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Abraham Wald's complete class theorem and Knightian uncertainty
Games and Economic Behavior 104 (2017) 666–673
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Abraham Wald’s complete class theorem and Knightian uncertainty Christoph Kuzmics 1 University of Graz, Austria
a r t i c l e
i n f o
Article history: Received 5 March 2014 Available online 5 July 2017 JEL classification: C72 C81 C90 D01 D03 D81
a b s t r a c t I study the implications of Wald’s (1947) complete class theorem for decision making under Knightian uncertainty (or ambiguity). Suppose we call someone who uses Wald’s approach to statistical decision making a Waldian. A Waldian may then have preferences over acts that are not in agreement with subjective expected utility but always chooses as if she was a subjective expected utility maximizer. In particular, even Wald’s (1945) minmax decision rule is consistent with subjective expected utility. © 2017 Elsevier Inc. All rights reserved.
Keywords: Ambiguity Decision theory Knightian uncertainty Experimental design
“Everything has been said but not everyone has said it.” [Morris Udall, 1922–1992, US American Politician] “Es ist schon alles gesagt, nur noch nicht von allen.” [Karl Valentin, 1882–1948, Bavarian Comedian] 1. Introduction Ellsberg (1961) conducted a thought experiment, asking individuals about their preferences over potential choices in his famous two- and three-color urn decision problems. The answers individuals gave to these questions demonstrate that many individuals have preferences that are inconsistent with the subjective expected utility (SEU) models of Savage (1954)
E-mail address: [email protected]. The three previous versions of this paper were circulated under the title “Inferring preferences from choices under uncertainty”, “An alternative subjective expected utility representation theorem”, and “A rational ambiguity averse person will never display her ambiguity aversion” respectively, and are still available as SSRN working papers. I would like to thank seminar participants at the Universities of Bielefeld, Bristol, Exeter, Heidelberg, Oslo, and Simon Fraser University as well as Herbert Dawid, Christoph Diehl, Jürgen Eichberger, David Freeman, Michael Greinecker, Faruk Gul, Chiaki Hara, David Kelsey, Peter Klibanoff, Sujoi Mukerji, Jörg Oechssler, Phil Reny, Frank Riedel, Arthur Robson, Zvi Safra, and particularly the anonymous associate editor and his or her assistant as well as the anonymous referees at Games and Economic Behavior for many helpful comments and suggestions. I am also particularly grateful to Jonathan Weinstein who pointed me to the work of Abraham Wald. 1
http://dx.doi.org/10.1016/j.geb.2017.06.012 0899-8256/© 2017 Elsevier Inc. All rights reserved.
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and Anscombe and Aumann (1963).2 There is now a large literature, beginning at least with Gilboa and Schmeidler (1989), including also Klibanoff et al. (2005), Maccheroni et al. (2006), Seo (2009), and Cerreia-Vioglio et al. (2013) in which an axiomatic foundation for such non-SEU preferences is given. Note that this is all about preferences. It is not immediately clear that this is also about choices. In a related problem, that of eliciting preferences from choices for models that violate the von Neumann and Morgenstern (1944) expected utility axioms for objective lotteries, we know, from Holt (1986), Karni and Safra (1987), and Segal (1988) that certain experiments such as those using the Becker et al. (1964) mechanism do not generally allow the correct elicitation of non-expected utility preferences. This literature, thus, demonstrates that eliciting preferences is not straightforward when the decision maker has preferences that deviate from standard expected utility. In this paper, I discuss whether it is possible to elicit preferences that do not agree with SEU. I do this under the assumption, also made in e.g. Gilboa and Schmeidler (1989), Klibanoff et al. (2005), Maccheroni et al. (2006), and Seo (2009), that the decision maker evaluates objective lotteries according to expected utility. As I am interested in individuals’ choices and the elicitation of their preferences, I need to discuss three additional assumptions that are not directly related to preferences. First, I assume that individuals can actively and objectively randomize when making their choices.3 Second, I assume that individuals can commit to following the realization of their random choice in case they choose randomly. Third, I assume that, when faced with more than one decision problem, individuals form a global plan (a detailed plan specifying a choice in every subproblem). These assumptions are not made because of their empirical plausibility but because of their normative appeal.4 I discuss their role for the result and their plausibility in the discussion section of this paper. To give a preview of the discussion below, note that there are two distinct issues that Ellsberg’s (1961) thought experiments raise. To see these, consider a slight variation of the two-color Ellsberg (1961) urn thought experiment.5 There are two urns. One urn, the risky or unambiguous urn, holds 49 green and 51 black balls. The other urn, the ambiguous urn, holds 100 balls, all of which are either red or white, but the exact composition is not known. Consider the following three bets. In bet zero, the decision maker (DM) receives a monetary prize if a (uniformly) randomly drawn ball from the risky urn is green and receives nothing otherwise. In bet one, the DM receives the same prize if a (uniformly) randomly drawn ball from the ambiguous urn is red and receives nothing otherwise. In bet two, finally, the DM receives the same prize if a (uniformly) randomly drawn ball from the ambiguous urn is white and receives nothing otherwise. Suppose we ask the DM how she would choose if she were presented with a choice between bets zero and one only. She might state that she prefers bet zero. Suppose we ask her then how she would choose if she were presented with a choice between bet zero and bet two. She might state that she prefers bet zero. These are hypothetical decision problems. What would happen if we give her an actual choice between various bets? Suppose first, and this is issue number one, we ask her to choose among all three bets. After she chooses we then perform the drawing of balls and pay her accordingly. Will she choose bet zero? And if she does, what does this tell us about her? Raiffa (1961) provided the following argument that she “should” not choose bet zero. She should consider choosing objectively randomly by flipping a fair coin. If the coin comes up heads she should choose bet one, if it comes up tails she should choose bet two. Then, regardless of the color of the ball drawn from the ambiguous urn, she has a probability of 1/2 of winning the prize, while bet zero only gives her a 49/100 probability of winning. In the language of game theory, the random bet (1/2 on each of the two bets one and two) strictly dominates bet zero, as it provides a strictly higher winning probability (and thus expected utility, provided the DM values the prize more than receiving nothing) than bet zero in every possible state of nature (i.e., for any possible composition of the ambiguous urn). If the DM avoids dominated strategies she will not choose bet zero, when presented with the three bets, and will thus not fully reveal her “preferences”. That is, she does not reveal that she would have chosen bet zero if she were only presented with bets zero and one, and bet zero again if she were only presented with bets zero and two. Giving the DM just one decision problem with the three bets is, however, only one way to try to elicit the DM’s preferences. Suppose, and this is issue number two, we give her two decision problems. We ask her to choose one of bets zero and one, and one of bets zero and two. We then need to specify how exactly we pay the DM in this case. We have many options. I will explore only one here. For further possibilities see the more general treatment in Section 3. Suppose we ask her to make her choices in both problems, then we choose one of the two problems objectively randomly and equally likely with the understanding that this decision problem is then used to pay her. Note that now the DM does not even have to randomize herself. The choice of bet one in problem one and bet two in problem two gives her a winning probability of 1/2
2 Note that there are objective probability distributions in the Ellsberg (1961) experiment, in the form of the risky urn and the uniformly random draws of balls from both urns. While objective probabilities are present in the Anscombe and Aumann (1963) model, the Savage (1954) model strictly speaking has no objective uncertainty. 3 Note that Gilboa and Schmeidler (1989), Klibanoff et al. (2005), and Maccheroni et al. (2006) each provide an axiomatization of preferences over acts only, not over the set of objective randomizations over acts. Battigalli et al. (2017) provide a unifying framework in which the above-mentioned preference models can be embedded and that allows for objective randomizations over acts. 4 One could probably argue that the literature on ambiguity aversion at least implicitly rejects some of these three assumptions. Empirically this rejection may well be warranted. 5 The variation is used to avoid payoff-ties.
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in each state, while choosing bet zero in both problems gives her a winning probability of 49/100 in both states. Choosing bet zero in both problems is again strictly dominated. Again, the decision maker does not reveal her true “preferences”. The result that choosing a non-SEU option is a dominated strategy, is a general phenomenon and it follows from a result due to Wald (1947), the so-called complete class theorem (see also Wald, 1950). I call a decision maker Waldian if she 1) evaluates objective lotteries according to expected utility, 2) considers objectively random choices and has the ability to commit to these choices, 3) forms a global plan for all decision problems, and 4) avoids dominated strategies. I do not want to claim that every ambiguity averse decision maker, with preferences axiomatized in e.g. Gilboa and Schmeidler (1989), Klibanoff et al. (2005), Maccheroni et al. (2006), and Seo (2009), should be automatically considered to be Waldian. I want to make the point that when talking about a decision maker’s choices and not her preferences we need to think about her attitude towards or ability to actively and objectively randomize, to commit to random choices (if these are made), and to form global plans. Then Pearce’s (1984) Lemma 3, which is a strict dominance version of the complete class theorem of Wald (1947), see also Wald (1950), implies that for a single given decision problem, any non-Bayesian (i.e., non-SEU) random act is strictly dominated by a Bayesian (i.e., SEU) random act. One random act strictly dominates another if in every possible state of nature the former provides strictly higher objective expected utility than the latter. Thus, a Waldian, in a single decision problem, behaves as if she was an SEU maximizer. I then also consider choices that a “pessimistic” Waldian decision maker would make. Wald (1945, 1950) proposes as an “optimal” choice the random minmax (or maxmin) choice, i.e., the random act that minimizes the maximal expected risk, where risk is simply the negative of utility. This is always a Bayesian (i.e., SEU) random act, see e.g. Wald (1945, Theorem 5.3). This is seemingly in contrast to the statement by Gilboa and Schmeidler (1989) that they provide “an axiomatic foundation of Wald’s [minmax] criterion.” The way to reconcile these two findings is as follows. While a pessimistic Waldian may have preferences that are in violation of SEU and that may, for instance, be as provided in Gilboa and Schmeidler (1989), the pessimistic Waldian’s actual choice in a single decision problem is as if she was an SEU maximizer. I then argue that if a Waldian DM, pessimistic or not, is faced with multiple decision problems and if she treats her set of decision problems as one big decision problem, Wald’s complete class theorem again implies that the DM behaves as if she was an SEU maximizer. In particular in every single one of her multiple decision problems she chooses an act that is optimal given the same single belief. 2. Setup and main result
ω ∈ all remaining α ( y ) = 1 be the set of y ∈Y
Let denote a finite set of states of nature with the interpretation that conditional on any state
uncertainty is objective. Let Y be a finite set of final outcomes and let (Y ) =
α : Y → R+ |
all probability distributions over Y , where R+ denotes the set of non-negative real numbers. An Anscombe and Aumann (1963) act is defined as a mapping f : → (Y ). For any act f , state ω ∈ , and final outcome y ∈ Y , let f (ω) y denote the probability that act f in state ω results in final outcome y ∈ Y . Let F denote a non-empty set of acts. Let v : Y → R denote the decision maker’s utility function over final outcomes. The decision maker evaluates objective lotteries over Y by computing their expected utility. That is, the decision maker’s preference relation, for objective lotteries, satisfies the von-Neumann Morgenstern axioms. Example. Consider the Ellsberg (1961) two-color urn setup from the introduction. We have one (ambiguous) urn with 100 balls, all of which are white or red. The exact composition of red and white balls is unknown. We eventually (objectively and uniformly) randomly draw one ball from this urn. The biggest possible set of states one might want to consider in this setup is the set of all Bernoulli-distributions with i the probability of a red ball given by some 100 for i ∈ {0, ..., 100}.6
49 Consider three acts. Act f 0 pays out 1 monetary unit with objective probability 100 in all states.7 This is, therefore, a constant act, as its payout does not depend on the composition of the ambiguous urn. Act f 1 pays out 1 monetary unit if we draw a red ball from the ambiguous urn and pays out 0 otherwise. Act f 2 pays out 1 monetary unit if we draw a white ball from the ambiguous urn and pays out 0 otherwise. Thus, Y = {0, 1} and F consists of the three given acts. The utility function is an arbitrary function v that satisfies v (1) > v (0). The decision maker, thus, simply aims to maximize the probability of winning the unit monetary prize.
For a given set of acts F , a Waldian is assumed to be able to choose any objectively random act ϕ ∈ ( F ) with probability that the random act assigns to pure act f ∈ F . For any random act ϕ ∈ ( F ) and any state ω ∈ let
ϕ ( f ) the
6 If we draw a ball only once the state can simply be the color of the drawn ball. If we draw balls multiple times the state must be the proportion of red balls in the urn. 7 One may want to think of this act as betting on a green ball that is taken from another (unambiguous) urn with a known composition of 49 green, and 51 black balls.
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u (ϕ , ω) =
669
ϕ ( f ) f (ω) y v ( y )
f ∈ F y ∈Y
be the (objective) expected utility induced by random act ϕ in state ω .8 A decision problem is then given by the tuple (, F , u ), where Y is implicit in F and u is derived from v. We say that one random act ϕ ∈ ( F ) strictly dominates another random act ϕ ∈ ( F ) if the objective expected utility induced by ϕ strictly exceeds the expected utility induced by ϕ in every state ω , i.e., if u (ϕ , ω) > u (ϕ , ω) for all ω ∈ . A random act ϕ ∈ ( F ) is undominated if there is no other random act ϕ ∈ ( F ) that strictly dominates ϕ . A random act ϕ ∈ ( F ) weakly dominates another random act ϕ ∈ ( F ) if u (ϕ , ω) ≥ u (ϕ , ω) for all ω ∈ and there is a state ω ∈ such that u (ϕ , ω) > u (ϕ , ω). A random act ϕ ∈ ( F ) is admissible if there is no other random act ϕ ∈ ( F ) that weakly dominates ϕ . Let () = μ : → R+ | ω∈ μ(ω) = 1 denote the set of all possible subjective beliefs over states of nature.9 Theorem 1. For any finite decision problem (, F , u ) a random act ϕ ∈ ( F ) is undominated if and only if it is Bayesian, that is, if and only if there is a (subjective) belief μ ∈ () such that ϕ ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). This result, stated as “a mixed strategy in a finite game is undominated if and only if it is a best reply to some (possibly correlated) belief about the opponents’ strategy profile”, has been proven by Pearce (1984, Appendix B, Lemma 3) using the minmax theorem and by Gale and Sherman (1950) directly using techniques from linear programming (see also van Damme, 1991, Lemma 3.2.1 and Theorem 3.2.2, and Myerson, 1991, Theorems 1.6 and 1.7). This result is very close to Wald’s (1947) complete class theorem, which can be translated into the language of modern game theory as follows. It states that “every mixed strategy that is not a best reply to some (possibly correlated) belief about the opponents’ strategy profile is weakly dominated by a mixed strategy that is a best reply to some (possibly correlated) belief about the opponents’ strategy profile”.10 Pearce’s Lemma 3 is, thus, a little bit stronger than Wald’s complete class theorem.11 As the proof of this theorem is well-known it is omitted. An insightful sketch of the proof based on the separating hyperplane theorem is given in Fudenberg and Tirole (1991, Theorem 2.2). For a recent proof of a generalized version of this result see Battigalli et al. (2016). For decision making under uncertainty, we have the following immediate implication. Consider a finite decision problem (, F , u ). Let a Waldian be a decision maker who evaluates objective lotteries according to expected utility, who can choose acts actively and objectively randomly, who can commit to her random choice, and who does not choose strictly dominated random acts. Then a Waldian’s random choice, ϕ ∈ ( F ), assigns total probability one on pure acts that are Bayesian for a single belief, i.e., there is a (subjective) belief μ ∈ () such that if ϕ ( f ) > 0 then f ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). Wald’s complete class theorem does not provide a unique optimal random act. Wald (1945, 1950) then provides an essentially unique undominated random act by offering us the minmax random act, i.e., the random act that maximizes the minimal expected utility.12 This minmax random act is undominated and Bayesian (see e.g. Wald, 1945, Theorem 5.3). It is a (mixed) minmax strategy in the zero-sum game against nature.13 3. Multiple decision problems Let again denote the finite set of states of nature. Suppose the DM has to make choices from a collection of decision problems F 1 , ..., F n , where each F i is a set of acts with implicit outcome set Y i . The DM is asked to announce her choices for all of these decision problems. Let F = ×ni=1 F i and let f = ( f 1 , ..., f n ) be a typical element of F , a profile of individual acts. To specify how the DM is paid out in the end, let an experimental design be identified by a tuple (G , Z , t ), where Z is a set of final outcomes, G is the set of global acts, which is a subset of the set of all acts, and t : F → G is a function that determines how profiles of acts in all individual decision problems translate into global acts. Two examples of experimental designs (based on the main example of this paper) are given in Section 4.3.
When a random act assigns probability one to a single pure act f ∈ F , I shall abuse notation slightly by identifying this random act also by f ∈ ( F ). In the literature, such as Gilboa and Schmeidler (1989), beliefs are often referred to as “priors”. 10 Wald’s setup is somewhat more complicated as he was interested in statistical decision rules that are functions of data. In Wald’s setup the state ω ∈ typically represents a parameter of some distribution function. The decision maker then observes a series of realizations, drawn in an iid fashion from this distribution, and before she observes these realizations chooses a decision rule, i.e., a decision (in some set of possible decisions) as a function of the data. Wald’s complete class theorem then states that any non-Bayesian decision rule is weakly dominated by a Bayesian decision rule. It is thus without loss of generality to consider only Bayesian decision rules. In this sense the set of Bayesian decision rules form a “complete class”. 11 Pearce (1984), in his Lemma 4, additionally provides a full characterization of weakly dominated mixed strategies (in finite games) as those mixed strategies that are not a best reply to any full support belief. 12 If there are multiple minmax random acts then they are all payoff equivalent. 13 While it seems that Abraham Wald, e.g. in Wald (1939), initially was unaware of von Neumann’s minmax theorem and the theory of games, in Wald (1945, Section 6) on the “Relationship to von Neumann’s theory of games” he interprets the statistical decision problem exactly in this way: as a zero-sum game against nature. 8 9
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Then, let v : Z → R denote the DM’s utility over final outcomes. Assume, as before, that the DM evaluates objective lotteries over Z by computing their expected utility. A Waldian is assumed to choose one of all objectively random global plans, i.e., an element ϕ ∈ ( F ). Finally, for any ϕ ∈ ( F ) let
u (ϕ , ω) =
ϕ ( f )t ( f )(ω) y v ( y ).
f ∈ F y ∈Y
With all this in place we can now state the main implication of Theorem 1 for multiple decision problems. Consider a set of decision problems (, ( F i )ni=1 ), with F = ×ni=1 F i , with experimental design given by (G , Z , t ) and resulting utility function u. Let a Waldian be a decision maker who evaluates objective lotteries according to expected utility, who chooses a global plan (a profile of acts), who can choose plans actively and objectively randomly, who can commit to her random choice, and who does not choose strictly dominated random plans. Then a Waldian’s random global choice, ϕ ∈ ( F ), assigns total probability one to the set of pure global acts that are Bayesian for a single belief, i.e., there is a (subjective) belief μ ∈ () such that if ϕ ( f ) > 0 then f ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). For f ∈ F and any i ∈ {1, ..., n} we can write f = ( f i , f −i ). Note that the above corollary can then be strengthened as follows. There is a (subjective) belief μ ∈ () such that, if ϕ ( f ) > 0 then f i ∈ argmax f ∈ F i ω∈ μ(ω)u ( f i , f −i , ω) for all i i ∈ {1, ..., n}. 4. Discussion and conclusion I have argued that when considering decision making under Knightian uncertainty, i.e., uncertainty that one cannot assign objective probabilities to, it is not enough to consider the decision maker’s (DM’s) preferences over acts if we are interested in the DM’s actual choices. A Waldian can “hedge” ambiguity away through active randomization and any actual choice a Waldian makes is Bayesian or in other words is “as if” she has a subjective belief, a probability distribution over the states of nature, according to which she maximizes her expected utility. This result follows from a variation of Wald’s (1947) complete class theorem due to Pearce (1984): a mixed strategy in a game is undominated if and only it is a best reply to some (possibly correlated) belief about the opponents’ strategy. In the remainder of this paper I discuss the three assumptions on the DM’s ability to randomize, to commit, and to form global plans in the definition of a Waldian and their role for the result. 4.1. On randomizing and the interpretation of mixed strategies Random choice in form of mixed strategies has a long history in game theory starting with the seminal work by Borel and von Neumann. While in von Neumann’s maxmin strategies players were assumed to actively randomize themselves, in the modern interpretations of mixed strategies, individuals do not have to do so. The randomization is either in opponent beliefs, see e.g. Aumann and Brandenburger (1995), or given by some exogenous uncertainty in the environment: the payoff types in Harsanyi’s (1973) purification argument or the random matching (from a large population of players) implicit in the mass action interpretation of John Nash’s PhD thesis, Nash (1950).14 In these interpretations of mixed strategies, however, the question is not so much whether people can actively randomize but whether they want to and, if so, how they want to randomize. In the unique Nash equilibrium of the game of matching pennies, for instance, each player is supposed to randomize in such a way as to make their respective opponent indifferent between all strategies. In such an equilibrium a player does obviously not have a strict incentive to randomize, which then leads back to the question as to why they would want to do it.15 The alternative interpretations of mixed strategies were developed as a response to this question. Note, however, that in the context of this paper, mixed strategies are only used to remove strategies (from the DM’s consideration) that are strictly dominated. Given the choice between one strategy and a mixed strategy that dominates the former the DM is not indifferent between the two and in fact has a strict incentive to use the mixed strategy, i.e., to randomize. For further discussions of randomization under Knightian uncertainty see Klibanoff (2001), Saito (2015), and Eichberger et al. (2016). 4.2. On randomizing and committing to random acts The role of the DM’s ability to randomize and her ability to commit to the outcome of this randomization is best discussed by means of a single decision problem based on the main example of this paper.16
14
See also Reny and Robson (2004) for a “unification of the classical and Bayesian views”. The debate, for instance, whether one should use maxmin strategies or Nash equilibrium strategies in non-zero sum games (that are in some sense close to zero-sum games) by Aumann and Maschler (1972) is not about whether one should randomize, but how one should do it. The point is that both the Nash equilibrium and maxmin strategies, although different, are both undominated and in fact good choices under some belief about the opponent’s play. In contrast, a non-SEU choice for a “Waldian” is a choice that is strictly dominated by a (possibly random) SEU choice. We cannot rank maxmin and Nash equilibrium strategies in the Aumann and Maschler (1972) example in such a clear way. 16 See also Machina (1989, p. 1643) for a discussion of the role of commitment to a random choice. As a concrete example, he discusses the problem a mother faces when considering to flip a coin to choose which one of her two children is allowed an indivisible treat. 15
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A single decision problem for the given example. Consider the case where the DM is asked to choose one of the three acts f 0 , f 1 , f 2 . This decision problem can be formulated in terms of Anscombe and Aumann’s (1963) framework in multiple ways. As we are only drawing a ball once, we can reduce the state space to just two elements, i.e., = { R , W }. We do not care about the exact composition of the urn. All we need to know is the color of the ball that is drawn from the urn. We then have the following Anscombe and Aumann (1963) representation of the problem, where the three acts are described as vectors of probabilities of winning the prize. State
f0
f1
f2
R
49 100 49 100
1
0
0
1
W
In this example, the set of undominated (and, hence, Bayesian) random acts is given by ({ f 1 , f 2 }), the set of all probability distributions over f 1 and f 2 . While act f 0 is the optimal choice for an ambiguity averse DM who cannot or does not contemplate to actively randomize her choice, for a DM who can actively randomize, act f 0 is dominated by the random act that places probability 12 on each of f 1 and f 2 , as already pointed out in Raiffa (1961). A pessimistic Waldian, who uses a minmax strategy, would choose exactly that equal mixture of the two acts f 1 and f 2 . Turning to the question of commitment, now suppose that this DM is ambiguity averse but can actively and objectively randomize. Suppose that this DM understands that the random choice that assigns probability one half on f 1 and f 2 each, results in an expected probability of winning the prize of one half in each state. Suppose that the DM therefore actually prefers this random act over the act f 0 . Suppose the randomization is executed and the realized choice is act f 1 . The DM now has a commitment problem. When comparing the two acts f 1 and f 0 she prefers f 0 . She would then like to switch to choose f 0 . Contemplating the choice f 0 she would then prefer to randomize again. When she observes the outcome of the randomization she would again prefer to opt for f 0 again, and then again to randomize again. It is, thus, not clear what choice such a DM would make if she does not have the ability to commit to a random choice. 4.3. On global plans The role of the “big picture” assumption, i.e., that the DM forms a global plan and evaluates this globally, is also best discussed by means of examples. These also help to illustrate the role that the experimental design plays for the DM’s problem. A randomly chosen “active” decision problem. Suppose the DM is asked to choose an act in each of the two decision problems { f 0 , f 1 } and { f 0 , f 2 }. Then, each of the two decision problems has an equal probability of 12 to be “active”, meaning the DM is paid according to her choice for the active decision problem only. Note that the DM has now four “pure strategies”: choose f 0 or f 1 for the first decision problem and f 0 or f 2 in the second. Denote the four pure strategies by ( f 0 , f 0 ), ( f 0 , f 2 ), ( f 1 , f 0 ), and ( f 1 , f 2 ). Each of these pure strategies is then equivalent to a random act. Strategy ( f 0 , f 0 ) is equivalent to act f 0 , ( f 0 , f 2 ) is equivalent to the random act that places equal probability 12 on acts f 0 and f 2 , ( f 1 , f 0 ) is equivalent to the random act that places equal probability 12 on acts
f 0 and f 1 , and finally ( f 1 , f 2 ) is equivalent to the random act which places equal probability 12 on acts f 1 and f 2 . The experimental design is then given by (G , Z , t ) with G the set of acts that are equivalent to the four given random acts, the set Z is given by Z = Y 1 = Y 2 = {0, 1} and t is given in the following table: State
t( f0, f0)
t( f0, f2)
t( f1, f0)
t( f1 , f2)
R
49 100 49 100
49 200 149 200
149 200 149 200
1 2 1 2
W
The set of undominated mixed strategies is then given by all probability distributions over ( f 0 , f 2 ) and ( f 1 , f 2 ) and all probability distributions over ( f 1 , f 0 ) and ( f 1 , f 2 ). All proper mixtures of ( f 0 , f 2 ) and ( f 1 , f 0 ) are dominated and so are any mixtures that include ( f 0 , f 0 ). A pessimistic Waldian, using a minmax strategy, would choose ( f 1 , f 2 ). A DM who does not form a global plan, but instead considers each decision problem in isolation, could choose f 0 in both problems. Tickets. Suppose the DM is again asked to choose an act in each of the two decision problems { f 0 , f 1 } and { f 0 , f 2 }. The DM’s choices are now “tickets” that both pay out (based on one draw of the ball). In this case the set of final outcomes is 49 Z = {0, 1, 2}, as the DM can get zero, one, or two (identical) prizes. Let p = 100 . The DM has again four pure strategies that again can be seen as acts through the translation function t as given here: R
W
Payoff
t( f0, f0)
t( f0, f2)
t( f1, f0)
t( f1, f2)
Payoff
t( f0, f0)
t( f0, f2)
t( f1 , f0)
t( f1, f2)
2 1 0
p2 2p (1 − p ) (1 − p )2
0 p 1− p
p 1− p 0
0 1 0
2 1 0
p2 2p (1 − p ) (1 − p )2
p 1− p 0
0 p 1− p
0 1 0
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Note that, as p <
1 , 2
pure strategy ( f 0 , f 0 ) is first order stochastically dominated in every state, and thus strictly dom-
inated, by the mixed strategy that assigns equal weight of 12 to the two pure strategies ( f 0 , f 2 ) and ( f 1 , f 0 ). No Waldian, pessimistic or not, would choose act f 0 in both problems. A DM who does not form a global plan, but instead considers each decision problem in isolation, could choose f 0 in both problems. 4.4. On Ellsberg behavior Given the main result, there are now at least two possible explanations of the behavior of real decision makers experimentally found by Ellsberg (1961) and many others, some of it reviewed in Camerer and Weber (1992). These experiments clearly demonstrate that many individuals do not behave in accordance with SEU theory. But as they are also clearly not Waldian, the explanation of their behavior becomes more difficult.17 It is not sufficient to say that they have preferences over the set of acts as given, for instance, by Gilboa and Schmeidler (1989), Klibanoff et al. (2005), and Maccheroni et al. (2006). One would have to add one of the following additional pieces to this explanation. Either 1) the decision maker does not contemplate choosing randomly, or 2) does not see the big picture (i.e., does not form a global plan for all her decision problems), or 3) does do both but does not evaluate random acts by applying the reduction of compound lotteries (or reversal of order). Note that this latter part of the explanation is very consistent with the empirical finding of Halevy (2007) that the individuals who appear ambiguity averse are those that do not perform the reduction of compound lotteries. Suppose that we accept that the empirically observed choices of real decision makers in Ellsberg (1961)-like experiments require these additional explanations. This then raises the question of whether the theoretically proposed models of behavior with ambiguity aversion are normatively very appealing, i.e., whether we can declare them models of “rational” decision making. If these individuals’ choices are not “rational” then these choices are subject to change if we teach these individuals Wald’s complete class theorem. 4.5. Alternative experimental designs There are at least three alternative experimental designs that I rule out. First, I do not allow surprising the DM, such as giving the DM one problem and then, without first telling her, giving her another problem. It has been shown (see Zambrano, 2005 and Shmaya and Yariv, 2016) that, in such cases, a DM can have beliefs about the possible sequence of decision problems given to her for each state, such that essentially all possible choices she can make can be rationalized by some such belief. Second, I rule out that the DM chooses sequentially and is paid out after each choice and before making her next one. The reason for this is that such a design would enable the DM to learn about the nature of uncertainty. It would be interesting to study such situations in the light of Wald’s complete class theorem. A third experimental design one might want to try is to ask the DM to make a choice in one environment (with say one ambiguous urn) and then another in another environment (with another ambiguous urn).18 The DM, however, is then free to have a belief that is any joint probability distribution over all state-tuples. After all, how are many urns with an “unknown” composition filled? Thus, this is nothing but a single experiment with a big state-space and the DM will choose as if she was an SEU maximizer. References Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability. Ann. Math. Stat. 34, 199–205. Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63 (5), 1161–1180. Aumann, R.J., Maschler, M., 1972. Some thoughts on the minimax principle. Manag. Sci. 18 (5), 54–63. Bade, S., 2015. Independent randomization devices and the elicitation of ambiguity averse preferences. J. Econ. Theory 159, 221–235. Battigalli, P., Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., 2016. A note on comparative ambiguity aversion and justifiability. Econometrica 84 (5), 1903–1916. Battigalli, P., Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., 2017. Mixed extensions of decision problems under uncertainty. Econ. Theory 63 (4), 827–866. Becker, G.M., DeGroot, M.H., Marshak, J., 1964. Measuring utility by a single response sequential method. Behav. Sci. 9, 226–232. Camerer, C., Weber, M., 1992. Recent developments in modelling preferences: uncertainty and ambiguity. J. Risk Uncertainty 5 (4), 325–370. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L., 2013. Ambiguity and robust statistics. J. Econ. Theory 148 (3), 974–1049. Eichberger, J., Grant, S., Kelsey, D., 2016. Randomization and dynamic consistency. Econ. Theory 62 (3), 547–566. Ellsberg, D., 1961. Risk, ambiguity, and the Savage axioms. Quart. J. Econ. 75 (4), 643–669. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. Gale, D., Sherman, S., 1950. Solutions of Finite Two-Person Games. Contributions to the Theory of Games, vol. 1, pp. 37–49. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. J. Math. Econ. 18 (2), 141–153. Halevy, Y., 2007. Ellsberg revisited: an experimental study. Econometrica 75 (2), 503–536. Harsanyi, J.C., 1973. Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points. Int. J. Game Theory 2 (1), 1–23.
17 There are, in fact, other interpretations possible, based on the idea that individuals may have a bigger state space in mind such as in e.g. Shmaya and Yariv (2016). 18 If the reader believes this can be done, the reader may want to consult Bade (2015) and Stoye (2015).
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Holt, C.A., 1986. Preference reversal and the independence axiom. Amer. Econ. Rev. 76, 508–515. Karni, E., Safra, Z., 1987. “Preference reversal” and the observability of preference by experimental methods. Econometrica 55 (3), 675–685. Klibanoff, P., 2001. Stochastically independent randomization and uncertainty aversion. Econ. Theory 18 (3), 605–620. Klibanoff, P., Marinacci, M., Mukerji, S., 2005. A smooth model of decision making under ambiguity. Econometrica 73 (6), 1849–1892. Maccheroni, F., Marinacci, M., Rustichini, A., 2006. Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74 (6), 1447–1498. Machina, M.J., 1989. Dynamic consistency and non-expected utility models of choice under uncertainty. J. Econ. Lit. 27 (4), 1622–1668. Myerson, R.B., 1991. Game Theory: Analysis of Conflict. Harvard University Press. Nash, J., 1950. Non-Cooperative Games. PhD dissertation, Princeton University Department of Mathematics. Pearce, D.G., 1984. Rationalizable strategic behavior and the problem of perfection. Econometrica 52, 1029–1050. Raiffa, H., 1961. Risk, ambiguity, and the Savage axioms: comment. Quart. J. Econ. 75 (4), 690–694. Reny, P.J., Robson, A.J., 2004. Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views. Games Econ. Behav. 48 (2), 355–384. Saito, K., 2015. Preferences for flexibility and randomization under uncertainty. Amer. Econ. Rev. 105 (3), 1246–1271. Savage, L., 1954. Foundations of Statistics. John Wiley and Sons, New York. Segal, U., 1988. Does the preference reversal phenomenon necessarily contradict the independence axiom? Amer. Econ. Rev. 78, 233–236. Seo, K., 2009. Ambiguity and second-order belief. Econometrica 77, 1575–1605. Shmaya, E., Yariv, L., 2016. Experiments on decisions under uncertainty: a theoretical framework. Amer. Econ. Rev. 106 (7), 1775–1801. Stoye, J., 2015. Choice theory when agents can randomize. J. Econ. Theory 155, 131–151. van Damme, E.E.C., 1991. Stability and Perfection of Nash Equilibria. Springer-Verlag, Berlin, Heidelberg. von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior. Princeton University Press. Wald, A., 1939. Contributions to the theory of statistical estimation and testing hypotheses. Ann. Math. Stat. 10 (4), 299–326. Wald, A., 1945. Statistical decision functions which minimize the maximum risk. Ann. Math., 265–280. Wald, A., 1947. An essentially complete class of admissible decision functions. Ann. Math. Stat. 18 (4), 549–555. Wald, A., 1950. Statistical Decision Functions. John Wiley and Sons, New York. Zambrano, E., 2005. Testable implications of subjective expected utility theory. Games Econ. Behav. 53, 262–268.
Contents lists available at ScienceDirect
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Note
Abraham Wald’s complete class theorem and Knightian uncertainty Christoph Kuzmics 1 University of Graz, Austria
a r t i c l e
i n f o
Article history: Received 5 March 2014 Available online 5 July 2017 JEL classification: C72 C81 C90 D01 D03 D81
a b s t r a c t I study the implications of Wald’s (1947) complete class theorem for decision making under Knightian uncertainty (or ambiguity). Suppose we call someone who uses Wald’s approach to statistical decision making a Waldian. A Waldian may then have preferences over acts that are not in agreement with subjective expected utility but always chooses as if she was a subjective expected utility maximizer. In particular, even Wald’s (1945) minmax decision rule is consistent with subjective expected utility. © 2017 Elsevier Inc. All rights reserved.
Keywords: Ambiguity Decision theory Knightian uncertainty Experimental design
“Everything has been said but not everyone has said it.” [Morris Udall, 1922–1992, US American Politician] “Es ist schon alles gesagt, nur noch nicht von allen.” [Karl Valentin, 1882–1948, Bavarian Comedian] 1. Introduction Ellsberg (1961) conducted a thought experiment, asking individuals about their preferences over potential choices in his famous two- and three-color urn decision problems. The answers individuals gave to these questions demonstrate that many individuals have preferences that are inconsistent with the subjective expected utility (SEU) models of Savage (1954)
E-mail address: [email protected]. The three previous versions of this paper were circulated under the title “Inferring preferences from choices under uncertainty”, “An alternative subjective expected utility representation theorem”, and “A rational ambiguity averse person will never display her ambiguity aversion” respectively, and are still available as SSRN working papers. I would like to thank seminar participants at the Universities of Bielefeld, Bristol, Exeter, Heidelberg, Oslo, and Simon Fraser University as well as Herbert Dawid, Christoph Diehl, Jürgen Eichberger, David Freeman, Michael Greinecker, Faruk Gul, Chiaki Hara, David Kelsey, Peter Klibanoff, Sujoi Mukerji, Jörg Oechssler, Phil Reny, Frank Riedel, Arthur Robson, Zvi Safra, and particularly the anonymous associate editor and his or her assistant as well as the anonymous referees at Games and Economic Behavior for many helpful comments and suggestions. I am also particularly grateful to Jonathan Weinstein who pointed me to the work of Abraham Wald. 1
http://dx.doi.org/10.1016/j.geb.2017.06.012 0899-8256/© 2017 Elsevier Inc. All rights reserved.
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and Anscombe and Aumann (1963).2 There is now a large literature, beginning at least with Gilboa and Schmeidler (1989), including also Klibanoff et al. (2005), Maccheroni et al. (2006), Seo (2009), and Cerreia-Vioglio et al. (2013) in which an axiomatic foundation for such non-SEU preferences is given. Note that this is all about preferences. It is not immediately clear that this is also about choices. In a related problem, that of eliciting preferences from choices for models that violate the von Neumann and Morgenstern (1944) expected utility axioms for objective lotteries, we know, from Holt (1986), Karni and Safra (1987), and Segal (1988) that certain experiments such as those using the Becker et al. (1964) mechanism do not generally allow the correct elicitation of non-expected utility preferences. This literature, thus, demonstrates that eliciting preferences is not straightforward when the decision maker has preferences that deviate from standard expected utility. In this paper, I discuss whether it is possible to elicit preferences that do not agree with SEU. I do this under the assumption, also made in e.g. Gilboa and Schmeidler (1989), Klibanoff et al. (2005), Maccheroni et al. (2006), and Seo (2009), that the decision maker evaluates objective lotteries according to expected utility. As I am interested in individuals’ choices and the elicitation of their preferences, I need to discuss three additional assumptions that are not directly related to preferences. First, I assume that individuals can actively and objectively randomize when making their choices.3 Second, I assume that individuals can commit to following the realization of their random choice in case they choose randomly. Third, I assume that, when faced with more than one decision problem, individuals form a global plan (a detailed plan specifying a choice in every subproblem). These assumptions are not made because of their empirical plausibility but because of their normative appeal.4 I discuss their role for the result and their plausibility in the discussion section of this paper. To give a preview of the discussion below, note that there are two distinct issues that Ellsberg’s (1961) thought experiments raise. To see these, consider a slight variation of the two-color Ellsberg (1961) urn thought experiment.5 There are two urns. One urn, the risky or unambiguous urn, holds 49 green and 51 black balls. The other urn, the ambiguous urn, holds 100 balls, all of which are either red or white, but the exact composition is not known. Consider the following three bets. In bet zero, the decision maker (DM) receives a monetary prize if a (uniformly) randomly drawn ball from the risky urn is green and receives nothing otherwise. In bet one, the DM receives the same prize if a (uniformly) randomly drawn ball from the ambiguous urn is red and receives nothing otherwise. In bet two, finally, the DM receives the same prize if a (uniformly) randomly drawn ball from the ambiguous urn is white and receives nothing otherwise. Suppose we ask the DM how she would choose if she were presented with a choice between bets zero and one only. She might state that she prefers bet zero. Suppose we ask her then how she would choose if she were presented with a choice between bet zero and bet two. She might state that she prefers bet zero. These are hypothetical decision problems. What would happen if we give her an actual choice between various bets? Suppose first, and this is issue number one, we ask her to choose among all three bets. After she chooses we then perform the drawing of balls and pay her accordingly. Will she choose bet zero? And if she does, what does this tell us about her? Raiffa (1961) provided the following argument that she “should” not choose bet zero. She should consider choosing objectively randomly by flipping a fair coin. If the coin comes up heads she should choose bet one, if it comes up tails she should choose bet two. Then, regardless of the color of the ball drawn from the ambiguous urn, she has a probability of 1/2 of winning the prize, while bet zero only gives her a 49/100 probability of winning. In the language of game theory, the random bet (1/2 on each of the two bets one and two) strictly dominates bet zero, as it provides a strictly higher winning probability (and thus expected utility, provided the DM values the prize more than receiving nothing) than bet zero in every possible state of nature (i.e., for any possible composition of the ambiguous urn). If the DM avoids dominated strategies she will not choose bet zero, when presented with the three bets, and will thus not fully reveal her “preferences”. That is, she does not reveal that she would have chosen bet zero if she were only presented with bets zero and one, and bet zero again if she were only presented with bets zero and two. Giving the DM just one decision problem with the three bets is, however, only one way to try to elicit the DM’s preferences. Suppose, and this is issue number two, we give her two decision problems. We ask her to choose one of bets zero and one, and one of bets zero and two. We then need to specify how exactly we pay the DM in this case. We have many options. I will explore only one here. For further possibilities see the more general treatment in Section 3. Suppose we ask her to make her choices in both problems, then we choose one of the two problems objectively randomly and equally likely with the understanding that this decision problem is then used to pay her. Note that now the DM does not even have to randomize herself. The choice of bet one in problem one and bet two in problem two gives her a winning probability of 1/2
2 Note that there are objective probability distributions in the Ellsberg (1961) experiment, in the form of the risky urn and the uniformly random draws of balls from both urns. While objective probabilities are present in the Anscombe and Aumann (1963) model, the Savage (1954) model strictly speaking has no objective uncertainty. 3 Note that Gilboa and Schmeidler (1989), Klibanoff et al. (2005), and Maccheroni et al. (2006) each provide an axiomatization of preferences over acts only, not over the set of objective randomizations over acts. Battigalli et al. (2017) provide a unifying framework in which the above-mentioned preference models can be embedded and that allows for objective randomizations over acts. 4 One could probably argue that the literature on ambiguity aversion at least implicitly rejects some of these three assumptions. Empirically this rejection may well be warranted. 5 The variation is used to avoid payoff-ties.
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in each state, while choosing bet zero in both problems gives her a winning probability of 49/100 in both states. Choosing bet zero in both problems is again strictly dominated. Again, the decision maker does not reveal her true “preferences”. The result that choosing a non-SEU option is a dominated strategy, is a general phenomenon and it follows from a result due to Wald (1947), the so-called complete class theorem (see also Wald, 1950). I call a decision maker Waldian if she 1) evaluates objective lotteries according to expected utility, 2) considers objectively random choices and has the ability to commit to these choices, 3) forms a global plan for all decision problems, and 4) avoids dominated strategies. I do not want to claim that every ambiguity averse decision maker, with preferences axiomatized in e.g. Gilboa and Schmeidler (1989), Klibanoff et al. (2005), Maccheroni et al. (2006), and Seo (2009), should be automatically considered to be Waldian. I want to make the point that when talking about a decision maker’s choices and not her preferences we need to think about her attitude towards or ability to actively and objectively randomize, to commit to random choices (if these are made), and to form global plans. Then Pearce’s (1984) Lemma 3, which is a strict dominance version of the complete class theorem of Wald (1947), see also Wald (1950), implies that for a single given decision problem, any non-Bayesian (i.e., non-SEU) random act is strictly dominated by a Bayesian (i.e., SEU) random act. One random act strictly dominates another if in every possible state of nature the former provides strictly higher objective expected utility than the latter. Thus, a Waldian, in a single decision problem, behaves as if she was an SEU maximizer. I then also consider choices that a “pessimistic” Waldian decision maker would make. Wald (1945, 1950) proposes as an “optimal” choice the random minmax (or maxmin) choice, i.e., the random act that minimizes the maximal expected risk, where risk is simply the negative of utility. This is always a Bayesian (i.e., SEU) random act, see e.g. Wald (1945, Theorem 5.3). This is seemingly in contrast to the statement by Gilboa and Schmeidler (1989) that they provide “an axiomatic foundation of Wald’s [minmax] criterion.” The way to reconcile these two findings is as follows. While a pessimistic Waldian may have preferences that are in violation of SEU and that may, for instance, be as provided in Gilboa and Schmeidler (1989), the pessimistic Waldian’s actual choice in a single decision problem is as if she was an SEU maximizer. I then argue that if a Waldian DM, pessimistic or not, is faced with multiple decision problems and if she treats her set of decision problems as one big decision problem, Wald’s complete class theorem again implies that the DM behaves as if she was an SEU maximizer. In particular in every single one of her multiple decision problems she chooses an act that is optimal given the same single belief. 2. Setup and main result
ω ∈ all remaining α ( y ) = 1 be the set of y ∈Y
Let denote a finite set of states of nature with the interpretation that conditional on any state
uncertainty is objective. Let Y be a finite set of final outcomes and let (Y ) =
α : Y → R+ |
all probability distributions over Y , where R+ denotes the set of non-negative real numbers. An Anscombe and Aumann (1963) act is defined as a mapping f : → (Y ). For any act f , state ω ∈ , and final outcome y ∈ Y , let f (ω) y denote the probability that act f in state ω results in final outcome y ∈ Y . Let F denote a non-empty set of acts. Let v : Y → R denote the decision maker’s utility function over final outcomes. The decision maker evaluates objective lotteries over Y by computing their expected utility. That is, the decision maker’s preference relation, for objective lotteries, satisfies the von-Neumann Morgenstern axioms. Example. Consider the Ellsberg (1961) two-color urn setup from the introduction. We have one (ambiguous) urn with 100 balls, all of which are white or red. The exact composition of red and white balls is unknown. We eventually (objectively and uniformly) randomly draw one ball from this urn. The biggest possible set of states one might want to consider in this setup is the set of all Bernoulli-distributions with i the probability of a red ball given by some 100 for i ∈ {0, ..., 100}.6
49 Consider three acts. Act f 0 pays out 1 monetary unit with objective probability 100 in all states.7 This is, therefore, a constant act, as its payout does not depend on the composition of the ambiguous urn. Act f 1 pays out 1 monetary unit if we draw a red ball from the ambiguous urn and pays out 0 otherwise. Act f 2 pays out 1 monetary unit if we draw a white ball from the ambiguous urn and pays out 0 otherwise. Thus, Y = {0, 1} and F consists of the three given acts. The utility function is an arbitrary function v that satisfies v (1) > v (0). The decision maker, thus, simply aims to maximize the probability of winning the unit monetary prize.
For a given set of acts F , a Waldian is assumed to be able to choose any objectively random act ϕ ∈ ( F ) with probability that the random act assigns to pure act f ∈ F . For any random act ϕ ∈ ( F ) and any state ω ∈ let
ϕ ( f ) the
6 If we draw a ball only once the state can simply be the color of the drawn ball. If we draw balls multiple times the state must be the proportion of red balls in the urn. 7 One may want to think of this act as betting on a green ball that is taken from another (unambiguous) urn with a known composition of 49 green, and 51 black balls.
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u (ϕ , ω) =
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ϕ ( f ) f (ω) y v ( y )
f ∈ F y ∈Y
be the (objective) expected utility induced by random act ϕ in state ω .8 A decision problem is then given by the tuple (, F , u ), where Y is implicit in F and u is derived from v. We say that one random act ϕ ∈ ( F ) strictly dominates another random act ϕ ∈ ( F ) if the objective expected utility induced by ϕ strictly exceeds the expected utility induced by ϕ in every state ω , i.e., if u (ϕ , ω) > u (ϕ , ω) for all ω ∈ . A random act ϕ ∈ ( F ) is undominated if there is no other random act ϕ ∈ ( F ) that strictly dominates ϕ . A random act ϕ ∈ ( F ) weakly dominates another random act ϕ ∈ ( F ) if u (ϕ , ω) ≥ u (ϕ , ω) for all ω ∈ and there is a state ω ∈ such that u (ϕ , ω) > u (ϕ , ω). A random act ϕ ∈ ( F ) is admissible if there is no other random act ϕ ∈ ( F ) that weakly dominates ϕ . Let () = μ : → R+ | ω∈ μ(ω) = 1 denote the set of all possible subjective beliefs over states of nature.9 Theorem 1. For any finite decision problem (, F , u ) a random act ϕ ∈ ( F ) is undominated if and only if it is Bayesian, that is, if and only if there is a (subjective) belief μ ∈ () such that ϕ ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). This result, stated as “a mixed strategy in a finite game is undominated if and only if it is a best reply to some (possibly correlated) belief about the opponents’ strategy profile”, has been proven by Pearce (1984, Appendix B, Lemma 3) using the minmax theorem and by Gale and Sherman (1950) directly using techniques from linear programming (see also van Damme, 1991, Lemma 3.2.1 and Theorem 3.2.2, and Myerson, 1991, Theorems 1.6 and 1.7). This result is very close to Wald’s (1947) complete class theorem, which can be translated into the language of modern game theory as follows. It states that “every mixed strategy that is not a best reply to some (possibly correlated) belief about the opponents’ strategy profile is weakly dominated by a mixed strategy that is a best reply to some (possibly correlated) belief about the opponents’ strategy profile”.10 Pearce’s Lemma 3 is, thus, a little bit stronger than Wald’s complete class theorem.11 As the proof of this theorem is well-known it is omitted. An insightful sketch of the proof based on the separating hyperplane theorem is given in Fudenberg and Tirole (1991, Theorem 2.2). For a recent proof of a generalized version of this result see Battigalli et al. (2016). For decision making under uncertainty, we have the following immediate implication. Consider a finite decision problem (, F , u ). Let a Waldian be a decision maker who evaluates objective lotteries according to expected utility, who can choose acts actively and objectively randomly, who can commit to her random choice, and who does not choose strictly dominated random acts. Then a Waldian’s random choice, ϕ ∈ ( F ), assigns total probability one on pure acts that are Bayesian for a single belief, i.e., there is a (subjective) belief μ ∈ () such that if ϕ ( f ) > 0 then f ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). Wald’s complete class theorem does not provide a unique optimal random act. Wald (1945, 1950) then provides an essentially unique undominated random act by offering us the minmax random act, i.e., the random act that maximizes the minimal expected utility.12 This minmax random act is undominated and Bayesian (see e.g. Wald, 1945, Theorem 5.3). It is a (mixed) minmax strategy in the zero-sum game against nature.13 3. Multiple decision problems Let again denote the finite set of states of nature. Suppose the DM has to make choices from a collection of decision problems F 1 , ..., F n , where each F i is a set of acts with implicit outcome set Y i . The DM is asked to announce her choices for all of these decision problems. Let F = ×ni=1 F i and let f = ( f 1 , ..., f n ) be a typical element of F , a profile of individual acts. To specify how the DM is paid out in the end, let an experimental design be identified by a tuple (G , Z , t ), where Z is a set of final outcomes, G is the set of global acts, which is a subset of the set of all acts, and t : F → G is a function that determines how profiles of acts in all individual decision problems translate into global acts. Two examples of experimental designs (based on the main example of this paper) are given in Section 4.3.
When a random act assigns probability one to a single pure act f ∈ F , I shall abuse notation slightly by identifying this random act also by f ∈ ( F ). In the literature, such as Gilboa and Schmeidler (1989), beliefs are often referred to as “priors”. 10 Wald’s setup is somewhat more complicated as he was interested in statistical decision rules that are functions of data. In Wald’s setup the state ω ∈ typically represents a parameter of some distribution function. The decision maker then observes a series of realizations, drawn in an iid fashion from this distribution, and before she observes these realizations chooses a decision rule, i.e., a decision (in some set of possible decisions) as a function of the data. Wald’s complete class theorem then states that any non-Bayesian decision rule is weakly dominated by a Bayesian decision rule. It is thus without loss of generality to consider only Bayesian decision rules. In this sense the set of Bayesian decision rules form a “complete class”. 11 Pearce (1984), in his Lemma 4, additionally provides a full characterization of weakly dominated mixed strategies (in finite games) as those mixed strategies that are not a best reply to any full support belief. 12 If there are multiple minmax random acts then they are all payoff equivalent. 13 While it seems that Abraham Wald, e.g. in Wald (1939), initially was unaware of von Neumann’s minmax theorem and the theory of games, in Wald (1945, Section 6) on the “Relationship to von Neumann’s theory of games” he interprets the statistical decision problem exactly in this way: as a zero-sum game against nature. 8 9
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Then, let v : Z → R denote the DM’s utility over final outcomes. Assume, as before, that the DM evaluates objective lotteries over Z by computing their expected utility. A Waldian is assumed to choose one of all objectively random global plans, i.e., an element ϕ ∈ ( F ). Finally, for any ϕ ∈ ( F ) let
u (ϕ , ω) =
ϕ ( f )t ( f )(ω) y v ( y ).
f ∈ F y ∈Y
With all this in place we can now state the main implication of Theorem 1 for multiple decision problems. Consider a set of decision problems (, ( F i )ni=1 ), with F = ×ni=1 F i , with experimental design given by (G , Z , t ) and resulting utility function u. Let a Waldian be a decision maker who evaluates objective lotteries according to expected utility, who chooses a global plan (a profile of acts), who can choose plans actively and objectively randomly, who can commit to her random choice, and who does not choose strictly dominated random plans. Then a Waldian’s random global choice, ϕ ∈ ( F ), assigns total probability one to the set of pure global acts that are Bayesian for a single belief, i.e., there is a (subjective) belief μ ∈ () such that if ϕ ( f ) > 0 then f ∈ argmaxϕ ∈( F ) ω∈ μ(ω)u (ϕ , ω). For f ∈ F and any i ∈ {1, ..., n} we can write f = ( f i , f −i ). Note that the above corollary can then be strengthened as follows. There is a (subjective) belief μ ∈ () such that, if ϕ ( f ) > 0 then f i ∈ argmax f ∈ F i ω∈ μ(ω)u ( f i , f −i , ω) for all i i ∈ {1, ..., n}. 4. Discussion and conclusion I have argued that when considering decision making under Knightian uncertainty, i.e., uncertainty that one cannot assign objective probabilities to, it is not enough to consider the decision maker’s (DM’s) preferences over acts if we are interested in the DM’s actual choices. A Waldian can “hedge” ambiguity away through active randomization and any actual choice a Waldian makes is Bayesian or in other words is “as if” she has a subjective belief, a probability distribution over the states of nature, according to which she maximizes her expected utility. This result follows from a variation of Wald’s (1947) complete class theorem due to Pearce (1984): a mixed strategy in a game is undominated if and only it is a best reply to some (possibly correlated) belief about the opponents’ strategy. In the remainder of this paper I discuss the three assumptions on the DM’s ability to randomize, to commit, and to form global plans in the definition of a Waldian and their role for the result. 4.1. On randomizing and the interpretation of mixed strategies Random choice in form of mixed strategies has a long history in game theory starting with the seminal work by Borel and von Neumann. While in von Neumann’s maxmin strategies players were assumed to actively randomize themselves, in the modern interpretations of mixed strategies, individuals do not have to do so. The randomization is either in opponent beliefs, see e.g. Aumann and Brandenburger (1995), or given by some exogenous uncertainty in the environment: the payoff types in Harsanyi’s (1973) purification argument or the random matching (from a large population of players) implicit in the mass action interpretation of John Nash’s PhD thesis, Nash (1950).14 In these interpretations of mixed strategies, however, the question is not so much whether people can actively randomize but whether they want to and, if so, how they want to randomize. In the unique Nash equilibrium of the game of matching pennies, for instance, each player is supposed to randomize in such a way as to make their respective opponent indifferent between all strategies. In such an equilibrium a player does obviously not have a strict incentive to randomize, which then leads back to the question as to why they would want to do it.15 The alternative interpretations of mixed strategies were developed as a response to this question. Note, however, that in the context of this paper, mixed strategies are only used to remove strategies (from the DM’s consideration) that are strictly dominated. Given the choice between one strategy and a mixed strategy that dominates the former the DM is not indifferent between the two and in fact has a strict incentive to use the mixed strategy, i.e., to randomize. For further discussions of randomization under Knightian uncertainty see Klibanoff (2001), Saito (2015), and Eichberger et al. (2016). 4.2. On randomizing and committing to random acts The role of the DM’s ability to randomize and her ability to commit to the outcome of this randomization is best discussed by means of a single decision problem based on the main example of this paper.16
14
See also Reny and Robson (2004) for a “unification of the classical and Bayesian views”. The debate, for instance, whether one should use maxmin strategies or Nash equilibrium strategies in non-zero sum games (that are in some sense close to zero-sum games) by Aumann and Maschler (1972) is not about whether one should randomize, but how one should do it. The point is that both the Nash equilibrium and maxmin strategies, although different, are both undominated and in fact good choices under some belief about the opponent’s play. In contrast, a non-SEU choice for a “Waldian” is a choice that is strictly dominated by a (possibly random) SEU choice. We cannot rank maxmin and Nash equilibrium strategies in the Aumann and Maschler (1972) example in such a clear way. 16 See also Machina (1989, p. 1643) for a discussion of the role of commitment to a random choice. As a concrete example, he discusses the problem a mother faces when considering to flip a coin to choose which one of her two children is allowed an indivisible treat. 15
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A single decision problem for the given example. Consider the case where the DM is asked to choose one of the three acts f 0 , f 1 , f 2 . This decision problem can be formulated in terms of Anscombe and Aumann’s (1963) framework in multiple ways. As we are only drawing a ball once, we can reduce the state space to just two elements, i.e., = { R , W }. We do not care about the exact composition of the urn. All we need to know is the color of the ball that is drawn from the urn. We then have the following Anscombe and Aumann (1963) representation of the problem, where the three acts are described as vectors of probabilities of winning the prize. State
f0
f1
f2
R
49 100 49 100
1
0
0
1
W
In this example, the set of undominated (and, hence, Bayesian) random acts is given by ({ f 1 , f 2 }), the set of all probability distributions over f 1 and f 2 . While act f 0 is the optimal choice for an ambiguity averse DM who cannot or does not contemplate to actively randomize her choice, for a DM who can actively randomize, act f 0 is dominated by the random act that places probability 12 on each of f 1 and f 2 , as already pointed out in Raiffa (1961). A pessimistic Waldian, who uses a minmax strategy, would choose exactly that equal mixture of the two acts f 1 and f 2 . Turning to the question of commitment, now suppose that this DM is ambiguity averse but can actively and objectively randomize. Suppose that this DM understands that the random choice that assigns probability one half on f 1 and f 2 each, results in an expected probability of winning the prize of one half in each state. Suppose that the DM therefore actually prefers this random act over the act f 0 . Suppose the randomization is executed and the realized choice is act f 1 . The DM now has a commitment problem. When comparing the two acts f 1 and f 0 she prefers f 0 . She would then like to switch to choose f 0 . Contemplating the choice f 0 she would then prefer to randomize again. When she observes the outcome of the randomization she would again prefer to opt for f 0 again, and then again to randomize again. It is, thus, not clear what choice such a DM would make if she does not have the ability to commit to a random choice. 4.3. On global plans The role of the “big picture” assumption, i.e., that the DM forms a global plan and evaluates this globally, is also best discussed by means of examples. These also help to illustrate the role that the experimental design plays for the DM’s problem. A randomly chosen “active” decision problem. Suppose the DM is asked to choose an act in each of the two decision problems { f 0 , f 1 } and { f 0 , f 2 }. Then, each of the two decision problems has an equal probability of 12 to be “active”, meaning the DM is paid according to her choice for the active decision problem only. Note that the DM has now four “pure strategies”: choose f 0 or f 1 for the first decision problem and f 0 or f 2 in the second. Denote the four pure strategies by ( f 0 , f 0 ), ( f 0 , f 2 ), ( f 1 , f 0 ), and ( f 1 , f 2 ). Each of these pure strategies is then equivalent to a random act. Strategy ( f 0 , f 0 ) is equivalent to act f 0 , ( f 0 , f 2 ) is equivalent to the random act that places equal probability 12 on acts f 0 and f 2 , ( f 1 , f 0 ) is equivalent to the random act that places equal probability 12 on acts
f 0 and f 1 , and finally ( f 1 , f 2 ) is equivalent to the random act which places equal probability 12 on acts f 1 and f 2 . The experimental design is then given by (G , Z , t ) with G the set of acts that are equivalent to the four given random acts, the set Z is given by Z = Y 1 = Y 2 = {0, 1} and t is given in the following table: State
t( f0, f0)
t( f0, f2)
t( f1, f0)
t( f1 , f2)
R
49 100 49 100
49 200 149 200
149 200 149 200
1 2 1 2
W
The set of undominated mixed strategies is then given by all probability distributions over ( f 0 , f 2 ) and ( f 1 , f 2 ) and all probability distributions over ( f 1 , f 0 ) and ( f 1 , f 2 ). All proper mixtures of ( f 0 , f 2 ) and ( f 1 , f 0 ) are dominated and so are any mixtures that include ( f 0 , f 0 ). A pessimistic Waldian, using a minmax strategy, would choose ( f 1 , f 2 ). A DM who does not form a global plan, but instead considers each decision problem in isolation, could choose f 0 in both problems. Tickets. Suppose the DM is again asked to choose an act in each of the two decision problems { f 0 , f 1 } and { f 0 , f 2 }. The DM’s choices are now “tickets” that both pay out (based on one draw of the ball). In this case the set of final outcomes is 49 Z = {0, 1, 2}, as the DM can get zero, one, or two (identical) prizes. Let p = 100 . The DM has again four pure strategies that again can be seen as acts through the translation function t as given here: R
W
Payoff
t( f0, f0)
t( f0, f2)
t( f1, f0)
t( f1, f2)
Payoff
t( f0, f0)
t( f0, f2)
t( f1 , f0)
t( f1, f2)
2 1 0
p2 2p (1 − p ) (1 − p )2
0 p 1− p
p 1− p 0
0 1 0
2 1 0
p2 2p (1 − p ) (1 − p )2
p 1− p 0
0 p 1− p
0 1 0
672
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Note that, as p <
1 , 2
pure strategy ( f 0 , f 0 ) is first order stochastically dominated in every state, and thus strictly dom-
inated, by the mixed strategy that assigns equal weight of 12 to the two pure strategies ( f 0 , f 2 ) and ( f 1 , f 0 ). No Waldian, pessimistic or not, would choose act f 0 in both problems. A DM who does not form a global plan, but instead considers each decision problem in isolation, could choose f 0 in both problems. 4.4. On Ellsberg behavior Given the main result, there are now at least two possible explanations of the behavior of real decision makers experimentally found by Ellsberg (1961) and many others, some of it reviewed in Camerer and Weber (1992). These experiments clearly demonstrate that many individuals do not behave in accordance with SEU theory. But as they are also clearly not Waldian, the explanation of their behavior becomes more difficult.17 It is not sufficient to say that they have preferences over the set of acts as given, for instance, by Gilboa and Schmeidler (1989), Klibanoff et al. (2005), and Maccheroni et al. (2006). One would have to add one of the following additional pieces to this explanation. Either 1) the decision maker does not contemplate choosing randomly, or 2) does not see the big picture (i.e., does not form a global plan for all her decision problems), or 3) does do both but does not evaluate random acts by applying the reduction of compound lotteries (or reversal of order). Note that this latter part of the explanation is very consistent with the empirical finding of Halevy (2007) that the individuals who appear ambiguity averse are those that do not perform the reduction of compound lotteries. Suppose that we accept that the empirically observed choices of real decision makers in Ellsberg (1961)-like experiments require these additional explanations. This then raises the question of whether the theoretically proposed models of behavior with ambiguity aversion are normatively very appealing, i.e., whether we can declare them models of “rational” decision making. If these individuals’ choices are not “rational” then these choices are subject to change if we teach these individuals Wald’s complete class theorem. 4.5. Alternative experimental designs There are at least three alternative experimental designs that I rule out. First, I do not allow surprising the DM, such as giving the DM one problem and then, without first telling her, giving her another problem. It has been shown (see Zambrano, 2005 and Shmaya and Yariv, 2016) that, in such cases, a DM can have beliefs about the possible sequence of decision problems given to her for each state, such that essentially all possible choices she can make can be rationalized by some such belief. Second, I rule out that the DM chooses sequentially and is paid out after each choice and before making her next one. The reason for this is that such a design would enable the DM to learn about the nature of uncertainty. It would be interesting to study such situations in the light of Wald’s complete class theorem. A third experimental design one might want to try is to ask the DM to make a choice in one environment (with say one ambiguous urn) and then another in another environment (with another ambiguous urn).18 The DM, however, is then free to have a belief that is any joint probability distribution over all state-tuples. After all, how are many urns with an “unknown” composition filled? Thus, this is nothing but a single experiment with a big state-space and the DM will choose as if she was an SEU maximizer. References Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability. Ann. Math. Stat. 34, 199–205. Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63 (5), 1161–1180. Aumann, R.J., Maschler, M., 1972. Some thoughts on the minimax principle. Manag. Sci. 18 (5), 54–63. Bade, S., 2015. 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17 There are, in fact, other interpretations possible, based on the idea that individuals may have a bigger state space in mind such as in e.g. Shmaya and Yariv (2016). 18 If the reader believes this can be done, the reader may want to consult Bade (2015) and Stoye (2015).
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