Statistical Decisions under Ambiguity: An ... - Semantic Scholar

Statistical Decisions under Ambiguity: An Axiomatic Analysis Jörg Stoye∗ Northwestern University February 5, 2005

Abstract Consider a decision maker who faces a number of possible models of the world. Every model generates objective probabilities, but no probabilities of models are given. Applications of this framework include statistical decision making with model uncertainty, e.g. due to concerns for misspecification, or Robust Bayesian decision analysis. I characterize a number of decision rules including Bayesianism, maximin expected utility, minimax expected regret, and the Hurwicz criterion. The main contributions are the unified axiomatization of many rules in a framework tailored to statistical decision making, an axiomatic system that relaxes transitivity as well as menu-independence of preferences, and the introduction of several new decision criteria. JEL classification codes: C44, D81. ∗I

am greatly indebted to Peter Klibanoff and Chuck Manski for their comments, suggestions, and corrections.

I am also grateful to Kim Border and Itai Sher for helpful discussions. Of course, all errors are mine. Financial support through the Robert Eisner Memorial Fellowship, Department of Economics, Northwestern University, as well as a Dissertation Year Fellowship, The Graduate School, Northwestern University, is gratefully acknowledged. Address: Jörg Stoye, Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208-2600, [email protected].

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1

Introduction

“[T]here are two types of uncertainty: one as to the hypothesis, which is expressed by saying that the hypothesis is known to belong to a certain class or model, and one as to the future events or observations given the hypothesis, which is expressed by a probability distribution.” (Arrow 1951, p. 418) This paper revisits statistical decision theory, that is, decision theory aimed at normatively guiding statistics- or econometrics-based decision making. Since the first wave of this literature in the 1950’s1 , there has been dramatic progress in decision theory, but many of these developments were in a form that is more directly relevant for economic theorists than for statisticians. The present paper differs from these treatments both in the framework used and in the type of results generated. As to the framework, I assume that a decision maker simultaneously faces nonprobabilistic ambiguity (about the true model) and probabilistic uncertainty (given the true model). This is modelled by an “Anscombe/Aumann” setup2 which is sometimes extended to also reveal the state space underlying roulette wheels. Within this environment, I characterize not only Bayesian and maximin utility decision rules, but also criteria that are defined with respect to regret, in particular minimax regret and a criterion I call “pairwise minimax regret,” as well as generalizations of all of these that correspond to the “Hurwicz criterion” generalization of maximin utility. To my knowledge, this constitutes the first characterization of minimax regret in an Anscombe/Aumannsetup, and pairwise minimax regret as well as the generalizations of minimax regret and pairwise minimax regret are essentially new to the literature. To capture the specific properties of these rankings, I relax some of the most standard axioms and allow preference orderings to be potentially intransitive as well as dependent on the choice set. A very quick summary of results goes as follows: • If expected utility is accepted as decision criterion absent ambiguity, then some rather weak axioms — short of transitivity and independence of irrelevant alternatives — imply that an act is characterized by its induced mapping from (ambiguous) states of the world to expected utilities 1 Contributions

that are directly relevant to this paper include the books by Luca and Raiffa (1957), Savage (1954),

and Wald (1950) as well as articles by Arrow (1951), Arrow and Hurwicz (1972, written in the 1950s) Chernoff (1954), Milnor (1954), and Savage (1951). 2 See Anscombe and Aumann (1963).

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or expected value. • Maximin utility has the feature of focusing on the objectively worst state of nature even if acts have infitesimal effect in that state but large effect in others. The intuitive argument for regretbased decision rules is that they avoid this feature. Indeed, an axiom designed to prevent it essentially enforces regret-based decision making. • Conditional on this axiom, an insistence on transitivity leads one to minimax regret. An insistence on menu-independence, in conjunction with as weak as possible a relaxation of transitivity, leads to pairwise minimax regret. Insisting on both features means that one is generally unable to rank admissible acts. The remainder of this paper is structured as follows. Section 2 is devoted to explaining the setup and notation and to further elaborating its relation to statistical decisions as well as differences to the existing literature. Section 3 contains the axiomatic treatment. Section 4 concludes, and the appendix collects all proofs.

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The Decision Theoretic Framework

2.1

Setup and Notation

Consider a decision maker who faces a number of possible models of the world. Every model gives rise to an objective probability distribution over a cardinally measurable outcome, but the models themselves do not have objective probabilities. Intuitively, the uncertainty about models is supposed to capture what is usually called “model uncertainty” and the uncertainty given a model to capture “estimation uncertainty.” Technically, in modern decision theoretic parlance, only the latter is uncertainty at all, whereas the true model is ambiguous. To model this, I need two state spaces, even though one of them will be invisible in most of the analysis. Thus, let the state of the world be (r, s) ∈ R × S, where R (as in roulette) captures probabilistic uncertainty and S captures ambiguity. Elements of R are realized according to a probability distribution Fr ∈ ∆R which is known and objective, or at least “realist” in the sense described below. No such information exists for S. There exists a set A of possible pure acts (e.g. treatments or policy choices) a, b, etc. Each combination of states (r, s) with an act a gives rise to an outcome v(a, r, s) ∈ R; 3

since Fr is known, acts can be thought of as mappings from S into ∆R.3 The decision maker can choose from a finite subset A ⊆ A. Furthermore, she has an objective randomization device at hand and may randomize over acts, thus her action space is really some menu M ∈ M ≡ {∆A : A ⊆ A, A finite}. To simplify matters, let S be finite; this is with loss of generality, but conveniently allows one to identify every act a with a finite vector of distributions (Fr (v(a, r, si )))i=1,...,#S . I will generally impose that absent ambiguity, von-Neumann-Morgenstern expected utility would apply, thus there exists U : R → R s.t. if S were a singleton, all acts would be ranked according to u(a, s) ≡ Er (U (v(a, r, s))). For most of the below analysis, the reader will want to think of s ∈ S as the significant source of uncertainty and of u(a, s) as summarizing the outcome of action a in state s. Finally, the decision maker has an objective randomization device at hand and may randomize over acts, so the action space is really ∆M . Mixtures between acts are written in the usual way, i.e. pa + (1 − p)b is the act generated by performing a with probability p and b otherwise. An act is called admissible if it is not dominated by another act with respect to u(a, s). It is potentially (uniquely) optimal if it (uniquely) maximizes u(a, s) for some s ∈ S. A potentially optimal act is always admissible, but not vice versa. Randomized acts are typically not potentially optimal; more formally, the class of pure acts is complete for potentially optimal acts. Finally, a potential bestresponse or Bayes act maximizes EΠ u(a, s) for some Π ∈ ∆S. This class is of less intrinsic interest here, but coincides with the admissible acts since the action space is a mixture space. There are several types of applications for this setup. In the most conventional one, the model is identified in the usual econometric sense, so that ambiguity about it will asymptotically vanish, and the question is essentially how to treat finite sample problems. Indeed, Arrow used the aforecited words to describe the decision problem that Neyman and Pearson purported to solve. Regret-based decision criteria have been used occasionally in such situations. For example, Droge (1998) uses minimax regret to choose between “selection” and “shrinkage” estimators, DasGupta and Studden (1991) use it in regression design, and Manski (2004a, 2004c) as well as Stoye (2004) apply it to treatment choice in identified models but with finite samples. In other applications, the true model is principally unknowable. The reason for this could be pragmatic, namely a sharp upper limit on sample size as encountered in macroeconomic policy planning. In this context, proper consideration of model uncertainty has recently gained some prominence. Exam3 The

ordered field structure of R is strictly needed only when an axiom called EO-independence is invoked, but is

also helpful in motivating EP-independence. For other results, v could also map into some outcome space V.

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ples include Hansen and Sargent’s research project on robust control (see in particular Hansen, Sargent, Turmuhambetova, and Williams 2002), Brock, Durlauf, and West’s (2003) work on policy choices under model uncertainty, or the literature on Bayesian model averaging (e.g., Sala-i-Martin, Doppelhofer, and Miller 2004). All of these consider maximin-type approaches at least tangentially, and Sargent and Hansen do so prominently. Also related is the technique of “scenario planning” in Operations Research and business economics. See the textbook by Kouvelis and Yu (1997) for explorations of minimax regret criteria in this context and Loulou and Kanudia (1999) for an application. The true model will also be unknowable if decision-relevant variables are incompletely observable. One example of this is the literature on partial identification as surveyed by Manski (2003).4 The problem here is that due to set-valued identification of parameters, a set of actions will remain admissible even as samples grow large. In normative analyses, Manski (2000, 2002) refrains from committing to any tighter solution concept than admissibility, but this typically leaves one with a large set of acts whose choice could be considered reasonable. In a descriptive analysis (2004b), he tentatively uses the Hurwicz criterion but without any claim about its normative merits, and in current research (2004c), he normatively applies minimax regret. A very similar problem is encountered in Robust Bayesian Analysis, where the situation given any prior is one of uncertainty, but a nonsingleton set of admissible priors introduces ambiguity and leads to a set of Bayes acts.5 Zen and DasGupta (1993) write that the literature “does not as yet contain substantial work on how exactly a specific action should be chosen,” and Arias, Hernández, Martín, and Suárez (2003) state that “the solution concept is the set of non-dominated alternatives,” suggesting demand for the present analysis. Chamberlain (2000) explores the use of minimax regret in such a setting, exploring the maximal regret caused by different priors but not attempting to identify minimax regret priors or to generally justify minimax regret. 4 For

example, consider an estimation problem, i.e. s is a true parameter, any possible act is a guess sb of s, and

the goal is to minimize the expectation of a loss function L(s, sb) ≥ 0 with L(s, s) = 0. Then in a situation of partial

identification, the set of potentially optimal acts is the identification region for s. See Horowitz, Manski, Ponomareva, and Stoye (2003) for an application. 5 The situation maps onto the one considered here by equating S not with possible states of the physical world, but with admissible priors.

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2.2

The Contenders

I am going to consider a collection of decision rules. The following two constitute benchmarks in that they only rank options whose comparison is uncontroversial. Definition 1 Weak Statewise Dominance (WSD) a ÂW SD b ⇐⇒

u(a, s) ≥ u(b, s), ∀s ∈ S u(a, s) > u(b, s), ∃s ∈ S

a ∼W SD b ⇐⇒

a ¨W SD b ∧ b ¨W SD a

Definition 2 Strict Statewise Dominance (SSD) a ÂSSD b ⇐⇒

u(a, s) > u(b, s), ∀s ∈ S

a ∼SSD b ⇐⇒

a ¨SSD b ∧ b ¨SSD a

WSD only excludes weakly dominated and SSD only strictly dominated acts; they prescribe indifference over the set of admissible acts. Give or take this set’s boundary, this is just how far one can go without committing to some attitude about ambiguity. A polar attitude is given by Bayesianism: Definition 3 Bayesianism There exists Π ∈ ∆S (the “prior”) s.t. a º b ⇐⇒ EΠ u(a, s) ≥ EΠ u(b, s). The focus of this paper is on criteria that avoid probabilistic treatment of ambiguity and are rather concerned with uniformity (in some sense) of performance across states. The best-known among them is the following: Definition 4 Maximin Utility (MU) a ºMU b ⇐⇒ min u(a, s) ≥ min u(b, s). s∈S

s∈S

Maximin utility was introduced to statistical decision theory by Wald (1950) and received a famous philosophical endorsement in Rawls’ “Theory of Justice” (1999). Its criticisms, which must date back almost as far, are usually centered around examples like Rawls’ own (p. 136):

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Rawls’ Example Against Maximin Utility

Let there be two states of the world α and β and

acts a and b inducing u(a, s) as follows (for some n > 1): α

β

a

0

n

b

1/n

1

In this example, maximin utility always picks action b. As no other than Rawls points out, this becomes egregiously implausible as n → ∞. It is eventually absurd to focus attention exclusively on the worst possible state of nature even if one has essentially no influence on things in that state yet a lot of influence in some other state. It has therefore been suggested to focus attention on the state of nature in which a decision is potentially most consequential. The idea, first formalized as Savage’s (1951) reading of Wald (1950), is to minimize maximum regret, where regret is defined as the loss incurred by not having chosen the ex post optimal act. Definition 5 Minimax Regret (MR) with Respect to M a ºMR(M) b ⇐⇒

max

s∈S,a∗ ∈M

{u(a∗ , s) − u(a, s)} ≤

max {u(b∗ , s) − u(b, s)}

s∈S,b∗ ∈M

The advantages of minimax regret over maximin utility are so clear that many authors consider it the obvious “real” maximin utility (see again Savage (1951), but also Berger (1985) or recently Manski (2004a)). However, it has the perhaps unpleasant feature that the preference between a and b depends on the menu from which the two can be chosen. In particular, it can happen that a is chosen from the menu {a, b}, yet b is chosen from {a, b, c}.6 This is avoided by the following criterion. Definition 6 Pairwise Minimax Regret (PMR) a ºP MR b ⇐⇒ max{u(b, s) − u(a, s)} ≤ max{u(a, s) − u(b, s)}, s∈S

s∈S

or equivalently, a ºP MR b ⇔ a ºMR({a,b}) b. 6 For

a striking example, assume there are three states and consider the actions a ≡ (1, 2, 3) and b ≡ (3, 4, 2). It is

easily verified that b ºM R({a,b}) a, that the MR-action is to choose b with probability 2/3 and that the pure action a is the least preferred choice in ∆({a, b}). If one adds action c ≡ (−10, −10, 5) to the picture, then the ranking is a ºM R({a,b,c}) b ºMR({a,b,c}) c and the MR-action is to choose a with certainty, i.e. an action that was feasible yet worst absent c. Arrow (1951), Chernoff (1954), and Milnor (1954) provide further examples.

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Pairwise minimax regret is related to Loomes and Sugden’s (1982) “regret theory” of decision under uncertainty and its generalization by Fishburn respectively Fishburn and LaValle (1988 and references therein). It has in common with them that it is defined by direct comparison of any two options and that it is intransitive. This latter feature led to its immediate rejection by Luce and Raiffa (1957), who mention it briefly after discussing minimax regret; to my knowledge, this is the criterion’s only appearance in the literature so far. The following is a well-known generalization of maximin utility. Definition 7 α-Maximin Utility (α-MU, “Hurwicz Criterion”) There exists α ∈ [0, 1] s.t. a ºα−MU b ⇐⇒ α max u(a, s) + (1 − α) min u(a, s) ≥ α max u(b, s) + (1 − α) min u(b, s). s∈S

s∈S

s∈S

s∈S

As mentioned above, the Hurwicz criterion has already been applied in related settings. Its classic axiomatization is due to Arrow and Hurwicz (1972); Olszewski (2003), in an interesting paper whose motivation is quite similar to mine, axiomatizes a generalization of it. Both MR and PMR can be generalized in similar ways, generating criteria that are new to the literature. I will not give these criteria quite as much attention as MR and PMR, but apart from being natural analogs to the Hurwicz criterion, they will turn out to occupy prominent positions within the axiomatic structure. Definition 8 α-Minimax Regret (α-MR) with Respect to M There exists α ∈ [0, 1] s.t. a ºα−MR(M) b ⇐⇒ α min{ max u(a∗ , s) − u(a, s)} + (1 − α) max{ max u(a∗ , s) − u(a, s)} ≤ ∗ ∗ s∈S a ∈M

s∈S

a ∈M

∗

α min{ max u(b , s) − u(a, s)} + (1 − α) max{ max u(b∗ , s) − u(a, s)}. ∗ ∗ s∈S b ∈M

s∈S

b ∈M

Definition 9 Pairwise α-Minimax Regret (α-PMR) There exists α ∈ [0, 1] s.t. a Âα−P MR b ⇐⇒ (1 − α) max{u(b, s) − u(a, s)} < max{u(a, s) − u(b, s)} s∈S

s∈S

a ∼α−P MR b ⇐⇒ a ¨SSD b ∧ b ¨SSD a. 8

Observe that in the definition of α-MR, the max and min operators have been swapped as compared to MU. This, as well as the choice of (1 − α) as multiplier in the definition of α-PMR, insure analogous nesting relations: MU[MR,PMR] is α-MU[MR,PMR] with α = 0. To get an intuition for these criteria, it is useful to understand their limiting behavior. α-MR with α = 1 assigns a value of zero to all potentially optimal acts and negative value to all others, thus its choice correspondence is the set of potentially optimal acts. As α → 1, α-MR converges to picking the maximin utility act among potentially optimal ones — there is a discontinuity at the limit, intuitively because the limiting ordering is lexicographic. Pairwise α-minimax regret smoothly converges to SSD, thus its choice correspondence converges to the set of weakly admissible acts. To get a first impression of the behavior of these decision criteria, reconsider the above example. Recall that the choice set induced by a particular ordering is defined as C(M ) ≡ {a ∈ M : b ∈ M ⇒ a ºM b}, thus it contains the “best elements,” that is the recommended choice, from M . Proposition 1 Rawls’ Example Solved In Rawls’ example, the above decision criteria induce the below choice sets. These are parameterized by π, the set of optimal probabilities of picking a. πMU πMR

π P MR

π α−MU

= {0} ½ ¾ n2 − n = n2 − n + 1 ⎧ √ ⎪ ⎪ {0}, n < 1+2 5 ⎪ ⎨ √ = [0, 1], n = 1+2 5 ⎪ √ ⎪ ⎪ ⎩ {1}, n > 1+ 5 2 ⎧ q 1 1 ⎪ {0}, n < + − ⎪ ⎪ 2 ⎨ qα = [0, 1] n = 12 + α1 − ⎪ q ⎪ ⎪ ⎩ {1}, n > 1 + 1 − 2 α

3 4 3 4 3 4

In particular, PMR induces a transitive ranking.

The simple example picked from Rawls’ book generates a diverse range of recommendations, illustrating that the criteria are meaningfully different. It also illustrates a generic feature of minimax regret, namely that the optimal rule is randomized. In contrast, PMR does generically not randomize,

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whilst the nonrandomized nature of the MU- and α-MU-solutions is specific to the example. All criteria except MU exhibit reasonable behavior as n grows large. Under MR, the probability of choosing a quickly approaches 1 as n → ∞; PMR, the Hurwicz criterion, and the other α-generalizations (not displayed above) prescribe to switch from action b to a as n crosses some threshold.

2.3

Comparison to Related Literature

In this section, I contrast my results with previous contributions, in particular the axiomatic characterizations of MU by Gilboa and Schmeidler (1989) as well as Casadesus-Masanell, Klibanoff and Ozdenoren (2000). My treatment obviously differs by also characterizing several other rules, but I wish to stress a more subtle aspect, namely that the representations generated here are of fundamentally different type. An MU representation result states that acts are ranked according to a utility function U , where Z Z U (a) ≡ min (1) u(v)dFr (v(a, r, s))dΠ(s). Π∈B

Here, u has an intuitive interpretation as “utility of v,” Fr as “probability distribution over v induced by act a in state s,” Π as “a prior over states s,” and B as “set of decision-relevant priors.” However, any one of these objects can relate to its intuitive counterpart in two ways. The object could be “realistic,” by which I mean that its intuitive analog is supposed to exist and is indeed meant to be represented. Alternatively, it could be “as if,” by which I mean that its mathematical existence is implied by some behavioral axioms, but it need not map onto anything in the real world, including our introspection and agents’ beliefs. To understand the difference, recall that both von Neumann/Morgenstern (1944) and Savage (1951) axiomatize expected utility representations, but with von Neumann and Morgenstern, the probability measure is realistic — in particular, it maps onto objective probabilities —, whereas with Savage, it is “as if.”7 Characterizations of MU can be similarly compared. For example, GS work within an Anscombe/Aumann setup, thus outcomes have objective probabilities contingent on s and a, and the object F in their representation is indeed identified with these. In contrast, B (and by implication Π) is “as if.” GS 7I

deliberately avoid the nomers “objective” and “subjective.” Whilst it is true that the von-Neumann/Morgenstern

probabilities are objective and Bayesian probabilities are subjective, it is not true that the “Savage” probabilities necessarily map onto a subjective probability measure in the mind of the decision maker. On the contrary, Savage’s result is often read as an “as if”-characterization that makes no claims about the existence of subjective probabilities as representers of epistemic attitudes.

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never claim that a decision maker’s epistemic attitudes can be represented by some set of priors B ∗ . And even if such a representation existed, it would not follow that B = B ∗ because B confounds the perception of ambiguity and attitude towards it. For example, B ⊂ B ∗ if the decision maker finds some states of the world conceivable but would feel overly pessimistic in considering them. In contrast, I will assume not only that F is given, but also that there is an objectively possible set of states of the world S, and the representation will identify B with the set of probability distributions concentrated at exactly one of these states. Since that set can be identified with S, [1] really simplifies to U (a) ≡ min s∈S

Z

u(v)dFr (v(a, r, s)),

(2)

which is the representation I will use henceforth. If, as in some cases below, u(v) is identified with v itself, then [2] is completely pinned down by the structure of the situation.8 This obviously comes at a cost because it requires more structure as well as axioms. The benefit is that when statistical decision makers use MU in practice, they typically use criterion [2]. To cite GS as foundation for such applications entails a significant amount of interpretation or implicit assumption, whereas my axioms, if believable, provide an exact foundation for [2]. The axiomatizations of minimax regret and pairwise minimax regret are understood in analogy to this, i.e. U is given by U (a) ≡ −

max

a∗ ∈M,s∈S

µZ

∗

u(v)dFr (v(a , r, s)) −

Z

¶ u(v)dFr (v(a, r, s))

(3)

for MR (no parametric representation exists for PMR, but the idea is similar). This is again the expression actually used by practicioners, e.g. by Loulou and Kanudia (1999), Manski (2004a, 2004c), and throughout Kouvelis and Yu (1997). But it also points at further research topics since it leaves open a characterization of these decision rules in the spirit of GS or CKO, more on which in the conclusion. The idea of taking “sets of priors” to be realist has recently surfaced in related work. Gajdos, Tallon, and Vergnaud (2004) take as given a set of priors that contains a reference prior and then axiomatize MU with respect to an “as if”-object B that must however be a subset of S. Hayashi (2003) takes only sets of priors as given and axiomatizes maximin utility with respect to an object B centered on an endogenously identified reference prior. Ahn (2003) as well as Olszewski (2003) take the 8 The

reduction from [1] to [2] is orthogonal to the “as if”-issue. The CKO representation achieves [2] but with F as

well as S being “as if.”

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sets of objective lotteries induced by acts as the primitive objects of analysis, thus avoiding state spaces altogether; although the object B does not explicitly appear in their approach, a realist interpretation is implied.9

3

An Axiomatic Treatment

3.1

Linking Uncertainty and Ambiguity

The list of decision rules to be axiomatized is somewhat long, and consequently, there will be no less than 17 axioms. They will be grouped into clusters and arranged so that interesting results can be shown early on. As some decision criteria are menu-dependent, all axioms will be stated with respect to a menu-subscripted preference ordering ºM . Such subscripts will not any more be used to identify the orderings themselves. I will take weak preference as primitive and ÂM as well as ∼M as derived in the usual way. The most standard axiom is that there exists a nontrivial ordering to begin with. To admit intransitive preferences, it must be decomposed. To begin, I will maintain the following assumption: Axiom 1 Nontriviality ∃a, b, c, M : a ÂM b ÂM c. This axiom is slightly stronger than usual because it imposes the existence of c, but this strengthening is only needed at one point, which will be highlighted. Two basic restrictions of more substantive interest are the following. Axiom 2 Completeness a ºM b ∨ b ºM a. Axiom 3 Transitivity a ºM b ºM c ⇒ a ºM c. Here and henceforth, quantifiers — e.g. “∀a, b ∈ M, M ⊆ 2A ” for axiom 2 and “∀a, b, c ∈ M, M ⊆

2A ” for axiom 3 — are made explicit only where they are not obvious. 9 This

approach does not permit a discussion of regret-based criteria because these use information about states.

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Since transitivity will be relaxed, it deserves some discussion. I take for granted that other things equal, transitivity would be desirable. However, it here conflicts with other axioms for which one can claim the same; thus a closer look is required. On such reconsideration, transitivity is overwhelmingly compelling only if one perceives the statement “a º b” as comparing an intrinsic, one-dimensional property (“goodness”) of a and b, a perspective called the “Maximization Thesis” by Schwartz (1972). If the statement “a º b” is seen to describe a property fundamentally of the pair {a, b}, then transitivity

is not obvious and will be generically violated, whether by PMR or by its aforementioned relatives.10

Notice, however, that this argument can be taken further because in axiom 3, a’s and b’s “goodness” may depend on M , making it not so intrinsic after all. In other words, if one really adopts the perspective just outlined, then one should should also be inclinced to embrace Independence of Irrelevant Alternatives (IIA) — thus MR, which fulfils axiom 3 but not IIA, arguably is transitive only to a limited degree.11 The remainder of this subsection is devoted to establishing a link between the evaluation of uncertain but unambiguous acts and of ambiguous ones. A maintained assumption will be that if s were known, acts would be evaluated by expected utility. Consequently, the maximin-type decision criteria will refer to (some transformation of) this expectation.12 Axiom 4 Monotonicity There exists a utility function U : R → R inducing expectation u(a, s) ≡ Er (U (v(a, r, s))) such that u(a, s) ≥ u(b, s), ∀s ∈ S =⇒ a ºM b. This axiom really says two things: Firstly, unambiguous acts, i.e. acts for whom u(a, s) does not depend on s, are ranked according to expected utility. Secondly, the ordering of ambiguous acts is monotonic in the obvious way. To make treatment of ambiguous acts tractable, it would be convenient if the mapping from s to u(a, s) were a sufficient description of a. That this is the case might appear obvious: if u(a, s) = u(b, s) 1 0 For

in-depth discussions, see Fishburn (1987) and Fishburn and LaValle (1988), who show that intransitivity is a

generic feature of context-dependent preferences and go on to defend it normatively. Compare also Sugden’s (1985) defense of similar properties of his regret model and Fishburn’s (1991) survey. 1 1 However, Loomes and Sugden (1992) maintain that this specific weakening of transitivity is consistent with the psychological motivation of regret models. 1 2 Manski (2004a) points out that conceptually, one could also think of minimax regret with respect to the Median (say). This would raise technical problems here because the fact that the expectation is a linear operator is used repeatedly.

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for all s, then monotonicity implies a ∼M b. Surely this means that a and b will compare identically to any third option c? Yes — in the presence of IIA and transitivity. But these are not always assumed here, so things are somewhat more complicated. However, the desired reduction is implied by a set of weaker axioms which are fulfilled by all orderings. One of them is a variation on monotonicity. To formalize it, let aS b denote the act that performs a in any s ∈ S ⊆ S and b otherwise and furthermore let as b ≡ a{s} b. Axiom 5 Transitive Extension of Monotonicity For any s ∈ S, if u(a, s) ≥ u(b, s) and u(c, s) ≥ u(d, s), then bs e ºM cs f =⇒ as e ºM ds f, ∀M ⊇ {bs e, cs f, as e, ds f } For brevity, transitive extension of monotonicity will also be called transitive monotonicity below. It states that if conditional on s, a is better than b, then “upgrading” an act e from b to a for that state s cannot undo a preference for e over f , and similarly f cannot become better relative to e by “downgrading” it. To understand the subtleties of the setup, notice that strictly speaking, transitive monotonicity is not a strengthening of monotonicity but independent of it. Specifically, the following statement holds. Lemma 1 Axiom 4 follows from axiom 5 only in the presence of completeness as well as axiom INA below, and the reverse implication holds only given transitivity.

However, since completeness and INA are imposed throughout but transitivity is not, little harm is done by henceforth thinking of axiom 5 as a strengthening of axiom 4. I should therefore clarify why I find the transitive extension of monotonicity compelling even where I am willing to sacrifice the full force of transitivity. The idea is that transitivity is relaxed to allow for context sensitivity of rankings, but this sensitivity should not apply in cases of statewise dominance as between bs e and as e or cs f and ds f . Therefore, such orderings should, to some limited degree, transcend contexts. More loosely speaking: which aspect of the comparison between e and f could possibly be invalidated by unambiguously upgrading e? Consider now the issue of menu dependence. To avoid it altogether, one has to impose the following axiom.

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Axiom 6 Independence of Irrelevant Alternatives (IIA) a ºM b ⇐⇒ a ºN b. This can be weakened as follows: Axiom 7 Independence of Never-Optimal Alternatives (INA) Let c be not strictly potentially optimal in M . Then a ºM b ⇐⇒ a ºM∪{c} b. INA has been previously identified as the strongest IIA-like condition fulfilled by minimax regret.13 At the same time, it turns out to be just sufficient for theorem 1 below. As with transitivity, the arguments in favor of IIA are well known. Why would one want to argue against it? Sen (1993) cites the phenomena of positional choice (not wanting to take the largest slice of cake), choosing something mainly to display rejection of something else (as in fasting vs. starving), and situations where the menu has epistemic value (as when items on a restaurant’s menu signal quality). The first two clearly fail to apply here. The last one is relevant in some situations where minimaxtype criteria are employed. For example, Borodin and El-Yaniv (1998) use it to argue for minimax regret (in the guise of “competitive ratio”) in computer science, where the arrival of a new algorithm that performs well for certain request sequences is informative about the difficulty of a problem. This intuition neatly supports the weakening of IIA to INA since under INA, the addition of an act to a menu can reverse rankings between other options only if it affects the utility frontier. On the other hand, if it were possible to infer from the menu to properties or likelihoods of states, then this should ideally be modelled explicitly and not informally via twists on some axiom. The core result of this subsection is the following. Theorem 1 Reduced-Form Description of Ambiguous Acts Assume axioms 5 and 7 hold and that u(a, s) = u(a, s), ∀s ∈ S as well as u(b, s) = u(b, s), ∀s ∈ S. Then a ºM ∪{a,b} b ⇐⇒ a ºM∪{a,b} b, ∀M. 1 3 Compare

Borodin and El-Yaniv (1998), who label it “Independence of Dominated Alternatives.” I avoid this because

c need not be dominated by any feasible option but only by the ex-post utility frontier. Also, Milnor (1954) calls the same axiom “special row adjunction.”

15

3.2

Characterizing Transitive Rankings

Theorem 1 simplifies the situation quite a bit because for any combination of axioms considered here, the desirability of an act is fully determined by its induced mapping from states of the world s to expected outcomes u(a, s). Since S is finite, this mapping is furthermore a vector, and any act a can be identified with (u(a, si ))i=1,2,...,#S ∈ R#S , where (si )i=1,2,...,#S is a pre-assigned ordering of

S, and a menu M ≡ ∆A can be characterized by a matrix whose rows correspond to elements of A. To keep statements of axioms simple, this notation will be used with immediate effect. For example, monotonicity now reads a = b ⇒ a ºM b. The numbering of states is arbitrary since S was not ordered to begin with, but the notation is not supposed to erase the identity of states, i.e. the decision maker knows which components of the vectors refer to which state. I begin by stating independence, even though many decision criteria considered here violate it. In the presence of IIA, independence is generally defined as a º b ⇔ pa + (1 − p)c º pb + (1 − p)c for all {a, b, c} and p ∈ (0, 1). Without IIA, this needs to be adapted somehow. I suggest the following axiom: Axiom 8 Independence a ºM b ⇐⇒ pa + (1 − p)c ºpM+(1−p)c pb + (1 − p)c. To understand the idea behind this adaptation, recall that independence is frequently motivated by the following thought experiment. Imagine you prefer a over b, but then you are told that your choice will be actualized only if heads occur in a previous toin coss; otherwise, c will occur whatever your intentions. Then it can be argued that this information should not reverse your preferences, hence these should obey independence. But in the thought experiment, c would be mixed into all your options, so absent IIA, the argument really supports the above variation of independence. I will also consider the following weakening of independence. Axiom 9 C-Independence Let c be unambiguous, i.e. all its components are equal, then a ºM b ⇐⇒ pa + (1 − p)c ºpM+(1−p)c pb + (1 − p)c. C-independence requires that the ranking of acts is not affected by mixing the entire menu with some unambiguous act; this caveat is absent in GS but needed here because IIA is not always assumed. 16

The intuition of C-independence is that violations of independence should be due only to ambiguity and not uncertainty, i.e. they occur when mixing with c constitutes a hedging of bets across states with respect to a but not b. This effect cannot occur when c itself is unambiguous. C-independence is fulfilled by all criteria here, even though I can occasionally avoid imposing it. Now consider Rawls’ example against maximin utility or also the examples presented by Berger (1985). They are ruled out by the following axiom. Axiom 10 External Payoff Independence (EP-Independence) a ºM b ⇐⇒ a + u0 ºM+u0 b + u0 , ∀uo ∈ R#S EP-independence requires that the ranking of options is not affected by an additional payoff that depends on s but not a nor b and is present across the menu. The idea is that a decision maker should be concerned more with the impact of her decisions in different states than with the quality of these states per se, which I believe is the reason why maximin utility appears implausible in the aforementioned examples. EP-independence is also extremely useful in practice because it permits localized decision making. For example, an individual violating it would have to integrate her decision over overall wealth if this wealth varied with states. Worse still, a policy-maker who considers treatment recipients’ wealth the ultimate carrier of utility, who knows that recipients’ identities potentially differ between states, but who does not know recipients’ initial wealth, faces a value function with statewise arbitrary origin. For another take on this, observe that revelation of s can have implications outside the decision problem at hand. If an identification problem is caused by selection into college, knowledge of the true s would likely be informative about selection into private high schools and the like. Absent EPindependence, one would then have to integrate a decision over the entire education system, even if the policies compared only affect college schooling. Although in a first-best world, one would want to perform general equilibrium evaluations anyways, a decision criterion that is not even defined for partial equilibrium analysis would encounter obvious limitations. However, both the normative intuition and the practical points may not really suggest EP-independence but rather a robustness of judgment with respect to external outcomes along the following lines:

17

External Outcome Independence (EO-Independence) Define a⊕b as the act that results from simultaneous consumption of acts a and b, i.e. v(a ⊕ b, r, s) ≡ v(a, r, s) + v(b, r, s), then a ºM b ⇐⇒ a ⊕ c ºM⊕c b ⊕ c. A core argument for EO- over EP-independence is that the latter essentially presumes a substantive interpretation of differences in expected utility. This feature is inherited by EP-independent orderings, i.e. the regret-based orderings in this paper. However, it is of reduced importance here because of the following observations. Lemma 2 Assume that axiom 4 holds. Then: (i) EO-Independence implies that the decision criterion for uncertainty is expected value, i.e. preferences can be represented using U (v(a, r, s)) = v(a, r, s). (ii) If U (v(a, r, s)) = v(a, r, s) is imposed, EO- and EP-independence are equivalent.

In the context of this paper, EO-independence therefore equals “EP-independence plus expected value.” For a clear exposition of causal connections and to keep avoiding the state space representation of uncertainty, it is helpful to separate these issues, and I will therefore work with EP-independence. As a value judgment, I do however feel that EO-independence is more compelling and hence that the restrictions of regret-based orderings to expected value are somewhat privileged. For a full understanding of the relation between axioms, the following observation is crucial. Lemma 3 Axioms 9 (C-independence) and 10 (EP-independence) jointly imply independence.

I now turn to two axioms that squarely contradict independence; consequently, they play a core role in enforcing a nonprobabilistic treatment of ambiguity. Axiom 11 Reduction Let s, s0 ∈ S be s.t. u(a, s) = u(a, s0 ), ∀a ∈ M. Let M 0 be the decision problem generated from M

by dropping state s0 from consideration, where every a ∈ M maps onto a0 ∈ M 0 . Then a ºM b ⇐⇒ a0 ºM 0 b0 .

18

If in the reduced form, two states have identical consequences for any action, then reduction imposes that one of them can be ignored. It would fail if being connected to several states would make a certain pattern of outcomes more worthy of consideration, as is the case with non-null states in a Bayesian treatment. Axiom 12 Anonymity For any vector a ∈ Rk , let [a]ij be the permutation of a that exchanges positions i and j. Then a ºM b ⇐⇒ [a]ij º[M]ij [b]ij . This axiom states that if the consequences of two states are exchanged throughout the menu, then the ranking is not affected. To understand that it is nontrivial, remember that states are supposed to have identities, e.g. they might reflect different assumptions about unobservables. These identities are not supposed to be exchanged too. Thus anonymity effectively says that there is no pair of states such that one is considered intrinsically more important (read: probable) than the other. Neither of these restrictions is very plausible if one has available, and wishes to consider, sharp prior information about states. This is not surprising because a Bayesian analysis would then seem appropriate. On the other hand, if truly no prior information about states exists and one does not want to impose arbitrary beliefs, then both axioms make a lot of sense. In particular, if either one were violated, a decision criterion would be sensitive to arbitrary manipulations of state space, either by relabeling states (if anonymity fails) or by duplicating some via conditioning on trivial events (if reduction fails).14 The case appears more difficult when there exists vague prior information, not enough to commit to a prior yet sufficient to reject, for example, the idea that one state is superfluous in a problem’s description. In such cases, I believe that the decision problem should be respecified to reflect this prior knowledge. For example, if one is willing to commit to a set of priors but not to choose among them, i.e. a Robust Bayesian approach, then every such prior should be identified with a state. Or if it is believed that the selection process generating a problem of partial identification has a certain property, then any processes contradicting this should be ignored — this is the idea behind “partially identifying assumptions” in Manski (2003). In either case, it appears that after properly considering 1 4 These

ideas really originate with Arrow and Hurwicz (1972), who axiomatize a class of choice rules by restricting its

“comparative statics” across state spaces in the spirit of these axioms. Milnor (1954) adapted them to axiomatization of preferences.

19

prior information in the problem’s description, one is back to the case of no prior information, and the above two axioms should be desirable. The final two restrictions are well known. Axiom 13 Ambiguity Aversion a ∼M b =⇒ pa + (1 − p)b ºM b Under abiguity aversion, a randomization between two equally good treatments must be weakly preferred to either of them, intuitively because it constitutes a hedging of bets across states. This particular way of modeling ambiguity aversion via quasiconcavity has been introduced by Schmeidler (1989), used by GS, and adapted by CKO. Compare also GS for a more elaborate defense and CKO for an in-depth discussion of its relation to other conceptions of ambiguity aversion. Axiom 14 Mixture Continuity a ÂM b ÂM c =⇒ ∃γ, γ 0 ∈ (0, 1), γa + (1 − γ)c ÂM b ÂM γ 0 a + (1 − γ 0 )c. Mixture continuity is also known as the Archimedian property. It must be distinguished from the significantly stronger axiom of “sequential continuity,” namely an → a∧an ºM b, ∀n ⇒ a ºM b, which turns out to be implied here but is not imposed. The core result of this subsection will now be stated and then discussed. Theorem 2 Assume axiom 1 applies. Then: (i) A ranking fulfils axioms 2, 3, 4, 6, 9, and 14 iff it is Bayesian. (ii) A ranking fulfils axioms 2, 3, 4, 6, 11, 12, 13, and 14 iff it is the maximin utility ranking. (iii) A ranking fulfils axioms 2, 3, 4, 6, 9, 11, 12, and 14 iff it is the α-maximin utility ranking. (iv) A ranking fulfils axioms 2, 3, 4, 7, 10, 11, 12, 13, and 14 iff it is the minimax regret ranking. (v) A ranking fulfils axioms 2, 3, 4, 7, 9, 10, 11, 12, and 14 iff it is the α-minimax regret ranking. In sum, the following table holds, where + denotes compliance, − noncompliance, and ⊕ indicates axiomatic characterization, that is, for any column, the “⊕”-entries are reminders of the corresponding statement above. α-MU is here understood to refer to α ∈ (0, 1).

20

Bayes

MU

α-MU

MR

α-MR

(2) complete

⊕

⊕

⊕

⊕

⊕

(3) transitive

⊕

⊕

⊕

⊕

⊕

(4) monotonic

⊕

⊕

⊕

⊕

⊕

(5) . . . (transitive extension)

+

+

+

+

+

(6) IIA

⊕

⊕

⊕

−

−

(7) INA

+

+

+

⊕

⊕

(8) independent

⊕

−

−

+

+

(9) C-independent

+

+

⊕

+

⊕

(10) EP-independent

+

−

−

⊕

⊕

(11) fulfils reduction

−

⊕

⊕

⊕

⊕

(12) anonymous

−

⊕

⊕

⊕

⊕

(13) ambiguity averse

+

⊕

−

⊕

−

(14) continuous

⊕

⊕

⊕

⊕

⊕

*weakly so; compare the below elaboration

∗

In generating this result, I somewhat ride on the shoulders of giants. Given lemma 3, part (i) follows from Anscombe and Aumann. Parts (ii) through (iv) draw on Milnor (1954), who without further discussion takes the vector notation to be primitive. Apart from nesting it via theorem 1 and generating result (v), I strengthen his result by substantially weakening his axioms. The price is the slight strengthening of nontriviality, which is required in parts (iv) and (v) and only there.15 A main benefit is alignment and thus comparability with more recent contributions. For example, the GS characterization here implies that axioms 2, 3, 4, 6, 9, 13, and 14 jointly imply an MU-representation with an “as if”-set of priors. The identification of the set of priors with S can therefore be attributed to axioms 11 and 12. Independence implies ambiguity aversion but with “∼M ” in the conclusion, a property that is intutively better described as ambiguity neutrality. It follows that Bayesians are ambiguity neutral, thus Bayes acts are generically not randomized. In contrast, both MU and MR decision makers tend to 1 5 The

proof reveals that with conventional nontriviality, the axioms characterizing MR would also allow the following

class of preference orderings, indexed by elements of 2S : An option has utility 1 if it performs optimally (within M) for every state in some pre-assigned subset of S and has utility 0 otherwise.

21

be strictly ambiguity averse and hence often, although not always, randomize.16 The α-generalizations of these criteria can even be ambiguity seeking, the more so the higher α, because randomization also tends to attenuate best-case performance. The most interesting aspects of the result can be seen in the rightmost four columns of rows 6 and 10 through 14. If reduction and anonymity are imposed, the decision between MU and MR is essentially one between IIA and EP-independence. Committing to the latter and plausibly weakening IIA to INA results in a characterization of MR, whereas IIA enforces α-MU and ambiguity aversion then picks MU. It is also interesting to note that MR (as well as PMR below) is consistent with what I consider the appropriate adaptation of independence. Whether one is Bayesian or not therefore depends on more than one’s attitude toward the independence axiom.

3.3

Characterizing Intransitive Rankings

The above table will now be extended by characterizations of intransitive rankings, in particular variations on PMR. Recall that PMR’s motivation is to maintain a minimax perspective but to insist on IIA. This requires one to sacrifice transitivity, but without any transitivity-like restrictions whatsoever, exotic rankings abound. It is therefore necessary to find weakenings of transitivity that preserve its most compelling aspects yet are simultaneously consistent with IIA and EP-independence. One such restriction, namely transitive monotonicity, already turned up and will be maintained throughout. Not only is it needed to get theorem 1, but it also meaningfully — and, I believe, plausibly — restricts preferences over reduced-form acts as follows: a = b ºM c ∨ a ºM b = c =⇒ a ºM c. The major problem with rankings like WSD and SSD is that by recommending all admissible acts, they are not very decisive. To prevent this, one plainly needs axioms with strict preferences on the then-side. For example, one might be tempted to impose that a À b =⇒ a ÂM b, a condition I will call “increasingness” because “monotonicity” is already defined. Interestingly, increasingness is implied by all combinations of axioms used. However, axioms with significant cutting 1 6 To

get an example where the MU-action randomizes, invert the positions of 0 and n in Rawls’ example. Then the

MU-action is π a =

n+1 . n2 +n+1

The MU-act in the original example is nonrandomized because the worst-case state is the

same for both acts, precluding meanigful hedging; see Klibanoff (2001) for a rigorous discussion.

22

power can be generated by further strengthening it, making it either “more increasing” or “more transitive.” The latter version goes as follows: Axiom 15 Transitive Extension of Increasingness a À b ºM c =⇒ a ÂM c. The basic intuition of this matches the one of transitive monotonicity, namely that dominance relations should make for context-independent comparisons and thus for ones that have some degree of transitivity. However, transitive increasingness “feels” stronger and therefore requires more faith in this intuition. It essentially imposes that orderings may be context-dependent but must be “sharp”: if I am indifferent between b and c, then an infitesimal, certain payment will induce me to trade one for the other. This makes much sense if indifference is taken very literal and may be acceptable if decisiveness is called for, but is dubious if indifference at least partly stands in for noncomparability. The “more increasing” version of increasingness is self-explanatory: Axiom 16 Strict Increasingness a = b =⇒ a ÂM b. Whilst these restrictions are intended to avoid “too large” choice correspondences, it is also reasonable to be concerned about “too small,” namely empty ones. This consideration motivates the following axiom: Axiom 17 Acyclical Strict Preference There exists no strict preference cycle, that is, no M and {a, b, . . .} ⊆ M such that a ÂM b ÂM . . . ÂM a. Substantively, this is perhaps the weakening of transitivity that retains most of its spirit. Since it does not invoke dominance, its plausibility stands or falls with the aforementioned idea that rankings reflect degrees of “goodness” — as Schwartz (1972) writes, “to accept [...] Noncircularity is to accept the guts of the Maximization thesis.” However, acyclical strict preference is of analytical interest because it is well-known to insure non-empty choice functions for general choice sets, i.e. the existence of a

23

“best” option in every menu. In contrast, intransitive preference orderings and PMR in particular can generate empty choice functions.17 The above three additional axioms suffice to characterize all of WSD, SSD, α-PMR, and PMR. The result is summarized in the below table, which also restates the characterizations of MU and MR for reference. Theorem 3 Assume that axioms 1, 2, 5, 6, 10, 11, and 12 apply. Then: (i) A ranking fulfils axioms 9 and 14 iff it is α-PMR. (ii) A ranking fulfils axiom 15 iff it is PMR. (iii) A ranking fulfils axioms 16 and 17 iff it is WSD. (iv) A ranking fulfils axioms 14 and 17 iff it is SSD. All in all, the following table is true, where + denotes compliance, − noncompliance, and ⊕ indicates axiomatic characterization, and α-PMR is understood to refer to α ∈ (0, 1). 1 7 With

general choice sets, any preference cycle yields a menu for which the choice function is empty. The situation is

more complicated here because there exists a large class of cyclical orderings that generate well-behaved choice functions whenever agents are allowed to randomize (see Fishburn 1984). However, PMR is not in this class, and it can be verified to induce C(∆{(0, 1, 2), (1, 2, 0), (2, 0, 1)}) = ∅. I leave to future research the identification of a weaker axiom that insures nonempty choice functions for the menus considered here. Schwarz (1972) proposes a generalized choice function that coincides with the conventional one in well-behaved cases but is never empty. But of course, this function can be uninformative if preference cycles are pervasive.

24

MU

MR

α-PMR

PMR

WSD

SSD

(2) complete

⊕

⊕

⊕

⊕

⊕

⊕

(3) transitive

⊕

⊕

−

−

−

−

(4) monotone

⊕

⊕

+

+

+

+

(5) monotone (transitive extension)

+

+

⊕

⊕

⊕

⊕

(6) IIA

⊕

−

⊕

⊕

⊕

⊕

(7) INA

+

⊕

+

+

+

+

(8) independent

−

−

+

+

+

+

(9) C-independent

+

+

⊕

+

+

+

(10) EP-independent

−

⊕

⊕

⊕

⊕

⊕

(11) fulfils reduction

⊕

⊕

⊕

⊕

⊕

⊕

(12) anonymous

⊕

⊕

⊕

⊕

⊕

⊕

(13) ambiguity averse

⊕

⊕

+

+

+

+

(14) continuous

⊕

⊕

⊕

+

−

⊕

(15) increasing (transitive extension)

+

+

−

⊕

−

−

(16) strictly increasing

−

−

−

−

⊕

−

(17) acyclic Â

+

+

−

−

⊕

⊕

By lemma 3 or direct verification, all the new criteria fulfil independence. Contrary to prevalent perceptions, independence is therefore not fundamentally incompatible with maximin-type decision rules. As in the remarks following theorem 2, this also means that PMR is ambiguity neutral, explaining why PMR-acts are in general not randomized. The most important aspects of the table are the 4 rightmost columns of row 5 and the last 3 rows. Transitive monotonicity already generates quite some structure, because one does not need to impose any of the new axioms to arrive at α-PMR. A further sharpening of transitivity-like conditions immediately leads to very specific rankings. Transitive increasingness and acyclic  are particularly crucial: The former enforces PMR and the latter dominance-based criteria, with continuity picking SSD and strict increasingness WSD. An important message here is the fundamental tension between a desire to have decisive rankings and one to avoid preference cycles, or in other words, the difficulty of finding a middle ground between too large and too small choice sets. The only axiom that prevents empty choice sets single-handedly

25

enforces dominance rankings and hence insures that all admissible acts (give or take weak dominance) are in the choice correspondence.

4

Summary and Outlook

This paper explored the foundations of a statistical decision theory for situations of simultaneous (“model”) ambiguity and (“estimation”) uncertainty. The purpose was to explore the theoretical foundations of numerous decision criteria, mainly ones that treat uncertainty but not ambiguity in a probabilistic fashion. The axiomatic discussion differs from previous contributions by being more applied, using the full structure of a real-world problem to give results that are tightly specified for this problem. In particular, they link all objects in the decision rule to the decision maker’s environment and therefore characterize maximin-criteria as they are actually used in practice; this bridges a sometimes overlooked gap in the foundations of maximin-type statistical decision making. Furthermore, I introduced a number of new criteria and, to fully explore the properties of regret-based criteria, examined an axiomatic system that relaxes both transitivity and IIA. Some core insights are the following. Firstly, as long as probabilistic uncertainty is evaluated by expected value or utility, weak additional assumptions lead to a convenient link between uncertainty and ambiguity. Once this has been established, the standard criteria — Bayesianism, maximin utility, minimax regret, and the Hurwicz criterion — can be characterized by means of axioms that have been used before. It then turns out that if one is convinced by the intuition of regret criteria and hence by EP- or EO-independence, two other desirable properties, namely transitivity and IIA, are in conflict. Minimax regret resolves this by prioritizing transitivity over IIA. The effect of the opposite choice is explored, and it turns out that some plausible weakenings of transitivity characterize PMR, but also that there arises a fundamental conflict between avoiding preference cycles and having a criterion that actually excludes any admissible acts. Finally, it turns out that numerous non-Bayesian criteria are compatible with the (adapted) independence axiom. The examination of statistical decision rules of this type offers rich opportunities for further research. Firstly, it is clearly desirable to extend the results to infinite state spaces. Secondly, theorists should be interested in axiomatizing MR and PMR with respect to “as if”-sets of priors, i.e. in the spirit of GS or CKO. They could also do away with objective probabilities, in which case axiomatizations might build on results by Fishburn (????) and Sugden (1993). Another interesting variation

26

within the present framwork — currently investigated by the author — is to consider regret but not in terms of (u(a) − u(a∗ )) but rather u(a − a∗ ), i.e. utility of efficiency loss rather than efficiency loss in utility. If expected utility is not imposed over unambiguous acts, then the close connection between EO-independence and EP-independence breaks down, with the former really pushing one toward the latter representation. Finally, it turns out that in Rawls’ example but also many examples not contained here, pairwise minimax regret is acyclical and even transitive. To establish the practical importance of its potential intransitivity, it would be interesting to identify rather general conditions under which this is the case.

A

Proofs

Proposition 1 To see the MR-act, recall the following fact (e.g. section 5.2.2 of Berger 1985): If some probability Pr(α) maximizes expected regret given π a and πa minimizes it given Pr(α), then πa is an MR act. The hypothesis of this statement can be verified with πa = Pr(α) =

n2 −n n2 −n+1 .

PMR induces a transitive ordering whenever there are only two states. To prove this, order acts such that a1 − b1 ≥ a2 − b2 and write a ºP MR b ⇐⇒ max(a1 − b1 , a2 − b2 ) ≥ max(b1 − a1 , b2 − a2 ) ⇐⇒ a1 − b1 ≥ b2 − a2 ⇐⇒ a1 + a2 ≥ b1 + b2 . Thus, with two states, PMR agrees with 0.5-MU, and in the example, a ºP MR b iff n ≥ 1 + 1/n. The other results are straightforward. Lemma 1 Assume there are two states and describe an act a by (u(a, 1), u(a, 2)). Consider the menu M = {(2, 2), (1, 3), (3, 2), (0, 3)}. Then the judgments (2, 2) ºM (1, 3) and (3, 2) ≺M (0, 3) jointly contradict transitive monotonicity but not monotonicity itself, thus axiom 4 does not imply axiom 5. However, any transitive ordering that implies the above judgments must contradict monotonicity, and it is easy to generalize this observation. Now let M = {(1, 1), (0, 0)}, then (1, 1) ≺M (0, 0) violates monotonicity but not its transitive extension, thus axiom 5 does not imply 4. However, if INA holds, then (1, 1) ≺M (0, 0) ⇔ (1, 1) ≺M∪{(0,1)} (0, 0), but completeness implies that (0, 1) ºM∪{(0,1)} (0, 1) and transitive monotonicity then that 27

(1, 1) ºM∪{(0,1)} (0, 0). The generalization of this argument works by an induction just as in theorem 1 below. Theorem 1 Fix any a, b, a, and b that satisfy the hypothesis and define the extended menu M 0 ≡ S S M ∪ S∈2S {aS a, bS b}. Clearly M ∪ {a, b}, M ∪ {a, b} ⊆ M 0 . No element of S∈2S {aS a, bS b} is strictly potentially optimal in M ∪ {a, b} or M ∪ {a, b}. By axiom INA, it follows that a ºM∪{a,b} b ⇔ a ºM 0 b and a ºM∪{a,b} b ⇔ a ºM 0 b, thus it suffices to show a ºM 0 b ⇔ a ºM 0 b. Now fix any s ∈ S. By transitive monotonicity, a ºM 0 b implies that as a ºM 0 bs b, thus a ºM 0 b by induction over S. The argument for a ºM 0 b ⇒ a ºM 0 b is symmetric. Lemma 2 Consider a situation where there is no ambiguity, i.e. v(a, r, s) = v(a, r, s0 ), ∀s, s0 ∈ S; then s can as well be dropped from notation. Since axiom 4 holds, there exists U such that for any acts a and b, a º b ⇐⇒

Z

U (v(a, r))dFr ≥

Z

U (v(b, r))dFr .

Consider now an act c that is constant over R, i.e. a certain payment: v(c, r) = vc , ∀r. Then EOIndependence implies that Z Z Z Z U (v(a, r))dFr ≥ U (v(b, r))dFr ⇐⇒ U (v(a, r) + vc )dFr ≥ U (v(b, r) + vc )dFr , ∀a, b, which is the definition of constant absolute risk aversion. In particular, the preference ordering is either globally risk averse, globally risk neutral, or globally risk seeking. Consider now acts a and b s.t. a is constant over R, i.e. v(a, r) = va , ∀r, and b is uncertain but R with expectation v(b, r)dFr = va , i.e. b is a mean-preserving spread of a. Assume the preference ordering is weakly risk averse, then Z

U (v(b, r))dFr ≤

Z

U (v(a, r))dFr .

Now define an act eb by v(eb, r) = supr∈R {v(b, r)} − v(b, r). Then b ⊕ eb is the certain act that yields payment supr∈R {v(b, r)} in every state. On the other hand, a ⊕ eb is an uncertain act that has by

construction the same expected outcome as b ⊕ eb, thus it is a mean-preserving spread of the former.

Yet applying EO-Independence to the above inequality, one finds that Z Z e U (v(b, r) + v(b, r))dFr ≤ U (v(a, r) + v(eb, r))dFr , 28

thus the preference ordering is weakly risk seeking. Similarly, if the preference ordering is weakly risk seeking, it is also weakly risk averse. Hence it must be both, i.e. risk neutral, and can therefore be represented by setting U (v(a, r, s)) ≡ v(a, r, s). Lemma 3 1−p 1−p c ºM + 1−p c b + c p p p ⇐⇒ pa + (1 − p)c ºpM+(1−p)c pb + (1 − p)c,

a ºM b ⇐⇒ a +

where the first equivalence follows from EP-independence and the second one from C-independence → − where c in that axiom is identified with 0 . To see that the converse is not true, consider the Bayesian decision criterion. Theorem 2 Compliance and noncompliance are easy to check. The below arguments establish the “only if” statements. Theorem 1 holds under all axiomatic settings, so vector notation is appropriate. IIA implies a ºM b ⇔ a ºN b ⇔ a ºA b, hence for (i), (ii), and (iii), all comparisons can be understood to be with respect to A and the reference to a menu be dropped from notation. (i) This follows from Anscombe/Aumann (1963), as is conveniently established by verifying the hypotheses of theorem 7.17 in Kreps (1988). His axiom 7.1, that º be a preference relation on A, is implied by completeness, transitivity, and IIA; in particular, the latter must be assumed since the first two here refer to ºM only. Axiom 7.2 (classical independence) follows from independence as assumed here in conjucntion with IIA. Axioms 7.3 (continuity) and 7.14 (nontriviality) are directly imposed, and axiom 7.16, a{s} c º b{s} c ⇔ a{t} c º b{t} c, ∀s, t ∈ S, holds because by monotonicity, both sides of this equivalence are equivalent to a ≥ b. I will now prepare (ii) and (iii). Fix any act a ≡ (a1 , . . . , a#S ) and define a ≡ mini {ai } and a ≡ maxi {ai }. I am going to show that a º b ⇔ (a, a) º (b, b). Without loss of generality, assume

that states are ordered such that a is ascending, then (a, . . . , a, a) 5 a 5 (a, a, . . . , a) and hence (a, . . . , a, a) ¹ a ¹ (a, a, . . . , a) by monotonicity. Let M ≡ {(a, a, . . . , a), (a, . . . , a, a)}, then anonymity implies that (a, a, . . . , a) ºM (a, . . . , a, a) iff (a, . . . , a, a) ºM (a, a, . . . , a). Completeness then implies that (a, a, . . . , a) ∼M (a, . . . , a, a); by 29

IIA, this indifference holds in any menu. Using the previous paragraph’s result and transitivity, one finds that (a, a, . . . , a, a) ¹ a ¹ (a, . . . , a, a). But completeness also gives (a, a) ∼ (a, a), thus (a, . . . , a, a) ∼ (a, . . . , a, a) by reduction. Further uses of transitivity and reduction now yield a º b ⇔ (a, . . . , a, a) º (b, . . . , b, b) ⇔ (a, a) º (b, b) As further preparation, observe that increasingness as defined in the text is jointly implied by completeness, nontriviality, transitivity, monotonicity, IIA, and C-independence, and can therefore be assumed. Suppose by contradiction that increasingness fails, i.e. there exists a and b such that (a1 , a2 ) ¿ (b1 , b2 ) but not a ≺ b, thus a º b by completeness, thus a ∼ b by monotonicity. Let the affine mapping ρ : a 7→ α + βa be constructed s.t. ρ(a) = b; specifically, β = (b2 − b1 )/(a2 − a1 ) and α = b1 −βa1 = b2 −βa2 . If β ∈ (0, 1), then this mapping is equivalent to mixing all acts with a constant act c ≡ α/(1 − β) with mixture probability p ≡ β. Thus C-independence implies that orderings are preserved under the mapping, i.e. b = ρ(a) ∼ ρ(b) = b+β(b−a) and therefore a ∼ b+β(b−a). Iterating the argument, it follows that a ∼ ρn (a) for any natural number n. Observing that β is the mapping’s

norm and using the standard geometric sum formula, it follows that a ∼ a + γ(b − a), ∀γ < 1/(1 − β).

Consider now (c1 , c2 ) ≡ (a1 + min{b1 − a1 , b2 − a2 }, a2 + min{b1 − a1 , b2 − a2 }). Then a ¿ c 5 b,

so that completeness and monotonicity imply a ∼ c. If the previous paragraph’s argument could be applied for (c1 , c2 ) identified with (b1 , b2 ), the coefficient β in the above mapping would equal 1, and the iteration over n would diverge, hence a ∼ a + γ(b − a), ∀γ > 0. In fact, the conclusion does not obtain directly because if β = 1, C-independence cannot be used. Consider however a sequence {cn }

s.t. a ¿ cn 5 b, cn → c, and (c2n − c1n )/(a2 − a1 ) < 1. Then (c2n − c1n )/(a2 − a1 ) % 1, and as n increases, the largest point for which indifference to a can be concluded diverges toward (∞, ∞). It

follows that the indifference set containing a is not bounded above. By a similar argument, it is not bounded below. Thus any point in R2 is dominated by some point in this indifference set but also dominates some point in it; by completeness, monotonicity, and transitivity, it must then be included in it, contradicting nontriviality. (ii) The preparatory argument implies that (a, a, a) ∼ (a, a, a) ∼ (a, a, a), thus ambiguity aversion

a+a a+a a+a yields (a, a+a 2 , 2 ) º (a, a, a) ∼ (a, a, a) and monotonicity then (a, 2 , 2 ) ∼ (a, a, a). By induction

a−a over n, the argument can be extended to imply (a, a + a−a 2n , a + 2n ) ∼ (a, a, a) for any natural number

n. As the sequence {2−n } is dense at 0, monotonicity, transitivity, and reduction then jointly imply

that (a, a + γ(a − a)) ∼ (a, a) for any γ ∈ (0, 1]. 30

Now suppose by contradiction that (a, a) Â (a, a). Fix any b < a, then increasingness and transitivity jointly imply that (a, a) Â (a, a) Â (b, b). By continuity, there must then exist δ ∈ (0, 1) s.t. (δa + (1 − δ)b, δa + (1 − δ)b) Â (a, a). The previous paragraph’s result implies that (δa + (1 − δ)b, γ(δa + (1 − δ)b) + (1 − γ)(δa + (1 − δ)b)) ∼ (δa + (1 − δ)b, δa + (1 − δ)b), ∀γ > 0, thus the left side of the above indifference is strictly preferred to (a, a) for any γ > 0. But as γ → 0, one finds that (δa + (1 − δ)b, γ(δa + (1 − δ)b) + (1 − γ)(δa + (1 − δ)b)) → (δa + (1 − δ)b, δa + (1 − δ)b) ¿ (a, a), so increasingness is eventually contradicted. It follows that (a, a) ¹ (a, a) and, by monotonicity, (a, a) ∼ (a, a). Hence a º b ⇔ a ≥ b as required. (iii) Let α = inf{a : (a, a) º (0, 1)}, then monotonicity implies that α ≤ 1 and increasingness that α ≥ 0. Suppose by contradiction that (α, α) Â (0, 1). Then since (0, 1) Â (−1, −1) by increasingness, there exists γ > 0 s.t. (γa − (1 − γ) ,γa − (1 − γ)) Â (0, 1), contradicting the definition of α. It follows that (α, α) ∼ (0, 1). Let α ∈ (0, 1), then (α, α) ∼ (0, 1) and C-independence imply that (a + α(a − a), a + α(a − a)) ∼ (a, a); here, p in the axiom is identified with α and c is identified with a/(1 − α). It follows that a º b ⇔ a + α(a − a) ≥ b + α(b − b) as required. If α ∈ {0, 1}, the result is immediate. (iv, v) Revert to menu dependent notation, consider any acts a, b, any menu M ⊇ {a, b}, and define ∨M ≡ (maxa∈M ai )i=1,2,...,#S , the meet of M . Then by INA, a ºM b ⇔ a ºM∪{∨M} b, − b − ∨M . Hence ºM is determined therefore by EP-independence, a ºM b ⇔ a − ∨M ºM−∨M∪{→ 0} → − 0 by the preference ordering ºM 0 , where M ≡ M − ∨M ∪ { 0 } is called a normalized menu and every

a ∈ M is identified with a normalized act a0 ≡ a − ∨M . → − For any normalized menu M 0 , ∨M = 0 ∈ M 0 by construction, hence no nonpositive vector can be potentially strictly optimal within a normalized menu. It follows by INA that the preference ordering over the negative quadrant in R#S extends the ordering over any normalized menu. Thus, it is sufficient

to characterize the preference ordering over R#S − . But the arguments of (ii) and (iii) only invoke additional options that are dominated by (a, . . . , a), which is nonpositive if a is nonpositive; thus if increasingness could be presumed, they would characterize this ordering. In particular, the assumptions made in (iv) would imply that a ºM b ⇔ a0 ºM 0 b0 ⇔ mini {a0i } ≥ mini {b0i } ⇔ mini {ai − (∨M )i } ≥ 31

mini {bi − (∨M )i }, which is the MR criterion. The argument for α-MR is analog. It therefore remains to establish increasingness of ºR#S . −

Suppose again by contradiction that there exist a ¿ b with a ∼ b. Then there exists γ < 1 s.t.

γa 5 b, thus γa ¹ b by monotonicity, thus γa ¹ a by transitivity, thus γa ∼ a by monotonicity. → − Using C-independence, where c is identified with 0 , it now follows that γa ∼ a for any γ > 0 (the

implication for γ > 1 follows by using the “only if”-direction of the axiom), where I also used that → − #S 0 (1/γ)R#S − = R− and therefore INA applies. Now consider any c ¿ 0 , then there exist γ, γ > 0 s.t.

γa 5 c 5 γ 0 a, thus γa ∼ c ∼ γ 0 a by monotonicity and transitivity. Hence the interior of R#S − is an indifference set.

→ − → − → − → − Fix b on any axis and a ¿ 0 . By monotonicity, 0 º a and 0 º b. Suppose that 0 ∼ a, then → − → − → − the ranking would be trivial, thus 0  a. Now suppose that 0  b. If 0  b  a, then continuity implies that γa  b  a for some γ > 0, contradicting the previous paragraph’s conclusion. But there → − also exists γ > 0 s.t. γa 5 b; thus if 0  a  b, then monotonicity implies that a  b º γa for this choice of γ, again contradicting the previous paragraph. It follows that a ∼ b. Similar to the → − → − previous paragraph’s argument, 0  b furthermore implies that 0  γb for any γ > 0. All in all, the relative interior of the axis on which b is located must be in the same indifference set as the interior → − → − third quadrant. Alternatively, suppose that 0 ∼ b. The same argument then implies that 0 ∼ b for → − any b on the same axis, and transitivity that b  a for any such b and any a ¿ 0 . In sum, there exist exactly two indifference sets, one containing the origin and some subset of the axes, the other one the remainder of the third quadrant and being ranked below the first one. But any such ranking contradicts nontriviality. This is the only use of the strengthening of nontriviality. With conventional triviality, the class of orderings just identified is permitted. Clearly, said orderings contradict the stronger notion of continuity as well as increasingness, so that either assumption could substitute for the strengthening of nontriviality. Theorem 3 Again, I only elaborate “only if.” First of all, theorem 1 applies and IIA implies a ºM b ⇔ a ºN b ⇔ a ºA b, so that the reference to → − a menu can be dropped from notation. Now observe that by EP-independence, a º b ⇔ c ≡ a−b º 0 .

32

Let (c, c) = (mini {ci }, maxi {ci }), then the following holds: − → 0 ⇐⇒ (c, c) º (0, 0) → − c ¹ 0 ⇐⇒ (c, c) ¹ (0, 0). c º

To prove this, let states be numbered s.t. c is ascending. Then (c, . . . , c, c) 5 c 5 (c, c, . . . , c). Now → − assume (c, c) º (0, 0), then reduction implies (c, . . . , c, c) º 0 , but then transitive monotonicity yields → − → − → − c º 0 . Analog arguments give that (c, c) - (0, 0) implies c - 0 and furthermore that c º [¹] 0 implies (c, c) º [¹](0, 0). It thus suffices to characterize the ranking between (c, c) and (0, 0). Define the decision function ⎧ ⎪ ⎪ 1, (c, c) Â (0, 0) ⎪ ⎨ d(c, c) ≡ 0, (c, c) ∼ (0, 0) . ⎪ ⎪ ⎪ ⎩ −1, (c, c) ≺ (0, 0)

This function is well-defined due to completeness. The proof is completed by examining the isoquants of d in (c, c)-space. Firstly, as c ≤ c by construction, restrict attention to the halfspace above {(c, c) :

c = c}, i.e. the 45◦ line. d partitions this halfspace into d−1 (−1), d−1 (0), and d−1 (1). Furthermore, d(c, c) = −d(−c, −c) since d(c, c) = 1 ⇔ (c, c) Â (0, 0) ⇔ (0, 0) Â (−c, −c) ⇔ (0, 0) Â (−c, −c) ⇔ d(−c, −c) = −1, where in addition to EP-independence and IIA, the second last step used anonymity. Thus d−1 (−1) is the reflection of d−1 (1) about {(c, c) : c = −c}, i.e. the decreasing 45◦ line. Since d−1 (1) and d−1 (1) are disjoint, the fixed points of said reflection are in neither set, hence {(c, c) : c = −c} ⊆ d−1 (0).

Now increasingness needs to be established once again. Suppose it fails, then there exists (c, c) À (0, 0) with (c, c) ∼ (0, 0) (where I used completenss and monotonicity as before). Using C-independence in a similar manner to above, it follows that γ(c, c) ∼ (0, 0) for any γ > 0, thus {γ(c, c) : γ > 0} ⊆ d−1 (0). The previous paragraph’s symmetry result now implies that the reflection of this ray, {γ(c, c) : γ < 0}, is contained in d−1 (0) as well. Now fix any a ∈ R2 . Clearly there exist γ < 0 < γ 0

s.t. γ(c, c) ¿ a ¿ γ 0 (c, c). Transitive monotonicity then implies that (0, 0) º a but also (0, 0) ¹ a, thus (0, 0) ∼ a. Thus all acts are pairwise indifferent, contradicting nontriviality. It follows that

increasingness holds and thus that d−1 (1) contains the interior of the first quadrant and d−1 (−1) the interior of the third one (above the 45◦ line). From here, the proof takes different directions: 33

(i) Assume that C-independence holds. Then a º b ⇔ γa º γb, ∀γ > 0, but this implies (c, c) º (0, 0) ⇔ γ(c, c) º (0, 0), ∀γ > 0 and thus that the relative interior of any origin ray is an isoquant of d. Consider now two distinct origin rays, A and B say, within the fourth (=northwestern) quadrant. Clearly A and B do not intersect, thus assume w.l.o.g. that A lies above B. Transitive monotonicity then implies that B ⊆ d−1 (0) ⇒ A ⊆ [d−1 (0) ∪ d−1 (1)] and B ⊆ d−1 (1) ⊆ A ∈ d−1 (1). Thus the

intersections of d−1 (−1), d−1 (0), and d−1 (1) with the fourth quadrant are ordered as follows: tracing

the quadrant with origin rays in positive (counterclockwise) direction, one first traces its intersection with d−1 (1), then d−1 (0), then d−1 (−1). As d−1 (1) and d−1 (−1) also contain the first respectively third quadrant, it follows that d−1 (−1), d−1 (0), and d−1 (1) are convex cones with the same ordering. Now suppose by contradiction that d−1 (0) is open. Then fix some point c on the boundary of d−1 (−1) and some point c0 on the boundary of d−1 (1), observing that this choice implies γc+(1−γ)c0 ∈

d−1 (0) for any γ ∈ (0, 1). Fix any act b ∈ R2 , then by construction, b + c0  b  b + c. By continuity,

there must then exist some γ ∈ (0, 1) s.t. b + γc + (1 − γ)c0 Â b, thus γc + (1 − γ)c0 ∈ d−1 (1), a contradiction. Thus d−1 (0) is closed. It now follows that d−1 (−1) is the half-open cone below a

downward sloping origin ray with absolute slope α, where α ≥ 0 since d−1 (−1) contains the interior third quadrant and α ≤ 1 because of the symmetry between d−1 (−1) and d−1 (1). Also using symmetry

between d−1 (−1) and d−1 (1), there exists α ∈ [0, 1] s.t.

d−1 (−1) = {(c, c) : αc + c < 0} d−1 (0) = [d−1 (−1) ∪ d−1 (−1)]c d−1 (−1) = {(c, c) : c/α + c > 0}. Noticing that c = min(a − b) = − max(b − a) and c = max(a − b) = − min(b − a), it is now apparent that d characterizes the decision rule PMR(α). (ii) Recall that {(−c, c) : c ∈ R+ } ⊆ d−1 (0). Consider any (c, c) with c > −c, then there exists

c ∈ R+ s.t. (c, c) À (−c, c) ∼ (0, 0). Transitive increasingness then implies that (c, c) Â (0, 0). Similarly, c < −c ⇒ (c, c) ≺ (0, 0). As c = − max(b − a) and c = max(a − b), it is now apparent that the ranking must be PMR. (iii, iv) Assume acyclicity and suppose by contradiction that there exists c < 0 < c s.t. (c, c) Â (0, 0). Then a cycle of strict preference can be constructed along the lines of footnote 14. For example, 34

if (−1, 9) Â (0, 0), the following constitutes such a cycle: s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

a1

1

2

3

4

5

6

7

8

9

10

a2 .. .

2 .. .

3 .. .

4 .. .

5 .. .

6 .. .

7 .. .

8 .. .

9 .. .

10 .. .

1 .. .

a10

10

1

2

3

4

5

6

7

8

9

Here, a1 Â a2 Â . . . Â a10 Â a1 ; the example can clearly be adapted for any c < 0 < c. Thus the

interior fourth quadrant must be contained in d−1 (0). It has been shown previously that if continuity

holds, then d−1 (0) is closed, thus here contains the closed fourth quadrant. Since the interior first and third quadrant are already accounted for, this fully determines d. In particular, d(c, c) = 1[−1] iff c > 0[c < 0], which characterizes SSD. If, on the other hand, strict increasingness holds, then d−1 (1) must contain the closed first and d−1 (−1) the closed third quadrant. This again fully determines d; in particular, d(c, c) = 1[−1] iff c ≥ 0[c ≤ 0], which characterizes WSD.

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