Great Expectations. Part I: On the Customizability of Generalized Expected Utility

1 Introduction

Many decision rules have been proposed in the lit~ra~e. Per­ haps the best-known approach is based on l_llax1m1zmg ex­ pected utility (EU). This approach comes m a number of variants· the two most famous are due to von Neumann and Morgen~tern [1947] and Savage [1954]. They all follow the same pattern: they formalize the set of alternative~ among which the decision maker (DM) must choose (typ1cally as acts 1 or lotteries 2 ) . They then give a set of assumptions (often called postulates or axioms) such that the DM's preferences on the alternatives satisfY these assumptions iff the prefer­ ences have an EU representation, where an EU representation of a preference relation is basically a utility function (and a probability measure when acts are involved) such ~~at the re­ lation among the alternatives based on expected utlhty agrees with the preference relation. Moreover, they show th~t the representation is essentially unique. In . other words, ~f the preferences of a DM satisfY the assumptions, then she 1s be­ having as if she has quantified her tastes via a real-valued utility function (and her beliefs via a probability measure)

*Work supported in part by NSF under grants IIS-0090145 and CTC-0208535 and by the DoD Multidisciplinary University Re­ search Initiative (MURI) program administered by ONR under grant NOOOI4-0I-I-0795.

1 Formally, given a setS of states ofthe world and an~the~ ~etC of consequences, an act a is a function from S to C that~ mtUI~Ively, associates with each states the consequence ofperformmg am s.

2 Formally, a lottery is a probability distributio~ ove~ ~onse­ quences; intuitively, the distribution quantifies how likely It IS that each consequence occurs.

and she is relating the alternatives according to their expected utility. The assumptions are typically regarded as criteria for rational behavior, so these results also suggest that if a DM's beliefs are actually described by a probability measure and her tastes are described by a utility function, then she should relate the alternatives according to their expected utility (if she wishes to appear rational).

Despite the appeal of EU maximization, it is well known that people do not follow its tenets in general [Resnik, 1987]. As a result, a host of extensions of EU have been proposed that accommodate some of the more systematic violations (see, for example, [Gul, 1991; Gilboa and Schmeidler, 1989; Giang and Shenoy, 2001; Quiggin, 1993; Sc~eidler, 1_9~9; Yaari, 1987]). Again, the typical approach m the dec1s10n theory literature has been to prove representation theorems, showing that, given a suggested decision rule n, there ex­ ists a collection of assumptions such that a preference rela­ tion satisfies the assumptions iff it has an n representation, where an n representation of a preference relation is essen­ tially a choice of tastes (and beliefs) such that, given these as inputs, n relates the alternatives the way the preference relation does.

Given this plethora of rules, it would be useful to have a general framework in which to study decision making. The framework should also let us understand the relationship be­ tween various decision rules. We provide such a framework in this paper.

The basic idea of our approach is to generalize the notion of expected utility so that it applies in as general a context as possible. To this end, we introduce expectation domains, which are structures consisting of

• three (component) domains: a plausibility domain, a utility domain , and a valuation domain,
• two binary operators EB and 0, which are the analogues of+ and x over the reals, and
• a reflexive binary relation~ (which generalizes :::;).

Intuitively, 0 combines plausibility values and utility values much the same way that x combines probability and (real) utility, while EB combines the products to form the (generalized) expected utility. .

We have three domains because we do not want to reqmre that DMs be able to add or multiply plausibility values or util­ ity values, since these could be qualitative (e.g., plausibility

values could be {"unlikely", "likely", "very likely", ... } and utility values could be {"bad", "good", "better", ... } ). In general, we do not assume that j is an order (or even a pre­ order), since we would like to be able to represent as many preference relations and decision rules as possible.

Once we have an expectation domain, DMs can express their tastes and beliefs using components of the expectation domain. More specifically, the DMs express their beliefs using a plausibility measure [Friedman and Halpern, 1995], whose range is the plausibility domain of the expectation do­ main (plausibility measures generalize probability measures and a host of other representations of uncertainty, such as sets of probability measures, Choquet capacities, possibility mea­ sures, ranking functions, etc.) and they express their tastes using a utility function whose range is the utility domain of the expectation domain. In an expectation domain, it is pos­ sible to define a generalization of expected utility, which we call generalized EU (GEU). The GEU of an act is basically the sum (in the sense of EfJ) of products (in the sense of 0 ) of plausibility values and utility values that generalizes the standard definition of (probabilistic) expected utility over the reals in the obvious way.

We start by proving an analogue of Savage's result with respect to the decision rule (Maximizing) GEU. 3 We show that every preference relations on acts has a GEU represen­ tation (even those that do not satisfy any of Savage's postu­ lates), where a GEU representation of a preference relation basically consists of an expectation domain E, plausibility measure PI, and utility function u, such that the way acts are related according to their GEU agrees with the preference relation (Theorem 3 .I). In other words, no matter what the DM's preference relation on acts is, she behaves as if she has quantified her beliefs by a plausibility measure and her tastes via a utility function, and is relating the acts according to their (generalized) expected utility as defined by the EfJ and 0 of some expectation domain. That is, we can model any DM using GEU, whether or not the DM satisfies any rationality assumptions. An important difference between our result and that of Savage is that he was constructing EU representations, which consists of a real-valued utility function u and a prob­ ability measure Pr (and the expectation domain is fixed, so EfJ, 0, and j are just+, x , and :::;, respectively).

Given that GEU can represent all preference relations, it might be argued that GEU is too general-it offers no guide­ lines as to how to make decisions. We view this as a feature, not a bug, since our goal is to provide a general framework in which to express and study decision rules, instead of propos­ ing yet another decision rule. Thus the absence of "guide­ lines" is in fact an absence of limitations: we do not want to exclude any possibilities at the outset, even preference rela­ tions that are not transitive or are incomplete. Starting from a framework in which we can represent all preference rela­ tions, we can then consider what preference relations have "special" representations, in the sense that the expectation domain, plausibility measure, and utility function in the rep­ resentation satisfy some (joint) properties. This allows us to show how properties of expectation domains correspond to properties of preference relations. We can then "customize" GEU by placing just the constraints we want. We illustrate this by showing how each of Savage's postulates corresponds in a precise sense to an axiom on GEU.

3 Many decision rules involve optimizing (i.e., maximizing or minimizing) some value function on the acts. Sometimes it is ex­ plicitly mentioned whether the function is to be maximized or min­ imized (e.g., "Minimax Regret" says explicitly to "minimize the maximum regret") while other times only the function name is men­ tioned and it is implicitly understood what is meant (e.g., "EU" means "maximize EU"). In this paper we will use "Maximizing GEU" and "GEU" interchangeably.

Intuitively, a decision rule maps tastes (and beliefs) to pref­ erence relations on acts. Given two decision rules R 1 and R 2 , an R 1 representation ofR2 is basically a function r that maps inputs of R 2 to inputs of R 1 that represent the same tastes and beliefs, with the property that R 1(r(x)) = R2(x). Thus, r models, in a precise sense, a user of R 2 as a user of R 1 , since r preserves tastes (and beliefs).· In a companion paper [Chu and Halpern, 2003] we show that (almost) every decision rule has a GEU representation.

Although there has been a great deal of work on decision rules, there has been relatively little work on finding gen­ eral frameworks for representing decision rules. In partic­ ular, there has been no attempt to find a decision rule that can represent all preference relations. There has been work in the fuzzy logic community on finding general notions of integration (which essentially amounts to finding notions of expectation) using generalized notions of EfJ and 0 ; see, for example, [Benvenuti and Mesiar, 2000]. However, the expec­ tation domain used in this work is (a subset of) the reals; ar­ bitrary expectation domains are not considered. Luce [ 1990; 2000] also considers general addition-like operations applied to utilities, but his goal is to model joint receipts (which are typically modeled as commodity bundles in economics) as a binary operation, rather than to represent decision rules.

The rest of this paper is organized as follows. We cover some basic definitions in Section 2: plausibility domains, utility domains, expectation domains, decision problems, and GEU. We show that every preference relation on acts has a GEU representation in Section 3. In Section 4, we show that each of Savage's postulates corresponds to an_axiom on GEU. We conclude in Section 5. Proofs of theorems stated are available at http://www.cs.cornell.edu/home/halpern.

2 Preliminaries

2.1 Plausibility, Utility, and Expectation Domains

Since one of the goals of this paper is to provide a general framework for all of decision theory, we want to represent the tastes and beliefs of the DMs in as general a framework as possible. In particular, we do not want to force the DMs to linearly preorder all consequences and all events (i.e. , subsets of the set of states). To this end, we use plausibility measures to represent the beliefs of the DMs and (generalized) utility functions to represent their tastes.

A plausibility domain is a set P, partially ordered by ~P (so ~P is a reflexive, antisymmetric, and transitive re­ lation), with two special elements ..lp and T p, such that for all x E P, ..lp ~P x ~P T p. Given a setS, a function PI : 2 5 --+ P is a plausibility measure iff

Clearly plausibility measures are generalizations of prob­ ability measures. As pointed out in [Friedman and Halpern, 1995], plausibility measures generalize a host of other repre­ sentations of uncertainty as well. Note that while the proba­ bility of any two sets must be comparable (since lR is totally ordered), the plausibility of two sets may be incomparable.

We also want to represent the tastes of DMs using some­ thing more general than IR, so we allow the range of utility functions to be utility domains, where a utility domain is a set U endowed with a reflexive binary relation ;::Su. Intuitively, elements of U represent the strength of likes and dislikes of the DM while elements of P represent the strength of her be­ liefs. Note that we do not require the DM's preference to be transitive (although we can certainly add this requirement). Experimental evidence shows that DMs' preferences occa­ sionally do seem to violate transitivity.

Once we have plausibility and utility, we want to com­ bine them to form expected utility. To do this, we intro­ duce expectation domains, which have utility domains, plau­ sibility domains, and operators EB (the analogue of+) and ® (the analogue of x ). 4 More formally, an expectation do­ main is a tuple E = (U, P, V, ® , EB ), where (U, ;::Su) is a util­ ity domain, (P, =5P) is a plausibility domain, (V, ;::Sv) is a valuation domain (where ;::Sv is a reflexive binary relation), ® : P x U -> V, and EB : V x V-> V . (As usual, we omit subscripts when they are clear.) We have four requirements on expectation domains:

E l. (xEB y) EBz = xEB (y EBz );

E2. x EB y = y EB x;

E4. (U, ;::Su ) is a substructure of (V, ;::Sv ).

El and E2 say that EB is associative and commutative. E3 says that T is the left-identity of ® and E4 ensures that the expectation domain respects the relation on utility values.

Note that we do not require that EB be monotonic; that is, we do not require that for all x, y , z E V,

We say that E is monotonic iff (2.1) holds. It turns out that monotonicity does not really make a difference by itself; see the comments after the proof of Theorem 3 .1.

2.2 Decision Situations and Decision Problems

A decision situation describes the objective part of the cir­ cumstance that the DM faces (i.e., the part that is independent of the tastes and beliefs of the DM). Formally, a decision sit­ uation is a tuple A= (A, S, C) , where

• S is the set of states of the world,
• C is the set of consequences, and
• A is a set of acts (i .e. , a set of functions from S to C).

4 Sometimes we use x to denote Cartesian product; the context will always make it clear whether this is the case.

An act a is simple iff its range is finite. That is, a is simple if it has only finitely many consequences. Many works in the literature focus on simple acts (e.g., [Fishburn, 1987]). We assume in this paper that A contains only simple acts; this means that we can define (generalized) expectation using finite sums, so we do not have to introduce infinite series or integration for arbitrary expectation domains. Note that all acts are guaranteed to be simple if either S or C is finite, although we do not assume that here.

A decision problem is essentially a decision situation to­ gether with information about the tastes and beliefs of the DM; that is, a decision problem is a decision situation to­ gether with the subjective part of the circumstance that faces the DM. Formally, a (plausibilistic) decision problem is a tuple V = (A, E, u , PI), where

• A= (A, S, C) is a decision situation,
• E = (U, P, V, ®, EB) is an expectation domain,
• u : C -> U is a utility function, and
• PI : 2 8 -> P is a plausibility measure.

We say that V is monotonic iff E is monotonic.

2.3 Expected Utility

Let V = ((A, S, C) , E , u , PI) be a plausibilistic decision problem. Each a E A induces a utility random variable Ua : S-> U as follows: ua(s) = u(a(s)). In the standard setting (where utilities are real-valued and PI is a probability measure Pr), we can identify the expected utility of act a with the expected value ofua with respect to Pr, computed in the standard way:

We can generalize (2.2) to an arbitrary expectation domain E = (U, P, V, ®, EB) by replacing +, x , and Pr by EB, ®, and PI, respectively. This gives us

We call (2.3) the generalized EU (GEU) of act a. Clearly (2.2) is a special case of (2.3).

In the probabilistic case, if all singleton sets are measurable with respect to Pr (i.e., in the domain ofPr), then

The plausibilistic analogue of (2.4) is not necessarily equiv­ alent to (2.3). A decision problem ((A, S, C) , E, u , PI) is additive iff

for all c E C and nonempty X , Y ~ S such that X n Y = 0. Note that the notion of additivity we defined is a joint prop­ erty of several components of a decision problem (i.e., EB, ® , u, and PI) instead of being a property of PI alone. Additiv­ ity is exactly the requirement needed to make the analogue of (2.4) equivalent to (2.3). While decision problems involv­ ing probability are additive, those involving representations of uncertainty such as Dempster-Shafer belief functions are not, in general.

3 Representing Arbitrary Preference Relations

In this section, we show that every preference relation on acts has a GEU representation. GEU, like all decision rules, is formally a function from decision problems to preference re­ lations on acts. Thus a GEU representation of a preference relation ~A on the acts in A = (A, . .. ) is a decision problem V = (A, E, u, PI), where E = (U, P, V, EB, 0 ), such that a1 ~A a2 iffEpJ,£(Ua 1 ) ~V EPI,E(lia2 ) .

Theorem 3.1: Every ~A has a GE U representation. 5

Proof: Fix some A = (A, S, C) and ~A · We want to construct a decision problem V = (A, E, u, PI) such that GEU(V) =~A ·

The idea is to let each consequence be its own utility and each set be its own plausibility, and define 0 and EB such that each act is its own expected utility. For each c E C, let ac denote the constant act with the property that ac ( s) = c for all s E S. Let E = ( U, P, V, EB, 0 ) be defined as follows:

1. U = ( C, ~c) , where c ~c d iff c = d or ac, ad E A and ac ~A ad. (Note that Savage assumes that A contains all simple acts; in particular, A contains all constant acts. We do not assume that here.)

1. V = (2 5 xc, ~v), where x ~v y iff x = y or x , y E A and x ~A y. (Note that set-theoretically a function is a set of ordered pairs, so A ~ 2 5 x c.)

We can identify c E C with S x { c} in V; with this identification, (U, ~u) is a substructure of (V, ~v) and T 0 c = c for all c E U ( = C), as required. Furthermore, EB is clearly associative and commutative, so E is indeed an expecta­ tion domain. Let V = (A, E , u , PI), where u(c) = c and PI( X) = X . Note that

That is, each act is its own expected utility; by the definition of~v, it is clear that a ~A b iffEpJ,e(ua) ~v EPI,e(ub) · Thus GEU(V) = ~A, as desired. I

Note that the representation constructed in Theorem 3.1 is in fact additive, since if X n Y = 0, then

The representation is not necessarily monotonic. However, as we show in the full paper, a simple modification to the defi­ nition of ~v makes the representation monotonic, so mono­ tonicity by itself does not restrict the kind of preference rela­ tions that can be represented.

5 Note that, unlike most representation theorems, there is no uniqueness condition. However, we can show that our representa­ tion is canonical in a certain sense.

Theorem 3.1 holds in large part because of the flexibility we have. Given a decision situation (A, S, C) and a prefer­ ence relation ~A on A, we are able to construct an expecta­ tion domain and a relation ~v that is customized to capture the relation ~A on A. However, this depends (in part) on two features of our setup.

The first is that, following Savage [ 1954 ], we took acts to be functions from states to consequences. Suppose instead that we have a consequence function c : A x S ---. C that takes an act a and a state s and gives the consequence of a in s. Of course, in this setting, two distinct acts a1 and a2 could induce the same function from states to consequences; that is, we might have c(a1, s) = c(a2, s) for all s E S. It is easy to see that if a1 and a 2 induce the same function from states to consequences, then no matter what expectation domain, utility function, and plausibility measure we use, a1 and a 2 will have the same expected utility. Thus, if ~A does not treat a 1 and a 2 the same way, then ~A has no GEU representation.

A second reason that we do not need consistency con­ straints on ~A is that we have placed no constraints on ~v , and relatively few constraints on EB, 0 , u, and Pl. If, for ex­ ample, we required ~v to be transitive, then we would also have to require that ~A be transitive. The lack of constraints on EB, 0 , u, and Pl is important because it gives us enough freedom to ensure that distinct acts have different expected utility. In the next section, we investigate what happens when we add more constraints.

4 Representing Savage's Postulates

Theorem 3.1 shows that GEU can represent any preference relation. We are typically interested in representing prefer­ ence relations that satisfy certain constraints, or postulates. The goal of this section is to examine the effect of such con­ straints on the components that make up GEU. For definite­ ness, we focus on Savage's postulates. For ease of exposition, we restrict to additive decision problems in this section; recall that this restriction does not affect Theorem 3. L

A set 'Pe of axioms about (i .e., constraints on) plausibilis­ tic decision problems represents a set of postulates 'Pr about decision situations and preference relations iff

Theorem 3.1 can be viewed as saying that the empty set of axioms represents the empty set of postulates.

Before we present Savage's postulates, we first introduce some notation that will make the exposition more succinct. Suppose that f : X ---. Y, g : X ---. Y, and Z ~ X. Let (f, Z, g) denote the function h such that '!l x) = f ( x) for all x E Z and h(x ) = g(x) for all x E Z. For example, if X= Y = lR and Z ={xI x < 0}, then (- x ,Z,x) is the absolute value function. In the intended application, the functions in question will be acts (i.e., functions from the set of states S to consequences C). So a = (a1, X , a2) is the act such that a(s) = a1(s ) for all sEX and a(s) = a2(s ) for all s E X. For brevity, we identify the consequence c E C with the constant act ac such that ac( s) = c for all s E S . So for c1 , c2 E C, (c1 , X , c2 ) is the act with the property that a(s) = c1 for all s E X and a(s) = c2 for all s EX. Recall that X1, ... , Xn is a partition ofY iff the X/s are nonempty and pairwise disjoint, and Ui Xi = Y.

Fix some decision situation (A, S, C). Readers familiar with [Savage, 1954] will recall that Savage implicitly as­ sumes that A consists of all possible functions from S to C, since the DM can be questioned about any pair of func­ tions. Throughout this paper, we have assumed that A could be any nonempty subset of the set of all simple acts. It is possible to maintain that assumption here, though some of the postulates would fail, not because .::SA does not relate cer­ tain members of A, but because A does not contain certain pairs of acts. We could adapt the postulates by "relativizing" them (so acts not in A are not required to be related) as we do in the full paper; however, that involves changing the state­ ments of Savage's postulates somewhat, and the presentation becomes more complicated. To simplify the exposition here, we assume in this section only that A consists of all simple acts. Here are Savage's first six postulates (stated under the assumption that A consists of all simple acts from S to C, so that, for example, if a and b are in A, and X ~ S, then (a, X , b) E A):

• Pl. For all a1, a2, a3 E A,
• (a) a1 .::SA a2 or a2 ;:SA a1, and
• (b) if a1 ;:SA a2 and a2 ;:SA a3, then a1 ;:SA a3.
• P2. For all a1, a2, b1, b2 E A and X~ S,

• P3. For all X ~ S, if there exist some a1, a2 E A such that for all bE A, (a1, X, b) -(A (a2, X, b), then for all c1, c2 E C, c1 .::SA c2 iff
• for all a E A, (c1, X, a) -(A (c2, X , a), or
• forallaE A, (c1,X,a) "'A (c2,X,a).
• P4. For all xl,x2 ~ S, C!,dl,C2,d2 E C, if d1 -(A c1 and d2 -(A c2, then

• P5. There exist c1, c2 E C such that c1 -(A c2.
• P6. For all a, b E A, c E C, if a -(A b, then there ex­ ists a partition Z1, ... , Zn of S, such that for all Zi, (c, zi, a) -(A band a -(A (c, Zi, b).

P 1 is the standard necessary condition for representation by EU (and many of its generalizations), since the reals are linearly ordered; it basically says that .::SA is a total preorder. More specifically, Pl(a) says that .::SA is total (from which reflexivity follows) and Pl(b) says that it is transitive. P2 is the sure-thing principle, which says that the way two acts are related depends only on their differences (that is, the part on which they agree can be ignored, since that is the "sure thing"). Savage defines for each subset X ~ S a conditional preference relation as follows: a1 .::5:4 a2 iff

• for all a E A, (a1,X,a) -(A (a2,X,a),or
• for all a E A, (a1,X,a) "'A (a2,X,a).

Intuitively, a1 .::5:4 a2 iff when X occurs, the DM would find a2 at least as good as a1. In the presence of P 1, P2 holds iff .::5:4 is a total preorder for all X. Using .::5:4, Savage defines what it means for X to be null : X is null iff a1 .::5:4 a2 for all a1, a2 E A. P3 says that if X is not null, then .::5:4 and ;:SA agree on the consequences. Savage defines a relation .::Ss on events as follows: X ;:Ss Y iff for all c, d E C, if d -(A c then (c, X, d) .::SA (c, Y, d). Intuitively, we expect the DM to pre­ fer a binary act that is more likely (according to her beliefs) to yield the more desirable consequence. P4, in the presence of Pl-P3, basically ensures that .::Ss is a total preorder. P5 asserts that Sis not null. Pl-P5 by themselves do not allow the construction of a unique EU representation. 6 For this we need P6, which roughly says that for all pairs of acts a, b E A and consequences c E C, if a -(A b then we can partition S into events such that the DM's preference about a and b is unaffected if c were to happen in any element ofthe partition. Savage also has a seventh postulate, but it is relevant only for general (nonsimple) acts; since we consider only simple acts, we omit it here.

Since we focus on additive decision problems in this sec­ tion, we can define a notion of conditional GEU, which greatly simplifies the presentation of the GEU axioms. Given a plausibilistic decision problem D = ((A, S, C), E, u , PI) and 0 =/=- Z ~ S, define the GEU of act a restricted to Z as follows:

Note that EpJ,E(Ua f S) = EpJ,E(ua).

Since we are restricting to additive decision problems, it is easy to check that, given a partition X 1, . .. , X n of Y ~ S, we have that

Also, it is easy to check that for all nonempty proper subsets X ofS,

Let Ev(X) = {EPI,E(ua f X) I a E A}. (We omit the subscript D if it is clear from context.) Intuitively, Ev(X) consists of all the expected utility values of acts in A re­ stricted to X. To simplify the statement of one of the axioms, let ((u, X, v)) =PI( X) ® u EB Pl(X) ® v if0 =/=-X =/=- S, and let ((u, 0, v)) = v and ((u, S, v)) = u.

• AI. For all x, y, z E E(S),
• (a) x ;:Sv y or y ;:Sv x, and
• (b) if x .::Sv y andy ;:Sv z, then x ;:Sv z.

A2. For all 0 =/=-X =/=- S, x1, x2 E c(X), Yl, Y2 E c(X),

• x1 EB Y1 .::Sv x2 EB Y1 iff x1 EB Y2 .::Sv x2 EB Y2·
• A3. For all 0 =I- X =I- S, if there exist x1, x2 E £(X) such that for all y E E(X) x1 EB y -(v x2 EB y, then for all u1, u2 E ran(u), u1 ;:Sv u2 iff

6 They do, however, ensure that ~s is a qualitative probability (which is a binary relation on 2 8 with certain properties) [Savage, 1954].

• for ally E c(X), Pl(X) 0 u1 EB y -<v Pl(X) 0 u2 EB y, or
• for ally E c(X), Pl(X) 0 Ul EB y "-'V Pl(X) 0 U2 EB y.
• A4. For all X1, X2 <::;: S, u1, v1, ·u2, v2 E ran(u), ifv1 -<v u1 and v2 -<v u2, then
• ((u1 , X1 , v1)) .::5v ((u1,X2 , v1)) iff ((u2, X1, V2)) _::5v ((u2, X2, V2))
• AS. For some u1, u2 E ran(u), u1 -<v u2.
• A6. For all x, y E c(S), u E ran(u), if x -<v y, then for all a, b E A, c E C, such that EPI,E(ua) = x, EPI,E(ub) = y, and u(c) = u, there exists a parti­ tion Z1, ... , Zn of S, such that x can be expressed as x1 EB · · · EB Xn andy can be expressed as Y1 EB · · · EB Yn, where Xi = EpJ,E(Ua r Z;) and Yi = EP!,E(Ub r z,) for 1 :::; i :::; n, and for all 1 :::; k :::; n,
• Pl(Zk) 0 u EB EBi# x; -<v y, and
• x -<v Pl(Zk) 0 u EB EBi# y;.

AI says that the expected utility values are linearly pre­ ordered; more specifically, Al(a) says that they are totally preordered and Al(b) says that the relation is transitive. Note that AI does not say that the whole valuation domain is linearly preordered: that would be a sufficient but not a necessary condition for GEU(V) to satisfy Pl. Since we want necessary and sufficient conditions for our represen­ tation results, some axioms apply only to expected utility values rather than to arbitrary elements of the valuation do­ main. As the following theorem shows, every subset of {A1(a), A1(b), ... , A6} represents the corresponding subset of {Pl(a), Pl(b), ... , P6}.

Theorem 4.1: For all I<::;: {l(a), 1(b), ... , 6}, {Ai I i E I} represents {Pi I i E I}.

Theorem 4.1 is a strong representation result. For example, if we are interested in capturing all of Savage's postulates but the requirement that .::5A is totally ordered, and instead are willing to allow it to be partially ordered (a situation explored by Lehmann [1996]), we simply need to drop axiom A1(b). Although we have focused here on Savage's postulates, it is straightforward to represent many of the other standard pos­ tulates considered in the decision theory literature in much the same way.

5 Conclusion

We have introduced GEU, a notion of generalized EU and shown that GEU can (a) represent all preference relations on acts and (b) be customized to capture any subset of Savage's postulates. In [Chu and Halpern, 2003], we show that GEU can be viewed as a universal decision rule in an even stronger sense: almost every decision rule has a GEU representation. Thus, the framework of expectation structures together with GEU provides a useful level of abstraction in which to study the general problem of decision making and rules for decision making.