On the Rejectability of the Subjective Expected Utility Theory

1 Introduction

Like the majority of theories in the literature on uncertainty/ambiguity, the Subjective Expected Utility (SEU) theory developed by Savage (1972) begins with a state space Ω with a generic element ω, and a set of consequences X with a generic element x. Then, there is a set of acts (functions from Ω to X) F with a generic element f. A binary order ≿ is defined over F, and the axioms imposed on ≿ guarantee the existence of an SEU representation of ≿. To test the SEU theory means to verify whether ≿ satisfies the Savage axioms.

Unfortunately, as it has been noted in the literature, ^[1] the state space is not observable and, consequently, is not part of the researcher’s (hereafter, referred to as “he”) data. This implies that the researcher is unable to test the SEU theory because with no state space there is no set of acts, and with no set of acts there is no binary order to be tested.

If the researcher assumes a specific choice theory, then he might be able to derive the state space from the agent’s (hereafter, referred to as “Ann” or “she”) observable choices. For example, Proposition 3 in Schipper (2013) might help to conduct such an exercise under the assumption that the agent satisfies the SEU theory. But assuming a theory to derive the agent’s state space, and then using that state space to test the very theory that has just been assumed is, obviously, an inappropriate approach.

In what I call the standard approach to testing the SEU theory, the researcher implicitly assumes some state space Ω. But what state space should he assume? Economic theory provides no answer to this question, a problem Savage already noticed sixty years ago: “I am unable to formulate criteria for selecting these small worlds and indeed believe that their selection may be a matter of judgment and experience about which it is impossible to enunciate complete and sharply defined general principles” (Savage (1972), 16).

If the researcher follows the standard approach and rejects the SEU theory, then his conclusion is not unconditionally valid. Rather, his conclusion depends on the state space Ω that he assumed at the start, and it would be more appropriate to say that the SEU theory is rejected conditional on the state space being Ω. It is possible that there exist a different state space Ω˜ and a probability measure λ˜ over Ω˜ which together do not lead to the rejection of the SEU theory.

For example, consider the two-urn Ellsberg experiment. With a typically assumed state space, this experiment rejects the SEU theory. However, Kadane (1992) shows that the experiment fits within the SEU framework as long as the agent assigns an arbitrarily small probability to the experimenter being malevolent. It does not seem unreasonable to assume that, from the agent’s perspective, the experimenter is malevolent. After all, there is nothing in this experiment’s design that prevents the agent from having such a belief. Hence, with a different state space, the two-urn Ellsberg experiment does not reject the SEU theory.

Since the state space Ω is assumed rather than derived (from data or theory), it is only appropriate to conduct a robustness check. That is, the researcher should verify whether it is possible to construct a new state space Ω˜ and a probability λ˜ over Ω˜ in such a way that the agent’s choices appear to be in accordance with the SEU theory. If the researcher constructs such Ω˜ and λ˜, then I say that he is able to SEU-rationalize the agent’s behavior. The standard approach does not include any robustness check. In this paper, I fill the gap by showing how to conduct the process of SEU-rationalization and determining when an SEU-rationalization is possible.

In Section 2, I present and analyze the process of SEU-rationalization. In Section 3, I apply my procedure to the classical three-color Ellsberg experiment, which is the most popular example of a violation of the SEU theory. I show how to SEU-rationalize the behavior observed in this experiment. Section 4 discusses the relevant literature. Section 5 concludes.

2 SEU-Rationalization

I consider the researcher who follows the standard approach to testing the SEU theory, rejects the SEU theory, and would like to conduct a robustness check. The quadruple (Ω,X,F,≿)— a state space Ω, a set of consequences X, a set of acts F, and a binary order ≿ over F — is taken as given and constitutes the researcher’s benchmark data set.

The objective of SEU-rationalization is to translate the original (Ω,X,F,≿) into a new (Ω˜,X˜,F˜,≿˜) and construct a probability measure λ˜ over Ω˜ such that ≿˜ has an SEU representation with respect to λ˜. This new (Ω˜,X˜,F˜,≿˜) is a re-interpretation of the original (Ω,X,F,≿). Obviously, allowing for any (Ω˜,X˜,F˜,≿˜) turns the procedure of SEU-rationalization into a trivial and uninteresting exercise. I allow to change only the state space. This implies that several conditions must be satisfied.

First, I require to keep the same set of consequences (condition 1 in Definition 2.1). Second, F and ≿ must be consistently translated into F˜ and ≿˜, respectively (conditions 2 and 3 in Definition 2.1).

Consistency between F˜ and F means that the perception of acts does not change. Since there is a new state space Ω˜, there is a new set of acts ˜F with a generic element f˜:Ω˜→X. Take f that is translated into f˜. I require that f˜ and f are perceived in the same way. Observe that f and f˜ are defined on different domains. Hence, in order to compare f and f˜ it is necessary to look at their common codomain X. Let Xf, a subset of the range of f, be the collection of all possible consequences associated with act f. If x∈Xf, then the agent perceives that such a consequence can be obtained if she chooses f. I assume that the researcher’s data set includes one Xf for each f∈F. Note that f˜ and λ˜ generate the probability measure λ˜f on X. The support of that measure is the collection of all possible consequences associated with act f˜. I say that f is consistently translated into f˜ if the support of λ˜f is precisely Xf.

Condition 3 in Definition 2.1 requires that ≿˜ be consistent with ≿; that is, f˜≿˜g˜ if and only if f≿g. This implies that the ranking of acts does not change.

Let ≻ and ∼ denote the strict and indifference part of ≿, respectively. Let fx be a constant act that generates x (i. e., f(ω)=x for each ω). Let fmaxf denote the constant act that yields the best possible consequence associated with f. Formally, maxf∈Xf and fmaxf≿fx for each x∈Xf. Let fminf be defined in a similar way.

Below, I present the assumptions imposed on the triple (X,F,≿) which allow to SEU-rationalize a given binary order ≿. There are no restrictions on Ω because it is the state space that is to be changed.

Assumption 1.1 is a rather natural assumption given that data sets are finite. Nevertheless, in the appendix, I replicate my result allowing for infinite sets of consequences and acts. Assumption 1.2, combined with Assumption 1.1, guarantees the existence of utility representation of ≿, V. The purpose of having constant acts (Assumption 1.3) is, obviously, to construct a utility function u on the set of consequences; u(x):=V(fx). Note that Assumption 2 can be seen as an extension of the Simple Dominance Axiom introduced by Barberà (1977), which holds that {x}≻{y} implies {x}≻{x,y}≻{y}.

The following proposition is the main result of this paper.

I prove Proposition 1 by construction. I develop a two-stage procedure of SEU-rationalization and show that a given binary order ≿ can be SEU-rationalized if Assumptions 1 and 2 are satisfied.

First stage (from acts to options): In the first stage, transform the set of acts F into the set of options, denoted by Fˆ. An option fˆ is defined as a pair (f,Xf), where f is the act (the option’s name) and Xf is the collection of consequences associated with act f.

To understand why f is transformed into (f,Xf) rather than Xf alone, consider a bet on the next winner of the U.S. presidential elections. There are two acts: a Democrat, denoted by fD, and a Republican, denoted by fR. Suppose that picking the winner yields $100, while picking the loser costs the agent $50. Each act is associated with the same set of consequences, that is, XfD=XfR={$100,−$50}. If the acts were to be represented only by the associated sets of consequences, then fD and fR would be perceived as the same object, or it would be necessary to assume that the agent is indifferent between two acts whenever their associated sets of consequences are the same. As the betting example suggests, this is generally not a correct approach; hence, a name is included in the definition of an option.

Next, construct the binary order ≿ˆ on Fˆ in a consistent way from ≿: fˆ ≿ˆ gˆ if and only if f≿g. Let ≻ˆ and ∼ˆ denote the strict and indifference part of ≿ˆ, respectively. Let fˆx be an option generated by a constant act fx.

Note that constructing (X,Fˆ,≿ˆ) from (X,F,≿) is always possible; that is, the assumptions presented above are irrelevant to accomplish the first stage of SEU-rationalization. However, they are important in the second stage.

Second stage (from options to acts): The second stage begins with the triple (X,Fˆ,≿ˆ) as derived in the first stage. First, construct a utility function Vˆ defined over Fˆ, which exists because of Assumption 1. This assumption also makes it possible to uniquely define a utility function u over X as u(x):=Vˆ(fˆx).

For each option fˆ, construct a probability measure λˆf defined over X such that (i) the support of λˆf is Xf, and (ii) Vˆ can be reconstructed as Vˆ(fˆ)=∑x∈Xfu(x)λˆf(x). The construction of λˆfs is based on the repetitive use of the Intermediate Value Theorem. Take fˆ such that Xf={x1,...,xK}. Without loss kof generality, assume that fˆx1 ≻ˆ ... ≻ˆ fˆxK. Assumption 2 indicates that fˆx1 ≻ˆ fˆ ≻ˆ fˆxK. However, because of Assumption 1, it is known how the agent ranks fˆ in comparison with each fˆxn for n=1,...,K. Next, without loss of generality, assume that fˆx1 ≻ˆ fˆ ≻ˆ fˆx2. Hence, it is true that fˆx1 ≻ˆ fˆ ≻ˆ fˆxn for each n=2,..,K. The Intermediate Value Theorem says that, for each n=2,..,K, there is an αn∈(0,1) such that Vˆ(fˆ)=u(x1)αn+u(xn)(1−αn). Now, take β2,...,βK∈(0,1) such that β2+...+βK=1, and observe that β2[u(x1)α2+u(x2)(1−α2)]+...+βK[u(x1)αK+u(xK)(1−αK)]=Vˆ(fˆ). This gives Vˆ(fˆ)=∑x∈Xfu(x)λˆf(x), where λˆf(x1)=β2α2+...+βKαK and, for n=2,...,K, λˆf(xn)=βn(1−αn). Each element of the set λˆf={λˆf(x1),...,λˆf(xK)} is a number in (0,1), and their sum is one. That is, λˆf denotes a desired probability.

The state space Ω˜ is defined as a product of Xfs. That is, Ω˜=×f∈FXf. Defining a state space in this way implies that a state is a function from the set of acts to the set of consequences (see Gilboa (2009), Karni and Schmeidler (1991), Karni and Vierø (2013), and Schmeidler and Wakker (1987)). The probability measure λ˜ over Ω˜ is built as a product measure of the probability measures λˆfs. The set of new acts F˜ is derived from the set of options Fˆ by defining f˜:Ω˜→X as a projection function. The binary order ≿˜ on F˜ is constructed in a consistent way from ≿ˆ: f˜≿˜g˜ if and only if fˆ≿ˆgˆ. Finally, define V˜:F˜→R as V˜(f˜):=∑ω˜∈Ω˜u(f˜(ω˜))λ˜(ω˜).

Since λ˜ is a product measure and f˜ is a projection function, they induce a probability measure on X that is the same as λˆf (condition 2 in Definition 2.1). Note that ≿˜ is constructed in a consistent way from ≿ (condition 3 in Definition 2.1). By construction of λ˜ and f˜, it follows that V˜(f˜)=Vˆ(fˆ). This implies that ≿˜ has an SEU representation (condition 4 in Definition 2.1). This ends the proof of Proposition 1.

3 SEU-Rationalization of the Three-Color Ellsberg Experiment

Consider the three-color Ellsberg experiment. In the urn, there are three balls, and only one of them is red. Each of the remaining two can be either green or blue. However, the number of green balls in the urn is unknown. One ball will be drawn from the urn. The typically assumed state space is Ω={R,G,B}, where the letter indicates which ball will be drawn, and the set of consequences X consists of two elements, S(uccess) and F(ailure). The agent has two decisions to make: choose between a bet on a red ball fr and a bet on a green ball fg (the agent picks fr), and choose between a bet on a red or blue ball frb and a bet on a green or blue ball fgb (the agent picks fgb).

Assuming that the state space is Ω={R,G,B} implies that the binary order depicted in eq. [1] is inconsistent with the SEU theory (Table 1).

The procedure of SEU-rationalization requires, first, to take Table 1 and translate it into the set of options. This yields fˆr, fˆg, fˆgb, and fˆrb such that Xr=Xg=Xrb=Xgb={S,F}. Define Vˆ and u in the way that agrees with eq. [1]: Vˆ(fˆr)=3, Vˆ(fˆg)=1, Vˆ(fˆgb)=4, Vˆ(fˆrb)=2, u(S)=5, and u(F)=0. Define the probabilities associated with each option: λˆfr(S)=0.6, λˆfg(S)=0.2, λˆfgb(S)=0.8, and λˆfrb(S)=0.4.

Next, construct the state space Ω˜={S,F}×{S,F}×{S,F}×{S,F}. Each state consists of four coordinates. Also, construct the probability λ˜ over Ω˜ as the product measure of λˆfr, λˆfg, λˆfgb, and λˆfrb. Finally, transform the set of options into the set of new acts by defining the new acts as projection functions. For example, option fˆr becomes act f˜r: Ω˜→X, defined as f˜r(ω˜)=S if and only if the first coordinate of ω˜ is S. This yields Table 2.

Table 2 has sixteen rows, with each denoting a state. Columns f˜r, f˜g, f˜gb, and f˜rb show the realizations of bets at a given state. For example, at ω˜1 all bets yield S. The probability column depicts the probability that a given state will occur.

However, Table 2 allows for states which are not reasonable. For example, it is not reasonable that a ball is simultaneously red and green (state ω˜1). In the suggested SEU-rationalization, there are 13 unreasonable states and 3 reasonable states. The three reasonable states are ω˜7 (i. e., red ball is drawn), ω˜10 (i. e., green ball is drawn), and ω˜13 (i. e., blue ball is drawn). These are the states R, G, and B, respectively, from the state space Ω (see Table 1). The total mass assigned to these three states is only 0.1792. For this reason, this SEU-rationalization is not satisfactory. Therefore, the obvious question to ask is: What is the smallest probability assigned to the unreasonable states that would still allow for the SEU-rationalization of the Ellsberg experiment? The answer is: As small as desired. Let ε<0.3 be the total mass assigned to unreasonable states.

In Table 3, there is only one unreasonable state Z. It is easy to verify that this SEU-rationalization captures the choices in eq. [1]: V˜(f˜r)=510+103ε>V˜(f˜g)=510−53ε; and V˜(f˜gb)=4510+53ε>V˜(f˜rb)=4510−103ε.

The existence of state Z in Table 3 could be due to the agent’s being unaccustomed to decision problems expressed in terms of an urn and some balls. As Gilboa (2009) writes, “David Schmeidler often says, ‘Real life is not about balls and urns.’ Indeed, important decisions involve war and peace, recessions and booms, diseases and cures.” An ε-small probability assigned to Z can be interpreted as a manifestation of the agent’s misunderstanding the problem or miscalculating the probability.

If the experiment’s subjects are college students, which is often the case in economics experiments, then the analysis conducted by psychologists suggests that such a misunderstanding or miscalculation is not unlikely. For example, Standing (2006) finds that only 33 % of undergraduate students at one liberal arts college achieved a perfect score in a test that focused on very elementary skills: the hardest task was to compute 92×32 (and students were permitted to use unlimited time and write a draft work). In Standing, Sproule, and Leung (2006), a similar test was applied to undergraduate students majoring in business and economics. The hardest question was to compute (36×7)+(33×7), and only 40 % of subjects answered all questions correctly. From LeFevre et al. (2014), we learn that basic arithmetic skills among college students have been declining from 1993 to 2005. If the students struggle with adding or multiplying natural numbers, it only seems natural to expect that they make mistakes in computing subjective probabilities.

A typical explanation of the three-color Ellsberg experiment allows only for the reasonable states (i. e., Ω={R,G,B}) but assumes that the agent behaves as if following a mathematically sophisticated theory (at least, more sophisticated than the SEU theory). Here, I suggest an alternative explanation: the agent behaves as if following the SEU theory but she assigns an arbitrarily small probability to an unreasonable state (Table 3). Which explanation of the agent’s behavior is “better”? Unfortunately, there is no clear answer. Gilboa and Samuelson (2012) observe that in situations like this, “people typically bring subjective criteria to bear in making this choice (i. e., choosing among theories), tending to select theories that seem a priori reasonable, intuitive, simple, elegant, familiar, or that satisfy a variety of other considerations.” Consequently, in the case of the three-color Ellsberg experiment, the reader is left to use subjective criteria to decide which explanation seems “better.”

However, I need to stress that from the empirical perspective – that is, from a perspective that bases conclusions only on observable data – the Ellsberg experiment does not reject the SEU theory. Although it is possible to test whether or not Z is a Savage-null event (and if it is, then the agent does not assign even an ε-small probability to Z), such a test does not contradict my claim. Note that when I use the phrase the Ellsberg experiment, I am referring to an experiment consisting of the four alternatives and two choices described above in eq. [1]. For clarity, I call this the original Ellsberg experiment. Testing whether or not Z is Savage-null requires that the researcher expand the experiment by adding new alternatives and observing more than two choices. This would not be the original Ellsberg experiment (which my claim is about) but rather the extended Ellsberg experiment.

Suppose that the extended Ellsberg experiment shows that not only Z but also all the remaining unreasonable states are Savage-null. This would indicate that the procedure of SEU-rationalization proposed in this paper cannot defend the SEU theory against the evidence provided by the original Ellsberg experiment. This also indicates how to design a robust experiment to test the SEU theory: in order to survive the procedure of SEU-rationalization, an experiment that is to reject the SEU theory must be nested in a bigger experiment which rules out all states that can be used by SEU-rationalization.

4 Related Literature

What differentiates my paper from the literature on choice theory under uncertainty/ambiguity is that my ambition is not to introduce a new theory, but rather to develop a procedure which allows the researcher to not reject the SEU theory. One of the main objectives of the theories is to have the unique representation of the agent’s belief (additive probability, non-additive probability, set of probabilities). In contrast, neither the state space Ω˜ nor the probability λ˜ constructed to SEU-rationalize the agent’s behavior is unique. There could be another state space Ω′ and another probability λ′ that would also do the job. In fact, even for a given Ω˜, the probability λ˜ is not unique because there could be a distinct λ˜∗ such that Ω˜ and λ˜∗ also lead to an SEU-rationalization. The non-uniqueness is a result of constructing both a state space and a probability rather than only a probability. However, the lack of uniqueness is not a concern in this paper. This is because the main goal is to find some state space with some probability such that the agent appears to be following the SEU theory. In order to emphasize that I am not constructing a new theory, I have been employing the term assumption, rather than the term axiom.

Two papers which are closely related to this paper are Gilboa and Schmeidler (1994) and Lipman (1999). Gilboa and Schmeidler (1994) considers an agent represented by the Choquet Expected Utility (Schmeidler (1989)) and shows that “the non-additivity of the “probability” υ may be explained by “omitted” states of the world. If those were introduced into the model explicitly, the non-additivity would disappear.”. The first difference between my paper and Gilboa and Schmeidler (1994) is that my key requirement (Assumption 2) is weaker than assuming the Choquet Expected Utility theory. In addition, the state space Ω˜ in my paper is constructed from the subsets of the set of consequences, while in Gilboa and Schmeidler (1994) the state space is an extension of a pre-assumed state space. Lipman (1999) considers a construction of state space that begins with the set of pieces of information (“propositions in logic, statements in English or another language, or mathematical formulas”) denoted by Φ. A state is defined as a subset of Φ. Lipman (1999) relaxes the assumption of the agent being logically omniscient, and determines conditions under which the agent’s behavior fits within the SEU theory. My paper differs from Lipman (1999) since, in general, my procedure need not depend on the violation of logical omniscience. Moreover, in contrast to my paper, Lipman (1999) assumes that the researcher knows the set Φ which implies that the researcher knows how the agent constructs her state space.

Although it does not develop a new theory, this paper associates alternatives with subsets of the set of consequences X and, consequently, is related to the literature concerned with an agent who ranks sets of objects. (See Barberà, Bossert, and Pattanaik (2004) for the most complete review of this literature.) The models in this literature consist of two stages. In the first stage, the agent chooses a subset of X. What happens at the second stage divides these models into the following two groups. Group A consists of models in which, at the second stage, Nature picks the final consequence (see Section 3 in Barberà, Bossert, and Pattanaik (2004)). Group B consists of models in which, at the second stage, the agent picks the final consequence (see Section 4 in Barberà, Bossert, and Pattanaik (2004)).

The models in both of these groups share one feature, which separates them from the setup I propose. They all consider the agent’s binary order to be defined over the subset of the power set of X, a construction that implicitly assumes that two alternatives with the same set of consequences should be perceived as the same object or, at least, that the agent should be indifferent between them. However, as noted in Section 2, this approach is not appropriate for this paper. Thus, in this paper, the agent ranks options, rather than subsets of X.

The models in Group A are concerned with “complete uncertainty” which, according to Barberà, Bossert, and Pattanaik (2004), “refers to a situation where the agent knows the set of possible consequences of an action but cannot assign probabilities to those outcomes.” For this reason, these models are not related to this paper. However, one important set of models on complete uncertainty can be interpreted as if they were considering an agent who constructs probabilities over the subsets of X. These are the models which build on the concept of Hurwicz’s α-criterion (see Arrow and Hurwicz 1972; Hurwicz 1951) and develop the idea of evaluating sets by taking only their best and worst consequences into consideration. ^[2] A subset Y of X is evaluated by Vα(Y)=α(Y)⋅u(maxY)+(1−α(Y))⋅u(minY), where α is a function from the set of subsets of X to (0,1), and where maxY denotes the best element of Y, and minY the worst. The function α can be interpreted as a probability on Y. This functional representation resembles the result obtained by Ghirardato (2001). In his model, the state space Ω is fixed, and the agent perceives acts as set-valued functions. In fact, Vα can be considered to be an extreme version of Ghirardato (2001), in which the agent perceives no state space. However, such a representation is not appropriate for the purpose of this paper, since it does not capture the agent’s perception of alternatives. Since Y represents all consequences deemed possible, I focus on the probability measures with support Y. In Vα, however, the support of α is not Y but rather {maxY,minY}.

For Group B, the relevant research is in the literature on “preference for flexibility” that begins with Kreps (1979) and examines the agent who is not sure about her future preferences. This uncertainty is captured by a state space S in which a state s represents a possible future binary order over set X. My paper implicitly assumes that the agent does not face uncertainty regarding her future self. Consequently, the construction of the state space here differs from the construction proposed in Kreps (1979), in which the agent evaluates a subset Y of X by V(Y)=∑s∈Smaxy∈Yu(s,y), where u(s,y) is an ex-post, state-dependent utility function. In contrast, in my paper a menu is evaluated by its subjective expected utility, and the utility over X is state-independent. Kreps (1992) reinterprets the original “preference for flexibility” model as representing the agent who faces unforeseen contingencies – that is, the agent who is unable to account for all possible future uncertainties. In contrast, my paper does not represent the agent who lacks a complete picture of uncertainty, but rather the researcher who does not know which uncertainties the agent has taken into account.

5 Conclusions

If the researcher, who tests the SEU theory, assumes some state space Ω and rejects the SEU theory, then his conclusion is not unconditionally valid. There may exist an alternative state space Ω˜ and a probability measure λ˜ over Ω˜ which together do not lead to the rejection of the SEU theory. If such Ω˜ and λ˜ exist, then the agent’s behavior is SEU-rationalizable. This paper shows how to SEU-rationalize the agent’s behavior and when an SEU-rationalization is possible.

I believe that the assumptions which allow to SEU-rationalize the agent’s behavior are not demanding. It is difficult to consider violations of Assumption 1 – incompleteness and intransitivity – as valid reasons to reject the SEU theory. I leave it for the reader to decide whether possible violations of Assumption 2 constitute convincing or interesting arguments against the SEU theory.

I would like to stress that if the agent’s behavior can be SEU-rationalized, then this does not mean that the SEU theory has been accepted and non-SEU theories ^[3] rejected. Rather, the correct conclusion would be that the SEU theory has not been rejected and that other theories may also be able to explain the agent’s behavior by relying on the assumption of a distinct state space. Consequently, instead of saying that my procedure allows the researcher to accept the SEU theory, I say – more precisely – that it allows him to not reject it. For this reason, this paper should not be interpreted as a critique of non-SEU theories. Rather, it is a critique of motivating such theories by only indicating the evidence which rejects the SEU theory.