Most models of decisionmaking under uncertainty describe a decision environment with a set of states and a set of outcomes. Objects of choice are acts
, functions from states to outcomes. The decision maker (DM) holds a preference relation on the set of all such functions. Representation theorems characterize those preference relations with utility functions on acts that separate (more or less) tastes on outcomes from beliefs on states. The canonical example is Savage’s Savage characterization of those preference relations that have a subjective expected utility (SEU) representation: Acts are ranked by the expectation of a utility payoff on their outcomes with respect to a probability distribution on states.Choquet expected utility [Schmeidler89] maintains the separation between tastes and beliefs, but does not require that beliefs be represented by an additive measure. Tversky and Kahneman’s TK92 cumulative prospect theory relaxes the taste-belief separation by assessing gains and losses with different belief measures; WT93 discuss generalizations of SEU from this point of view. Modern attempts to represent ambiguity in choice theory relax both the meaning of likelihood and the separation of tastes and beliefs that characterize SEU. All of these generalizations of SEU, however, maintain the state-outcome-act description of objects of choice and, moreover, take this description of choice problems as being given prior to the consideration of any preference notion.
We, on the other hand, follow Ellsberg Ellsberg01 in locating the source of ambiguity in the description of the problem. For Savage [p. 9]Savage, the world is ‘the object about which the person is concerned’ and a state of the world is ‘a description of the world, leaving no relevant aspect undescribed.’ But what are the ‘relevant’ descriptors of the world? Choices do not come equipped with states. Instead they are typically objects described by their manner of realization, such as ‘buy 100 shares of IBM’ or ‘leave the money in the bank,’ ‘attack Iraq,’ or ‘continue to negotiate.’ In Savage’s account [sec. 2.3]Savage it is clear that the DM ‘constructs the states’ in contemplating the decision problem. In fact, his discussion of the rotten egg foreshadows this process. Subsequently, traditional decision theory has come to assume that states are given as part of the description of the decision problem. We suppose instead that states are constructed by the DM in the course of deliberating about questions such as ‘How is choice different from choice ?’ and ‘In what circumstances will choice turn out better than choice ?’. These same considerations apply (although here Savage may disagree) to outcomes. This point has been forcefully made by weibull.
There are numerous papers in the literature that raise issues with the state-space approach of Savage or that derive a subjective state space. Machina03 surveys the standard approach and illustrates many difficulties with the theory and with its uses. These difficulties include the ubiquitous ambiguity over whether the theory is meant to be descriptive or normative, whether states are exogenous or constructed by the DM, whether states are external to the DM, and whether they are measurable or not. Kreps92 and DLR01 use a menu choice model to deal with unforeseen contingencies—an inability of the DM to list all possible states of the world. They derive a subjective state space that represents possible preference orders over elements of the menu chosen by the DM. Ghir01 takes an alternative approach to unforeseen contingencies and models acts as correspondences from a state space to outcomes. GS04 and Karni06 raise objections to the state space that are similar to ours and develop decision theories without a state space. Both papers derive subjective probabilities directly on outcomes. Ahn07 also develops a theory without a state space; in his theory, the DM chooses over sets of lotteries over consequences. AE07 allow for the possibility that there may be different descriptions of a particular event, and use this possibility to capture framing. For them, a ‘description’ is a partition of the state space. They provide an axiomatic foundation for decision making in this framework, built on Tversky and Koehler’s tverskykoehler notion of support theory.
Our approach differs significantly from these mentioned above. The inspiration for our approach is the observation that objects of choice in an uncertain world have some structure to them. Individuals choose among some simple actions: ‘buy 100 shares of IBM’ or ‘attack Iraq’. But they also perform various tests on the world and make choices contingent upon the outcome of these tests: ‘If the stock broker recommends buying IBM, then buy 100 shares of IBM; otherwise buy 100 shares of Google.’ These tests are written in a fixed language (which we assume is part of the description of the decision problem, just as Savage assumed that states were part of the description of the decision problem). The language is how the DM describes the world. We formalize this viewpoint by taking the objects of choice to be (syntactic) programs in a programming language.
The programming language is very simple—we use it just to illustrate our ideas. Critically, it includes tests (in the context of if …then …else statements). These tests involve syntactic descriptions of the events in the world, and allow us to distinguish events from (syntactic) descriptions of events. In particular, there can be two different descriptions that, intuitively, describe the same event from the point of view of the modeler but may describe different events from the point of view of the decision maker. Among other things, this enables us to capture framing effects in our framework, without requiring states as AE07 do, and provides a way of dealing with resource-bounded reasoners.
In general, we do not include outcomes as part of the description of the decision problem; both states and outcomes are part of the DM’s (subjective) representation of the problem. We assume that the DM has a weak preference relation on the objects of choice; we do not require the preference relation to be complete. The set of acts for a decision problem is potentially huge, and may contain acts that will never be considered by the DM. While we believe that empirical validity requires considering partial orders, there are also theoretical reasons for considering partial orders. Our representation theorems for partial orders require a set of probabilities and utility functions (where often one of the sets can be taken to be a singleton). [p. 572]Schmeidler89 observes that using a set of probability distributions can be taken as a measure of a DM’s lack of confidence in her likelihood assessments. Similarly, a set of utilities can be interpreted as a lack of confidence in her taste assessments (perhaps because she has not had time to think them through carefully).
The rest of this paper is organized as follows. We begin the next section with a description of the syntactic programs that we take as our objects of choice, discuss several interpretations of the model, and show how syntactic programs can be interpreted as Savage acts. In Section 3, we present our assumptions on preferences The key postulate is an analogue of Krantz et al.’s KLST71 cancellation axiom. In Section 4 we present our representation theorems for decision problems with subjective outcomes and those with objective outcomes. Section 5 discusses how our framework can model boundedly rational reasoning. In Section 6 we discuss how updating works for new information about the external world as well as for new information about preferences. Section 7 concludes.
2 Describing Choices
We begin by describing the language of tests, and then use this language to construct our syntactic objects of choice. We then use the language of tests to describe theories of the world. We show how framing problems can be understood as ‘odd’ theories held by decision makers.
2.1 Languages for tests and choices
A primitive test is a yes/no question about the world, such as, ‘IBM’s price-earnings ratio is 5’, ‘the economy will be strong next year’ and ‘the moon is in the seventh house’. We assume a finite set of primitive tests. The set of tests is constructed by closing the set of primitive tests under conjunction and negation. That is, is the smallest set such that , and if and are in , so is and . Thus, the language of tests is just a propositional logic whose atomic propositions are the elements of .
We consider two languages for choices. In both cases, we begin with a finite set of primitive choices. These may be objects such as ‘buy 100 shares of IBM’ or ‘buy $10,000 worth of bonds’. The interpretation of these acts is tightly bound to the decision problem being modeled. The first language simply closes off under if …then …else. By this we mean that if is a test in and and are choices in , then if then else is also a choice in . When we need to be clear about which and were used to construct , we will write . Note that allows nesting, so that is also a choice.
The second languages closes off with if …then …else and randomization. That is, we assume that objective probabilities are available, and require that for any , if and are choices, so is . Randomization and if …then …else can be nested in arbitrary fashion. We call this language ( when necessary).
Tests in are elements of discourse about the world. They could be events upon which choice is contingent: If the noon price of Google stock today is below $600, then buy 100 shares, else buy none. More generally, tests in are part of the DM’s description of the decision problem, just as states are part of the description of the decision problem in Savage’s framework. However, elements of need not be complete descriptions of the relevant world, and therefore may not correspond to Savage’s states. When we construct state spaces, elements of will clearly play a role in defining states, but, for some of our representation theorems, states cannot be constructed out of elements of alone. Additional information in states is needed for both incompleteness of preferences and when the outcome space is taken to be objective or exogenously given.
The choices in and are syntactic objects; strings of symbols. They can be given semantics—that is, they can be interpreted—in a number of ways. For most of this paper we focus on one particular way of interpreting them that lets us connect them to Savage acts, but we believe that other semantic approaches will also prove useful (see Section 7). The first step in viewing choices as Savage acts is to construct a state space , and to interpret the tests as events (subsets of ). With this semantics for tests, we can then construct, for the state space and a given outcome space , a function that associates with each choice a Savage act , that is, a function from to . Given a state space , these constructions work as follows:
A test interpretation for the state space is a function associating with each test a subset of . An interpretation is standard if it interprets and in the usual way; that is
Intuitively, is the set of states where is true. We will allow for nonstandard interpretations. These are interpretations in which, for some test , there may be some state where neither nor is true; that is, there may be some state in neither nor ); similarly, there may be some state where both and are true. Such nonstandard interpretations are essentially what philosophers call ‘impossible possible worlds’ [Rant]
; they have also been used in game theory for modeling resource-bounded reasoning[Lip99]. A standard interpretation is completely determined by its behavior on primitive tests. This is not true of nonstandard interpretations. All test interpretations are assumed to be standard until Section 5. There we motivate nonstandard interpretations, and show how our results can be modified to hold even with them.
A choice interpretation for the state space and outcome space assigns to each choice a (Savage) act, that is, a function . Choice interpretations are constructed as follows: Let be a choice interpretation for primitive choices, which assigns to each a function from . We extend to a function mapping all choices in to functions from to by induction on structure, by defining
This semantics captures the idea of contingent choices; that, in the choice if then else , the realization of is contingent upon , while is contingent upon ‘not ’. Of course, and could themselves be compound acts.
Extending the semantics to the language , given , , and , requires us to associate with each choice an Anscombe-Aumann (AA) act [AA63], that is, a function from to probability measures on . Let denote the set of probability measures on and let be the subset of consisting of the probability measures that put probability one on an outcome. Let be a choice interpretation for primitive choices that assigns to each a function from . Now we can extend by induction on structure to all of in the obvious way. For if …then …else choices we use (1); to deal with randomization, define
That is, the distribution is the obvious mixture of the distributions and . Note that we require to associate with each primitive choice in each state a single outcome (technically, a distribution that assigns probability 1 to a single outcome), rather than an arbitrary distribution over outcomes. So primitive choices are interpreted as Savage acts, and more general choices, which are formed by taking objective mixtures of choices, are interpreted as AA acts. This choice is largely a matter of taste. We would get similar representation theorems even if we allowed to be an arbitrary function from to . However, this choice does matter for our interpretation of the results; see Example 4.3 for further discussion of this issue.
2.2 Framing and equivalence
Framing problems appear when a DM solves inconsistently two decision problems that are designed by the modeler to be equivalent or that are obviously similar after recognizing an equivalence. The fact that choices are syntactic objects allows us to capture framing effects.
Consider the following well-known example of the effects of framing, due to McNeil et al. MPST82. DMs are asked to choose between surgery or radiation therapy as a treatment for lung cancer. The problem is framed in two ways. In the what is called the survival frame, DMs are told that, of 100 people having surgery, 90 live through the post-operative period, 68 are alive at the end of the first year, and 34 are alive at the end of five years; and of 100 people have radiation therapy, all live through the treatment, 77 are alive at the end of the first year, and 22 are alive at the end of five years. In the mortality frame, DMs are told that of 100 people having surgery, 10 die during the post-operative period, 32 die by the end of the first year, and 66 die by the end of five years; and of 100 people having radiation therapy, none die during the treatment, 23 die by the end of the first year, and 78 die by the end of five years. Inspection shows that the outcomes are equivalent in the two frames—90 of 100 people living is the same as 10 out of 100 dying, and so on. Although one might have expected the two groups to respond to the data in similar fashion, this was not the case. While only 18% of DMs prefer radiation therapy in the survival frame, the number goes up to 44% in the mortality frame.
We can represent this example in our framework as follows. We assume that we have the following tests:
, which intuitively represents ‘100 people have radiation therapy’;
, which intuitively represents ‘100 people have surgery’;
, for and , which intuitively represents that out of 100 people live through the post-operative period (if ), are alive after the first year (if , and are alive after five years (if );
, for and , which is like , except ‘live/alive’ are replaced by ‘die/dead’.
In addition, we assume that we have primitive programs and that represent ‘perform surgery’ and ‘perform radiation theory’. With these tests, we can characterize the description of the survival frame by the following test :
(where, as usual, is an abbreviation for ); similarly, the mortality frame is characterized by the following test :
The choices offered in the McNeil et al. experiment can be viewed as conditional choices: what would a DM do conditional on (resp., ) being true. Using ideas from Savage, we can capture the survival frame as a decision problem with the following two choices:
where is an arbitrary choice. Intuitively, comparing these choices forces the DM to consider his preferences between and conditional on the test, since the outcome in these two choices is the same if the test does not hold. Similarly, the mortality frame amounts to a decision problem with the analogous choices with replaced by .
There is nothing in our framework that forces a DM to identify the tests and ; the tests and a priori are completely independent, even if the problem statement suggests that they should be equivalent. Hence there is no reason for a DM to identify the choices if then else and if then else . As a consequence, as we shall see, it is perfectly consistent with our axioms that a DM has the preferences and .
We view it as a feature of our framework that it can capture this framing example for what we view as the right reason: the fact that DMs do not necessarily identify and . Nevertheless, we would also like to be able to capture the fact that more sophisticated DMs do recognize the equivalence of these tests. We can do this by associating with a DM her understanding of the relationship between tests. For example, a sophisticated DM might understand that , for and . Formally, we add to the description of a decision problem a theory, that is, a set of tests. Elements of the theory are called axioms. A test interpretation for the state space respects a theory AX iff for all , . A theory represents the DM’s view of the world; those tests he takes to be axiomatic (in its plain sense). Different people may, however, disagree about what they take to be obviously true of the world. Many people will assume that the sun will rise tomorrow. Others, like Laplace, will consider the possibility that it will not.
Choices and are equivalent with respect to a set of test interpretations if, no matter what interpretation is used, they are interpreted as the same function. For example, in any standard interpretation, if then else is equivalent to if then else ; no matter what the test and choices and are, these two choices have the same input-output semantics. The results of the McNeil et al. experiment discussed in Example 2.2 can be interpreted in our language as a failure by some DMs to have a theory that makes tests stated in terms of mortality data or survival data semantically equivalent. This then allows choices, such as the choice of surgery or radiation therapy given these tests, to be not seen as equivalent by the DM. Thus, it is not surprising that such DMs are not indifferent between these choices.
For a set of test interpretations, choices and are -equivalent, denoted , if for all state spaces , outcomes , test interpretations , and choice interpretations , .
Denote by the set of all standard interpretations that respect theory AX. Then -equivalent and are said to be AX-equivalent, and we write . Note that equivalence is defined relative to a given set of interpretations. Two choices may be equivalent with respect to the set of all standard interpretations that hold a particular test to be true, but not equivalent to the larger set of all standard test interpretations.
3 The Axioms
This section lays out our basic assumptions on preferences. Since our basic framework allows for preferences only on discrete sets of objects, we cannot use conventional independence axioms. Instead, we use cancellation. Cancellation axioms are not well known, so we use this opportunity to derive some connections between cancellation and more familiar preference properties.
We assume that the DM has a weak preference relation on a subset of the sets (resp., ) of non-randomized (resp., randomized) acts. This weak preference relation has the usual interpretation of ‘at least as good as’. We take to be an abbreviation for and , even if is not complete. We prove various representation theorems that depend upon the language, and upon whether outcomes are taken to be given or not. The engines of our analysis are various cancellation axioms, which are the subject of the next section. At some points in our analysis we consider complete preferences:
The preference relation is complete.
The completeness axiom has often been defended by the claim that ‘people, in the end, make choices.’ Nonetheless, from the outset of modern decision theory, completeness has been regarded as a problem. [Section 2.6]Savage discusses the difficulties involved in distinguishing between indifference and incompleteness. He concludes by choosing to work with the relationship he describes with the symbol , later abbreviated as , which he interprets as ‘is not preferred to’. The justification of completeness for the ‘is not preferred to’ relationship is anti-symmetry of strict preference. Savage, Aumann62, Bewley and mandler argue against completeness as a requirement of rationality. eliazok have argued that rational choice theory with incomplete preferences is consistent with preference reversals. In our view, incompleteness is an important expression of ambiguity in its plain meaning (rather than as a synonym for a non-additive representation of likelihood). There are many reasons why a comparison between two objects of choice may fail to be resolved: obscurity or indistinctness of their properties, lack of time for or excessive cost of computation, the incomplete enumeration of a choice set, and so forth. We recognize indecisiveness in ourselves and others, so it would seem strange not to allow for it in any theory of preferences that purports to describe tastes (as opposed to a theory which purports to characterize consistent choice).
Axioms such as the independence axiom or the sure thing principle are required to get the requisite linearity for an SEU representation. But in their usual form these axioms cannot be stated in our framework as they place restrictions on preference relations over acts and we do not have acts. Moreover, some of our representation theorems apply to finite sets of acts, while the usual statement of mixture axioms requires a mixture space of acts. For us, the role of these axioms is performed by the cancellation axiom, which we now describe. Although simple versions of the cancellation axiom have appeared in the literature (e.g. scott64 and KLST71), it is nonetheless not well known, and so before turning to our framework we briefly explore some of its implications in more familiar settings. Nonetheless, some of the results here are new; in particular, the results on cancellation for partial orders. These will be needed for proofs in the appendix.
Let denote a set of choices and a preference relation on . We use the following notation: Suppose and are sequences of elements of . If for all , , we write . That is, the multisets formed by the two sequences are identical.
[Cancellation] The preference relation on satisfies cancellation iff for all pairs of sequences and of elements of such that , if for , then . Roughly speaking, cancellation says that if two collections of choices are identical, then it is impossible to order the choices so as to prefer each choice in the first collection to the corresponding choice in the second collection. The following proposition shows that cancellation is equivalent to reflexivity and transitivity. Although Krantz et al. [p. 251]KLST71, [p. 743]Fishburn92 have observed that cancellation implies transitivity, this full characterization appears to be new.
A preference relation on a choice set satisfies cancellation iff
is reflexive, and
First suppose that cancellation holds. To see that is reflexive, take and in the cancellation axiom. The hypothesis of the cancellation axiom clearly holds, so we must have . To see that cancellation implies transitivity, consider the pair of sequences and . Cancellation clearly applies. If and , then cancellation implies . We defer the proof of the converse to the Appendix. ∎
We use two strengthenings of cancellation in our representation theorems for and , respectively. The first, statewise cancellation, simply increases the set of sequence pairs to which the conclusions of the axiom must apply. This strengthening is required for the existence of additively separable preference representations when choices have a factor structure. Here we state the condition for Savage acts. Given are finite sets of states and of outcomes. A Savage act is a map . Let denote a set of Savage acts and suppose that is a preference relation on .
[Statewise Cancellation] The preference relation on a set of Savage acts satisfies statewise cancellation iff for all pairs of sequences and of elements of , if for all , and for , then . Statewise cancellation is a powerful assumption because equality of the multisets is required only ‘pointwise’. Any pair of sequences that satisfy the conditions of cancellation also satisfies the conditions of statewise cancellation, but the converse is not true. For instance, suppose that , and we use to refer to an act with outcome in state , . Consider the two sequences of acts and . These two sequences satisfy the conditions of statewise cancellation, but not that of cancellation.
In addition to the conditions in Proposition 1, statewise cancellation directly implies event independence, a condition at the heart of SEU representation theorems (and which can be used to derive the Sure Thing Principle). If , let be the Savage act that agrees with on and with on ; that is if and if . We say that satisfies event independence iff for all acts , , , and and subsets of the state space , if , then .
If satisfies statewise cancellation, then satisfies event independence.
Take and take . Note that for each state , , and for each state , . Thus, we can apply statewise cancellation to get that if , then . ∎
Proposition 1 provides a provides a characterization of cancellation for choices in terms of familiar properties of preferences. We do not have a similarly simple characterization of statewise cancellation. In particular, the following example shows that it is not equivalent to the combination of reflexivity and transitivity of and event independence.
Suppose that , . There are nine possible acts. Suppose that is the smallest reflexive, transitive relation such that
using the representation of acts described above. To see that satisfies event independence, note that
However, statewise cancellation does not hold. Consider the sequences
This pair of sequences clearly satisfies the hypothesis of statewise cancellation, that and , but .
For our representation theorems for complete orders, statewise cancellation suffices. However, for partial orders, we need a version of cancellation that is equivalent to statewise cancellation in the presence of A1, but is in general stronger.
[Extended Statewise Cancellation] The preference relation on a set of Savage acts satisfies extended statewise cancellation if and only if for all pairs of sequences and
of elements of such that
for all , if there exists some such that for , , and , then .
In the presence of A1, extended statewise cancellation and statewise cancellation are equivalent.
Suppose the hypotheses of extended statewise cancellation hold. If , we are done. If not, by A1, . But then the hypotheses of statewise cancellation hold, so again, . ∎
The extension of cancellation needed for is based on the same idea as extended statewise cancellation, but probabilities of objects rather than the incidences of objects are added up. Let denote a collection of elements from a finite-dimensional mixture space. Thus, can be viewed as a subspace of for some , and each component of any is a probability. We can then formally ‘add’ elements of , adding elements of pointwise. (Note that the result of adding two elements in is no longer an element of , and in fact is not even a mixture.)
[Extended Mixture Cancellation] The preference relation on satisfies extended mixture cancellation iff for all pairs of sequences and of elements of , such that , if there exists some such that for , , and , then . We can extend Proposition 1 to get a characterization theorem for preferences on mixture spaces by using an independence postulate. The preference order satisfies mixture independence if for all , , and in , and all , iff . The preference relation satisfies rational mixture independence if it satisfies mixture independence for all rational .
A preference relation on a finite-dimensional mixture space satisfies extended mixture cancellation iff is reflexive, transitive, and satisfies rational mixture independence.
Suppose that satisfies extended mixture cancellation. Then it satisfies cancellation, and so from Proposition 1, is reflexive and transitive. To show that satisfies rational mixture independence, suppose that and . Let and ; let and . Then , and so .
Similarly, if , then applying extended mixture cancellation to the same sequence of acts shows that .
For the converse, suppose that is reflexive, transitive, and satisfies rational mixture independence. Suppose that and are sequences of of elements of such that , for , , and . Then from transitivity and rational mixture independence we get that
Since and , we have that
Thus, by transitivity,
By rational mixture independence, it follows that . ∎
We can strengthen extended mixture cancellation just as we extended cancellation, by defining a statewise version of it appropriate for AA acts (i.e., functions from to ). For completeness, we give the definition here: [Extended Statewise Mixture Cancellation] The preference relation on a set of AA acts satisfies extended statewise mixture cancellation iff for all pairs of sequences and of elements of , such that for all states , if there exists some such that for , , and , then .
It turns out that we do not needed extended statewise mixture cancellation. As the following result shows, it follows from extended mixture cancellation.
satisfies extended statewise mixture cancellation iff satisfies extended mixture cancellation.
Clearly if satisfies extended statewise mixture cancellation, then it satisfies extended mixture cancellation. For the converse, suppose that satisfies extended mixture cancellation, for all states , there exists some such that , and . Then . By rational mixture independence (which follows from extended mixture cancellation, by Theorem 3.2), since , we have that . By a straightforward induction (using transitivity, which again follows from extended mixture cancellation), it follows that
Now from mixture independence, it follows that , as desired. ∎
3.3 The cancellation postulate for choices
We use cancellation to get a representation theorem for preference relations on choices. However, the definition of the cancellation postulates for Savage acts and mixtures rely on (Savage) states. We now develop an analogue of this postulate for our syntactic notion of choice. Given a set of primitive tests, an atom over is a test of the form , where is either or .
An atom is a possible complete description of the truth value of tests according to the DM. If there are primitive tests in , there are atoms. Let denote the set of atoms over . It is easy to see that, for all tests and atoms , and for all state spaces and standard test interpretations , either or . (The formal proof is by induction on the structure of .) We write if the former is the case. We remark for future reference that a standard test interpretation is determined by its behavior on atoms. (It is, of course, also determined completely by its behavior on primitive tests).
An atom (resp., test ) is consistent with a theory AX if there exists a state space and a test interpretation such that (resp., ). Let denote the set of atoms over consistent with .
Intuitively, an atom is consistent with AX if there is some state in some state space where might hold, and similarly for a test .
A choice in can be identified with a function from atoms to primitive choices in an obvious way. For example, if , , and are primitive choices and , then the choice can be identified with the function such that
Formally, we define by induction on the structure of choices. If , then is the constant function , and
We consider a family of cancellation postulates, relativized to the axiom system . The cancellation postulate for (given the language ) is simply statewise cancellation for Savage acts, with atoms consistent with playing the role of states.
If and are two sequences of choices in such that for each atom , , and there exists some such that for all , , and , then .
We drop the prime, and refer to when .
Axiom implies the simple cancellation of the last section, and so the conclusions of Proposition 1 hold: on will be transitive and reflexive. has another consequence: a DM must be indifferent between -equivalent choices.
Suppose that satisfies . Then implies .
Let , the set of atoms consistent with , let be , the set of primitive choices, and define to be the constant function for a primitive choice . It is easy to see that for all choices . If , then , so we must have . Now apply with and to get , and then reverse the roles of and . ∎
Proposition 5 implies that the behavior of and on atoms not in is irrelevant; that is, they are null in Savage’s sense. We define this formally:
A test is null if, for all acts , and , then else .
An atom (or test) inconsistent with the theory AX is null, but consistent tests may be null as well. The notion of a null test is suggestive of, more generally, test-contingent preferences. If is a test in , then for any acts and , iff for some , .
Proposition 2 shows that statewise cancellation implies that the choice of is irrelevant, and so test-contingent preferences are well-defined.
To get a representation theorem for , we use a mixture cancellation postulate, again replacing states by atoms. The idea now is that we can identify each choice with a function mapping atoms consistent with into distributions over primitive choices. For example, if is the only test in and , then the choice can be identified with the function such that
Formally, we just extend the definition of given in the previous section by defining
Consider the following cancellation postulate:
. If and are two sequences of acts in such that
for all atoms consistent with , and there exists such that for , , and , then .
Again, can be viewed as a generalization of .
is analogous to extended statewise mixture cancellation. It may seem strange that we need the cancellation to be statewise, since Proposition 4 shows that extended statewise mixture cancellation is equivalent to extended mixture cancellation. This suggests that we might be able to get away with the following simpler axiom:
If and are two sequences of acts in such that
and there exists such that for , , and , then .
is implied by , but they are not quite equivalent. To get equivalence, we need one more property:
If , then .
EQUIV says that all that matters about a choice is how it acts as a function from tests to primitive choices.
A preference order on choices satisfies iff it satisfies and EQUIV.
If satisfies , then clearly satisfies . To see that satisfies EQUIV, suppose that . A straightforward argument by induction on the structure of shows that, for all sets of states, sets of outcomes, test interpretations , and choice interpretations , we have that , where is the unique atom such that . Similarly, . Thus, since , . It follows that implies . It now follows from Proposition 5 that .
For the converse, since , it follows that . Applying EQUIV, we conclude that
. We can now continue as in the proof of Proposition 4. ∎
4 Representation Theorems
Having discussed our framework, we now turn to the representation theorems. Our goal is to be as constructive as possible. In this spirit we want to require that preferences exist not for all possible acts that can be described in a given language, but only for those in a given subset, henceforth designated . We are agnostic about the source of . It could be the set of choices in one particular decision problem, or it could be the set of choices that form a universe for a collection of decision problems. One cost of our finite framework is that we will have no uniqueness results. In our framework the preference representation can fail to be unique because of our freedom to choose different state and outcome spaces, but even given these choices, the lack of richness of may allow multiple representations of the same (partial) order.
4.1 A Representation Theorem for
By a representation for a preference order on we mean the following: A preference relation on a set has a constructive AX-consistent SEU representation iff there is a finite set of states , a finite set of outcomes, a set of utility functions , a set of probability distributions on , a subset , a test interpretation consistent with AX, and a choice interpretation such that iff
We are about to claim that satisfies if and only if it has a constructive AX-consistent representation. In the representation, we have a great deal of flexibility as to the choice of the state space and the outcome space . One might have thought that the space of atoms, , would be a rich enough state space on which to build representations. This is not true when preferences are incomplete. A rich enough state space needs also to account for the incompleteness. Given a partial order on a set of choices, let denote all the extensions of to a total order on satisfying . Our proof shows that we can take to be . Thus, in particular, if is complete, then we can take the state space to be . We later give examples that show that if is not complete then, in general, the state space must have cardinality larger than that of . While for some applications there may be a more natural state space, our choice of state space shows that we can always view the DM’s uncertainty as stemming from two sources: the truth values of various tests (which are determined by the atom) and the relative order of two choices not determined by (which is given by the extension of ). The idea of a DM being uncertain about her preferences is prevalent elsewhere in decision theory; for instance, in the menu choice literature [kreps79]. This uncertainty can be motivated in any number of ways, including both incomplete information and resource-bounded reasoning.
A preference relation on a set has a constructive AX-consistent SEU representation iff satisfies . Moreover, in the representation, either or can be taken to be a singleton and, if is a singleton , the state space can be taken to be . If, in addition, satisfies A1, then can be taken to be a singleton (i.e., both and can be taken to be singletons).
Theorem 4.1 is proved in the appendix. The proof proceeds by first establishing a state-dependent representation using as the outcome space, and then, by changing the outcome space, ‘bootstrapping’ the representation to an EU representation. This technique shows that, when the state and outcome spaces are part of the representation, there is no difference between the formal requirement for a state-dependent representation and that for a SEU representation. This does not mean that expected utility comes ‘for free’; rather, we interpret it to mean that the beliefs/desires formalism that motivates expected utility theory is sensible for the decision problems discussed in this subsection only if the particular outcome space chosen for the representation has some justification external to our theory. We note that if preferences satisfy A1, the theorem requires only the cancellation axiom rather than the stronger .
There are no uniqueness requirements on or in Theorem 4.1. In part, this is because the state space and outcome space are not uniquely determined. But even if A1 holds, so that the state space can be taken to be the set of atoms, the probability and the utility are far from unique, as the following example shows.
Take , , and . Suppose that is the reflexive transitive closure of the following string of preferences:
Every choice in is equivalent to one of these four, so A1 holds, and we can take the state space to be . Let , and define so that is the constant function and