A.8. Persistence of ConjecturesTable A.4Persistence of ConjecturesLVCM LC HCRoundNo. of New Overall No. of New Overall No. of new OverallConjectures Conjectures Conjectures Conjectures Conjectures ConjecturesPractice 314 314 471 471 332 3321 11 36 42 106 12 642 6 23 30 66 10 333 9 27 20 76 8 274 7 18 19 69 8 385 6 26 17 67 2 226 8 25 14 61 5 237 0 8 12 47 1 188 2 13 12 47 3 269 2 10 8 48 0 1510 1 4 9 27 1 1211 2 8 5 39 1 1412 2 8 5 40 3 1013 1 5 4 29 0 914 0 9 1 20 0 915 1 9 4 25 0 816 1 5 4 25 2 717 0 3 4 30 0 318 1 4 3 33 0 519 0 3 1 22 0 820 0 4 0 17 2 1292A.9. Intensity and Processing Speed on Calculator Usage4 6 8 10 12 14 16 18 200102030405060708090100Round%ofnewconjecturesLVCM LC HCFigure A.19. New conjectures as a percentage of overall con-jectures. The solid lines of the graph display the five-roundmoving average of the number of new conjectures as a per-centage of overall conjectures. Notice that for period 4 weinclude data from the practice round, for which the percent-age of new conjectures is 100%.A.9 Intensity and Processing Speed on Calculator UsageA.9.1 Intensity of calculator usageIt is not obvious that longer spells of time unambiguously imply higher effort or better informa-tion processing. Therefore we combine time measures with records of actual interactions with theinterface by counting how many times the calculator was activated before a choice was recorded.Looking at the responses of Type 1 subjects in the LVCM and HC treatments, we see that subjectsmake no use of the calculator in three quarters of the rounds. In addition, it takes an average of only9 seconds in the LVCM and 6 seconds in the HC to submit a choice.In the LC treatments, however, this finding is reversed, as the Type 1 subjects use the calculatorin 55% of the rounds. Their average time to submission is 42 seconds. Subjects who do not use thecalculator spend an average of only 6 seconds before committing to a choice.81 In general, it appearsthat Type 1 subjects tend to use the calculator only in challenging environments, when identifyingBR strategies is not trivial.81This value does not change if we focus on Type 2 subjects who do not activate the calculator. They spend an averageof only 7 seconds submitting their choices.93A.10. InstructionsA.10 InstructionsThe instructions distributed to subjects in all the treatments are reproduced on the following pages.A.10.1 Treatment Variations for Chapter 1All subjects received the same set of instructions except that those in the LVCM treatment receivedthe following explanation about how the income from the group account was calculated:The total group income depends on the investments of all group members, and it isshared equally among all group members. This means that each group member receivesone quarter (1/4) of the total group income. Some important points to keep in mind:a. The more you and others invest in the group account, the higher the total groupincome.b. The group income is obtained by multiplying the sum of the investments of all groupmembers by 1.6 (remember that the resulting group income is shared equally amonggroup members).Also, the exchange rate was adjusted so that the average expected payoff was the same across alltreatments.A.10.2 Treatment Variations for Chapter 2Subjects in the T = 30 and T = 60 treatments received the same set of instructions as in the T = 0treatment except for the following modifications:1. The second point in the Income calculation section was changed to this:Income from the group account. The investments of all group members willgenerate value in the group account. The group account value depends on theinvestments of all group members, and is shared equally among all of them. Thatis, each group member receives one quarter (1/4) of the total group account value.As explained below, the income from the group account will be positive only ifthe group account value is above a certain threshold specified below; if the valueis lower than the threshold, the income from the group account will be zero.94A.10. Instructions2. For the points to keep in mind section we made the following changes:(a) We added the following:The income from the investments in your group account will be paid to themembers of the group only if the group account value is higher than (120)240 tokens (which means at least (30) 60 tokens per group member). If thegroup account value is lower than (120) 240 tokens, the income each memberreceives from the group account will be zero irrespective of the investmenteach member made to the group account.(b) Point (b) was modified to this:Taking as given the investments of all other group members, consider twolevels for your investment in the group account (say, low investment and highinvestment). Next, increase both the low investment and the high investmentby 1 token. The group account value will increase in both cases. However, theincrease is smaller in the case of the higher investment level. Moreover, yourincome from the group account will increase only if the group account passesthe threshold of 120 (240) tokens. Also, keep in mind that when the groupaccount value is just below the threshold, if you invest one additional token inthe group account, the group account value may pass the threshold.(c) Point (e) was modified to this:If all other members in your group invest zero, the group account value will bedetermined by multiplying your investment in the group account by 1.6; Notethat in this case your income from the group account will be zero because thegroup account value is lower than the threshold of 120 (240) tokens.3. Finally, the second paragraph of the Using the calculator to compute your income section wasmodified as follows:To activate the calculator, you will be asked to fill in a hypothetical value foryour own investment and for the other group members’ investment; then, you will95A.10. Instructionsbe able to visualize 25% of the group account value and the resulting incomefor such hypothetical investment choices. You can consider as many hypotheticalinvestment combinations as you want.A.10.3 Treatment Variations for Chapter 3For the VCMC, all subjects received the same set of instructions except that those in the ShowPayoffs and No Link (Show Ranking) treatments received the following explanation about how theinformation would be displayed at the end of the rounds:At the end of each round (after all choices are submitted), you will see: (i) your invest-ment choice, (ii) the investment choices of the other members in your group, (iii) yourincome, and (iv) the income (ranking) of the other members in your group. Then, nextround starts automatically and you will receive a new endowment of 20 tokens.For the case of the linear treatments, participants received the following explanation about howthe income from the group account would be calculated:The total group income depends on the investments of all group members, and it isshared equally among all group members. This means that each group member receivesone quarter (1/4) of the total group income. Some important points to keep in mind:a. The more you and others invest in the group account, the higher the total groupincome.b. The group income is obtained by multiplying the sum of the investments of all groupmembers by 1.6 (remember that the resulting group income is shared equally amonggroup members).Also, the exchange rate was adjusted so that the average expected payoff was thesame across all treatments.96  January, 2015  Page 1 of 3  Vancouver School of Economics #997 - 1873 East Mall Vancouver, B.C. Canada V6T 1Z1 Tel: (604) 822-2876    Fax: (604) 822-5915 Website: http://www.econ.ubc.ca Instructions You are taking part in an economic experiment in which you will be able to earn money. Your earnings depend on your decisions and on the decisions of the other participants with whom you will interact.  It is therefore important to read these instructions with attention. You are not allowed to communicate with the other participants during the experiment.  All the transactions during the experiment and your entire earnings will be calculated in terms of tokens. At the end of the experiment, the total amount of tokens you have earned during this session will be converted to CAD and paid to you in cash according to the following rules:  1. The game will be played for 20 rounds. At the end of the experiment, the computer will randomly select one of your decision rounds for payment. That is, there is an equal chance that any decision you make during the experiment will be the decision that counts for payment.  2. The amount of tokens you get in the randomly selected round will be converted into CAD at the rate: 3 tokens = $1. 3. You will get $0.20 for every control question you answer correctly in the first attempt; $0.15 for every question you answer correctly in the second attempt; and $0.10 for every question you answer correctly in the third attempt. 4. In addition, you will get a show-up fee of $5. Introduction  This experiment is divided into different rounds. There will be 20 rounds in total. In each round you will obtain some income in tokens. The more tokens you get, the more money you will be paid at the end of the experiment. During all 20 rounds the participants are divided into groups of four. Therefore, you will be in a group with 3 other participants. The composition of the groups will change every round.  You will meet each of the participants only four times, in randomly chosen rounds. However, each time you are matched with a participant that you encountered before, the other group members will be different. This means that the group composition will never be identical in any two rounds. Moreover, you will never be informed of the identity of the other group members.  Description of the rounds  At the beginning of the rounds each participant in your group receives 20 tokens. We will refer to these tokens as the initial endowment. Your only decision will be on how to use your initial endowment. You will have to choose how many tokens you want to invest in a group account and how many of them 97  January, 2015  Page 2 of 3  Vancouver School of Economics #997 - 1873 East Mall Vancouver, B.C. Canada V6T 1Z1 Tel: (604) 822-2876    Fax: (604) 822-5915 Website: http://www.econ.ubc.ca you'll want keep for yourself in a private account. You can invest any amount of your initial endowment in the group account. The decision on how many tokens to invest is up to you. Each other group member will also make such a decision. All decisions are made simultaneously. That is, nobody will be informed about the decision of the other group members before everyone made his or her decision.  End of the rounds At the end of each round (after all choices are submitted), you will see:  (i) your investment choice, (ii) the investment choices of the other members in your group, and (iii) your income. Then, next round starts automatically and you will receive a new endowment of 20 tokens.  Income calculation Each round, your total earnings will be calculated by adding up the income from your private account and the income from the group account: 1. Income from your private account. You will earn 1 token for every token you keep in you private account. If for example, you keep 10 tokens in your private account your income will be 10 tokens. 2. Income from the group account. The total group income depends on the investments of all group members, and it is shared equally among all of them. That is, each group member receives one quarter (1/4) of the total group income. Some important points to keep in mind: a. The more you and others invest, the higher the total group income.  b. Taking as given the investments of all other group members, consider two levels for your investment in the group account (say, low investment and high investment). Next, increase both the low investment and the high investment by 1 token. The total group income will increase in both cases. However, the increase is smaller in the case of the higher investment level. c. When you increase your investment in the group account, the total income will not increase at a constant rate. The rate depends on the value of all participants’ investments in the group account. d. For the same average investment in the group account, the total group income would be higher if there is not much difference between the investments chosen by each one of the group members.  e. If all other members in your group invest zero, the total group income will be determined by multiplying your investment in the group account by 1.6; the resulting amount is the group income and it will be shared equally among all group members.    98  January, 2015  Page 3 of 3  Vancouver School of Economics #997 - 1873 East Mall Vancouver, B.C. Canada V6T 1Z1 Tel: (604) 822-2876    Fax: (604) 822-5915 Website: http://www.econ.ubc.ca Using the calculator to compute your income To calculate your potential income you will have access to a calculator (look at the picture below). To activate the calculator, you will be asked to fill in a hypothetical value for your own investment and for the other group members’ investment; then, you will be able to visualize your income for such hypothetical investment choices. You can consider as many hypothetical investment combinations as you want. Before the experiment starts you'll understand how to use the calculator; you will be able to practice with it; and finally, you will have to answer some control questions. For every correct answer you will get $0.20, $0.15, $0.10 if you give the correct answer in the first, second and third attempt, respectively.  Remember that your actual investment decision has to be entered on the right hand side of the screen. Every round you will have 95 seconds to do that.  Screen-shot of the experiment interface  99Appendix BAppendix for Chapter 2B.1 Derivation of the Best-Response FunctionPlayer i’s payoff ispii =ω−gi +β(∑ni=1 gρi)1/ρ if β (∑ni=1 gρi )1/ρ > Tω−gi if β(∑ni=1 gρi)1/ρ ≤ T ,where ρ ≤ 1 denotes the degree of complementarity, gi denotes individual i’s contribution in thegroup account, ω is the endowment, β is a constant, and T is the threshold. The BR of player i is aunique solution, g∗i (g−i), to the first order condition0 =∂pii∂gi= β(gρi +∑gρ−i) 1−ρρ(gρ−1i)−1β(gρi +∑gρ−i) 1−ρρ = g1−ρigρi +∑gρ−i = gρi βρρ−1gρi(βρρ−1 −1)= (n−1)∑gρ−in−1 .In the last line we multiply and divide the right hand side by (n− 1) so the BR of player i isdefined as a function of Mρ =(∑gρ−in−1 .)1/ρ. Finally, defining k ≡(n−1βρρ−1−1) 1ρyieldsg∗i (g−i) = k(∑gρ−in−1)1/ρ.The second order condition100B.1. Derivation of the Best-Response Function∂ 2pii∂g2i= (1−ρ)β (gρi +∑gρ−i) 1−ρρ −1 g2(ρ−1)i +(ρ−1)β (gρi +∑gρ−i) 1−ρρ gρ−2i= (ρ−1)β (gρi +∑gρ−i) 1−ρρ gρ−2i(1− gρigρi +∑gρ−i)< 0,implies concavity of pii.The next step is to make sure that given g∗i (g−i) and g−i the threshold is met:β((g∗i )ρ +∑gρ−i)1/ρ> Tβ(∑gρ−iβρρ−1 −1+∑gρ−i)1/ρ> T β ρ2ρ−1βρρ−1 −1∑gρ−i1/ρ > TMρ(g−i) >β ρ/(1−ρ)kT.If Mρ(g−i)≤(β ρ/(1−ρ)/k)T , then player i maximizes his or her pecuniary payoff by contribut-ing enough so the threshold is met(gi =[(Tβ)ρ−∑gρ−i]1/ρ)as long as the following conditionshold:1.[(Tβ)ρ −∑gρ−i]1/ρ ≤ ω . That is, gi does not exceed the endowment.2. pii([(Tβ)ρ −∑gρ−i]1/ρ ,g−i) ≥ pii (0,g−i). That is, the payoff associated with gi and g−i isequal to or higher than the payoff associated with gi = 0 and g−i.82The first condition implies that82The payoff associated with gi = 0 and g−i is ω , given that the threshold is not met.101B.1. Derivation of the Best-Response Function(Tβ)ρ−∑gρ−i < ωρ∑gρ−i >(Tβ)ρ−ωρMρ(g−i) >[1n−1((Tβ)ρ−ωρ)]1/ρ.The second condition implies thatω−[(Tβ)ρ−∑gρ−i]1/ρ+β(([(Tβ)ρ−∑gρ−i]1/ρ)ρ+∑gρ−i)1/ρ> ωω−[(Tβ)ρ−∑gρ−i]1/ρ+β[(Tβ)ρ]1/ρ> ωω−[(Tβ)ρ−∑gρ−i]1/ρ+T > ω.It follows thatT >[(Tβ)ρ−∑gρ−i]1/ρ∑gρ−i >(Tβ)ρ−T ρMρ(g−i) >[1n−1((Tβ)ρ−T ρ)]1/ρ.If either the first or the second condition does not hold, then g∗i = 0.Thus, player i’s BR function isg∗i (g−i) =ω if Mρ (g−i)≥( 1k)ωkMρ (g−i) if(β ρ/(1−ρ)k)T < Mρ (g−i)<( 1k)ω((Tβ)ρ −∑gρ−i)1/ρ + ε if ( 1n−1 [( Tβ )ρ −min{ωρ ,Tρ}])1/ρ + ε ≤Mρ (g−i)≤ ( β ρ/(1−ρ)k )T0 otherwise.102B.2. Additional equilibriaB.2 Additional equilibriaThe original threshold public good experiments used the LVCM as their framework. The LVCM is aparticular case of the VCMC (when ρ = 1). In these games, the introduction of thresholds increasesthe Nash equilibrium set. Any vector of contributions for which the associated public good output isequal to the threshold is a Nash equilibrium (see Isaac et al., 1989). Thus, for the case of the VCMCit is important to verify whether the introduction of thresholds increases the equilibrium set. Thisis especially relevant for situations in which, given the contributions of others, the BR is to make acontribution that guarantees that the threshold is met.For simplicity, we start by showing that for the 2-player case, when k > 1 (two-equilibria) theintroduction of thresholds does not change the equilibrium set; then we generalize the argument forthe case of n players.B.2.1 2-player caseAssume that there are 2 players denoted by i and j. Player i’s payoff ispii =ω−gi +β(gρi +gρj)1/ρif β(gρi +gρj)1/ρ> Tω−gi if β(gρi +gρj)1/ρ ≤ T.The BR function is:g∗i (g j) =ω if g j ≥( 1k)ωkg j if(βρ/(1−ρ)k)T ≤ g j <( 1k)ω((Tβ)ρ −gρj )1/ρ + ε if ([(Tβ )ρ −min{ωρ ,T ρ}])1/ρ + ε ≤ g j < (βρ/(1−ρ)k )T0 otherwise.Where k ≡(1/(βρρ−1 −1)) 1ρ.Figure B.1.a and B.1.b display graphically the BR functions of players i and j. Figure B.1.ashows an example of an environment in which complementarity is sufficiently high so there aretwo-equilibria, ρ < ln(2)ln(2/β ) . Whereas Figure B.1.b reports a case of low complementarity and oneequilibrium. Figure B.1.a suggests that the introduction of thresholds does not change the equi-librium set, geqi = {0,ω}, when complementarity is sufficiently high. However, for the case of103B.2. Additional equilibriaρ > ln(2)ln(2/β ) (Figure B.1.b), the introduction of thresholds increases the equilibrium set (where theblue and red lines overlap).Below we prove that thresholds does not introduce nonsymmetric equilibria when ρ < ln(2)ln(2/β )(k > 1). The intuition is the following: if g j is small enough so the BR of player i is to make a con-tribution that guarantees that the threshold is met, then it can be shown that when complementarityis high enough, k > 1, g j is not a BR.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789101112131415161718192045◦gi(gj)gj(gi)gjgi0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2002040ThresholdgjGroupAcc.Output(a) HC0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200123456789101112131415161718192045◦gi(gj)gj(gi)gjgi0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200102030ThresholdgjGroupAcc.Output(b) LCFigure B.1. Best-Response Functions. The x-axis shows the contributions of player j; the y-axis displays player i’scontributions. The figure plots the BR as a function of the other player contributions. The figure below displays the groupaccount output associated with g∗i (g j) and g j. The following parameter values were set for the left panel (HC): β = 0.8,ρ = 0.68, n = 2 and T = 23. Here the BR has the same slope as in the 4-player case with T = 30 and ρ = 0.54. Thefollowing parameter values were set for the right panel (LC): β = 0.8, ρ = 0.82, n = 2 and T = 23. Here the BR has thesame slope as in the 4-player case with T = 30 and ρ = 0.65.It is straightforward to verify that only symmetric equilibria exist. Suppose that there ex-ists a nonsymmetric equilibrium g∗ denoted by g∗i =[(Tβ)ρ −gρ−i]1/ρ and ((Tβ )ρ −ωρ)1/ρ ≤ g∗j <(1kβρ1−ρ)T . For the case of k > 1, it follows that kg∗i > g∗j , which is a contradiction. The proof isbelow.Proof. It is possible to find lower and upper bounds for g∗i given g∗j ,104B.2. Additional equilibria[(Tβ)ρ−(1k)ρβρ21−ρ T ρ]1/ρ< g∗i ≤[(Tβ)ρ−((Tβ)ρ−ωρ)]1/ρT(1β ρ−(1k)ρβρ21−ρ)1/ρ< g∗i ≤ ω.If the upper bound of g∗j is smaller than k times the lower bound of g∗i , this would be a contra-diction. Below we show that for ρ < ln(2)ln(2/β ) —the two-equilibria environment— g∗−i is not the BRto g∗i .kT(1β ρ−(1k)ρβρ21−ρ)1/ρ>1kβρ1−ρ T(kρβ ρ−βρ21−ρ)T ρ >1kρβρ21−ρ T ρ1(βρρ−1 −1)β ρ−βρ21−ρ >(βρρ−1 −1)βρ21−ρ1(βρρ−1 −1)β ρ> βρ−ρ2ρ−11(βρρ−1 −1)β ρ>1β ρβρρ−1 −1 < 1βρρ−1 < 2ρ <ln(2)ln(2/β ).Absence of Nash equilibrium in mixed strategiesA symmetric Nash equilibrium in mixed strategies is a distribution µ over g j such that i is indifferentbetween all gi ∈ supp(µ). In Chapter 1 we have proven that for the case of(βρ/(1−ρ)k)T ≤ g j < ωthere is no mixed-strategy equilibria. If player j assigns positive probabilities to strategies associatedto contributions smaller than(βρ/(1−ρ)k)T , then the BR of player i is g∗i =((Tβ)ρ−gρmin)1/ρ, wheregmin = min{g j}, as long as:1.[(Tβ)ρ −gρmin]1/ρ ≤ ω . That is, g∗i does not exceed the endowment.105B.2. Additional equilibria2. pii([(Tβ)ρ −gρmin]1/ρ ,´supp(µ) g jdµ (g j))≥ pii(0,´supp(µ) g jdµ (g j)). That is, the expected pay-off of player i is equal to or higher than the payoff than his expected payoff when contributeszero.If these two conditions do not hold, then the BR of player i is to contribute zero. For the case ofg∗i =((Tβ)ρ −gρmin)1/ρ we have proven that there exist a unique pure best response for player j,g∗j = kg∗i if ρ <ln(2)ln(2/β ) .B.2.2 n-player caseSuppose that there exists a nonsymmetric equilibrium g∗ denoted by g∗i =((Tβ)ρ −∑gρ−i)1/ρ andg∗−i such that( 1n−1)1/ρ ((Tβ)ρ −ωρ)1/ρ ≤Mρ(g∗−i)< (βρ/(1−ρ)k )T .Denote also g∗min = min{g∗−i}≤(βρ/(1−ρ)k )T . For the case of k ≥ 1, it follows that kMρ (g∗−min)>g∗min, which is a contradiction.83 The proof can be found below.Proof: It is possible to find lower and upper bounds for g∗i given g∗−i,[(Tβ)ρ−(1k)ρβρ21−ρ T ρ]1/ρ< g∗i ≤[(Tβ)ρ−((Tβ)ρ−ωρ)]1/ρT(1β ρ−(1k)ρβρ21−ρ)1/ρ< g∗i ≤ ω.If we prove that the lower bound of g∗i is greater than g∗min, then we can show that if k > 1, thiscannot be an equilibrium because kMρ(g∗−min)> g∗min, which is a contradiction. Below we showthat ρ < ln(n)ln(n/β ) implies that g∗min < g∗i . Thus, g∗min is not the BR to g∗−min.83Assume that ε → 0.106B.2. Additional equilibriaT(1β ρ−(1k)ρβρ21−ρ)1/ρ>(β ρ/(1−ρ)k)T1β ρ− (n−1)(1k)ρβρ21−ρ >(1k)ρβρ21−ρ1β ρ> n(1k)ρβρ21−ρ1 >nn−1(βρρ−1 −1)βρ1−ρn−1 > n(1−βρ1−ρ )nβρ1−ρ > 1βρ1−ρ >1nρ <ln(n)ln(n/β ).Notice that as the number of players increases it becomes easier to satisfy the last condition,which implies the degree of complementarity should be lower for the nonsymmetric equilibria toarise.Absence of Nash equilibrium in mixed strategiesA symmetric Nash equilibrium in mixed strategies is a distribution µn−1 over g−i such that i isindifferent between all gi ∈ supp(µ). In Chapter 1 we have proven that for the case of(βρ/(1−ρ)k)T <Mρ(g−i) < ω there is no mixed-strategy equilibria. If players −i assigns positive probabilities tostrategies associated with their average contribution smaller than(βρ/(1−ρ)k)T , then the BR of playeri is g∗i =((Tβ)ρ −gmin)1/ρ (where gmin = min{∑gρ−i}) as long as:1.[(Tβ)ρ −gmin]1/ρ ≤ ω . That is, g∗i does not exceed the endowment.2. pii([(Tβ)ρ −gmin]1/ρ ,´supp(µ) g−idµn−1 (g−i)) ≥ pii(0,´supp(µ) g−idµn−1 (g−i)). That is, the ex-pected payoff of player i is equal to or higher than the payoff than his expected payoff whencontributes zero.If these two conditions do not hold, then the BR of player i is to contribute zero. For the case ofg∗i =((Tβ)ρ−gmin)1/ρwe have proven that if ρ < ln(n)ln(n/β ) , there exist a unique pure best responsefor players −i.107B.3. Best-Response Range and ContributionsB.3 Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.2. Session 1 (T = 0)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.3. Session 2 (T = 0)108B.3. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.4. Session 3 (T = 0)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.5. Session 4 (T = 30)109B.3. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.6. Session 5 (T = 30)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.7. Session 6 (T = 30)110B.3. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.8. Session 7 (T = 60)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.9. Session 8 (T = 60)111B.3. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure B.10. Session 9 (T = 60)112B.4. Computer InterfaceB.4 Computer InterfaceFigure B.11. Main computer interfaceFigure B.12. Feedback113B.5. Control QuestionsB.5 Control QuestionsFigure B.13. Control question 1/8Figure B.14. Control question 2/8114B.5. Control QuestionsFigure B.15. Control question 3/8Figure B.16. Control question 4/8115B.5. Control QuestionsFigure B.17. Control question 5/8Figure B.18. Control question 6/8116B.5. Control QuestionsFigure B.19. Control question 7/8Figure B.20. Control question 8/8117Appendix CAppendix for Chapter 3C.1 History-Dependence of ContributionsTo analyze whether conjectures are a function of past contributions of others group members, webegin by looking at the evolution of conjectures over time for the different treatments. Figure C.1shows evidence of history dependence of conjectures, as there is a positive correlation betweenthe trajectory of conjectures and actual contributions within treatments. The average conjectureincreases over time in the Show Contr sessions, whereas it decreases over time in the rest of thetreatments. For the linear sessions (C.1.b), there is not much difference across treatments. However,in the last rounds, subjects’ beliefs about others’ contributions are lower for the Show Payoff Lineartreatment compared to those in the Show Contr Linear treatment.Practice Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16Rounds 17-2068101214161820Avg.ConjectureShow Contr Show Payoffs Show Ranking No Link(a) VCMCPractice Rounds 1-4 Rounds 5-8 Rounds 9-12 Rounds 13-16Rounds 17-2068101214161820Avg.ConjectureShow Contr Linear Show Payoffs Linear(b) LVCMFigure C.1. Average conjecture about others’ contributions. The lines display the average value for the generalized meanof the conjectures of others’ contributions.To further assess how many periods of history subjects consider when forming expectations, weregress the generalized mean of the conjectures on the generalized mean of other players’ contribu-tions over the previous five rounds. The results, shown in Table C.1, indicate that subjects’ beliefsare explained to a large extent by their more recent experience. Much of the variation in currentconjectures can be related to other players’ behavior in the two previous rounds.118C.2. Best-Response Range and ContributionsTable C.1Response of Subjects’ Conjectures to Others’ Contributions( 1n−1 ∑gρ−i)1/ρ 1n−1 ∑g−iF(g−i,t−1)0.512∗∗∗ 0.518∗∗∗(0.06) (0.07)F(g−i,t−2)0.291∗∗∗ 0.302∗∗∗(0.07) (0.07)F(g−i,t−3)0.082 0.079(0.05) (0.06)F(g−i,t−4)0.071 0.065(0.06) (0.06)F(g−i,t−5)-0.018 -0.045(0.06) (0.05)Constant 0.637∗∗∗ 0.822∗∗∗(1.91) (2.08)Observations 881 881Notes: We estimate the following least-squares specification: F ( ˆgi,t) = C +∑5L=1 ALF (g−i,t−L)+ ui,t , where gˆiis a vector of player i’s conjectures about other group members’ contributions, g−i,t−L contains the vector ofcontributions made by other members in round t−L, C is a common constant, and ui,t is an idiosyncratic error. Welet the function F(·) be either the arithmetic or the generalized mean of degree ρ . The standard errors (reported inparentheses) are clustered by individuals and obtained by bootstrap estimations with 1,000 replications. *p < 0.1,**p < 0.05, ***p < 0.01. As a robustness check, we also estimate this specification including dummy variablesto control for different treatments. Results look very similar.C.2 Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.2. Session 1 (Show Contr)119C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.3. Session 2 (Show Contr)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.4. Session 3 (Show Contr)120C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.5. Session 4 (Show Payoffs)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.6. Session 5 (Show Payoffs)121C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.7. Session 6 (Show Ranking)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.8. Session 7 (Show Ranking)122C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.9. Session 8 (No Link)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.10. Session 9 (No Link)123C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.11. Session 10 (Show Contr Linear)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.12. Session 11 (Show Contr Linear)124C.2. Best-Response Range and Contributions051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.13. Session 12 (Show Payoffs Linear)051015200510152005101520051015200 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Subject 1 Subject 2 Subject 3 Subject 4Subject 5 Subject 6 Subject 7 Subject 8Subject 9 Subject 10 Subject 11 Subject 12Subject 13 Subject 14 Subject 15 Subject 16 BR Range Myopic BR Contribution Use calculator Do not use calculatorTokensRoundFigure C.14. Session 13 (Show Payoffs Linear)125C.3. Main Computer InterfaceC.3 Main Computer InterfaceFigure C.15. Main computer interfaceC.4 Computer Interface (Feedback)Figure C.16. Feedback Show Contr treatment126C.4. Computer Interface (Feedback)Figure C.17. Feedback Show Payoffs treatmentFigure C.18. Feedback Show Ranking treatment127C.4. Computer Interface (Feedback)Figure C.19. Feedback No Link treatment128

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