Minimax and the Value of Information 
Evan Sadlery
 February 13, 2013
 Abstract
 In his discussion of minimax decision rules, Savage (1954, p. 170) presents an example purporting to
 show that minimax applied to negative expected utility (referred to by Savage as \negative income") is an
 inadequate decision criterion for statistics; he suggests the application of a minimax regret rule instead.
 The crux of Savage?s objection is the possibility that a decision maker would choose to ignore even
 \extensive" information. More recently, Parmigiani (1992) has suggested that minimax regret su ers
 from the same  aw. He demonstrates the existence of \relevant" experiments that a minimax regret
 agent would never pay a positive cost to observe. On closer inspection, I  nd that minimax regret is
 more resilient to this critique than would  rst appear. In particular, there are cases where no experiment
 has any value to an agent employing the minimax negative income rule, while we may always devise a
 hypothetical experiment that a minimax regret agent would pay for. The force of Parmigiani?s critique
 is further blunted by the observation that \relevant" experiments exist for which a Bayesian agent would
 never pay. I conclude by discussing the notion of pessimism in the context of minimax decision rules.
 1 Introduction
 In recent years there has been a revival of interest in the application of minimax procedures to statistical
 and econometric problems. Manski (2004) for instance, has spawned a literature on the application of
 minimax regret to treatment choice. One reason for this newfound popularity is surely the increase in
 computing power now available to researchers, rendering such methods feasible in more domains. With
 renewed interest has come increased scrutiny. In particular, concerns over the use of information by minimax
 procedures have resurfaced. It has long been known that minimax applied to negative expected utility
 (hereafter, \negative income") can entirely ignore available information in some decision problems. Savage
 (1954, p. 170) presented a simple example demonstrating this phenomenon, asserting that minimax applied
 to negative income was entirely inadequate as a criterion for statistics. His alternative was applying minimax
 to regret. While many credit Savage with the invention of the minimax regret criterion, Savage himself gave
 priority to Wald (1950), believing that Wald could not possibly have intended minimax negative income in
 his work. Savage clearly believed this critique inapplicable to regret, but more recent work has disputed this.
 Parmigiani (1992) analyzes an example in which minimax applied to negative income assigns positive
 value to an experiment, while minimax regret does not. He goes on to characterize the circumstances under
 which minimax rules refrain from \relevant" experimentation, even if the experiment is nearly costless.
 Parmigiani describes this phenomenon as ultrapessimism. The essence of his result is best understood by
 interpreting the statistician?s problem as a zero sum game played against nature. If nature has an equilirium
 strategy that is supported on relatively few states, then experiments that fail to distinguish these states will
 be worthless. If such cases can be expected to occur regularly outside of contrived examples, then this raises
 serious concerns about the use of minimax regret in applied work.
  I am grateful to Roy Radner for calling my attention to this problem as well as for numerous fruitful discussions. I also
 thank J org Stoye for pointing me to useful references. Any errors are mine alone.
 yStern School of Business, New York University { esadler@stern.nyu.edu
 1
One goal of the present paper is to controvert this criticism of minimax applied to regret. Savage?s
 example uncovered a more fundamental  aw in minimax negative income than that discussed by Parmigiani.
 In some decision problems, minimax negative income not only fails to utilize an observation that another
 procedure  nds valuable, but in fact, no conceivable experiment provides any value to an agent following
 a minimax negative income rule. This is in stark contrast to the situation we  nd with minimax regret,
 where a valuable experiment can always be constructed. Moreover, even when valuable experiments do exist
 for the minimax negative income rule, there may still be some states such that no amount of evidence in
 their favor will cause the decision maker to adopt an action better suited to those states. Again, minimax
 applied to regret avoids such issues. On further examination, the characterization of ultrapessimism given by
 Parmigiani can apply to Bayesian procedures just as easily|even when a minimax regret agent would pay for
 the information. Ultrapessimism, so de ned, fails to distinguish meaningfully between decision procedures,
 but Savage?s observation remains a substantive distinction between minimax regret and minimax negative
 income.
 A second goal is to critically examine other objections to minimax regret. There are other senses in
 which the procedure can be described as \pessimistic," and these may present challenges to researchers in
 applications. Many of the issues raised stem from prescriptions that go against what we intuitively expect
 from a \good" estimator. In a sense, regret is criticized for failing to conform to our prior information, but
 a major motivation for minimax procedures is to avoid subjective priors. An important direction for future
 work lies in devising ways to utilize such information alongside minimax procedures.
 In the next section I provide a detailed analysis of the examples given by Savage (1954) and Parmigiani
 (1992). The following section generalizes previous results that characterize when minimax values information.
 This section also establishes the key distinction between minimax applied to negative income and minimax
 applied to regret. Finally, I o er some thoughts on the problem of \pessimism" in minimax estimation and
 the e ective application of minimax regret.
 2 Examples
 In this section I present two examples;  rst, Savage?s (1954) example, followed by Parmigiani?s (1992).
 Consider a state space partitioned into two events, B1 and B2, with two primary acts, f1 and f2. Suppose
 f1 yields a payo of  1 utiles in both events, and f2 yields  10 if B1 obtains and 1 if B2 obtains.
 B1 B2
 f1  1  1
 f2  10 1
 Assuming the decision maker is capable of making randomized choices, there are derived acts of the form
 f =  f1 + (1  )f2 for  2 [0; 1]. The negative income from a particular act in each event is then,
  I(f ;B1) = 10 9 ;  I(f ;B2) = 2  1:
 Note that for all values of  in the given range,  I(f ;B1) is larger than  I(f ;B2), so if our decision maker
 applies a minimax negative income decision rule, the chosen act will be the one that minimizes  I(f ;B1),
 which is to simply choose f1 with probability 1.
 Now suppose the decision maker has the option to observe at no cost the random variable X taking
 values in f1; 2g with
 P(X = 1 jB1) = P(X = 2 jB2) = 1 g; P(X = 2 jB1) = P(X = 1 jB2) = g;
 for some g < 12 . There are now four primary acts as functions of X: g1(X) = f1 for each value of X,
 g2(X) = f2 for each value of X, g3 de ned by g3(1) = f1 and g3(2) = f2, and g4 de ned by g4(1) = f2 and
 g4(2) = f1. The set of derived acts available to the decision maker are the convex combinations of these
 four: f =  1g1 +  2g2 +  3g3 +  4g4, where  1 +  2 +  3 +  4 = 1. For such an act, the negative income in
 each event can be calculated as,
 2
 I(f ;B1) =  1 + 10 2 + (1 + 9g) 3 + (10 9g) 4;  I(f ;B2) =  1   2 + (2g  1) 3 + (1 2g) 4:
 Note that just as before,  I(f ;B1)   I(f ;B2) given the constraints on the parameters. In the expression
 for  I(f ;B1), the coe cients of the  2,  3, and  4 terms are all strictly larger than 1, so this is minimized
 by allocating all weight to  1. That is, the decision maker chooses f1 regardless of the observed X and
 regardless of the value g, earning utility of  1 in all cases. If observing X were costly, the agent would
 evidently decline the observation, no matter how small the cost. This is the basis of Savage?s criticism of
 the minimax negative income rule.
 Compare this result with that obtained by a decision maker using a minimax regret rule when faced with
 the same situation. Let L(f ;Bi) denote the regret of action f if Bi obtains. For the original problem in the
 absence of the observation,
 L(f ;B1) = 9(1  ); L(f ;B2) = 2 :
 The decision maker chooses  to minimize the maximum of L(f ;B1) and L(f ;B2). The two are equal when
  = 911 , and since L(f ;B1) is a strictly decreasing function of  and L(f ;B2) is a strictly increasing function
 of  , this gives the minimum. The regret at the minimum is 1811 .
 Given the same random variable X as before, and the same four primary acts, the situation changes.
 The regret in each event is,
 L(f ;B1) = 9 2 + 9g 3 + 9(1 g) 4; L(f ;B2) = 2 1 + 2g 3 + 2(1 g) 4:
 Note that since g < 12 , both of these are reduced by reallocating weight from  4 to  3, so we can take  4 = 0.
 Now consider the following observations. In the solution we must have  1  (9 2 + 7g 3)=2, otherwise the
 maximum can be reduced by reallocating weight from either  2 or  3 to  1. If  1  (9 2 + 7g 3)=2, then
 L(f ;B1)  L(f ;B2). Further, the solution must have L(f ;B1) = L(f ;B2) since if L(f ;B1) < L(f ;B2), the
 maximum can be reduced by reallocating weight from  1 to  2. Write  3 = 1  1   2, and this condition
 becomes
 (9 9g) 2  9g 1 + 9g = (2 2g) 1  2g 2 + 2g =)  1 =
 (9 7g) 2 + 7g
 2 + 7g
 :
 The regret is then
 (9 9g) 2  9g
 (9 7g) 2 + 7g
 2 + 7g
 + 9g =
 18 36g
 2 + 7g
  2 + 9g +
 63g2
 2 + 7g
 :
 Since g < 12 , the coe cient of  2 is positive, and the regret is minimized by choosing  2 = 0. Thus, the
 minimax regret act sets  1 = 7g=(7g + 2),  2 = 0,  3 = 2=(7g + 2). The regret is L(f ;B1) = L(f ;B2) =
 18g=(7g + 2), which goes to zero as g goes to zero (i.e. as X becomes more informative). The agent should
 be willing to pay as much as 1811  18g=(7g + 2) > 0 to observe X.
 For the second example, consider a box containing two ordered marbles in one of three arrangements.
 Either both are red, the  rst is blue with the second red, or both are blue. Two primary acts are available.
 The act f1 pays  2 utiles if both are red, 0 if the  rst is blue and the second red, and  4 if both are blue.
 The act f2 pays  4 if both are red or the  rst blue and the second red, and it pays 0 if both are blue. The
 payo table is depicted below.
 RR BR BB
 f1  2 0  4
 f2  4  4 0
 3
Considering mixed actions as before, the negative income from choosing f =  f1 + (1  )f2 in each state is
  I(f ;RR) = 4 2 ;  I(f ;BR) = 4 4 ;  I(f ;BB) = 4 :
 The minimax solution sets  = 23 , giving expected negative income of
 8
 3 in all states.
 If the decision maker can observe the  rst marble at no cost, there are four primary acts as a function
 of the observation. In fact, we need only consider two of these acts since f1 dominates f2 whenever the
 observed marble is red. If the  rst marble is blue, then clearly the minimax solution is to choose f1 and f2
 with equal probability, leading to expected negative income of 2 in each state. The decision maker should
 be willing to pay as much as 23 to observe the  rst marble.
 Now consider a decision maker using minimax regret in the same situation. The regret table is
 RR BR BB
 f1 0 0 4
 f2 2 4 0
 The regret from the act f =  f1 + (1  )f2 in each state is
 L(f ;RR) = 1  ; L(f ;BR) = 2 2 ; L(f ;BB) = 2 :
 The minimax regret strategy is to randomize equally between the two acts, yielding a maximum expected
 regret of 2.
 Given the opportunity to observe the  rst marble, a cursory examination of the payo table shows that
 zero regret can be achieved in state RR, but will be unchanged in the other two. The optimal strategy will
 be to randomize equally between the act that chooses f1 always and the act that chooses f1 only when the
 observed marble is red. The maximum regret is still 2, so the decision maker should be unwilling to pay any
 positive amount for the observation based on the minimax regret criterion. Parmigiani (1992) claims that
 this example shows minimax regret is vulnerable to the same criticism as minimax negative income. In the
 next section, I shall rebut this claim while providing some insight into when these two decision rules will
 ignore available information.
 3 The Value of an Experiment
 The phenomenon demonstrated by these examples can be well-understood through the correspondence
 between the decision maker?s problem and a two player zero sum game played between nature and the
 decision maker (Wald, 1950). Consider a decision problem with a  nite collection of states  and a  nite
 number of primary acts. Imagine the corresponding payo table, either a negative income or a regret table.
 If this is considered as a zero sum game where nature earns the negative of the decision maker?s payo ,
 then it is well-known that the Nash equilibria of the game share a common value, and this is the value the
 decision maker will achieve through application of a minimax decision rule.
 Let  = ( d;  n) denote an equilibrium strategy pro le in the game described above, and let   denote
 the collection of states that nature plays with positive probability under  . Consider a random variable X
 taking values in some arbitrary measure space (S;S); we shall call such a random variable an experiment
 when the decision maker can observe its value prior to acting. We obtain the following result.
 Proposition 1. In the decision problem described above, the experiment X has no value to a minimax
 decision maker if there exists an equilibrium strategy pro le  such that
 P(X 2 S j!1) = P(X 2 S j!2)
 for all !1; !2 2   and for all S 2 S.
 If there are only two primary acts, and the experiment X has no value, then such an equilibrium strategy
 pro le necessarily exists.
 4
Proof. Observe  rst that the value of the game to the decision maker can be no lower with the experiment
 since the decision maker may always choose to ignore it. If we have P(X 2 S j!1) = P(X 2 S j!2) for all
 !1; !2 2   and all S 2 S, then nature may play the strategy speci ed by  and be sure to obtain the same
 value from the game with or without the experiment X. The value of the game is therefore unchanged;
 hence, the attainable minimax value is unchanged with the introduction of the experiment.
 Conversely, suppose there are two primary acts and that after the introduction of the experiment, nature
 has a strategy  n that can ensure the same value of the game as before. Consider a minimal such strategy, in
 the sense that no strict subset of states played by nature with positive probability can be used by themselves
 to attain the same value. Nature could have played  n in the game without the experiment, obtaining a
 value at least as high; therefore, this strategy is part of an equilibrium strategy pro le ( d;  n) in the game
 without the experiment. However, if there exist !1; !2 2   and S 2 S with
 p1 = P(X 2 S j!1) 6= P(X 2 S j!2) = p2;
 then the decision maker can obtain a strictly higher payo if nature plays  n after the introduction of the
 experiment.
 To see why,  rst suppose that  n is minmal in the game without the experiment. Given there are two
 primary acts f1 and f2, we can relabel !1 and !2 such that the payo from f1 is at least as high as that from
 f2 in state !1, and the payo from f2 is at least as high as that from f1 in state !2; further, one of these
 inequalities is strict. If this is not the case, then  n is clearly not minimal (either !1 or !2 could be removed
 from the support of nature?s strategy). We can partition   into  1 and  2, with !1 2  1, !2 2  2, and
 P(X 2 S j! 2  1) 6= P(X 2 S j! 2  2):
 Without loss of generality, suppose the  rst probability is greater. Consider the strategy  (X)d which plays
 f1 if X 2 S and f2 if X 2 Sc, and note that for small enough  the strategy   
(X)
 d + (1  ) d will obtain a
 strictly higher payo than  d in response to  n.
 Alternatively, if  n is no longer minimal when the experiment is removed from the game, we can  nd a
 minimal equilibrium strategy  0n for nature to follow in the game without the experiment, which is supported
 on a strict subset of the states supporting  n. By the argument in the preceding paragraph, we cannot have
 !1; !2 2   0 and S 2 S with
 p1 = P(X 2 S j!1) 6= P(X 2 S j!2) = p2;
 otherwise the experiment allows the decision maker to make a strict improvement. However, by the argument
 in the  rst paragraph of this proof, this means that  0n is an equilibrium strategy of nature in the game with
 the experiment, guaranteeing the same payo . This contradicts the assumption that  n was minimal in the
 game with the experiment.
 I note in passing that the restriction to situations with two primary acts in the second part of the
 proposition is needed to rule out the situation depicted in the following regret table, where many (but not
 all!) informative experiments would be insu cient to cause a minimax agent to deviate from f3.
 !1 !2
 f1 0 10
 f2 10 0
 f3 1 1
 The key insight of this result is that an experiment will be worthless to a minimax agent if it does not
 discriminate between the states that nature plays in equilibrium. Put this way, it sounds rather unremarkable,
 but it explains the unwillingness of minimax agents to pay for information in the examples given previously.
 If nature makes use of only a small subset of available states in equilibrium, then the minimax agent?s
 decision process is entirely dominated by this subset of states, and experiments giving information about
 5
other states will have no value. In Savage?s example, nature has a dominant strategy in the negative income
 game, and consequently there can never be an experiment with value to a minimax negative income agent.
 In Parmigiani?s example, state RR is weakly dominated in the regret game, but not in the negative income
 game.
 Parmigiani (1992) proves a similar result under somewhat stronger conditions, and calls a decision rule
 ultrapessimistic about the experiment X if X is \relevant," yet has no value. An experiment is said to be
 relevant if there exist distinct states !1 and !2, and a set S 2 S, such that
 P(X 2 S j!1) 6= P(X 2 S j!2):
 As a corollary, it is shown that as long as there are at least three states, and nature has an equilibrium
 strategy that is supported on a strict subset of the state space, then there exist experiments for which the
 minimax rule is ultrapessimistic. The need for three distinct states stems from a feature of games in regret
 form: so long as the decision maker does not have a dominant strategy, nature must employ a mixed strategy
 in equilibrium. Thus, nature?s equilibrium strategy is supported on at least two states, and ultrapessimism
 cannot appear in the regret form of a decision problem with two states.
 This hints at a more signi cant distinction between minimax negative income and minimax regret. In
 the negative income game, nature may have strictly dominated strategies, and information in favor of these
 states will always be worthless. In a regret form game, no state ever represents a strictly dominated strategy
 for nature. This ensures that valuable experiments always exist, and further, a strong enough signal in favor
 of a given state will decrease the regret obtained in that state.
 Proposition 2. Consider a decision problem with  nitely many states and  nitely many primary acts. If
 the minimax regret is greater than zero, there exists an experiment with strictly positive value to a minimax
 regret agent.
 Note that the experiment need not reveal the true state with certainty. Further, Savage?s (1954) analysis
 of partition problems demonstrates that repeated observation will cause the minimax regret to approach
 zero.
 Proof. Let L denote the maximum regret from any act in any state, and let L denote the minimax regret
 of the decision problem in the absence of an experiment. Label the states of nature !1; !2; :::; !n, and let X
 be such that
 P(X = i j!i) =
 L L 
L
 ; P(X = i j!j ; j 6= i) =
 L 
(n 1)L
 :
 It is a simple exercise to verify that this experiment enables a strict improvement in minimax regret, and
 the signal i must increase the probability that the optimal action in state i is chosen.
 I contend that the spirit of Savage?s objection to minimax negative income lies in the existence of decision
 problems for which no such experiment can be devised, and in this sense, the critique clearly does not apply
 to the minimax regret decision rule. Indeed this claim is bolstered by the observation that a Bayesian decision
 maker is vulnerable to the charge of ultrapessimism as de ned above. Recall the payo table from Savage?s
 example.
 B1 B2
 f1  1  1
 f2  10 1
 Given the same experiment X described in section 2, a Bayesian with a prior that assigns equal probability to
 B1 and B2 will place zero value on the experiment so long as g  211 . Clearly, experiments with
 2
 11  g <
 1
 2
 are relevant according to the de nition given by Parmigiani (1992), so this would have us label the Bayesian
 as ultrapessimistic. Further, recall from our previous analysis that for this range of g, a minimax regret agent
 attaches a strictly positive value to the experiment. The existence of \relevant" but worthless experiments
 6
for a minimax regret agent fails to distinguish minimax regret from other decision rules. The existence of
 decision problems where no experiment is valuable stands out as a property of minimax applied to negative
 income.
 4 Discussion
 The analysis from the previous section suggests that the relevance of an experiment to a decision maker
 is best viewed in the context of the decision rule being applied: we could simply say that an observation
 is relevant if it has value to the decision maker, given the decision rule. The real question becomes what
 observations are relevant. For any pair of decision rules we have considered, examples can be constructed
 where an observation is relevant to one but not the other, or vice versa.
 Minimax applied to negative income has the peculiar feature of ignoring any conceivable experiment
 in some decision problems. Other examples of this are presented by Hodges and Lehmann (1950) and by
 Radner and Marschak (1954). Hodges and Lehmann examine the problem of estimating the parameter of
 a binomial distribution. Under one loss function they consider, the minimax negative income estimate is
 constant despite any observations made. Similarly, Radner and Marschak posit a hypothetical gamble on
 the outcome of  ipping a weighted coin. They show that a decision maker applying a minimax negative
 income rule never observes more than one  ip of the coin to obtain information, even if doing so is nearly
 costless. More recently, Manski (2004) has noted this problem in the context of treatment choice. In all of
 these examples, the minimax regret rule avoids the pitfall of failing to utilize relevant information; Savage
 (1954, p. 200) even addresses the example of Hodges and Lehmann (1950) directly.
 The Hurwicz criterion has been proposed as a way to address the pessimism of minimax applied to
 negative income. In its general form, the criterion suggests choosing an action f to maximize
 H(f) =  (sup
 Bi
 I(f ;Bi); inf
 Bi
 I(f ;Bi))
 where  is any monotonically increasing function of both arguments, and fBig is a partition of the state space.
 Unfortunately, as shown by Radner and Marschak (1954), the Hurwicz criterion falls victim to essentially the
 same problem as the minimax negative income rule. A small set of states can entirely dominate the decision
 making process, and there are examples where no conceivable experiment is relevant. In the simplest case,
 suppose there are three possible events, B1, B2, and B3. Suppose that if B1 obtains, then no matter what
 action is chosen, the payo will be higher than in all other events, and suppose that if B3 obtains, then no
 matter what action is chosen, the payo will be lower than in all other events. The Hurwicz criterion then
 simpli es to choosing an action which maximizes
 H(f) =  (I(f ;B1); I(f ;B3));
 and the consequences of an action in event B2 are completely ignored, even if available information strongly
 suggests that B2 obtains.
 In the context of treatment choice, Stoye (2009) has renewed concerns about the pessimism of minimax
 applied to regret. Speci cally, he examines the way minimax regret makes use of the information provided
 by covariates,  nding that in general, minimax regret is achieved by examining each value of the covariates
 separately. Pooling to any extent is never optimal by this criterion. Stoye?s Proposition 4 asserts that with
 a large enough covariate space|in particular, covariates taking uncountably many values|minimax regret
 will recommend a \no-data" rule. However, this result should be unsurprising. Given the state space under
 consideration, nature is free to independently choose a di erent outcome distribution for uncountably many
 values of the covariate. Under these circumstances, a Bayesian estimator can be shown to be inconsistent
 essentially everywhere1. A  nite sample cannot provide meaningful information without some restriction
 1Consistency in this context means that the estimator will converge in probability to the true parameter value. A pair
 comprised of a prior distribution on the parameter space, and a true value of the parameter, is consistent if that particular prior
 will converge in probability to that particular parameter value as the amount of available data increases. Freedman (1965) gives
 7
of the state space. This example hardly constitutes a criticism of minimax regret; rather, it acknowledges
 fundamental limitations of statistical inference.
 Stoye?s broader critique relates to the implicit assumption made by the minimax regret procedure that
 the outcome distributions are independent across each value of the covariates. I must also object to this
 criticism as the appeal of his argument rests on our intuition that such distributions should be related. It
 strikes me as improper to critique a procedure designed to eliminate the use of subjective prior information
 on the basis of its disagreement with subjective prior information. Stoye (2009) has himself suggested the
 possibility of incorporating such information into the minimax regret procedure by suitably restricting the
 state space under consideration. I emphasize that I do not wish to dismiss these concerns regarding minimax
 applied to regret. Indeed, they represent important challenges that must be addressed in any application of
 the procedure. My point is that far from being fundamental  aws, these issues represent a cautionary note
 on the proper implementation of minimax regret.
 Savage (1954, p. 203) acknowledged a distinct but related criticism of minimax decision rules, derived
 from another example given by Hodges and Lehmann (1950); the example again concerns estimating the
 parameter of a binomial distribution. If squared error is used as the loss function, then the minimax regret
 estimate su ers higher loss than the maximum likelihood estimate over most of the parameter space. As the
 number of trials grows without bound, the region in which minimax regret is superior becomes vanishingly
 small. In this example, minimax appears overly concerned with the possibility that the parameter lies in a
 small neighborhood of 12 , and it sacri ces performance elsewhere to reduce the risk of loss in this part of
 the parameter space. This phenomenon could also be described as pessimism. Stoye (2009) notes what I
 believe to be a related example in his concluding remarks, where a sample that is heavily skewed towards
 one treatment leads to a counter-intuitive recommendation.
 This may be an inevitable consequence of ensuring \uniform" performance across all possible states. In
 the case of the binomial distribution, it appears to be a result of the relatively high variance of a binomial
 random variable with parameter close to 12 . This feature of the binomial distribution renders it more
 di cult to estimate the parameter in this region, and obtaining uniform performance necessitates sacri cing
 performance in regions with lower variance. One might consider rescaling the loss function to obtain a
 more desirable estimate. Indeed, the choice of loss function for parameter estimation seems to have received
 little attention, yet it is fundamental to the behavior of the minimax regret procedure. With modern
 computational tools, it may be feasible to consider minimax regret estimates over a range of loss functions,
 providing another potential avenue for future research.
 References
 Freedman, D. (1965), \On the Asymptotic Behavior of Bayes Estimates in the Discrete Case II." The Annals
 of Mathematical Statistics, 36, 454{456.
 Hodges, J. and E. Lehmann (1950), \Some Problems in Minimax Point Estimation." Annals of Mathematical
 Statistics, 21, 182{197.
 Manski, C. (2004), \Statistical Treatment Rules for Heterogeneous Populations." Econometrica, 72, 1221{
 1246.
 Parasarathy, K. (1967), Probability Measures on Metric Spaces. Academic Press.
 Parmigiani, G. (1992), \Minimax, Information, and Ultrapessimism." Theory and Decision, 33, 241{252.
 an example with countably many parameters where the set of consistent prior-parameter pairs is of category 1. More generally,
 as a consequence of well-known results in measure theory (See Parasarathy, 1967, p. 29) and topology (See Schaefer, 1966, p.
 23), all measures on complete, separable metric spaces are tight, and hence put probability one on a countable union of compact
 sets. Moreover, all locally compact Hausdor topological vector spaces are  nite-dimensional. This implies that all measures
 on an in nite-dimensional Hausdor topological vector space assign probability one to a category 1 (\meagre") set. Therefore,
 with in nitely many parameters, a Bayesian estimator is inconsistent outside an insigni cant portion of the parameter space.
 8
Radner, Roy and Jacob Marschak (1954), \Note on Some Proposed Decision Criteria." In Decision Processes
 (R. Thrall, C. Coombs, and R. Davis, eds.), 61{69, John Wiley and Sons.
 Savage, L. (1954), The Foundations of Statistics, second edition. Dover Publications Inc. 2nd Ed. published
 1972.
 Schaefer, H. (1966), Topological Vector Spaces. MacMillan.
 Stoye, J. (2009), \Minimax Regret Treatment Choice with Finite Samples." Journal of Econometrics, 151,
 70{81.
 Wald, A. (1950), Statistical Decision Functions. John Wiley and Sons.
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