[{"key":"dc.contributor.author","value":"Chew, Soo Hong","language":null},{"key":"dc.date.accessioned","value":"2010-03-26T23:10:12Z","language":null},{"key":"dc.date.available","value":"2010-03-26T23:10:12Z","language":null},{"key":"dc.date.issued","value":"1980","language":null},{"key":"dc.identifier.uri","value":"http:\/\/hdl.handle.net\/2429\/22726","language":null},{"key":"dc.description.abstract","value":"This dissertation consists of two parts. Part I contains the statements\r\nand proofs of two representation theorems. The first theorem, proved in Chapter 1, generalizes the quasilinear mean of Hardy, Littlewood and Poly by weakening their axiom of quasilinearity. Given two distributions with the same means, quasilinearity requires that mixtures of these distributions with another distribution in the same proportions share the same mean, regardless of the distribution that they are mixed with. We weaken the quasilinearity axiom by allowing the proportions that give rise to the same means to be different, This leads to a more general mean, denoted by M[sub=\u03b1\u0424], which has the form:\r\nM[sub=\u03b1\u0424] = \u0424\u207b\u00b9(\u0283[sub=R] \u03b1\u0424F\/\u0283\u03b1dF), where \u03b1 is continuous and strictly monotone, a is continuous and strictly positive (negative) and F is a probability distribution. The quasilinear mean, denoted by M[sub=\u0424], results when the a function is constant. We showed, in addition, that the M[sub=\u03b1\u0424] mean has the intermediate value property, and can be consistent with the stochastic dominance (including higher degree ones) partial order. We also generalized a well known inequality among quasilinear means, via the observation that the M[sub=\u03b1\u0424] mean of a distribution F can be written as the quasilinear mean of a distribution F[sup=\u03b1], where F[sup=\u03b1] is derived from F via a as the Radon-Nikodym derivative of F[sup=\u03b1] with respect to F.\r\nWe noted that the M[sub=\u03b1\u0424] mean induces an ordering among probability distributions via the maximand, \u0283[sub=R] \u03b1\u0424F\/\u0283\u03b1dF, that contains the (expected utility) maximand, \u0283[sub=R] \u03b1\u0424F, of the quasilinear mean as a special case. Chapter 2 provides an alternative characterization of the above representation\r\nfor simple probability measures on a more general outcome set\r\n\r\n\r\nwhere mean values may not be defined. In this case, axioms are stated directly in terms of properties of the underlying ordering. We retained several standard properties of expected utility, namely weak order, solvability and monotonicity but relaxed the substitutability axiom of Pratt, Raiffa and Schlaifer, which is essentially a restatement of quasi-linearity in the context of an ordering.\r\nPart II of the dissertation concerns one specific area of application decision theory. Interpreting the M[sub=\u03b1\u0424](F) mean of Chapter 1 as the certainty equivalent of a monetary lottery F, the corresponding induced binary relation has the natural interpretation as 'strict preference' between lotteries. For non-monetary (finite) lotteries, we apply the representation theorem of Chapter 2. The hypothesis, that a choice agent's preference among lotteries can be represented by a pair of \u03b1 and \u0424 functions through the induced ordering, is referred to as alpha utility theory. This is logically equivalent to saying that the choice agent obeys either the mean value (certainty equivalent) axioms or the axioms on his strict preference binary relation.\r\nAlpha utility theory is a generalization of expected utility theory in the sense that the expected utility representation is a special case of the alpha utility representation. The motivation for generalizing expected utility comes from difficulties it faced in the description of certain choice phenomena, especially the Allais paradox. These are summarized in Chapter 3.\r\nChapter 4 contains the formal statements of assumptions and the derivations of normative and descriptive implications of alpha utility theory. We stated conditions, taken from Chapter 1, for consistency with\r\n\r\n\r\nstochastic dominance and global risk aversion and derived a generalized\r\nArrow-Pratt index of local risk aversion. We also demonstrated how\r\n\r\nalpha utility theory can be consistent with those choice phenomena that contradict the implications of expected utility, without violating either stochastic dominance or local risk aversion. The chapter ended with a comparison of alpha utility with two other theories that have attracted attention; namely, Allais' theory and prospect theory.\r\nSeveral other applications of the representation theorems of Part I are considered in the Conclusion of this dissertation. These include the use of the M[sub=\u03b1\u0424] mean as a model of the equally-distributed-equivalent level of income (Atkinson, 1970), and as a measure of asymmetry of a distribution (Canning, 1934). The alpha utility representation can also be used to rank social situations in the sense of Harsanyi (1977). We ended by pointing out an open question regarding conditions for comparative risk aversion and stated an extension of Samuelson's (1967) conjecture that Arrow's impossibility theorem would hold if individuals and society express their preferences by von Neumann-Morgenstern utility functions.","language":"en"},{"key":"dc.language.iso","value":"eng","language":"en"},{"key":"dc.publisher","value":"University of British Columbia","language":null},{"key":"dc.rights","value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","language":null},{"key":"dc.subject","value":"Statistical decision","language":"en"},{"key":"dc.title","value":"Two representation theorems and their application to decision theory","language":"en"},{"key":"dc.type","value":"Text","language":"en"},{"key":"dc.degree.name","value":"Doctor of Philosophy - PhD","language":"en"},{"key":"dc.degree.discipline","value":"Interdisciplinary Studies","language":"en"},{"key":"dc.degree.grantor","value":"University of British Columbia","language":"en"},{"key":"dc.date.graduation","value":"1980-11","language":"en"},{"key":"dc.type.text","value":"Thesis\/Dissertation","language":"en"},{"key":"dc.description.affiliation","value":"Graduate and Postdoctoral Studies","language":null},{"key":"dc.degree.campus","value":"UBCV","language":"en"},{"key":"dc.description.scholarlevel","value":"Graduate","language":"en"}]