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The Independence Axiom Versus the Reduction Axiom: Must We Have Both?

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The Independence Axiom Versus the Reduction Axiom: Must We Have Both? 7 THE INDEPENDENCE AXIOM VERSUS THE REDUCTION AXIOM: MUST WE HAVE BOTH? Uzi Segal Introduction Since the early 18th Century, expected utility theory became the leading theory in explaining behavior of decision makers under uncertainty. For most of this time it was used as a purely descriptive theory. For example, Ramsey (1931) explicitly assumed that people evaluate a risky prospect by its expected utility. He expressed this idea in the following words: I suggest that we introduce as a law of psychology that his Ithe decision maker's] behavior is governed by what is called the mathematical expectation: that is to say that, if p is a proposition about which he is doubtful. any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculations multiplied by the same fraction, which is called the 'degree of his belief in p'. This theory changed its status sometime in the late forties when different authors tried to give it an axiomatic basis, that is, a set of decision rules implying expected utility maximization. Some of the early axiomatizations were suggested by von Neumann and Morgenstern (1947), Marschak (1950), and Samuelson (1952). W. Ed ••..ards (ed.), UTILITY THEORIES: MEASUNEMEN7:\' AND AI'I'L1CATIONS. Copyright © 1992. Kluwcr AClIdemic Publishers. lIos/()/I. All riKht.1 reserl'ed. 166 UTILITY THEORIES: MEASUREMENTS AND APPLICATI()N~ These sets of axioms were usually considered normative. that is rules that must be obeyed by "rational" decision makers. The commor belief is (or at least was) that violating these rules implies that decision makers are willing to give something for nothing. a violations of the most sacred rule in Economics. that more is better. But axioms have another important role. which is their descriptive value. A certain theory mayor may not be used by decision makers. Although we may find by chance a behavioral violation of a theory, it may be considered an insignificant mistake. However. if we find that people systematically violate one of the rules that necessarily follow from a certain theory. then this evidence will probably carry more value. Moreover. it should be easier to find counterexamples to well-defined rules than to the theory itself. This approach has also a positive value. Obviously, it is impossible to verify the validity of a theory by any finite number of observations. However. if we can isolate some rules and test them directly. then we will be presumably more willing to accept them. If these rules imply a certain decision theory. we are then thus bound to accept the theory as well. Undoubtedly. the existence of an axiomatic basis for expected utility theory helped those who claimed that decision makers do not follow it to find behavioral departures from this theory. The first. and by far the most famous type of nonexpected utility behavior. was suggested by Allais (1953) shortly after the first axiomatizations were found. Since then a lot of other evidence against this theory was accumulated. (See. for example, ElIsberg. 1961; Kahneman and Tversky. 1979; MacCrimmon and Larsson. 1979.) Obviously. if a certain set of axioms implies a certain theory. and empirical evidence fails to support this theory, then one or more of these axioms must be violated. Some axioms have more appeal than ot-hers, thus, we would prefer to try to keep these axioms at the cost of the less appealing ones. Moreover. if a certain axiom is widely used, not only in theories of decision making under uncertainty, but in general decision problems as well, then we will probably be more reluctant to abandon it. This is one of the reasons why economists are so unwilling to give up the transitivity axiom. Since the continuity and completeness axioms carry a similar. or even a higher value. the only disposable axiom is the independence axiom. Let X. Y. and Z be lotteries. According to the independence axiom. the lottery aX + (I - a)Z is preferred to the lottery a Y + (I - a)Z if and only if X is preferred to Y. The normative justification for this axiom is that if X is preferred to Y, it should also be so when receiving these lotteries becomes uncertain. Recent authors in the field of decision theory are of course well aware of the normative appeal of this axiom, THE INDEPENDENCE AXIOM VERSUS THE REDUCTION AXIOM ]( but claim that whether we like it or not, decision makers do not accept i In other words, even if nonexpected utility theories cannot be used a normative grounds because they violate the independence or the trar, sitivity axioms (for the latter, see Fishburn, 1983; Loomes and Sugden 1986; MacCrimmon and Larsson, 1979), they are still superior to expectel utility as descriptive theories. But do decision makers really violate the independence axiom? Afte all, the idea that a preference relation should not depend on outcome that were never realized is very compelling even on a purely behaviora basis. Moreover, this rule appears in a lot of other areas of economil theory. Indeed, there is not much difference between the reasonin! behind this axiom and the justification for the axiom of revealed pre· ferences (Samuelson, 1938) or Arrow's rule of independence of irrelevanl alternatives. This does not mean that decision makers necessarily folio •••• this axiom, but it raises the theoretical price we have to pay to exclude it. I claim below that most of the evidence against expected utility theory, as well as some other evidence, can be analyzed as a violation of the reduction of compound lotteries axiom. Moreover, in an appropriate setting, this axiom has less normative value than is usually believed. This analysis follows closely the one presented in Segal (1990), where the model is more formally presented. Of course, it is nicer to have a model in which this axiom is accepted. But it is my belief that releasing the reduction axiom is the lesser of the many other evils we may choose. This point appears in several former works. Luce and Narens (1985) analyze, among other things, a model where multi-stage uncertainty is different from a single-stage uncertainty with the same compound prob• abilities. Luce (1988) gives an axiomatization of the rank dependent model (Quiggin, 1982) based on the assumption that in some cases, the order at which uncertainty is resolved makes no difference to the decision maker. For different axiomatizations of the same model, based on other versions of order indifference, see Segal (1989), and Chew (1989). For some references to the experimental literature, see Segal, 1990. The chapter is organized as follows. In the next section I discuss two• stage lotteries with the reduction of compound lotteries axiom and the (compound) independence axiom. Section three analyzes a Dutch book in support of the reduction axiom. In section four I show how this approach may be useful in analyzing some empirical evidence dealing with two• stage lotteries. The Ellsberg paradox is discussed in section five. Section six concludes with some further remarks on the independence axiom. 168 UTILITY THEORIES: MEASUREMENTS AND APPLICATIO~ Two-Stage Lotteries There are situations where uncertainty is resolved in more than on stage. This may be because some real time elapses between the poir when the uncertainty begins to be resolved and the point where all th uncertainty is resolved, or because the decision maker's concept of th uncertainty is multi-stage, even if no real time is involved. In this sectiol I describe the formal structure of such lotteries. For a more detaile( discussion see Segal (1990). Let LI be the set of lotteries with outcomes in a bounded interva [-M, MI. That is, L, = {(x" PI:"': x". PII): X" ...• XII E [-M. M] p), ... , PII ~ 0, and 'Lp; = I}. Elements of L, are denoted X, Y, etc. We usually assume that the outcomes of these lotteries are monetar) outcomes. Let ~I be a complete, transitive, monotonic. and continuous preference relation on L,. We say that X>, Y if X~, Y but not Y ~I X and that X -I Y if X ~I Yand Y ~I X. A function V: L, -> IR is said to represent the order ~I if VeX) ~ V(Y) if and only if X ~I Y. For X ELI, let CE(X), the certainty equivalent of X, be defined by X -I (CE(X). I). A two-stage lottery is a lottery where the outcomes are tickets for simple lotteries in LI' Formally, let L2 = {(X), q,: ... ; X"" q",): XI,' .. ,X", E L), q), ... ,q", ~ 0, L'q; = I}. Elements of L2 are denoted A, B, etc. A preference relation on L2 is denoted ~2' We assume throughout that such a preference relation is complete and transitive. Two subsets of Lz are of special interest. Let L1 = {(X. I): X E L,} C L2• LJ is the set of all two-stage lotteries which have no uncertainty in the first stage-the decision maker knows that with probability one he will receive a ticket for lottery X. In other words, all the uncertainty is resolved in the second stage. Let r = {((XI' 1), PI;"'; (x"' I), PII):(X" p,; ... ; X". PII) E L d. r is the set of all lotteries where all the uncertainty is resolved in the first stage. With probability P; the decision maker knows that he has won a ticket for the lottery (x;, I), that is. a sure gain of Xi dollars. In other words. both rand LJ are L, type spaces. Let X = (x,. PI;' .. : x"' p,,) E L, be a simple, one-stage lottery. Define the two lotteries Yx E r and ()x E LJ as follows: Yx = ((XI' I), PI;' ., : (XII' I), PII) ()x = ((XI' PI;"'; x"' p,,).
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