Great Expectations. Part II: Generalized Expected Utility as a Universal Decision Rule

1 Introduction

A great deal of effort has been devoted to studying decision making. A standard formalization decribes the choices a de­ cision maker (DM) faces as acts, where an act is a function from states to consequences. Many decision rules (that is, rules for choosing among acts, based on the tastes and beliefs of the DM) have been proposed in the literature. Some are meant to describe how "rational" agents should make deci­ sions, while others aim at modeling how real agents actually make decisions. Perhaps the best-known approach is that of maximizing expected utility (EU). Normative arguments due to Savage [ 1954] suggest that rational agents should behave as if their tastes are represented by a real-valued utility func­ tion on the consequences, their beliefs about the likelihood of events (i.e., sets of states) are represented by a probability measure, and they are maximizing the expected utility of acts with respect to this utility and probability.

Despite these normative arguments, it is well known that EU often does not describe how people actually behave when they make decisions [Resnik, 1987]; thus EU is of limited utility if we want to model (and perhaps predict) how peo­ ple will behave. As a result, many alternatives to EU have been proposed in the literature (see, for example, [Gul, 1991; Gilboa and Schmeidler, 1989; Giang and Shenoy, 2001; Quiggin, 1993; Schmeidler, 1989; Yaari, 1987]). Some of these rules ii:wolve representations of beliefs by means other than a (single) probability measure; in some cases, beliefs and tastes are combined in ways other than the standard way which produces expected utility; yet other cases, such as Maximin and Minimax Regret [Resnik, 1987], do not require a representation of beliefs at all.

*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 N00014-01-l-0795.

In a companion paper [Chu and Halpern, 2003], we pro­ pose a general framework in which to study and compare de­ cision rules. The idea is to define a generalized notion of expected utility (GEU), where a DM's beliefs are represented by plausibility measures [Friedman and Halpern, 1995] and the DM's tastes are represented by general (i.e., not necessar­ ily real-valued) utility functions. We show there that every preference relation on acts has a GEU representation. Here we show that GEU is universal in a much stronger sense: we show that essentially all decision rules have GEU representa­ tions. The notion of representing one decision rule using an­ other seems to be novel. Intuitively, decision rules are func­ tions from tastes (and beliefs) to preference relations, so a representation of a decision rule is a representation of a func­ tion, not a preference relation.

Roughly speaking, given two decision rules R1 and R2, an nl representation ofR2 is a function T that maps inputs of R2 to inputs of R1 that contain the same representation of tastes (and beliefs) such that Rl(r(x)) = n2(x). Thus, T models, in a precise sense, a user of R2 as a user of R 1, since r preserves tastes (and beliefs). We show that a large collec­ tion of decision rules have GEU representations and charac­ terize the collection. Essentially, a decision rule has a GEU representation iff it is uniform in a precise sense. It turns out that there are well-known decision rules, such as maximizing Choquet expected utility (CEU) [Schmeidler, 1989] (which essentially assumes that the DM is representing beliefs using a Dempster-Shafer belief function Bel, and then maximizing CEU with respect to Bel), that have no GEU representations. This is because r is not allowed to modify the representation of the tastes (and beliefs). We then define a notion of ordinal representation, in which r is allowed to modify the represen­ tation of the tastes (and beliefs), and is required to preserve only the ordinal aspect of the tastes (and beliefs). We show that almost all decision rules, including CEU, have ordinal GEU representations.

There seems to be no prior work in the literature that con­ siders how one decision rule can represent another. Perhaps the closest results to our own are those of Lehmann [200 1]. He proposes a "unified general theory of decision" that con­ tains both quantitative and qualitative decision theories. He considers a particular decision rule he calls Expected Quali­ tative Utility Maximization, which allows utilities to be non­ standard real numbers; he defines a certain preorder on the nonstandard reals and makes decisions based on maximiz­ ing expected utility (with respect to that preorder). That his framework has EU as a special case is immediate, since for the standard reals, his preorder reduces to the standard order on the reals. He argues informally that Maximin is a special case of his approach, so that his approach can capture aspects of more qualitative decision making as well. It is easy to see that Lehmann's approach is a special case ofGEU; his rule is clearly not universal in our sense.

2 Preliminaries

To make this paper self-contained, much of the material in the first three subsections of this section is taken (almost verba­ tim) from [Chu and Halpern, 2003].

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. To this end, we use plausibility measures to repre­ sent 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 l_p and T p, such that for all x E P, l_p ~P x ~P T p . A function Pl : 2 8 -> Pis a plausibility measure iff

As pointed out in [Friedman and Halpern, 1995], plausibil­ ity measures generalize, not only probability, but a host of other representations of uncertainty as well. A utility domain is a set U endowed with a reflexive binary relation ;:,u. Intu­ itively, elements of U represent the strength of likes and dis­ likes of the DM while elements of P represent the strength of her beliefs.

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). 1 More formally, an expectation do­ main is a tuple E = (U, P, V, ®, EB ), where (U, ;:,u) is a util­ ity domain, (P, ~P) is a plausibility domain, (V, .::$v) is a valuation domain (where ;:,v 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:

E4. (U, ;:,u) is a substructure of (V, ;:,v ).

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

E1 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.

The standard expectation domain, which we denote lE, is (JR., [0, 1], JR., +, x), where the ordering on each domain is the standard order on the reals.

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 offunctions from S to C).

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 nonplausibilistic decision problem is a tuple (A, U, u), where

• A = (A, S, C) is a decision situation,
• U is a utility domain, and
• u : C -> U is a utility function.

A plausibilistic decision problem is a tuple (A, E, u , Pl), 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
• Pl : 2 8 -> P is a plausibility measure.

We could have let a plausibilistic decision problem be simply a nonplausibilistic decision problem together with a plausi­ bility domain and a plausibility measure, without including the other components of expectation domains. However, this turns out to complicate the presentation (see below).

We say that Vis standard iff its utility domain is JR. (and, if V is plausibilistic, its plausibility measure is a probability measure and its expectation domain is lE).

2.3 Expected Utility

Let V be a decision problem with S as the set of states, U as the utility domain, and u as the utility function. Each act a of V induces a utility random variable Ua : S -> U as follows: ua(s) = u(a(s)). If in addition V is plausibilistic with P as the plausibility domain and Pl as the plausibility measure, then each a also induces a utility lottery e~J,u : ran(ua) ----; p as follows: £~ 1 · u ( u) = PI( u;; 1 ( u)). If V is in fact standard (so E = IE and PI is a probability measure Pr), we can iden­ tify the expected utility of act a with the expected value of Ua with respect to Pr, computed in the standard way:

We can generalize (2.1) 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 (2.2) the generalized EU (GEU) of act a. Clearly (2.1) is a special case of(2.2).

2.4 Decision Rules

Intuitively, a decision rule tells the DM what to do when facing a decision problem in order to get a preference rela­ tion on acts--e.g., compare the expected utility of acts. Just as we have nonplausibilistic decision problems and plausi­ bilistic decision problems, we have nonplausibilistic deci­ sion rules and plausibilistic decision rules. As the name suggests, (non)plausibilistic decision rules are defined on (non)plausibilistic decision problems.

We do not require decision rules to be defined on all deci­ sion problems. For example, (standard) EU is defined only on standard plausibilistic decision problems. More formally, a (non)plausibilistic decision rule R is a function whose do­ main, denoted dom(R), is a subcollection of the collection of (non)plausibilistic decision problems, and whose range, de­ noted ran(R), is a collection of preference relations on acts. IfV E dom(R) and a1 and a2 are acts in V, then we write

Here are a few examples of decision rules:

• GEU is a plausibilistic decision rule whose domain con­ sists of all plausibilistic decision problems. Given a decision problem V = (A, E , .. . ), where E = (U, P, V, EB , ® ),we have a1 .::5cEU(D) az iffEpJ,E(Ua 1 ) .::5v EpJ,e(uaJ for all acts a1 , a2 of A. Note that GEU would not be a decision rule according to this definition if plausibilistic decision problems contained only a utility function and a plausibility measure, and did not include the other components of expectation domams.
• Of course, standard EU is a decision rule (whose domain consists of all standard plausibilistic decision problems).
• Maximin is a nonplausibilistic decision rule that orders acts according to their worst-case consequence. It is a con­ servative rule; the "best" act according to Maximin is the one with the best worst-case consequence. Intuitively, Maximin views Nature as an adversary that always pick a state that realizes the worst-case consequence, no matter what act the DM chooses. The domain of (standard) Maximin consists of nonplausibilistic decision problems with real-valued util­ ities. Given an act a and a real-valued utility function u, let wu(a) = minsES ua(s). Then given a decision problem v = (A, JR, u), al .::5Maximin(D) az iffwu(al)::; Wu(az).
• Minimax Regret (REG) is based on a different philoso­ phy. It tries to hedge a DM's bets, by doing reasonably well no matter what the actual state is. It is also a nonplausibilistic rule. As a first step to defining it, given a nonplausibilistic de­ cision problem V = ((A, S, C), JR, u), for each states E S, let Us = SUPaEA ua(s); that is, Us is the least upper bound of the utilities in state s. The regret of a in state s, denoted r(a, s ), is 'ils - ua(s ); note that no act can do better than a by more than r(a, s) in states. Let r(a) = supsES r(a, s). For example, suppose that r(a) = 2 and the DM picks a. Sup­ pose that the DM then learns that the true state is s0 and is offered a chance to change her mind. No matter what act she picks, the utility of the new act cannot be more than 2 higher then ua(so). REG orders acts by their regret and thus takes the "best" act to be the one that minimizes r(a). Intuitively, this rule tries to minimize the regret that a DM would feel if she discovered what the situation actually was: the "I wish I had done az instead of a1" feeling. Thus, a1 .::5REG(D) az iff r(a1) ?: r(a2) . Like Maximin, Nature is viewed as an adversary that would pick a state that maximizes regret, no matter what act the DM chooses. It is well known that, in general, Maximin, REG, and EU give different recommenda­ tions [Resnik, 1987].
• The Maxmin Expected Utility rule (MMEU) [Gilboa and Schmeidler, 1989] assumes that a DM's beliefs are repre­ sented by a set P of probability measures. Act a1 is pre­ ferred to az if the worst-case expected utility of a1 (taken over all the probability measures in P) is at least as large as the worst-case expected utility of a2 . Thus MMEU is, in a sense, a hybrid of EU and Maximin. To view MMEU as a function on decision problems, we must first show how to represent a set of probability measures as a single plausibil­ ity measure. We do this using an approach due to Halpern [2001]. Let the plausibility domain P = [0, 1]1', that is, all functions from P to [ 0, 1], ordered pointwise; in other words, p ~P q iff p(Pr) ::; q(Pr) for all Pr E P. Thus, in this domain, _l is the constant function 0 and T is the constant function 1. For each X ~ S, let fx E P be the function that evaluates each probability measure in P at X; that is, fx(Pr) = Pr(X) for all Pr E P. Let Plp(X) = fx; it is easy to verify that Pip is a plausibility measure. We view Pip as a representation of the set P of probability measures; clearly P can be recovered from Pip. The domain ofMMEU consists of all plausibilistic decision problems of the form V = ((A,S,C),(JR,[O,l]P,V, EB , ® ),u,Plp), where Pis a set of probability measures on 2 8 , and a1 .::5MMEU(D) az iffinfPrEPEPr(Ua 1 ) ::; infPrEPEPr(Ua 2 ) . Note that this definition ignores EB, ® ,and V.
• (Dempster-Shafer) belief functions [Dempster, 1967] are a representation of uncertainty that generalize probability. That is, every probability measure is a belief function, but the converse is not necessarily true. 2 Given a belief func­ tion Bel, it is well-known that there exists a set PBeJ of probability measures such that for all X ~ S, Bel(X) = infPrEPBet Pr(X) [Dempster, 1967]. A notion of expected

2 Due to lack of space, we assume that the reader is familiar with belief functions .

utility for belieffunctions was defined by Choquet [ 1953] as follows. Given an act a and a real-valued utility function u , ifran(ua) = {u1 , ... , 'Un } and 11.1 < · · · < 1Ln , then

where X; = u;:;- 1 ( { u;, ... , un} ). It is easy to check (2.3) agrees with (2.1) if Bel is a probability measure. Moreover,

in general. The Choquet expected utility (CEU) rule has as its domain decision problems of the form V = (A, IE. , u , Bel) . Define a1 ~CEU('D) a2 iff EBel ( Ua1 ) ::; EBel ( U a2 ). It is immediate from (2.4) that if Vp 801 is the decision problem that results from V by replacing Bel by Plp 8 · 1 , and replacing the plausibility domain [0, 1] in the expectation domain by [0, 1jPsel, then a1 ~CEU('D) a2 iff a1 ~MMEU('Dp 801 ) a2.

3 Representing Decision Rules

Given a decision rule n and a preference relation ~A on the set of acts A, an R representation of~A is basically a deci­ sion problem V E dom(R) such that R(V) = ~A (and the set of acts in V is A). In other words, an R representation of ~A makes R relate acts in A the way ~A relates them, so we can model a DM whose preference relation is ~A as a user of R . In [ Chu and Halpern, 2003] we show that every pref­ erence relation on acts has a GEU representation. Here we want to extend the notion of representation to decision rules; intuitively, we want an R 1 representation of R2 to allow us to model a user ofR2 as a user ofR1. To make this precise, we need a few definitions.

Two (plausibilistic) decision problems V1 and V2 are con­ gruent, denoted V1 ~ V2, iff they involve the same decision situation, utility domain, and utility function (and, if both are plausibilistic, the same plausibility domain and plausibility measure as well). Note that if V1 ~ V2, then they agree on the tastes (and beliefs) of the DM, so if they are both nonplau­ sibilistic, then V1 = V2, and if they are both plausibilistic, then they differ only in the V, ~v, EB, and ® components of their expectation domains.

A decision rule transformation T is a function that maps inputs of one decision rule R2 to the inputs of another rule nl . A decision rule transformation Tis an n1 representation ofR2 iff dom(r) = dom(R2) and for all V E dom(R2),

Thus a DM that uses R 2 to relate acts based on her tastes (and beliefs) behaves as if she is using R 1, since r(V) ~ V and R 1(r(V)) = R2(V).

Note that r(V) = V is a GEU representation of EU. We now consider some less trivial examples.

Example 3.1: To see that Maximin has a GEU representa­ tion, let Emax = (lR, {0,1},lR,min, x ), and let Plmax be the plausibility measure such that Plmax (U) is 0 if X = 0 and 1 otherwise. If V = (A, JR., u) , where A = (A, S, C), then it is easy to check that Eptmax,EmaJua) = Wu(a). Take r(V) = (A, Emax> u, Plmax)· Clearly r(V) ~ V: the util­ ity function has not changed. Moreover, it is immediate that GEU(r(V)) = Maximin(V) . I

Example 3.2: To see that Minimax Regret (REG) has a GEU representation, for ease of exposition, we take dom(REG) to consist of standard decision problems V = ( (A, S, C) , IR, u) such that Mv = supaEA,sES ua(s) < oo. (If Mv = oo, given the restriction to simple acts, it is easy to show that all acts have infinite regret.) Let Ereg = (JR., [0, 1], IR, min, ® ), where x ® y = y - log(x) if x > 0, and x ® y = 0 if x = 0. Note that l_ = 0 and T = 1. Clearly, min is associative and commutative, and T ® r = r - log(1) = r for all r E R Thus, Ereg is an expectation domain.

For 0 =1- X ~ S, define Mx = supaEA,sEX ua(s ). Note that Ms = Mv < oo; also if X~ Y, then Mx ::; My. Let Plv(0) = 0 and Plv(X) = eM x-Ms. It is easy to verify that Plv is a plausibility measure.

It is easy to check that EPtv,E,.s (ua) = Mv - ru (a) for all acts a E A. Let r(V) = (A, Ereg, u , Plv ). Since the utility function has not changed, r(V) ~ V ; further­ more, higher expected utility corresponds to lower regret, so GEU(r(V)) = REG(V) . I

Example 3.3: To see that MMEU has a GEU representation, p ~ ~ ~ let V =(A, (JR., [0, 1] , V, EB, ® ) , u, Plp) E dom(MMEU). Let Ep = (JR., [0, 1]P, JRP, EB , ® ),where EB is pointwise func­ tion addition, ® is scalar multiplication, and

Note that we can identify JR. with the constant functions in JRP, so JR. can be viewed as a substructure of JRP. With these definitions, Ep is an expectation domain.

Let r(V) = (A, Ep, u , Pip) . It is immediate from the definition of ~JK'P that

Thus GEU(r(V)) = MMEU(V); furthermore, it is clear that r(V) ~ V, since the utility function and plausibility measure have not changed. I

Although it can represent many decision rules, GEU cannot represent CEU. We can in fact characterize the conditions under which a decision rule is representable by GEU.

There is a trivial condition that a decision rule must satisfy in order for it to have a GEU representation. Intuitively, a de­ cision rule n respects utility if n relates acts of constant util­ ity according to the relation between utility values. Formally, a decision ruleR respects utility iff for all V E dom(R) with A as the set of acts, S as the set of states, U as the utility domain, and u as the utility function, for all a1, a2 E A, if U a; ( s) = 71.; for all states s E S, then

We say that R weakly respects utility iff (3.5) holds for all constant acts (but not necessarily for all acts of constant util­ ity). It is easy to see that GEU respects utility, since T® u = u for all u E U and (U, ;Su) is a substructure of (V, ;Sv ). Thus if R does not respect utility, it has no GEU representation. While respecting utility is a necessary condition for a deci­ sion rule to have a GEU representation, it is not sufficient. It is also necessary for the decision rule to treat acts that behave in similar ways similarly.

Two acts a1, a2 in a decision problem V are indistinguish­ able, denoted a1 ~v a2 iff either

• D is nonplausibilistic and Ua 1 = Ua 2 , or
• Dis pJausibiJistic and £PI,u = £PI,u ' a 1 a2 '

where u is the utility function of D and PI is the plausibil­ ity measure of D. In the nonplausibilistic case, two acts are indistinguishable if they induce the same utility random vari­ able; in the plausibilistic case, they are indistinguishable if they induce the same utility lottery.

A decision rule R is uniform if it respects indistinguisha­ bility. More formally, R is uniform iff for all D E dom(R) and a1, a2, b1, b2 acts ofD such that ai "'D bi,

Intuitively, we can think of utility random variables and utility lotteries as descriptions of what an act a does in terms of the tastes (and beliefs) of the DM. IfR is uniform, we can view R as relating the acts indirectly by relating their descriptions.

As the following theorem shows, all uniform decision rules have GEU representations.

Theorem 3.4: For all decision rules R, R has a GEU repre­ sentation if!R is uniform and R respects utility.

Proof: The "if' direction is somewhat similar in spirit to the proof of Theorem 3.6, given below; due to the lack of space, we omit this direction.

For the "only if' direction, suppose that T is a GEU rep­ resentation of R and let Do E dom(R) be arbitrary. Sup­ pose that a 1, a2, b1, b2 are acts of Do such that ai "'Do bi. It is easy to check that if D = (A, E, u, PI) ~ Do, then EpJ,E(Ua;) = EpJ,E(ubJ. Thus for all plausibilistic D, if D ~ Do, then a1 ;::::GEU(D) a2 iff b1 ;::::GEU(D) b2. Since T is a GEU representation of R, T(D0 ) ~ D0 and R(Do) = GEU(T(Do)). It follows then that a1 ;::::R(Do) a2 iff b1 ;::::R(Do) b2; thus R is uniform.

Now suppose that a1 and a2 are two acts of constant util­ ity, say ·u1 and u2 , respectively, ofD0 . Since T(D0 ) ~ D0 , ai is still an act of constant utility Ui in T(D0 ). Note that a1 ;::::R(Do) a2 iff a1 ;::::GEU(r(Do)) a2 iff U1 ;::::u u2, where U is the utility domain of D0 , since T is a GEU representation of R. Thus R respects utility. I

Most of the decision rules we have discussed are uniform. However, CEU is not, as the following example shows:

Example 3.5: Let D. = ((A, S, C), JE, u, Bel), where

• A= {a1, a2}; S = {81, 82, 83}; C = {1, 2, 3};
• u(j) = j, for j = 1, 2, 3;
• a1(8j) = j and a2(81) = 3- j, for j = 1, 2, 3; and
• Bel is the belief function such that Bel(X) = 1 if { 8 1 , 82} ~ X and Bel(X) = 0 otherwise.

Since u ;:;-/(j) is a singleton, Bel(u;:;-/(j)) = 0 fori = 1, 2 and j = 1, 2, 3; thus a1 "'D. a2 . On the other hand, by definition,

EseJ(uaJ = 1 + Bel(X23)(2- 1) + Bel(X3)(3- 2) = 1, while

EseJ(uaJ = 1 + Bel(X12)(2- 1) + Bel(XI)(3- 2) = 2, where Xij = { 8i, 8j} and xk = { 8k}. It follows that CEU is not uniform, and so has no GEU representation. I

The reader may have noticed an incongruity here. Ex­ ample 3.3 shows that MMEU has a GEU representation; moreover, as shown earlier, MMEU produces essentially the same order on acts as CEU. However, CEU has no GEU representation. There is no contradiction to Theorem 3.4 here: There is no decision problem D such that D ~ D. (from Example 3.5) and GEU(D) = CEU(D.). However, GEU((A,S,C),Ep 8 .t>u,Plp 8 .,) = CEU(D.). Ofcourse, ((A, S, C), EpBel' u, PlPa.J '=~- D.; PlpBel and Bel are not the same, and they in fact represent related but different be­ liefs. (It is easy to show that sets are partially preordered by Plp 8 · 1 but totally preordered by Bel.)

The key reason that GEU cannot represent nonuniform de­ cision rules is because they do not respect the indistinguisha­ bility relations imposed by the utility function (and the plau­ sibility measure). Recall that we require that T(D) ~ D be­ cause we want a user of one decision rules to appear as if she were using another, without pretending that she has different tastes (and beliefs). So we want T to preserve the tastes (and beliefs) of its input.

There is a long-standing debate in the decision-theory liter­ ature as to whether preferences should be regarded as ordinal or cardinal. If they are ordinal, then all that matters is their order. If they are cardinal, then it should be meaningful to talk about the differences between preferences, that is, how much more a DM prefers one consequence to another. Simi­ larly, if representations of likelihood are taken to be ordinal, then all that matters is whether one event is more likely than another. As we show below, if we require only that T(D) and D describe the same ordinal tastes (and beliefs), then we can in fact express almost all decision rules, including CEU, in terms of GEU.

Two utility functions u1 : C -t U1 and u 2 : C -+ U2 represent the same ordinal tastes if for all c1, c2 E C, u1(c1) ;::::u, u1(c2) iff u2(ci) ;::::u 2 u2(c2). Similarly, two plausibility measures Ph : 2 8 -+ P1 and Pl2 : 2 8 -+ P2 represent the same ordinal beliefs iff for all X, Y ~ S, Pl1(X) ~P, Pl1(Y) iffPl2(X) ~p2 Pl2(Y). Finally, two decision problems D1 and D2 are similar, denoted D1 ~ D2, iff they involve the same decision situations, their utility func­ tions represent the same ordinal tastes, and their plausibil­ ity measures represent the same ordinal beliefs. Note that Dl ~ D2 implies Dl ~ D2, but the converse is false in generaL A decision rule transformation T is an ordinal R 1 representation of R2 iff dom( T) = dom(R2) and for all DE dom(R2), T(D) ~ D and R1(T(D)) = R2(D).

We want to show next that almost all decision rules have an ordinal GEU representation. Doing so involves one more subtlety. Up to now, we have assumed that plausibility do­ mains are partially ordered. This implies that two plausibil­ ity measures that represent the same ordinal beliefs necessar­ ily induce the same indistinguishability relation (because of antisymmetry). Thus, in order to distinguish sets that have equivalent plausibilities when computing expected utility us­ ing ® and ®, we need to allow plausibility domains to be partially preordered. So, for this result, we assume that ;::::P is a reflexive and transitive relation that is not necessarily anti­ symmetric (i.e., we could have that P1 ;::::P P2 and P2 ;::::P Pl but PI =/:. P2).

Theorem 3.6: A decision ruleR has an ordinal GEU repre­ sentation if!R weakly respects utility.

Proof: There are two cases, plausibilistic and nonplausibilis­ tic. They are almost identical; we just do the plausibilistic case here. (Also, the "only if' direction is quite similar to the one in the proof of Theorem 3.4, so we omit it here.)

Suppose that R is a plausibilistic decision rule that weakly respects utility. Fix a plausibilistic decision problem D = ((A, S, C), E1, u1 , Pll) E dom(R) . Let U1 and P1 be the utility domain and plausibility domain of E1, respectively. Let E2 = ( U2 , P2 , V, 87, ® ) be defined as follows (note that x denotes Cartesian product in this proof):

• u2 = (Ul X C, ~u2), where (u1,c1) ~u2 (u2,c2) iffu1 ~u, u2,
• P2 = (P1 x 2 5 , ~p2 ), where (Pl, X1) ~P2 (P2, X2) iffp1 ~P, P2·

(Note that ~p2 is a partial preorder, although it is not a partial order.)

• V = (2sxu 2 ,~v), where x ~v y itfx = y or

1. x = S x {(u1 , c1)}, y = S x {(u2 , c2)}, and

• (u1, c1) ~u2 (u2, c2), or

2. x = {(s, (u1(a(s)), a(s))) Is E S},

y = {(s, (u1(b(s)), b(s))) I s E S}, and

a ~'R.(V) b, for some a, b E A,

• (p,X) ® (u,c)=Xx{(u,c)}
• X 87 y = X U y for all X, y E V.

Note that (J_p 1 , 0) ~p2 (p, X) ~p2 (T p 1 , S), so we have l_p 2 = (l_p,, 0) and T p 2 = (T p,, S); thus, P2 is a plausi­ bility domain. Since R weakly respects utility, 1 and 2 are consistent with one another. We identify ( u, c) E U2 with S x { (u, c)} in V; with this identification, T ® (u, c) = (u, c) for all (u, c) E U2 and, given 1 in the definition of ~v, (U2,~u2 ) is a substructure of (V,~v). Furthermore, 87 is clearly associative and commutative, so E2 is indeed an ex­ pectation domain.

Now we need to define a utility function and a plausibil­ ity measure. Let u2(c) = (u1(c),c) for all c E C and let Pl2(X) = (Ph(X),X) for all X s;; S. Note that

Pl2(X) ~p2 Pl2(Y) iffPh(X) ~P, Ph(Y). (3.6) Thus Ph is a plausibility measure, since Pl1 is a plausibility measure. Also,

Let T(D) = ((A,S,C),E2,u2 ,Ph). Note that, by (3.6) and (3.7), T(D) ~ D; furthermore, it is easy to check that Ept 2 ,E 2 ((u2)a) = {(s, (u1(a(s)),a(s))) Is E S}; it easily follows that GEU(T(D)) = R(D). Thus Tis an ordinal GEU representation of n. I

Theorem 3.6 shows that GEU can emulate essentially all decision rules. Thus, there is a sense in which GEU can be viewed as a universal decision rule. (We remark that although we have focused here on alternatives that are acts, in the sense of Savage, that is, functions from states to consequences, it is not hard to show-and we do in the full paper-that the same results hold if alternatives are taken to be horse lotteries, in the sense of Anscom be and Aumann [ 1963].)