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Article Excerpt
Summary. This paper considers the applicability of the standard separability axiom for both risk and other-regarding preferences, and advances arguments why separability might fail. An alternative axiom, which is immune to these arguments, leads to a preference representation that is additively separable in a reference variable and the differences between the other variables and the reference variable. For other-regarding preferences the reference variable is the decision-maker's own payoff, and the resulting representation coincides with the Fehr-Schmidt model. For risk preferences the reference variable is initial wealth, and the resulting representation is a generalization of prospect theory.
Keywords and Phrases: Other-regarding preferences, Risk, Separability, Axiomatic foundation, Prospect theory.
JEL Classification Numbers: D81, D64.
1 Introduction
This paper presents a new preference axiom called self-referent separability. When combined with the usual axioms of completeness, transitivity, and continuity, it guarantees the existence of a preference representation that is additively separable in a reference variable and the difference between the other variables and the reference variable. In other words, the self-referent separability axiom generates reference-dependent preferences, and such preferences arise in the literature. Most prominently, prospect theory (Kahneman and Tversky, 1979) is a reference-dependent representation of preferences toward risk, and so self-referent separability can be used as part of a system of axioms for prospect theory. In a different branch of the literature, Fehr and Schmidt (1999) propose a reference-dependent representation for other-regarding preferences, and self-referent separability is the key axiom for generating their functional form.
The paper begins by stating the axiom and the main representation theorem. Since the applicability of the axiom depends on the choice setting, its rationale is left for two later sections. It is first applied to other-regarding (or interdependent or social) preferences, which arise from the voluminous literature on ultimatum, dictator, and trust games. (1) The upshot of this literature is that players in these games care not just about their own payoffs, but also about the payoffs of their opponents/partners in the game. Thus far, most of the attention on other-regarding preferences has been on constructing new experiments to identify their existence and characteristics and on generating highly-parameterized models to fit the data from the experiments. (2) From a purely decision-theoretic perspective, though, the possibility of preferences being other-regarding raises some interesting issues. In particular, do the standard preference axioms that are used in so many other areas of decision theory make sense in an other-regarding setting, or must they be replaced by something else? If they do need to be replaced, what should they be replaced with? (3)
The standard separability axiom states that if two bundles are identical on some dimensions but differ on others then preferences depend only on those dimensions that differ between the two bundles, and its primary appeal is that it has been used fruitfully in a variety of settings. (4) It does not, however, allow preferences to depend on the ordering or rank of the different dimensions. It seems reasonable that in an interpersonal setting a decision-maker cares about the rank of his own payoff relative to the payoffs of others affected by his decision, and Kahneman and Tversky (1979) established that when faced with risk a decision-maker cares about whether his new wealth level is above or below his previous wealth level. Self-referent separability allows position to matter. (5)
The new axiom does place one restriction on preferences that may or may not be deemed restrictive, depending on the setting. For risk preferences, self-referent separability implies constant absolute risk aversion. Since it does not allow for the standard expected utility formulation with asset integration (i.e. in which the carrier of value in the utility function is the final wealth level), however, constant absolute risk aversion places no restrictions on the functional forms of the underlying utility functions. For other-regarding preferences, self-referent separability implies constant absolute reallocation preferences, which is a generalization of the idea that adding $100 to everyone's payoff should have no effect on the decision-maker's willingness to take $20 away from one opponent and give it to another.
Section 2 introduces the self-referent separability axiom and presents the main representation theorem. Section 3 discusses the applicability and applications of the axiom to other-regarding preferences, and Section 4 does the same for risk preferences. Section 4 also shows that in the setting of risk preferences, self-referent separability implies constant absolute risk aversion, and Section 5 discusses constant absolute reallocation preferences. The paper concludes in Section 6.
2 The axiom and the representation theorem
Let N = {0,..., n}, and let x = ([x.sub.0],..., [x.sub.n]) denote a vector of real numbers with [x.sub.i] [member of] [X.sub.i] for each i [member of] N. Define X = [[PI].sub.i=0.sup.n] [X.sub.i]. The vector x is referred to as an allocation, and the components are referred to as payoffs.
Let S be a subset of N, and let ~ S be its complement. Let ([x.sub.S], [y.sub.~S]) denote the allocation z [member of] X such that [z.sub.i] = [x.sub.i] when i [member of] S and [z.sub.i] = [y.sub.i] when i [member of]~ S. For the special case when S contains only a single element, so that S = {i}, use the notation ([x.sub.i], [y.sub.~i]) to denote the allocation ([y.sub.0],..., [y.sub.i-1], [x.sub.i],...
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