Expected utility without full transitivity

Transcription

1 Expected utility without full transitivity Walter Bossert Department of Economics and CIREQ University of Montreal P.O. Box 6128, Station Downtown Montreal QC H3C 3J7 Canada FAX: (+1 514) and Kotaro Suzumura School of Political Science and Economics Waseda University Nishi-Waseda Shinjuku-ku, Tokyo Japan FAX: (+81 3) Preliminary version February 5, 2014 Financial support from a Grant-in-Aid for Specially Promoted Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan for the Project on Economic Analysis of Intergenerational Issues (grant number ), the Fonds de Recherche sur la Société et la Culture of Québec, and the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

2 1 Introduction We generalize the classical expected-utility criterion by weakening transitivity to the conjunction of transitive indifference and Suzumura consistency. In the absence of full transitivity, completeness no longer follows as a consequence of the system of axioms employed and a richer class of rankings of probability distributions results. The indifference relation corresponding to the expected-utility criterion is preserved but its strict counterpart allows for some non-comparability. The set of decision rules is characterized by means of standard expected-utility axioms in addition to transitive indifference and Suzumura consistency. It is illustrated how the use of some members of this generalized class may allow us to soften the negative impact of well-known paradoxes (such as the Allais paradox) at least in some instances without abandoning the expected-utility framework altogether. 2 Definitions Suppose there is a fixed finite set of alternatives X = {x 1,..., x n }, where n N \ {1, 2}. We exclude the one-alternative and two-alternative cases because they are trivial. The set = {p R n + n p i = 1} is the unit simplex in R n +, interpreted as the set of all probability distributions on X. For all i {1,..., n}, the i th unit vector in R n is denoted by e i. A (binary) relation on is a set 2 with asymmetric and symmetric parts and. For notational convenience, we write p q instead of (p, q) etc. The transitive closure of is defined by letting, for all p, q, p q there exist K N and r 0,..., r K such that p = r 0 and r k 1 r k for all k {1,..., K} and r K = q. Here are some standard properties of a relation on. Reflexivity. For all p, p p. Completeness. For all p, q such that p q, p q or q p. Transitivity. For all p, q, r, [p q and q r] p r. Solvability. For all p, there exists α [0, 1] such that p (α, 0,..., 0, 1 α). Monotonicity. For all α, β [0, 1], (α, 0,..., 0, 1 α) (β, 0,..., 0, 1 β) α β. Independence. For all p, q, r and for all α [0, 1], p q αp + (1 α)r αq + (1 α)r. 2

3 Transitive indifference. The relation is transitive. Suzumura consistency. For all p, q, p q (q p). The special role played by the alternatives labeled x 1 and x n in the axioms of solvability and monotonicity does not involve any loss of generality. All that matters is that there are two alternatives that are strictly ranked and we simply assume that these alternatives are given by x 1 and x n. The properties of reflexivity and completeness are usually not imposed when formulating one of the versions of the classical expected-utility theorem; they are implied by the set of axioms employed in the requisite result when transitivity is one of these axioms. This is true, in particular, for the representation result that we present and prove in the next section (which is based on a theorem in Kreps, 1988). We do not require reflexivity and completeness in our generalization of this expected-utility theorem either. Because the full force of transitivity is not imposed in our result, a more general class of (not necessarily complete) decision criteria can be obtained. Reflexivity continues to be implied by our axioms. Transitivity implies transitive indifference and Suzumura consistency. In the presence of reflexivity and completeness, Suzumura consistency and transitivity are equivalent but, because our new result does not involve completeness, transitivity is not implied by the conjunction of transitive indifference and Suzumura consistency in the present setting. See Suzumura (1976) and Bossert and Suzumura (2010) for discussions. Unlike Kreps (1988) and other authors, we treat as the primitive concept rather than. As is standard when is considered to be the primary relation, Kreps (1988) imposes asymmetry and negative transitivity on, the latter property being defined as follows. Negative transitivity. For all p, q, r, [ (p q) and (q r)] (p r). If the relation is required to be reflexive, complete and transitive, it does not matter whether we start out with or with the associated asymmetric part because the conjunction of asymmetry and negative transitivity of implies that the corresponding relation is reflexive, complete and transitive; see Kreps (1988, pp.9 10). Thus, adopting Kreps s (1988) setting would result in an immediate conflict with our main objective of examining the consequences of weakening the transitivity requirement in the context of expected-utility theory. 3 Classical expected utility The formal treatment of expected-utility theory goes back as far as von Neumann and Morgenstern s (1944; 1947) seminal contribution. The following result is a corollary of Kreps s (1988, p.46) theorem on the existence and uniqueness of an expected-utility representation. We provide a proof (inspired by the proof employed by Kreps) to link this classical result to our generalized expected-utility theorem to be established in the next section. To this end, we use transitive indifference and Suzumura consistency instead of transitivity where 3

4 possible in the proof and, moreover, indicate where the full force of transitivity cannot be dispensed with. Theorem 1 A relation on satisfies transitivity, solvability, monotonicity and independence if and only if there exists a function U: X [0, 1] such that (i) U(x 1 ) = 1 and U(x n ) = 0; (1) (ii) (iii) p q p q p i U(x i ) = q i U(x i ) for all p, q ; (2) p i U(x i ) > q i U(x i ) for all p, q. (3) The reason why we separate the indifference and strict preference parts in the theorem statement (and in the proof) is that this allows us to draw precise parallels when establishing our generalization in the following section. Clearly, the conjunction of (2) and (3) is equivalent to the corresponding if-and-only-if statement that involves the relation. Proof of Theorem 1. It is straightforward to verify that the existence of a function U such that (1), (2) and (3) are satisfied guarantees that has the properties stated in the theorem. Conversely, suppose that satisfies the axioms of the theorem statement. Let p and α [0, 1] be such that p (α, 0,..., 0, 1 α). (4) The existence of α is guaranteed by solvability. Furthermore, α is unique. To see this, suppose, by way of contradiction, that there exists β [0, 1] \ {α} such that If α > β, we have by monotonicity and, by (4) and (5), p (β, 0,..., 0, 1 β). (5) (α, 0,..., 0, 1 α) (β, 0,..., 0, 1 β) (6) (β, 0,..., 0, 1 β) (α, 0,..., 0, 1 α) which, together, with (6), leads to a contradiction to Suzumura consistency. The same argument applies if β > α. Thus, α must be unique for p and we can write it as a function ϕ: [0, 1], that is, Define U: X [0, 1] by letting p (ϕ(p), 0,..., 0, 1 ϕ(p)) for all p. (7) U(x i ) = ϕ(e i ) for all i {1,..., n}. (8) 4

5 Substituting p = e 1 in (7) and using (8), it follows that e 1 (U(x 1 ), 0,..., 0, 1 U(x 1 )). (9) Suppose that U(x 1 ) 1, which implies U(x 1 ) < 1 because U maps into the interval [0, 1] by definition. By monotonicity, e 1 (U(x 1 ), 0,..., 0, 1 U(x 1 )), contradicting (9). Thus, U(x 1 ) = 1. That U(x n ) = 0 follows analogously and, thus, the proof of (1) is complete. Let p, q and α [0, 1]. Because p (ϕ(p), 0,..., 0, 1 ϕ(p)), independence implies αp + (1 α)q α(ϕ(p), 0,..., 0, 1 ϕ(p)) + (1 α)q. Because q (ϕ(q), 0,..., 0, 1 ϕ(q)), independence implies α(ϕ(p), 0,..., 0, 1 ϕ(p)) + (1 α)q α(ϕ(p), 0,..., 0, 1 ϕ(p)) + (1 α)(ϕ(q), 0,..., 0, 1 ϕ(q)). Transitive indifference implies Because and αp + (1 α)q α(ϕ(p), 0,..., 0, 1 ϕ(p)) + (1 α)(ϕ(q), 0,..., 0, 1 ϕ(q)). (10) α(ϕ(p), 0,..., 0, 1 ϕ(p)) = αϕ(p)e 1 + α(1 ϕ(p))e n = α(ϕ(p)e 1 + (1 ϕ(p))e n ) (1 α)(ϕ(q), 0,..., 0, 1 ϕ(q)) = (1 α)ϕ(q)e 1 + (1 α)(1 ϕ(q))e n (10) is equivalent to = (1 α)(ϕ(q)e 1 + (1 ϕ(q))e n ), αp + (1 α)q α(ϕ(p)e 1 + (1 ϕ(p))e n ) + (1 α)(ϕ(q)e 1 + (1 ϕ(q))e n ). By definition of ϕ, this means that Next, we show that ϕ(αp + (1 α)q) = αϕ(p) + (1 α)ϕ(q). (11) ϕ(p) = p i U(x i ) for all p. (12) We do so by induction on the number of positive components of p, that is, on the cardinality of the set {i {1,..., n} p i > 0}. 5

6 If this set contains only one element j, it follows that p = e j. Clearly, ϕ(e j ) = n p iu(x i ) = U(x j ) in this case and (12) is satisfied. Now let 1 < m n and suppose (12) is satisfied for all probability distributions in with m 1 positive components. Let p be such that p has m positive components and let i {1,..., n} be such that p i > 0. Define a distribution q by { 0 if j = i, q j = p j 1 p i if j i. By definition, q has m 1 positive components and we can express p as By (11), p = p i e i + (1 p i )q. ϕ(p) = p i ϕ(e i ) + (1 p i )ϕ(q) and, using (8) and applying the induction hypothesis to q, it follows that ϕ(p) = p i U(x i ) + (1 p i ) j=1 j i p j 1 p i U(x i ) = p i U(x i ), as was to be shown. Now we establish (2). Let p, q. Using (7), transitive indifference and monotonicity, we obtain p q (ϕ(p), 0,..., 0, 1 ϕ(p)) (ϕ(q), 0,..., 0, 1 ϕ(q)) ϕ(p) = ϕ(q). Substituting (12) in the above equivalence, we obtain (2). For future reference, note that (1) and (2) are obtained as a consequence of solvability, monotonicity, independence, transitive indifference and Suzumura consistency; full transitivity need not be invoked. In contrast, transitivity is required for the next and final step. To complete the proof, we show that (3) must be true. Let p, q. By (7), transitivity and monotonicity, we obtain p q (ϕ(p), 0,..., 0, 1 ϕ(p)) (ϕ(q), 0,..., 0, 1 ϕ(q)) ϕ(p) > ϕ(q). Substituting (12) in the above equivalence, (3) results. Reflexivity and completeness of are implied by the axioms used in Theorem 1. The reason is that the restriction of to pairs of the form ((α, 0,..., 0, 1 α), (β, 0,..., 0, 1 β)) is reflexive and complete by monotonicity and these properties spread to all of by virtue of solvability and transitivity. 6

7 4 Weakening the transitivity requirement We now turn to the main result of this paper. In particular, we show that if transitivity is weakened to the conjunction of transitive indifference and Suzumura consistency, a class of generalized expected-utility criteria is characterized. These relations retain the indifference part of the classical expected-utility framework (because transitive indifference is sufficient to guarantee this feature) but allow for non-comparability in some situations where traditional expected-utility theory demands a strict preference. The strict preferences imposed by the monotonicity property continue to be required but, because full transitivity is no longer imposed, any additional pairs that belongs to the strict expected-utility relation may or may not be included in the relation. Pairs of probability distributions that do not respect the strict preference according to the classical expected-utility criterion are ruled out (because Suzumura consistency is sufficient to do so and the full force of transitivity is not needed). Thus, the new class contains as special cases the standard (complete) expectedutility criterion as the maximal relation and the one where the only strict preferences are those imposed by monotonicity as the minimal relation satisfying the axioms. In particular, this means that any pair that is strictly ranked by the expected-utility criterion may either be strictly ranked or non-comparable according to our generalization. Thus, undesirable observations such as the Allais paradox (Allais, 1953) can be ameliorated by replacing the offending counter-intuitive strict preference with non-comparability. We illustrate this feature after our statement and proof of the theorem. Theorem 2 A relation on satisfies transitive indifference, Suzumura consistency, solvability, monotonicity and independence if and only if there exists a function U: X [0, 1] such that (i) U(x 1 ) = 1 and U(x n ) = 0; (13) (ii) p q p i U(x i ) = q i U(x i ) for all p, q ; (14) (iii) p q p i U(x i ) > q i U(x i ) for all p, q ; (15) (iv) (α, 0,..., 0, 1 α) (β, 0,..., 0, 1 β) for all α, β [0, 1] such that α > β. (16) Note that (iii) in the statement of Theorem 2 differs from (iii) of Theorem 1 in that, owing to the absence of full transitivity, only an implication rather than an equivalence is obtained. Furthermore, unlike in the case of Theorem 1, an explicit statement of (iv) is now required because this observation is not covered by part (iii) of Theorem 2. Proof of Theorem 2. That the relations of the theorem statement satisfy transitive indifference, solvability, monotonicity and independence is straightforward to verify. To see that Suzumura consistency is satisfied, suppose, by way of contradiction, that p, q are such that p q and q p. Thus, by definition of the transitive closure, there exist K N 7

8 and r 0,..., r K such that p = r 0 and r k 1 r k for all k {1,..., K} and r K = q. Consider any k {1,..., K}. If r k 1 r k, it follows that r k 1 i U(x i ) = because of (14). If r k 1 r k, (15) implies Thus, for all k {1,..., K}, we have r k 1 i U(x i ) > r k 1 i U(x i ) ri k U(x i ) ri k U(x i ). ri k U(x i ). Combining these inequalities for all k and using p = r 0 and r K = q, it follows that p i U(x i ) q i U(x i ). (17) By (15), q p implies q i U(x i ) > p i U(x i ), contradicting (17). Thus, Suzumura consistency is satisfied. Conversely, suppose that satisfies the axioms of the theorem statement. The proofs of (13) and (14) are identical to the corresponding proofs in Theorem 1 because the arguments employed there apply when satisfies transitive indifference and Suzumura consistency. Statement (16) follows immediately from monotonicity. This leaves us with (15) to prove. By way of contradiction, suppose p, q are such that p q and q i U(x i ) p i U(x i ). (18) We know from (7) and (12) that ( p p i U(x i ), 0,..., 0, 1 ) p i U(x i ) (19) and ( q q i U(x i ), 0,..., 0, 1 ) q i U(x i ). (20) 8

9 By monotonicity, (18) implies that ( ) ( q i U(x i ), 0,..., 0, 1 q i U(x i ) p i U(x i ), 0,..., 0, 1 ) p i U(x i ) which, together with (19) and (20), implies q p. Because p q, we obtain a contradiction to Suzumura consistency. Clearly, as is the case for the classical expected-utility criterion, U is unique up to increasing affine transformations as long as a higher number is assigned to x 1 than to x n. Note that completeness no longer is implied if transitivity is weakened to the conjunction of transitive indifference and Suzumura consistency. However, reflexivity still results as a consequence of the axioms in Theorem 2. Because only transitive indifference rather than full transitivity is used, the reflexivity of the restriction of to pairs of the form ((α, 0,..., 0, 1 α), (β, 0,..., 0, 1 β)) is, as a consequence of solvability and transitive indifference, inherited by the entire relation but strict rankings within this restriction do not extend due to the absence of full transitivity. As mentioned earlier, the classical expected-utility criterion is a special case of the class characterized in Theorem 2. Another special case is obtained if no strict preferences are added to those imposed by (iv), that is, the case where p q p i U(x i ) = q i U(x i ) and p q there exist α, β [0, 1] such that α > β and p = (α, 0,..., 0, 1 α) and q = (β, 0,..., 0, 1 β). To see how some members of the class characterized in Theorem 2 allow us to ameliorate the supposedly disagreeable consequences of observations such as the Allais paradox, consider the following example. Suppose we have a set of alternatives X = {5, 1, 0}, where x 1 = 5 stands for receiving five million dollars, x 2 = 1 means receiving one million dollars and x 3 = 0 is an alternative where the decision-maker receives zero. Experimental evidence (see Kahneman and Tversky, 1979, for instance) suggests that numerous subjects express something resembling the following rankings of specific probability distributions. Consider the distributions p = (0, 1, 0) and q = (0.1, 0.89, 0.01) on the one hand, and the pair of distributions p = (0, 0.11, 0.89) and q = (0.1, 0, 0.9) on the other. Many subjects appear to rank p as being better than q and q as being better than p. This is inconsistent with classical expected utility theory because, for any function U: X [0, 1] such that U(5) = 1 and U(0) = 0, p q entails U(1) > U(1) and, thus, U(1) > 1/11, whereas q p implies 0.1 > 0.11 U(1) and, thus, U(1) < 1/11. If, however, a generalized expected-utility criterion that allows for incompleteness is employed and the subjects are given the option of treating two distributions as non-comparable, it may very well be the case that, according to the decisions of some experimental subjects, p is ranked higher than q and p and 9

10 q are non-comparable (or p and q are non-comparable and q is preferred to p, or both pairs are non-comparable). Of course, the so-called paradox persists if the original strict rankings are retained even in the presence of a non-comparability option. But it seems to us that the use of an incomplete generalized expected-utility criterion may considerably reduce the instances of dramatically conflicting pairwise rankings without abandoning the core principles of expected-utility theory altogether. References Allais, M. (1953), Le comportement de l homme rationnel devant le risque: critique des postulats et axiomes de l école Americaine, Econometrica 21, Bossert, W. and K. Suzumura (2010), Consistency, Choice, and Rationality, Harvard University Press, Cambridge, MA. Kahneman, D. and A. Tversky (1979), Prospect theory: an analysis of decision under risk, Econometrica 47, Kreps, D. (1988), Notes on the Theory of Choice, Westview Press, Boulder, CO. Suzumura, K. (1976), Remarks on the theory of collective choice, Economica 43, von Neumann, J. and O. Morgenstern (1944; second ed. 1947), Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ. 10