Decision Making under Uncertainty and Subjective Probabilities Edi Karni Johns Hopkins University September 7, 2005 Abstract This paper presents two axiomatic models of decision making under uncertainty that avoid the use of a state space The first is a subjective expected utility model with action-dependent subjective probabilities and effect-dependent preferences (the case of effect-independent preferences is obtained as a special instance) The second is a nonexpected utility model involving well-defined families of action-dependent subjective probabilities on effects and a utility representation that is not necessarily linear in these probabilities (a probabilistic sophistication version of this model, with actiondependent subjective probabilities is obtained as a special case) I am grateful to the National Science Foundation for financial support under grant SES-0314249 1
1 Introduction The notion of states of nature is a cornerstone of modern theories of decision-making under uncertainty Introduced by Savage (1954), a state of nature formalizes the idea of complete resolution of uncertainty that is, the assignment of a unique consequence to each conceivable course of action 1 For example, when betting on the outcome of a horse race, the states of nature correspond to the orders in which the horses may cross the finish line (these are the states of nature in Anscombe and Aumann [1963]) Theories of decision-making under uncertainty that invoke the notion of a state space (that is, the set of all states of nature) require that the states be so defined as to render the likely realization of alternative events (that is, subsets of the state space) independent of the decision maker s choice, and the valuation of the consequences independent of the states in which they may obtain Consequently, if the likely outcomes of a horse race may be affected by actions taken by a decision maker (for example, a jockey throwing a race) then the outcomes of the race no longer qualify as states of nature Similarly, taking out health insurance policy is betting on one s state of health The uncertainty involved is resolved once the true state of the insured s health becomes known However, the likely realization of alternative states of health is not independent of the life style (eg, diet and exercise regimen) adopted by the insured Moreover, in general, the insured s valuation of the indemnity is not independent of his state of health Hence the states of the decision maker s health do not 1 Formally, a state of nature is a function on the set of courses of action, or acts, to the set of consequences 2
qualify as states of nature If the framework of Savage is to be maintained, a different, more abstract, formulation of a state space is called for in which the outcomes of the horse race and the states of the decision maker s health become random variables However and here is the rub in many situations involving decision-making under uncertainty, the state space that meets these conditions is too abstract and/or too complex to contemplate Except in special circumstances, the state space does not correspond to an image of the world that decision makers invoke when making decisions under uncertainty 2 In Karni (2005) I advanced a new approach to modeling decision making under uncertainty This approach dispenses with the idea of a state space In this paper I pursue this approach using the Anscombe and Aumann (1963) device of roulette lotteries Taking advantage oftherichnessof thechoicespaceafforded by the availability of roulette lotteries, I develop axiomatic subjective expected utility models with unique family of action-dependent subjective probabilities and a family of effect-dependent utility functions that are simple and transparent Moreover, following Machina and Schmeidler (1992, 1995), I also develop an axiomatic model of decision-making under uncertainty in which action-dependent subjective probabilities are defined but the utility representation is not necessarily linear in these probabilities As a special case, I obtain a version of Machina and Schmeidler s probabilistically sophisticated choice model with action-dependent subjective probabilities A remarkable aspect of this new approach is that the family of subjective probabilities that figure in the representation is a proper representation of the decision makers introspec- 2 Additional examples, a more detailed discussion and references, are provided in Karni (2005) 3
tive beliefs, that is, his beliefs about the likely realization of events conditional on his actions that he uses to assess the alternative courses of action In the next section I describe the analytical framework The subjective expected utility theory is the subject matter of Section 3 Section 4 includes an exposition of the more general nonexpected utility models Concluding remarks appear in Section 5 and the proofs are given in Section 6 2 The Model 21 The Analytical Framework Let Θ be a finite set whose elements are effects In the examples cited earlier, effects are possible outcomes of a horse race or the states of a person s health Let A be a connected separable topological space set whose elements, referred to as actions, correspond to initiatives by which a decision maker believes that he might affect the likely realization of alternative effects (eg, effort level) Let Z (θ) be an arbitrary set of prizes that are feasible if the effect θ obtains To simplify the exposition assume that > Z (θ) 3 for every θ, and denote by (Z (θ)) the set of all probability distributions on Z (θ) Elements of (Z (θ)) are referred to as roulette lotteries, or simply lotteries Assume that, for each θ Θ, (Z (θ)) is endowed with the R n topology 4
Bets are effect-contingent lottery payoffs Formally, a bet, b, is a function on Θ such that b (θ) (Z (θ)) Denote by B the set of all bets (that is, B := Π θ Θ (Z (θ))) The choice set is the product set C := A B whose generic element, (a, b), is an action-bet pair Action-bet pairs represent conceivable alternatives among which decision makers may have to choose Assume that C is endowed with the product topology The set of consequences C consists of the prize-effect pairs, that is, C := {(z,θ) z Z (θ),θ Θ} Decision makers are characterized by preference relations, <, on C that have the usual interpretation 3 In other words, decision makers are supposed to be able to choose, or express preferences, among action-bet pairs presumably taking into account their beliefs regarding the influence that their choice of action may exert on the likely realization of alternative effects and, consequently, on the desirability of the corresponding bets The strict preference relation, Â, and the indifference relation,, are defined as usual For all b, b 0 B and α [0, 1], defined (αb +(1 α) b 0 )(θ) =αb (θ) +(1 α) b 0 (θ), for all θ Θ For p (Z (θ)) Iusethenotationb θ p to denote the bet that result from replacing the θ coordinate of b with the lottery p Denote by δ z the (degenerate) lottery that yields the prize z with probability one The prize z (θ; p) Z (θ) is said to be the certainty equivalent of p (Z (θ)) in θ, if 3 A preference relation, <, is a binary relation on C; (a, b) < (a 0,b 0 ) has the interpretation (a, b) is at least as desireable as (a 0,b 0 ) 5
a, b θ δ z(θ;α) (a, b θ p) I assume throughout that for all θ Θ and p (Z (θ)), the certainty equivalent of p in θ exists The preference relation < is said to be nondegenerate if there are b 0,b B such that b 0 Â b, otherwise the preference relation is degenerate Given <, an effect θ Θ is null given the action a if (a, b θ p) (a, b θ q) for all p, q (Z (θ)), and b B, otherwise it is nonnull given the action a Note that, given <, an effect may be null under some actions and nonnull under others Denote by Θ (a; <) the subset of effects that are nonnull given a 22 Continuous weak orders The first two axioms are part of all the models below These axioms are familiar and require no further explanation (A1) (Weak order) < is complete and transitive (A2) (Continuity) For all (a, b) C the sets {(a 0,b 0 ) (a 0,b 0 ) < (a, b)} and {(a 0,b 0 ) (a, b) < (a 0,b 0 )} are closed 6
23 Constant utility bets In general the welfare implications of the choice of actions are direct (for example, exertion of effort may be unpleasant), and indirect, through their perceived impact on the likely realization alternative effects Broadly speaking, a constant-utility bet is a bet whose effectcontingent payoffs entail compensating variations that render uniform the valuation of the ensuing effect-lottery pairs Thus, having accepted a constant utility bet, the decision maker is indifferent to which particular effect obtains Consequently, constant utility bets neutralize the indirect well-being implications of the actions The notion of constant-utility bet, introduced in Karni (2005a), is key to the theory presented here and to the claim that the subjective probabilities defined in this framework represent the decision makers beliefs The following additional axioms separate the direct well-being implications of actions and those of the constant-utility bets Let B be a convex subset of B and suppose that the restriction of < to A B satisfies the following well-known axioms: (A3) (Essentiality) There are a, a 0 A and b, b 0 B such that a, b  a 0, b and a, b  a, b0 (A4) (Coordinate independence) Foralla, a 0 A and b, b 0 B, a, b < a, b 0 if and only if a 0, b < a 0, b 0 and a, b < a 0, b if and only if a, b 0 < a 0, b 0 (A5) (Hexagon condition) For all a, a 0,a 00 A and b, b 0, b 00 B if a, b 0 a 0, b and a, b00 a 0, b 0 a 00, b then a 0, b 00 a 00, b 0 7
Theelementsof B have the interpretation of constant-utility bets The assumption that B is convex merits elaboration Denote by δ z := δ z(θ) θ Θ B the bet that assigns to θ the sure outcome z (θ) Suppose that δ z,δ z 0 B and (a, δ z 0) Â (a, δ z ) The the convexity of B implies that αδ z +(1 α) δ z 0 B, for all α [0, 1] Let z (θ, α) be the certainty equivalent of αδ z(θ) +(1 α) δ z 0 (θ), then, by the richness assumption below, δ z(α) := δ z(θ;α) θ Θ B for all α [0, 1] The convexity of B is responsible for the particular relation among the effect-dependent utility function in Theorem 1 below and, more importantly, the uniqueness of the action-dependent subjective probabilities I assume throughout that the constant utility bets are well-defined and for every given action each bet has an equivalent constant utility bet To formalize these assumption I introduce the following additional notations For every b B and a A, let I b; a = {b B (a, b) a, b } and let I (p; θ, b, a) ={q (Z (θ)) (a, b θ q) (a, b θ p)} Richness assumption: (a) b 0 a A I b; a if and only if b0 (θ) I b (θ);θ, b, a for all θ Θ and a A (b) For all (a, b) C there is b B such that a, b (a, b) To grasp the meaning of this assumption, note that distinct actions correspond to distinct beliefs regarding the likely realization of the different effects If A is not sufficiently rich, then given < there may exist b B and bets b a A I b; a (that is, a, b (a, b) for all a A) andyetb (θ) / a A I b (θ);θ, b, a for some θ Θ In this case, it is easy to conceive 8
of an action, ā, such that ā, b (ā, b) Hence if ā is added to A, then b/ a A {ā} I b; a The assumption states that the set of actions is sufficiently rich that there exist no additional actions that, if added to the set A, would reduce the set of bets that are indifferent to b conditional on every action in A The second part of the richness assumption requires that every bet has a certain-utility equivalent under all actions 3 Subjective Expected Utility Theory In this section I explore alternative subjective expected utility models 31 Effect-Dependent Preferences Conditional independence is the application of the independence axiom of expected utility theory to elements of the choice set that have the same action (A6) (Conditional Independence) Foralla A, b, b 0,b 00 B and α (0, 1], (a, b) < (a, b 0 ) if and only if (a, (αb +(1 α) b 00 )) < (a, (αb 0 +(1 α) b 00 )) Next I assume that, conditional of the effects, the risk attitudes displayed by the decision maker are independent of the actions This requires that the ranking of bets whose payoffs agree on all nonnull effects except one, be independent of the action (that is, the conditional preferences on lotteries given the effect are action-independent) Formally, 9
(A7) (Action independent risk attitudes) For all a, a 0 A, b B, θ Θ (a; <) Θ (a 0 ; <) and p, q (Z (θ)), (a, (b θ p)) < (a, (b θ q)) if and only if (a 0, (b θ p)) < (a 0, (b θ q)) 32 Subjective expected utility with effect-dependent preferences The first representation theorem below asserts that if the richness assumption holds then the axiomatic structure depicted by (A1) (A7) is necessary and sufficient for the existence of subjective expected utility representation with action-dependent probabilities and effectdependent risk attitudes Theorem 1 Let < be a nondegenerate preference relation on C and suppose that the richness assumption holds, then (a) The following conditions are equivalent: (ai) < satisfies (A1) (A7) (aii) There exists a family of probability measures {π ( ; a)} a A on Θ, continuous, jointly cardinal, real-valued functions u on C and v on A such that, for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) if and only if X π (θ; a) X π (θ; a 0 ) X u (z; θ) b 0 (z; θ)+v (a 0 ) (1) θ Θ u (z; θ) b (z; θ)+v (a) X z Z(θ) θ Θ 10 z Z(θ)
where u (λ (θ 0 ) z; θ 0 )=u (z; θ),λ(θ 0 ) > 0 for all θ 0 Θ {θ} and z Z (θ) (b) Thefamilyofprobabilitymeasures{π ( ; a) a A} on Θ is unique and π (θ; a) =0if and only if θ is null given a 33 Subjective expected utility with effect-independent preferences Consider next the case of effect-independent risk attitudes, that is, the case in which the ranking of lotteries is the same across effects This is analogous to the Anscombe and Aumann (1963) state-independence, or monotonicity, axiom Formally, (A8) (Effect independence) For all a A, b B, θ,θ 0 Θ (a; <), and p, q (Z (θ)) (Z (θ 0 )), (a, b θ p) < (a, b θ q) if and only if (a, b θ 0p) < (a, b θ 0q) The next theorem establishes that if the preference relation satisfies effect independence in addition to the other axioms, then the utility functions that figures in Theorem 1 are effect independent That is, not only do they display uniform attitudes towards risk, they also represent identical evaluation of the prizes Theorem 2 Let < be a preference relation on C and suppose that the richness assumption holds, then (a) The following conditions are equivalent: 11
(ai) < satisfies (A1) (A8) (aii) There exists a family of probability measures {π ( ; a) a A} on Θ, and jointly cardinal, continuous, real-valued functions u on θ Θ Z (θ) and v on A such that, for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) if and only if X π (θ; a) X π (θ; a 0 ) X u (z) b 0 (z; θ)+v (a 0 ) (2) θ Θ z Z(θ) u (z) b (z; θ)+v (a) X θ Θ z Z(θ) (b) Thefamilyofprobabilitymeasures{π ( ; a) a A} on Θ is unique and π (θ; a) =0if and only if θ is null given a Unlike in the traditional formulation of the subjective expected utility model, effectindependent preferences (or risk attitudes) imply effect-independent utility functions Hence in this case Ramsey s (1931) idea of defining the degree of belief in the truth of a proposition, or the likely realization of events, by the decision maker s betting behavior makes sense Otherwise, the decision maker s beliefs are confounded with his tastes in a way that renders the representation of these beliefs by subjective probabilities, representing the odds that he will accept, meaningless 4 4 For a detailed discussion of this issue see Karni (2005a) 12
4 Subjective Probabilities without Expected Utility In this section I weaken the conditional independence axiom to obtain a more general choicebased definition of subjective probabilities In particular, the model does not imply that the utility representation is linear in the probabilities 41 Motivation Machina and Schmeidler (1992, 1995) argue, convincingly, that a choice-theoretic definition of subjective probabilities does not require that the preference relation satisfy the axioms of subjective expected utility Invoking the analytical frameworks of Savage (1954) and Anscombe and Aumann (1963), respectively, they show that a decision maker may be probabilistically sophisticated, in the sense that his betting behavior implies the existence of unique subjective probabilities on the state space, his choice among acts is determined by his preferences among the induced distributions on the set of outcomes, and his preferences are representable by a utility function that is not necessarily linear in the probabilities However, as in expected utility theory, the definition of subjective probabilities in the theory of probabilistically sophisticated choice of Machina and Schmeidler is based in an implicit unverifiable assumption of state-independent outcome valuation My next objective is to develop, within the analytical framework of Section 2, a decision theory that yields a definition of action-dependent subjective probabilities on the effects 13
without requiring that the utility representation be linear in the probabilities Unlike the models of Machina and Schmeidler, the theory I propose does not require that the implicit valuation of the consequences be effect independent However, if constant bets are constant utility bets, a version of this model analogous to probabilistic sophistication is obtained Some modification of the lottery structure is introduced to simplify the exposition Let Θ = {θ 1,, θ n } and, for each θ Θ, let Z (θ) :=[zθ,z θ monetary prizes Let b = ³ δ z θ 1,,δ z θn and b = maximal and minimal elements of C, respectively 5 ³ δ z θ1,, δ z θ n ] be a real interval representing be bets in B that are the 42 Axioms To obtain the desired generalization, I replace conditional independence, (A6), that is responsible for the linear structure of the preference relation with two axioms that are analogous to Machina and Schmeidler s (1995) axioms 5 and 6 Given θ Θ and a lottery p (Z (θ)), denote by F p the cumulative distribution function corresponding to p Then, as usual, p is said to first-order stochastically dominate q if F q (z) F p (z) for all z Z (θ) This dominance relation is denoted p 1 q If, in addition, the inequality is strict for some z Z (θ), then the dominance relation is strict and is denoted p> 1 q 5 This assumption is analogous to the assumption of Machina and Schmeidler (1995) that the set of outcomes includes a best and worst outcome 14
The axiom of conditional monotonicity asserts that, given a A, b B, and θ Θ (a; <), the restriction of < to {(a, b θ p) p (Z (θ))} is strictly monotonic with respect to first-order stochastic dominance Formally, (A9) (Conditional monotonicity) For all a A, θ Θ, b B, and p, q (Z (θ)), if p 1 q, then (a, b θ p) < (a, b θ q) If p> 1 q and θ is nonnull under a then (a, b θ p) Â (a, b θ q) To grasp the meaning of the next axiom, let Y T Θ Then given a A and b 00, b 0 B such that b 00 Â b 0, a betony conditional on T isabetb satisfying b (θ) = b 00 (θ) if θ Y and b (θ) = b 0 (θ) if θ T The conditional replacement axiom below requires that for every given action, if the decision maker is indifferent between betting on Y conditional on T and betting on the outcome of a coin flip with probability of winning α conditional on T, then he must remain indifferent when the constant utility bets that figure in the bet on Y conditional on T change, and/or when the lotteries on any other effect change Formally, (A10) (Conditional replacement) For any a A and Y T Θ, if a, b (θ) b (θ) b (θ) θ Y α b (θ)+(1 α) b (θ) θ Y θ T Y a, α b (θ)+(1 α) b (θ) θ T Y θ Θ T b (θ) θ Θ T 15
then, for all b 0, b 00 B such that b 00 Â b 0 and b B, a, b0 (θ) b (θ) b00 (θ) θ Y α b 00 (θ)+(1 α) b 0 (θ) θ T Y a, α b 00 (θ)+(1 α) b 0 (θ) θ Θ T b (θ) θ Y θ T Y θ Θ T 43 Action-dependent subjective probabilities without expected utility The following theorem asserts that a preference relation is a continuous weak order satisfying conditional monotonicity, and conditional replacement and is additively separable on A B if and only if there exist a unique family of action-dependent subjective probabilities, {µ (a) a A}, on Θ and utility functions, V on B and v on A such that, for any action-bet pair ³ (a, b) C, (a, b) V Σ θ Θ µ (θ; a) b θ b (θ) + v (a), where b θ b (θ) denotes the element of B whose θ coordinate is b (θ) The following matrix is useful in trying to understand the meaning of the expression Σ θ Θ µ (θ; a) b θ b (θ) Given a bet b, and θ i Θ, b θi b (θ i ) B is 16
depicted in the i th column of the following matrix: Θ θ 1 θ 2 θ n θ 1 b1 (θ 1 )=b (θ 1 ) b2 (θ 1 ) bn (θ n ) θ 2 b1 (θ 2 ) b2 (θ 2 )=b (θ 2 ) bn (θ 2 ) θ n b1 (θ n ) b2 (θ n ) b n (θ n )=b (θ n ) A real-valued function, f, on a convex subset, S, of a linear space is mixture continuous if f (αp +(1 α) q) is continuous in α for all p, q S Theorem 3 Let < be a nondegenerate preference relation on C and suppose that the richness assumption holds, then (a) The following conditions are equivalent: (ai) < satisfies (A1), (A2), (A9) and (A10) and its restriction to A B satisfies (A3) (A5) (aii) There exist a family of probability measures {µ (a) a A} on Θ, continuous strictly monotonic, real valued function V on B and a continuous real-valued v on A such that, V and v are jointly cardinal and, for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) 17
if and only if V ³ Σ θ Θ µ (θ; a) b θ b (θ) + v (a) V ³ Σ θ Θ µ (θ; a 0 ) b θ b 0 (θ) + v (a 0 ) (3) (b) Thefamilyofprobabilitymeasures{µ (a) a A} on Θ is unique and µ (θ; a) =0if and only if θ is null given a 44 Probabilistically sophisticated choice Like all theories based on the analytical framework of Savage (1954), the models of probabilistically sophisticated choice of Machina and Schmeidler (1992, 1995), and Grant (1995) assume, implicitly, that constant acts are constant-utility acts Adding this assumption to the model of the preceding section yields a model of probability sophisticated choice with action-dependent subjective probabilities Note that in this case Z (θ) =Z for all θ Θ Let B c := {b B b (θ) =b (θ 0 ) θ, θ 0 Θ} be the set of constant bets Theorem 4 Let < be a nondegenerate preference relation on C and suppose that the richness assumption holds, then (a) The following conditions are equivalent: (ai) < satisfies (A1), (A2), (A9) and (A10) and its restriction to A B c satisfies (A3) (A5) 18
(aii) There exist a family of probability measures {µ ( ; a) a A} on Θ, acontinuous, strictly monotonic, real-valued function, V, on (Z) and a continuous real valued function v on A such that, V and v are jointly cardinal and for all (a, b), (a 0,b 0 ) C, Ã! Ã! X X (a, b) < (a 0,b 0 ) V µ (θ; a) b (θ) + v (a) V µ (θ; a 0 ) b 0 (θ) + v (a 0 ) (4) θ Θ θ Θ (b) Thefamilyofprobabilitymeasures{µ ( ; a) a A} on Θ is unique and µ (θ; a) =0if and only if θ is null given a 5 Concluding Remarks Using the analytical framework of Karni (2005) and invoking the device of roulette lotteries, this paper examines the choice-based foundations of subjective probabilities and decisionmaking under uncertainty The choice set consists of actions-bet pairs, where bets are lottery-valued functions on a set of effects and actions represent initiatives that decision makers believe might affect the likely realization of different effects The decision makers preferences over actions and bets reveal these beliefs as well as their risk preferences The main results of this paper are: (a) A general subjective expected utility theory with action-dependent subjective probabilities and effect-dependent risk preferences The cases of effect-independent risk preferences is obtained as special instance (b) A nonexpected utility theory involving well-defined families of action-dependent sub- 19
jective probabilities on effects The utility assigned to action-bet pairs depends on the convex combination of constant-utility bets corresponding to the coordinates of the original bet, where the weights are the action-dependent subjective probabilities A probabilistic sophistication version of this model is obtained as a special case in which constant bets are the constant-utility bets (c) As argued in Karni (2005a), the canonical probabilities defined in these models constitute a numerical representation of a decision maker s degree of belief regarding the likely realization of events conditional on his choice of action 6 Proofs 61 Proof of Theorem 1 (ai) (aii) The restriction of < to A B is a continuous weak order satisfying axioms (A3) (A5) Hence there exist continuous, jointly cardinal, real-valued functions V on B and v on A such that the restriction of < to A B has an additive representation, a, b 7 V b + v (a) (see Wakker 1989, Theorem III41) Let b, b B satisfy a, b  a, b (that such bets exist is an implication of nondegeneracy of < and the richness assumption) Invoking the uniqueness properties of V and v normalize these functions so that V b =0 and V b =1 20
Axiom (A2) implies the Archimedean axiom Hence, by the von Neumann Morgenstern theorem, < satisfies (A1), (A2) and (A6) if there exist jointly cardinal, continuous functions {w a (,θ):z (θ) R θ Θ} such that, for every given a A, and all b, b 0 B, (a, b) < (a, b 0 ) X θ Θ w a (b (θ),θ) X θ Θ w a (b 0 (θ),θ), (5) where w a (b (θ),θ)= P z Z(θ) w a (z, θ) b (z,θ) If θ is null given a then w a (,θ) is a constant function Invoking the joint cardinality normalize the additive-valued functions {w a (,θ)} θ Θ as follows: Let w a b (θ),θ = v (a) / Θ, for all θ Θ and a A, and P θ Θ wa b (θ),θ (v (a) / Θ ) = 1 (Note that if θ is null given a then w a (,θ)=v (a) / Θ ) Because < is nondegenerate Axiom (A7) implies that, for all a, a 0 A and θ Θ (a; <) Θ (a 0 ; <), there are numbers β (θ; a, a 0 ) > 0 and γ (θ; a, a 0 ) such that w a (,θ)=β (θ; a, a 0 ) w a 0 (,θ)+γ (θ; a, a 0 ) (6) Probabilities and utilities: Define π (θ; a) =w a b (θ);θ v (a) Θ for all θ Θ and a A (7) Then, by the normalization of w a, for all a A, π (θ; a) 0 for all θ Θ,and P θ Θ π (θ; a) = 1 But w a ( ; θ) =β (θ; a, a 0 ) w a 0 ( ; θ)+γ (a, a 0,θ) and, by the normalization of w a,forevery 21
a A, w a b (θ),θ = v (a) / Θ, for all θ Θ Hence Equations (7) and (8) imply that β (θ; a, a 0 )= γ (a, a 0,θ)= v (a) Θ β (θ; a, a0 ) v (a0 ) Θ (8) π (θ; a) π (θ; a 0 ) for all a, a0 A and θ Θ satisfying π (θ; a 0 ) > 0 (9) For any given p (Z (θ)), θ Θ and a A, define U (p; θ, a) = w a (p; θ) v (a) Θ /π (θ; a) if π (θ; a) > 0 and U (p; θ, a) =ū otherwise (10) Note that, for all a A and θ Θ (a 0 ; <) Θ (a; <), h i h i w a U (p; θ, a 0 0 (p; θ) v(a0 ) w Θ a (p; θ) v(a) Θ )= = π (θ; a 0 ) β (a,a,θ)π (θ; a 0 ) 0 = h i w a (p; θ) v(a) Θ π (θ; a) = U (p; θ, a), (11) where the third inequality is implied by equation (9) Hence U (p; θ, a) :=U (p; θ) for all a A and θ Θ (a; <) Note that U b (θ),θ =1and U b (θ),θ =0for all θ Θ By definition, w a (p; θ) =π (θ; a) U (p; θ)+ v (a) Θ for all a A, θ Θ and p (Z (θ)) (12) Hence X w a (b (θ),θ)=x U (b (θ),θ) π (θ; a)+v (a) (13) θ Θ θ Θ By von Neumann-Morgenstern theorem, U (,θ) if affine Hence, by the standard argument, there is a real-valued function, u, on C such that U (b (θ),θ)= P z Z(θ) u (z,θ) b (z, θ) 6 6 See Kreps (1988) 22
Next observe that, by the uniqueness properties of the additive representations and the normalizations, for every a A and b B, X X u (z,θ) b (z, θ) π (θ; a)+v (a) =V b + v (a) (14) θ Θ z Z(θ) For every given b B, let b ³ δ z( b), where δ z( b) = δ z(θ; b(θ)) (that is, z θ; b (θ) is the θ Θ certainty equivalent of b (θ) in θ) Then δ z ( b) B and, because B is convex, z θ; b (θ) = λ (θ) z θ 0 ; b (θ 0 ),λ(θ) > 0, θ Θ Hence X X u (z,θ) b (z, θ) π (θ; a) = X u λ (θ) z θ 0 ; b (θ 0 ),θ π (θ; a) θ Θ (15) θ Θ z Z(θ) But equations (14) and (15) imply that P θ Θ u z θ; b (θ),θ π (θ; a) is independent of a Hence u λ (θ) z θ 0 ; b (θ 0 ),θ = u λ (θ 00 ) z θ 0 ; b (θ 0 ),θ 00 for all θ, θ 00 Θ (a; <) Thus, by equation (14), u λ (θ) z θ 0 ; b (θ 0 ),θ = V b for all θ Θ (a; <) Representation of <: For all (a, b) and (a 0,b 0 ) in C, (a, b) < (a 0,b 0 ) if and only if X X u (z, θ) b (z,θ) X u (z,θ) b 0 (z,θ) π (θ; a 0 )+v(a 0 ) π (θ; a)+v (a) X θ Θ θ Θ z Z(θ) z Z(θ) To see this observe that, by the richness assumption, there are constant utility bets b and b 0 such that a, b (a, b) < (a 0,b 0 ) a 0, b 0 By transitivity, a, b < a 0, b 0 But a, b < a 0, b 0 if and only if V b + v (a) V b0 + v (a 0 ) And, by equation (14) and (15), (16) V b + v (a) V b0 + v (a 0 ) if and only if X X u (z, θ) b (z,θ) π (θ; a)+v (a) X X θ Θ θ Θ z Z(θ) 23 z Z(θ) u (z,θ) b 0 (z,θ) π (θ; a 0 )+v(a 0 ) (17)
But a, b (a, b) if and only if P θ Θ w P a b (θ),θ = θ Θ w a (b (θ),θ) Thus, by equations and (13), X a, b (a, b) X and θ Θ (a 0,b 0 ) a 0, b 0 X θ Θ z Z(θ) X z Z(θ) u (z,θ) b (z, θ) π (θ; a) = X X θ Θ z Z(θ) u (z, θ) b 0 (z, θ) π (θ; a 0 )= X X θ Θ z Z(θ) u (z,θ) b (z, θ) π (θ; a), (18) u (z,θ) b 0 (z,θ) π (θ; a 0 ) Equation (16) follows from equations (17) (19) and transitivity This completes the proof that (ai) (aii) (19) (aii) (ai) That (aii) implies (A1) (A2) is immediate That (aii) implies (A6) follows from the von Neumann Morgenstern theorem The proof of (A7) is immediate That (A3) (A5) are implied by Wakker (1989) Theorem III41 The functions (V,v) are jointly cardinal and u z θ; b (θ),θ = V b for all θ Θ Moreover, by the richness assumption, for every z Z (θ) there is b θ δ z B such that u (z,θ) =V b θ δ z (To see this let λ (θ 0 ) z satisfy u (λ (θ 0 ) z,θ 0 )=u (z,θ) and let b θ δ z = δλ(θ)z θ Θ ) Thus u and v are jointly cardinal (b) To prove the uniqueness of {π ( ; a)} a A suppose, by way of negation, that for some a A there exists a probability measure, µ ( ; a), on Θ and µ ( ; a) 6= π ( ; a) Then there are effects θ 0,θ 00 Θ such that µ (θ 0 ; a) >π(θ 0 ; a) and π (θ 00 ; a) >µ(θ 00 ; a) Without essential loss of generality suppose that µ (θ; a) = π (θ; a) for all θ Θ {θ 0,θ 00 } (Note 24
that µ (θ; a) >π(θ; a) and π (θ 0 ; a) >µ(θ 0 ; a) imply that θ and θ 0 are nonnull given a) Let b B and r 0,r 00 R such that u (r 0,θ 0 ) > P z Z(θ) u(z,θ0 ) b (z; θ 0 ) and u (r 00,θ 00 ) < P ³ z Z(θ 00 ) u(z, θ00 ) b (z; θ 00 ) satisfy b θ 0δ r 0 δ θ 00 r 00 b Then the representation implies that u (r 0,θ 0 ) π (θ 0 ; a)+u (r 00,θ 00 ) π (θ 00 ; a) =V b [π (θ 0 ; a)+π (θ 00 ; a)] and, since π (θ 0 ; a)+π (θ 00 ; a) =µ (θ 0 ; a)+µ (θ 00 ; a), u (r 0,θ 0 ) µ (θ 0 ; a)+u (r 00,θ 00 ) µ (θ 00 ; a) >V b [µ (θ 0 ; a)+µ (θ 00 ; a)] A contradiction Hence µ ( ; a) =π ( ; a) By definition π(θ; a) =0for all θ that is null given a 62 Proof of Theorem 2 (ai) (aii) By Theorem 1, for all a A, θ, θ 0 Θ (a; <), and p, q (Z (θ)) (Z (θ 0 )), (a, b θ p) < (a, b θ q) X u (z; θ) p (z) X u (z; θ) q (z) (20) z Z(θ) z Z(θ) and (a, b θ 0p) < (a, b θ 0q) X u (z; θ 0 ) p (z) X u (z; θ 0 ) q (z) (21) z Z(θ 0 ) z Z(θ 0 ) By axiom (A8) X u (z; θ) p (z) X u (z; θ) q (z) X u (z; θ 0 ) p (z) X u (z; θ 0 ) q (z) (22) z Z(θ) z Z(θ) z Z(θ 0 ) z Z(θ 0 ) 25
Let u ( ; θ 0 ):=u ( ), then, by the uniqueness of the von Neumann Morgenstern utility function, for all a A and θ Θ (a; <), u ( ; θ) =σ (θ) u ( )+κ (θ),σ(θ) > 0 (23) But, by Theorem 1, u (λ (θ) z,θ) =u (λ (θ 0 ) z,θ 0 ) for all a A and θ, θ 0 Θ (a; <) Hence, by equation (23),for all z, [σ (θ) σ (θ 0 )] u (λ (θ) z,θ) =κ (θ 0 ) κ (θ) (24) But u ( ; θ) is non-constant Hence σ (θ) σ (θ 0 )=κ(θ 0 ) κ (θ) =0 Invoking the uniqueness of the von Neumann Morgenstern utility function, let σ (θ) =1and κ (θ) =0for all θ Θ Thus, by equation (23), u ( ; θ) =u ( ) That u and v are continuous and jointly cardinal is implied by Theorem 1 This complete the proof that (ai) implies (aii) The proof that (aii) implies axioms (A1) (A7) follows from Theorem 1 The proof that it also implies axiom (A8) is straightforward The proof of part (b) follows from that of part (b) in Theorem 1 63 Proof of Theorem 3 (ai) (aii) The restriction of < to A B is a continuous weak order satisfying axioms (A3) (A5) Hence there exist continuous, jointly cardinal, real-valued functions V on B and v on A such that the restriction of < to A B has an additive representation, a, b 7 V b + v (a) (see Wakker 1989, Theorem III41) 26
By the richness assumption, for all (a, b), (a 0,b 0 ) C, thereexist b, b 0 B such that a, b (a, b) and a 0, b 0 (a 0,b 0 ) Transitivity implies that, for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) a, b < a 0, b 0 V b + v (a) V b0 + v (a 0 ) (25) Because < is nondegenerate axioms (A2) and (A9) imply that for every given (a, b) C there exists a unique number ϕ (a, b) [0, 1], defined by ϕ (a, b) =Sup{ϕ (a, b) < a, ϕ b +(1 ϕ) b }, (26) such that (a, b) a, ϕ (a, b) b +(1 ϕ (a, b)) b (For a proof see Kreps (1988) using the fact that (A2) implies the Archimedean axiom) Moreover, by the convexity of B, ϕ (a, b) b +(1 ϕ (a, b)) b B Clearly, by axioms (A2) and (A9), ϕ(a, ) is mixture continuous and monotonic with respect to first order stochastic dominance For every a A and θ Θ define µ (θ; a) by a, b θ b (θ) (a, µ (θ; a) b +(1 µ (θ; a)) b ) (27) By the preceding discussion µ (θ; a) [0, 1], is well-defined, and µ (θ; a) =0if and only if θ is nullthis proves part(b) of the theorem Fix a A and, without loss of generality, let θ 1 Θ (a; <) (that is, θ 1 is nonnull under a) Consider two constant utility bets, b 00 and b 0, such that b 00 Â b 0 For every i =1,,n 1 and a A let α i (a) be given by a, b00 (θ 1 ),, b 00 (θ i ), b 0 (θ i+1 ),, b 0 (θ n ) a, α i (a) b 00 +(1 α i (a)) b 0 (28) 27
Then α i (a) [0, 1] is well defined and, by axiom (A10) with T = Θ, is independent of the constant utility bets b 00 and b 0 assigned to the subset of effects {θ 1,, θ i } and {θ i+1,, θ n } provided that b 00 Â b 0 Let α 0 (a) =0and define τ i (a) =α n 1 (a) α n 2 (a) α i (a)(1 α i 1 (a)), i =1,,n 1, τ n (a) =(1 α n 1 (a)) ThenΣ n i=1τ i (a) =1 Note that b θ1 b (θ 1 )(θ 1 )=b (θ 1 ) Hence by repeated application of axiom (A10), for any (a, b) C, a, b (θ 1 ) on θ 1 b (θ 2 ) on θ 2 b (θ 3 ) on θ 3 b (θ 4 ) on θ 4 b (θ n ) on θ n a, α 1 (a) b (θ 1 )+(1 α 1 (a))b θ2 b (θ 2 )(θ 1 ) on θ 1 α 1 (a) b θ1 b (θ 1 )(θ 2 )+(1 α 1 (a))b (θ 2 ) on θ 2 b (θ 3 ) on θ 3 b (θ 4 ) on θ 4 b (θ n ) on θ n 28
a, α 2 (a) h α 1 (a) b θ1 b (θ 1 )+(1 α 1 (a))b θ2 b (θ 2 ) i (θ 1 )+(1 α 2 (a)) b θ3 b (θ 3 )(θ 1 ) on θ 1 α 2 (a) h α 1 (a) b θ1 b (θ 1 )+(1 α 1 (a))b θ1 b (θ 2 ) i (θ 2 )+(1 α 2 (a)) b θ3 b (θ 3 )(θ 2 ) on θ 2 α 2 (a) h α 1 (a) b θ1 b (θ 1 )+(1 α 1 (a)) b θ2 b (θ 2 ) i (θ 3 )+(1 α 2 (a)) b (θ 3 ) on θ 3 b (θ 4 ) on θ 4 b(θ n ) on θ n a, τ 1 (a) b (θ 1 )+τ 2 (a) b θ2 b (θ 2 )(θ 1 )+ + τ n (a) b θn b (θ n )(θ 1 ) on θ 1 τ 1 (a) b θ1 b (θ 1 )(θ 2 )+τ 2 (a) b (θ 2 )+ + τ n (a) b θn b (θ n )(θ 2 ) on θ 2 τ 1 (a) b θ1 b (θ 1 )(θ n )+τ 2 (a) b θ2 b (θ 2 )(θ n )+ + τ n (a) b (θ n ) on θ n = ³ a, ³ τ 1 (a) b θ1 b (θ 1 )+τ 2 (a) b θ2 b (θ 2 )+ + τ n (a) b θn b (θ n ) = ³ a, Σ n i=1τ i (a) b θi b (θ i ) Thus any action-bet pair (a, b) is indifferent to the action-bet pair a, b where b B is a convex combination of the constant utility bets corresponding to the coordinates of the original bet b But ³ a, b θ j b (a, τ j (a) b +(1 τ j (a)) b ), (29) 29
³ Hence τ j (a) =µ (θ j ; a) for all θ j,j=1,, n, and a A, and (a, b) a, Σ θ Θ µ (θ; a) b θ b (θ) By (25), for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) V ³ Σ θ Θ µ (θ; a) b θ b (θ) + v (a) V ³ Σ θ Θ µ (θ; a 0 ) b 0 θ b0 (θ) + v (a 0 ) (30) This completes the proof that (i) (aii) The proof that (aii) (ai) is immediate 64 Proof of Theorem 4 (ai) (aii) Since constant bets are constant utility bets, b θ p =[p,, p] for all θ Θ Identify [p,, p] with p and, with slight abuse of notation, let V ([p,, p]) = V (p) Then P θ Θ µ (θ; a) b θp =[p,, p] and, by Theorem 3 and the identification above, for all (a, b), (a 0,b 0 ) C, (a, b) < (a 0,b 0 ) V Ã! X µ (θ; a) b (θ) + v (a) V θ Θ Ã! X µ (θ; a) b 0 (θ) + v (a 0 ) (31) θ Θ The rest of the proof of the theorem follows from the proof of Theorem 3 30
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