# Open Collections

## UBC Theses and Dissertations

### Two representation theorems and their application to decision theory Chew, Soo Hong

#### Abstract

This dissertation consists of two parts. Part I contains the statements and 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=αФ], which has the form: M[sub=αФ] = Ф⁻¹(ʃ[sub=R] αФF/ʃαdF), where α 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=Ф], results when the a function is constant. We showed, in addition, that the M[sub=αФ] 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=αФ] mean of a distribution F can be written as the quasilinear mean of a distribution F[sup=α], where F[sup=α] is derived from F via a as the Radon-Nikodym derivative of F[sup=α] with respect to F. We noted that the M[sub=αФ] mean induces an ordering among probability distributions via the maximand, ʃ[sub=R] αФF/ʃαdF, that contains the (expected utility) maximand, ʃ[sub=R] αФF, of the quasilinear mean as a special case. Chapter 2 provides an alternative characterization of the above representation for simple probability measures on a more general outcome set where 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. Part II of the dissertation concerns one specific area of application decision theory. Interpreting the M[sub=αФ](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 α and Ф 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. Alpha 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. Chapter 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 stochastic dominance and global risk aversion and derived a generalized Arrow-Pratt index of local risk aversion. We also demonstrated how alpha 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. Several 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=αФ] 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.

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