The well-known Blackwell's theorem states the equivalence of statistical informativeness and economic valuableness. Celen (2012) generalizes this theorem, which is well-known for subjective expected utility (SEU), to maxmin expected utility (MEU) preferences. We demonstrate that the underlying definition of the value of information used in Celen (2012) is in contradiction with the principle of recursively defined utility. As a consequence, Celen's

framework features dynamic inconsistency. Our main contribution consists in the definition of a value of information for MEU preferences that is compatible with recursive utility and thus respects dynamic consistency

Heyen, Daniel

Wiesenfarth, Boris R.

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                                                          U n iv e rs i t y  o f  H e i de l b er g  Discussion Paper Series    No. 572 482482 Department of Economics Informativeness of Experiments for MEU – A Recursive Definition  Daniel Heyen and  Boris R. Wiesenfarth October 2014 Informativeness of Experiments for meu –A Recursive DefinitionDaniel Heyen ∗Department of EconomicsHeidelberg UniversityBoris R. Wiesenfarth†Department of EconomicsHeidelberg UniversityOctober 9, 2014Abstract The well-known Blackwell’s theorem states the equivalence of sta-tistical informativeness and economic valuableness. C¸elen (2012) generalizesthis theorem, which is well-known for subjective expected utility (seu), tomaxmin expected utility (meu) preferences. We demonstrate that the under-lying definition of the value of information used in C¸elen (2012) is in contradic-tion with the principle of recursively defined utility. As a consequence, C¸elen’sframework features dynamic inconsistency. Our main contribution consists inthe definition of a value of information for meu preferences that is compatiblewith recursive utility and thus respects dynamic consistency.Keywords Value of information; Maxmin expected utility; Recursive utilityJEL D81; D83; D841 IntroductionFor decades economists have been studying the relationship between decision-making under uncertainty and the so-called value of information. A famousand well-known result in this context is Blackwell’s theorem (Blackwell 1953)stating that an experiment is more valuable than another if and only if thesame experiment is more informative than the latter. In order to obtain this∗Corresponding Author: heyen@eco.uni-heidelberg.de†Boris.Wiesenfarth@awi.uni-heidelberg.de1equivalence (e.g. Cre´mer 1982), a standard assumption has been that decision-makers are subjective expected utility (seu) maximizers, cf. Savage (1954).The objective of C¸elen (2012) is to extend the Blackwell theorem to meupreferences, which were axiomatized by Gilboa and Schmeidler (1989). In thisnote we demonstrate that C¸elen’s proof relies on a value of information for meupreferences that is not defined via backward induction and thus is incompatiblewith the intertemporal extension of meu by Epstein and Schneider (2003). Inparticular, optimal strategies in C¸elen’s framework prescribe decisions condi-tional on signal realizations that an meu decision-maker will not find optimalto adhere to once those signal realizations have been observed. In this sense,C¸elen’s framework features dynamic inconsistency. Our contribution is to de-fine a value of information that is compatible with the recursive intertemporalformulation of meu preferences by Epstein and Schneider (2003).2 Framework and definition of the value of informationin C¸elen (2012)In the following, we adopt C¸elen’s framework and notation. Let Ω := {ω1, . . . , ωn}be the finite set of states and X := {a1, . . . , aχ} the finite set of actionsavailable to a decision-maker. Moreover, let ∆(Ω) and ∆(X) be the set ofall probability distributions defined on Ω and X, respectively. Let furtheru : Ω×X → R be a utility function and u with uij = u(ωi, aj) the correspond-ing utility matrix. An seu decision-maker is characterized by (pi,u), wherepi ∈ ∆(Ω) is a prior over the states.An experiment is a tuple (S,p) with the signal space S = {s1, . . . , sσ} andthe Markov matrix p with pij = Pr(sj|ωi) for sj ∈ S. C¸elen introduces astrategy as a vector valued mapping f : S → ∆(X), thus characterizing all(mixed) actions the decision maker plans to take after observing certain signalrealizations s. The σ×χ-matrix f is defined such that (fi1, · · · , fiχ) := f(si).In this framework, C¸elen determines the value of the experiment (S,p) fora given strategy f asUf(pi,u)(S,p) =∑jPr(sj)∑iPr(ωi|sj)∑kfjku(ωi, ak) (1)=∑j∑ipijpii∑kfjkuik (by Bayes’ rule) . (2)With a strategy f ∗ maximizing (2), C¸elen defines U∗(pi,u)(S,p) = Uf∗(pi,u)(S,p) asthe value of the experiment for an seu decision-maker.2Building on this, C¸elen extends the definition of the value of an experimentto the class of meu preferences. For that purpose, he characterizes an meudecision-maker by (A, u), where A ⊂ ∆(Ω) is a convex and compact set ofpriors. As a counterpart of U∗(pi,u)(S,p), he definesW∗(A,u) = maxfminpi∈AUf(pi,u)(S,p) (3)as the value of an experiment (S,p) for an meu decision-maker. It is expression(3) that C¸elen relies on in his proof of the generalized Blackwell’s theorem.3 A recursively defined MEU value of informationIt is insightful to note that C¸elen’s framework essentially constitutes an in-tertemporal setting with two periods. In the second period, after observinga signal realization, the decision maker takes a (mixed) action. In the firstperiod, before observing the signal realization, the value of the experiment(S,p) is determined. C¸elen accounts for the intertemporal structure insofar ashe considers strategies, that is complete contingent plans for appropriate playafter observing signal realizations.His formulation, however, is in contrast to the usual intertemporal formu-lation of meu preferences that is provided by Epstein and Schneider (2003).One of the main characteristics of their recursive definition of intertemporalmeu is the compatibility with backward induction. We follow the approachin Epstein and Schneider (2003) and develop an alternative definition of thevalue of information for meu preferences.According to backward induction, the first step to define a value of informa-tion is to determine the value of the final decision given that the decision makerchooses an optimal action for all possible signal realizations sj, j = 1, . . . , σ.For meu preferences this value isV (Mj) = maxg∈∆(X)minµ∈MjEµ[u](g) , (4)where Eµ[u](g) =∑k,i gkµi uik denotes the expected utility under action g ∈∆(X) and ex-post belief µ. The optimal action is determined considering theworst posterior µ ∈Mj. Formally, the set of posteriors isMj = {pi(·|sj) : pi ∈ A},where pi(·|sj) denotes the conditional probability of the prior pi ∈ ∆(Ω) giventhe signal sj. We obtain pi(·|sj) via Bayes’ rule and update each prior pi inthis way.11Epstein and Schneider (2007) show that further restrictions on the setM can be made.For the sake of simplicity, you may think of full Bayesian updating.3Building on (4), we can define the value of the experiment (S,p) for meupreferences asV(A,u) = minpi∈A∑i,jpiipijV (Mj) . (5)As usual, the value of information is obtained by taking the expectation overall possible signal realizations. Due to meu preferences the value of the ex-periment is the worst of those expectations. This alternative way of definingthe value of an experiment is in line with the intertemporal model of recursiveutility under multiple priors as introduced in Epstein and Schneider (2003,2007).The key characteristic of (5) is that optimal actions are determined withthe maxmin rule for each signal realization sj individually. In particular, theworst posterior in (4) in general depends on the signal realization sj. Thisis in contrast to (3). By following the derivation of the seu counterpart,essentially the step from (1) to (2), C¸elen silently assumes that the worstprior from the ex-ante perspective coincides with the preimage of all worstposteriors, irrespective of the signal realization. For the seu decision-makerthis argumentation is innocent as there is a unique prior, and thus a uniqueposterior as well. For the meu decision-maker, however, this argument is inconflict with backward induction.In the appendix, we demonstrate that the conflict of C¸elen’s frameworkwith intertemporal recursive utility can be made even more concrete. Weprovide an example in which the optimal strategy derived in C¸elen’s frameworkprescribes actions that are different from what an meu decision-maker willactually do after observing those signals realizations.2 This supports our claimthat the value of information for meu preferences should be defined by (5).By construction, our definition of the value of information is compatible withdynamic consistency.2One could think that the reason we observe this form of dynamic inconsistency is themissing assumption of rectangularity of the prior set, a key assumption in Epstein andSchneider (2003) to ensure dynamic consistency within an intertemporal setting of recursiveutility. But this is not the case. Even though C¸elen’s setting is not fully transferable to thesetting of Epstein and Schneider, in particular the analysis in Epstein and Schneider (2007)suggests that rectangularity is no issue in this setting, simply because the learning processis defined via conditional one-step-ahead conditionals, as required by Epstein and Schneider(2003). The reason for the violation of dynamic consistency in C¸elen’s framework is thatintertemporal utility is defined in a non-recursive way, thus incompatible with dynamicconsistency right from the start.44 Results and DiscussionWe have shown that C¸elen’s proof of Blackwell’s theorem only applies to avalue of information that is defined in a non-recursive utility framework. Wehave offered a definition for the value of information derived via backwardinduction, thus compatible with the dynamic consistent intertemporal axiom-atization of Epstein and Schneider (2003). Consequently, we argue that theproof of Blackwell’s theorem should deal with expression (5) as the definitionof the value of information for meu preferences. This proof is still pending.AcknowledgementsWe would like to acknowledge helpful comments by Ju¨rgen Eichberger, Jean-Philippe Lefort and Bog˘ac¸han C¸elen. The first author acknowledges the sup-port of the German Science Foundation, grant no. GO 1604/2-1.ReferencesBlackwell, D., 1953. Equivalent comparison of experiments. The Annals ofMathematical Statistics 24, 265–272.C¸elen, B., 2012. Informativeness of experiments for meu. Journal of Mathe-matical Economics 48, 404–406.Cre´mer, J., 1982. A simple proof of blackwell’s comparison of experimentstheorem. Journal of Economic Theory 27, 439–443.Epstein, L.G., Schneider, M., 2003. Recursive multiple-priors. Journal ofEconomic Theory 113, 1–31.Epstein, L.G., Schneider, M., 2007. Learning under ambiguity. Review ofEconomic Studies 74, 1275–1303.Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with a non-uniqueprior. Journal of Mathematical Economics 18, 141–153.Savage, L.J., 1954. The foundations of statistics. Wiley, New York.5A Example demonstrating the dynamic inconsistencyof C¸elen’s frameworkWe restrict the number of states of the world Ω = {ω1, ω2}, actions X = {a1, a2} and signalrealizations S = {s1, s2} to two. Payoffs are specified by u11 = 1, u12 = −1, u22 = 2 andu21 = 0. This is a simple example of a setting in which the decision maker wants to learnthe true ω because action a1 is optimal if ω = ω1 and action a2 is optimal if ω = ω2.For the signal likelihood λ = p11 = p22 we assume 1/2 < λ < 3/4 and specify the set ofpriors as A = {(pi1, 1− pi1) : 1/4 ≤ pi1 ≤ 3/4}.Using equation (2), we obtain that the optimal strategy f∗ in C¸elen’s framework isf∗(s1) = (1, 0) , f∗(s2) = (1/2, 1/2) . (6)In words, the optimal strategy in the C¸elen framework consists of taking action a1 if s = s1and mixing over actions a1 and a2 with equal weights if s = s2.Next, we demonstrate that an meu decision-maker operating with the principle of back-ward induction would deviate from f∗ as soon as the signal materializes. The decision ruleafter observing a signal realization sj is given in (4), where g is a randomization over actionsa1 and a2, and Mj ⊂ ∆(Ω) is the set of posteriors that depends on the set of priors A, thelikelihood p, and the signal realization sj observed. After simple algebra we getg∗(s1) = (3/4, 1/4) and g∗(s2) = (3/4, 1/4) . (7)In words, the optimal action of the meu decision-maker, both after receiving s = s1 ands = s2, is to mix over actions a1 and a2 with the ratio 3 to 1. This is in contrast to thebehavior prescribed in (6). Our example thus demonstrates the dynamic inconsistency inC¸elen’s framework.6

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