aa r X i v : . [ ec on . T H ] D ec A Theory of Updating Ambiguous Information ∗ Rui Tang † December 29, 2020
Abstract
We introduce a new updating rule, the conditional maximum likelihood rule (CML)for updating ambiguous information. The CML formula replaces the likelihood termin Bayes’ rule with the maximal likelihood of the given signal conditional on thestate. We show that CML satisfies a new axiom, increased sensitivity after updating,while other updating rules do not. With CML, a decision maker’s posterior isunaffected by the order in which independent signals arrive. CML also accommodatesrecent experimental findings on updating signals of unknown accuracy and has simplepredictions on learning with such signals. We show that an information designer canalmost achieve her maximal payoff with a suitable ambiguous information structurewhenever the agent updates according to CML.
Keywords : ambiguous information; conditional maximum likelihood; non-dilation;increased sensitivity after updating; under-reaction to ambiguous information
JEL Codes : D01, D81, D93 ∗ I am deeply indebted to Faruk Gul, Pietro Ortoleva and Wolfgang Pesendorfer for their invaluable advice,guidance and encouragement. I thank Roland Bénabou, Sylvain Chassang, Xiaoyu Cheng, Xiaosheng Mu,John K.-H. Quah, Satoru Takahashi, Mu Zhang and seminar participants at Princeton MicroeconomicTheory Student Lunch Seminar for helpful comments and discussions. All errors are my own. † Department of Economics, Princeton University, [email protected] Introduction
In decision theory, the term ambiguity refers either to an event with unknown probabilityor to information that has multiple probabilistic interpretations. Since the seminal workof Ellsberg (1961), a variety of models have been proposed to rationalize decision makers’(henceforth DM) choices over bets on ambiguous events. In contrast, relatively few papersfocus on updating ambiguous information. The growing literature on mechanism andinformation design under ambiguity has highlighted the importance of the latter topic.In many applications (e.g., Bose and Renou 2014 and Beauchêne, Li, and Li 2019),the DM’s prior over the payoff-relevant state space is unambiguous but the information hereceives is ambiguous. Our focus in this paper is on such applications. As in most existingmodels of choice under uncertainty, our DM is a max-min expected utility maximizer. Thatis, if the DM has a set of priors, he evaluates each ambiguous prospect according to itsminimal expected utility over all possible priors. In this paper, we offer an alternativeto full-Bayesian updating (henceforth FB) where the set of posteriors is the set of Bayes’updates of the priors.One consequence of FB is that a DM’s set of posteriors may be a superset of his setof priors. This dilation of beliefs may occur even when the DM has a single prior; thatis, even if there is no payoff-relevant ambiguity. Shishkin and Ortoleva (2019) (henceforthSO19) offer experimental evidence indicating that ambiguity averse DMs do not dilate afterreceiving ambiguous information. This finding is inconsistent with FB and other well-knownupdating rules such as the maximum likelihood rule (henceforth ML). Our new updatingrule, CML, does not create dilation and is consistent with experimental evidence presentedin recent papers. See, for instance, Gilboa and Schmeidler (1989), Schmeidler (1989),Maccheroni, Marinacci, and Rustichini (2006), Chew and Sagi (2008), Gul and Pesendorfer (2014),etc. For more discussions of dilation, see Wasserman and Kadane (1990).
2o see how our updating rule works, consider the following example: there is a coinand an urn. The urn contains 100 balls; each ball is either red or blue. It is not known howmany balls there are of either color. The coin is tossed, and a ball is drawn from the urn.Consider a bet that yields $20 to the DM if the coin shows tails and $0 if the coin showsheads. Before taking the bet, the DM receives a message about the outcome of the toss. Themessage is either “tails” or “heads”. The message matches the outcome of the coin toss if ared ball is drawn and does not match the outcome if a blue ball is drawn. After receivingthe message, the DM is asked if he would be willing to give up the bet in exchange for aobjective lottery that yields $20 with probability 1 / / S and a signal space Θ. We call S × Θ3he extended state space. A function mapping from the extended state space to payoffsis an extended act. A function mapping from the state space to payoffs is an act. Anextended evaluation function, V , describes the DM’s ex-ante preferences over extended acts.An evaluation function, V θ , describes the DM’s ex-post preferences over acts after observingsignal θ . Section 2 provides the necessary formalism and an axiomatic characterization ofmax-min (extended) evaluation functions.An updating rule specifies for each extended evaluation function V and each signal θ anevaluation function V θ . CML is defined as follows. Consider an extended evaluation function V , which admits a max-min representation by a set of priors P over S × Θ. Throughoutthe paper, we assume that P has no ambiguity over S , i.e., the DM has a single prior overthe state space. For any signal θ , the evaluation function V θ specified by CML admits anexpected utility representation by the posterior µ θ over S , where µ θ satisfies that µ θ ( s ) = max p ∈P p ( s, θ ) P s ′ ∈ S max p ∈P p ( s ′ , θ ) , ∀ s ∈ S. That is, following CML, a DM who has prior set P updates his belief over S to µ θ afterobserving θ . The maximization in the formula captures the conditional maximum likelihoodof the signal on each state.In Section 3, we introduce a new axiom, increased sensitivity after updating (henceforthISU), which distinguishes CML from existing updating rules. The idea of the axiom can beillustrated through the following example. Consider a state space containing three states { s , s , s } with prior (1 / , / , / { s , s } be realized. The Bayes’ posterior isgiven by (1 / , / s increases by ǫ >
0, his ex-ante expected payoff increases by 1 / ǫ , and his ex-postexpected payoff increases by 1 / ǫ . Obviously, he is more sensitive to payoff changes on s That is, for any extended act f ∗ , V ( f ∗ ) = inf p ∈P P ( s,θ ) ∈ S × Θ p ( s, θ ) f ∗ ( s, θ ) ! . P has no ambiguity over S if for any p, p ′ ∈ P , p ( { s } × Θ) = p ′ ( { s } × Θ) for all s ∈ S . If V θ admits an expected utility representation by µ θ , then for each act f , V θ ( f ) = P s ∈ S µ θ ( s ) f ( s ). { s , s } is realized. Axiom ISU is motivated by this observation.We state axiom ISU in our framework. Consider two extended acts f ∗ and g ∗ . Supposethat g ∗ ( s, θ ) > f ∗ ( s, θ ) for some ( s, θ ) ∈ S × Θ, and g ∗ ( s ′ , θ ′ ) = f ∗ ( s ′ , θ ′ ) for all ( s ′ , θ ′ ) differentfrom ( s, θ ). Let f and g be two acts satisfying f ( s ) = f ∗ ( θ, s ) and g ( s ) = g ∗ ( θ, s ) for each s ∈ S . An updating rule satisfies ISU if for any extended evaluation function V and anysignal θ , V ( f ∗ ) = V θ ( f ) implies V ( g ∗ ) ≤ V θ ( g ) , where V θ is the evaluation function specified by the updating rule. We interpret this axiomas follows. If the extended act changes from f ∗ to g ∗ , the DM has a payoff increase in ( s, θ ).The payoff increase should affect the DM more after θ is observed, since it rules out event S × (Θ \{ θ } ) and increases the chance that ( s, θ ) occurs. Therefore, the difference betweenthe evaluations of f ∗ and g ∗ at the ex-post stage should be higher than that at the ex-antestage. Since f has the same evaluation as f ∗ , g must have a higher evaluation than g ∗ . Thecondition V ( f ∗ ) = V θ ( f ) further rules out the possibility that different sensitivities to payoffchanges are caused by different ex-ante and ex-post utility levels. Axiom ISU is satisfied byCML but violated by other updating rules.In Section 4, we characterize CML. We show that an updating rule is CML if and only ifit satisfies axiom ISU, axiom independence of irrelevant signals, and axiom ratio consistency(Theorem 1). The latter two axioms are also satisfied by FB. When the DM’s extendedevaluation function is not observed, we provide a characterization of CML rationalizablebelief profiles. Specifically, a belief profile contains the DM’s ex-ante belief over the statespace and ex-post belief after observing each signal. We provide a sufficient and necessarycondition for the existence of an information structure under which the belief profile isconsistent with CML updating.In Section 5, we consider three applications of CML. First, we show that CML isdivisible. That is, the DM’s posterior is not affected by the order in which independent5ignals arrive. This makes CML suitable for the analysis of Wald-type problems underambiguity. Second, we relate CML to experimental evidence involving signals of unknownaccuracy. We show that a DM who updates according to CML under-reacts to ambiguousinformation, as observed by Liang (2019) (henceforth L19). We also investigate the learningbehavior of a DM who updates signals of unknown accuracy with CML.Finally, we apply CML to a Kamenica and Gentzkow (2011)-style information designproblem. We consider a designer who wants to persuade an agent to take an action thataffects the designer’s payoff. The designer can choose any ambiguous or unambiguousinformation structure. We show that the designer can almost achieve her maximal payoffwhen facing a CML agent. That is, for any ǫ >
0, there exists an ambiguous informationstructure such that the designer’s payoff is no less than her maximal payoff minus ǫ. Section 6 contains our comparisons of CML with existing alternatives. We show thatwithin the max-min expected utility framework, no updating rule satisfies axiom ISU if weallow multiple priors over the payoff-relevant state space. Hence, we show that the ISUassumption is appropriate only when ambiguity arises from information.
Related Literature.
A couple of experimental works directly test how subjects reactto ambiguous infomation, e.g., Epstein, Halevy et al. (2019), Kellner, Le Quement, and Riener(2019), etc. Our study is mostly related to the two experimental papers of SO19 and L19. Inboth SO19 and L19, the state space is finite. Both papers test how DMs react to ambiguousinformation when there is no ambiguity over the state space. Our theory can partiallyaccommodate their findings.Another stream of literature studies ambiguous information in game-theoretical frame-works, including Blume and Board (2014), Bose and Renou (2014), Kellner and Le Quement(2017), Beauchêne, Li, and Li (2019), Kellner and Le Quement (2018), etc. In this paper,we also apply CML to study the information design problem. We characterize the set of See Section 5.2 for the definition of under-reaction to ambiguous information. CML differs from this rule in two respects. First, CML satisfiesconsequentialism. Second, the CML posterior of a DM may not be contained in his FBposterior set (Example 1).A new updating rule, named the proxy rule, is proposed and axiomatized byGul and Pesendorfer (2018). Gul and Pesendorfer (2018) introduce axiom “not all newsis bad news” and show that the axiom is satisfied by the proxy rule but not FB or ML. Acommon feature of the proxy rule and CML is that information does not render unambiguousevents ambiguous. The key difference between CML and the proxy rule is that CML doesnot satisfy “not all news is bad news” while the proxy rule does not satisfy ISU. Consequentialism says that the ex-post evaluation of a choice does not depend on its payoffs on statesnot contained in the realized event. See Section 6.2 for a detailed discussion.
We define notations used in the paper. For an arbitrarily nonempty set H , let ∆( H ) denotethe set of finitely supported probability distributions over H , i.e., d ∈ ∆( H ) if and onlyif there exists a nonempty and finite subset H ′ ⊆ H such that d ( H ′ ) = 1 . If H is finite,let ∆ o ( H ) be the relative interior of ∆( H ), i.e., d ∈ ∆ o ( H ) if and only if d ( H ′ ) > H ′ ⊆ H . Any probability distribution over H ′ ⊆ H is considered as aprobability distribution over H that has support H ′ , and any probability distribution over H that has support H ′ is considered as a probability distribution over H ′ . For any two setsof probability distributions D and D ′ over H and any α ∈ [0 , , let α D + (1 − α ) D ′ be the α -convex combination of the two sets: α D + (1 − α ) D ′ = { αd + (1 − α ) d ′ : d ∈ D , d ′ ∈ D ′ } . For any function f : H → R and any d ∈ ∆( H ) , define E d ( f ) = P h ∈ H d ( h ) f ( h ) asthe d -evaluation of f . For any nonempty D ⊆ ∆( H ) , define E D ( f ) = inf d ∈D E d ( f ) as the D -evaluation of f . For any two functions f and g , we write f ≥ g if f ( h ) ≥ g ( h ) for all h ∈ H. For any subset H ′ ⊆ H, we write f = H ′ g if f ( h ) = g ( h ) for all h ∈ H ′ . f [ H ′ ] g denotes the function that agrees with f on H ′ and agrees with g on H \ H ′ . For any α ∈ [0 , ,αf + (1 − α ) g denotes the function satisfying that ( αf + (1 − α ) g )( h ) = αf ( h ) + (1 − α ) g ( h )for all h ∈ H. For any partition Π = { H , ..., H n } of H , f is said to be measurable withrespect to Π if f ( h ) = f ( h ′ ) whenever h and h ′ are in the same block of the partition.Let H = H × H . For any d ∈ ∆( H ) and any h ∈ H , if d ( h , h ) > h ∈ H , then d | h ∈ ∆( H ) denotes the conditional probability of d on h : d | h ( h ′ ) =8 ( h ′ , h ) / ( P ˆ h ∈ H d (ˆ h , h )) for all h ′ ∈ H . For any
D ⊆ ∆( H ) and any h ∈ H , define theset of conditional distributions of D on h as D| h = { d | h : d ∈ D , ∃ h ∈ H s.t. d ( h , h ) > } . For any function f : H → R , let f | h denote the function mapping from H to R satisfying that f | h ( h ) = f ( h , h ) , ∀ h ∈ H . For convenience, we write h for the singletonset { h } throughout the paper when there is no confusion. In this section, we introduce evaluation functions and extended evaluation functions, bothof which are assumed to have max-min representations. Each updating rule is defined as afunction that maps extended evaluation functions and signals to evaluation functions.Let S be the state space. We assume that S contains at least three states and is finite.Let Θ be the set of signals. Θ is countably infinite. Let K be the payoff space, which isan interval of R containing a nonempty interior. An act is a function f : S → K , and anextended act is a function f ∗ : S × Θ → K . We use x ∈ K to denote the constant act as wellas the constant extended act that equals x everywhere. Let F denote the set of all acts and F ∗ the set of all extended acts.The DM’s priors over S × Θ can be revealed from his choices over extended acts. Choicesover extended acts can be described by an extended evaluation function. An extendedevaluation function is a map V : F ∗ → K satisfying that for any x ∈ K and any f ∗ , g ∗ , h ∗ ∈F ∗ : (i) (Identity): V ( x ) = x . (ii) (Continuity): the sets { α ∈ [0 ,
1] : V ( αf ∗ + (1 − α ) g ∗ ) ≥ V ( h ∗ ) } and { α ∈ [0 ,
1] : V ( αf ∗ + (1 − α ) g ∗ ) ≤ V ( h ∗ ) } are closed. (iii) (Certainty Independence): ∀ α ∈ (0 , V ( f ∗ ) > V ( g ∗ ) if and only if V ( αf ∗ +(1 − α ) x ) > V ( αg ∗ + (1 − α ) x ). (iv) (Monotonicity): f ∗ ≥ g ∗ implies V ( f ∗ ) ≥ V ( g ∗ ).9 v) (Uncertainty Aversion): V ( f ∗ ) = V ( g ∗ ) implies V ( f ∗ + g ∗ ) ≥ V ( f ∗ ). (vi) (Finite Support): There exists a nonempty and finite subset Θ ′ ⊆ Θ such that V ( f ∗ ) = V ( g ∗ ) whenever f ∗ = S × Θ ′ g ∗ . (vii) (Non-ambiguity over State Space): If f ∗ and g ∗ are measurable with respectto the partition { s × Θ } s ∈ S and V ( f ∗ ) = V ( g ∗ ), then V ( f ∗ + g ∗ ) = V ( f ∗ ).Let V be the set of all extended evaluation functions. Conditions (i)-(v) are standard fora max-min expected utility representation. Together with condition (vi), we know that thereexists a unique nonempty, convex and closed set
P ⊆ ∆( S × Θ ′ ) such that V ( f ∗ ) = E P ( f ∗ )for each extended act f ∗ . P is said to represent V. Conditions (vii) ensures that P induces aunique prior over S , i.e., for each p, p ′ ∈ P and each s ∈ S, p ( s × Θ) = p ′ ( s × Θ). Such a setof priors over S × Θ is said to be simple. We discuss more general prior sets in Section 6.1.For any P that is simple, we can decompose it to a prior µ over S and a set of conditionalprobabilities { c t ( ·| s ) s ∈ S } t ∈ T ⊆ (∆(Θ)) S satisfying that:(1) µ ( s ) = p ( s × Θ) for each s ∈ S and each p ∈ P ,(2) ∀ t ∈ T , there exists p ∈ P such that µ ( s ) c t ( θ | s ) = p ( s, θ ), ∀ s ∈ S and ∀ θ ∈ Θ, and(3) ∀ p ∈ P , there exists t ∈ T such that p ( s, θ ) = µ ( s ) c t ( θ | s ), ∀ s ∈ S and ∀ θ ∈ Θ.We write P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ) if the above conditions hold. Based on the reformulation,a DM who has a simple prior set has no ambiguity over the state space but may havemultiple probabilistic interpretations over the signals, where each t ∈ T denotes one possibleinterpretation.After observing some signal θ , the DM’s ex-post beliefs can be revealed from his choicesover acts. The DM’s choices over acts are described by an evaluation function. An evaluationfunction is a map U : F → K satisfying identify, continuity, certainty independence,monotonicity and uncertainty aversion. These conditions ensure that U admits a max-min representation by a unique nonempty, convex and closed set of probability distributions See Gilboa and Schmeidler (1989) for more details. ⊆ ∆( S ). When Q = { µ ∗ } , we say that U is represented by µ ∗ . Q is the DM’s set of beliefsover the state space after observing signal θ . Let U be the set of all evaluation functions.We proceed to define updating rules. We only consider signals that happen with non-zero probabilities. For this purpose, we define non-null signals for an arbitrary extendedevaluation function V . ( s, θ ) ∈ S × Θ is said to be V null if V ( f ∗ ) = V ( g ∗ ) for any f ∗ , g ∗ ∈ F ∗ satisfying f ∗ = ( S × Θ) \ ( s,θ ) g ∗ . If ( s, θ ) is not V null, it is V non-null. It can be easily shownthat if V is represented by P , ( s, θ ) is V null if and only if for all p ∈ P , p ( s, θ ) = 0 . A signal θ is said to be V non-null if there exists s ∈ S such that ( s, θ ) is V non-null. The set of all V non-null signals is denoted by Θ V . V non-null states are defined similarly. In the examplesand applications, we omit states and signals that are not non-null.An updating rule is a functionΓ : [ V ∈V ( { V } × Θ V ) → U . For simplicity, for any V ∈ V and θ ∈ Θ V , we write V θ for Γ( V, θ ). To interpret, the extendedevaluation function V characterizes the DM’s prior beliefs over the state space and how heinterprets signals. Given V , an updating rule maps each observed signal to the DM’s ex-post evaluation function over acts, which reflects the DM’s ex-post beliefs after observingthe signal. We introduce CML in this section. Let the DM’s prior set over S × Θ be a simple set P .Consider a signal θ satisfying that p ( s, θ ) > p ∈ P and s ∈ S . When signal θ is That is, we do not model how DMs react to unexpected information. For theories of updating eventswith zero probability, see, for example, Ortoleva (2012). { µ θ } ⊆ ∆( S ), where µ θ ( s ) = max p ∈P p ( s, θ ) P s ′ ∈ S max p ∈P p ( s ′ , θ ) , ∀ s ∈ S. (1)Equivalently, if P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ), µ θ can be defined as µ θ ( s ) = µ ( s ) max t ∈T c t ( θ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈T c t ( θ | s ′ ) , ∀ s ∈ S. (2)We interpret the above formulation as the DM uses the signal’s maximal conditionalprobability on each state to update his belief. The DM’s CML posterior set is always asingleton and thus is not dilated. With CML, information does render unambiguous eventsambiguous.We note that the CML posterior may not be the Bayes’ posterior of any one of theDM’s priors over S × Θ, i.e., µ θ may not be contained in P| θ . This can be seen by thefollowing example. Example 1.
Let S = { s, s ′ , s ′′ } and Θ = { θ, θ ′ } . The DM’s prior set over S × Θ is asimple set P , which consists of all convex combinations of p and p . p and p are shown inTable 1. Table 1 p θ θ ′ s /
15 1 / s ′ /
15 1 / s ′′ /
15 4 / p θ θ ′ s /
15 1 / s ′ /
15 4 / s ′′ /
15 1 / p ∈{ p ,p } p ( s, θ ) = max p ∈{ p ,p } p ( s ′ , θ ) = max p ∈{ p ,p } p ( s ′′ , θ ) = 4 / , the DM’s CML posterior is (1 / , / , /
3) when signal θ is observed. Note that P| θ consists of all convex combinations of p | θ and p | θ , where p | θ = (4 / , / , /
9) and p | θ = (4 / , / , / / , / , /
3) is not a convex combination of p | θ and12 | θ . That is, (1 / , / , /
3) is not contained in P| θ . Since the DM updates his belief basedon the conditionally most likely scenario for each state, correlations among different states’conditional probabilities are neglected. Hence, the DM’s posterior can be outside of P| θ .The following is the formal definition of CML. Definition 1.
An updating rule is CML if for any V ∈ V represented by P and any θ ∈ Θ V ,the evaluation function V θ specified by the updating rule can be represented by µ θ , where µ θ satisfies condition (1). In this section, we introduce axiom ISU. This axiom distinguishes CML from existingupdating rules. To start with, we introduce two existing updating rules: FB and ML.Let V be the DM’s extended evaluation function represented by P . Let a V non-null signal θ be observed. (FB.) A DM who follows FB updates P to the posterior set P| θ . That is, given signal θ , he updates each prior p in P following Bayes’ rule. With FB, the DM’s ex-post evaluationfunction V θ is represented by the closure of P| θ . (ML.) A DM who follows ML first selects priors P θ,M from P that maximize theprobability of S × θ : P θ,M = { p ∈ P : p ( S × θ ) ≥ p ′ ( S × θ ) , ∀ p ′ ∈ P} . Then, he updates each prior in P θ,M following Bayes’ rule, which leads to the posterior set P θ,M | θ . With ML, the DM’s ex-post evaluation function V θ is represented by P θ,M | θ .We motivate axiom ISU in an unambiguous environment, in which the DM updatesaccording to Bayes’ rule and is an expected utility maximizer. Let S = { s, s ′ } and Θ = If min p ∈P p ( S × θ ) >
0, then P| θ is nonempty, convex and closed. If min p ∈P p ( S × θ ) = 0, then P| θ isnonempty and convex but may not be closed. θ, θ ′ } . The DM has a single prior p over S × Θ satisfying that p ( s, θ ) = p ( s ′ , θ ) = p ( s, θ ′ ) = p ( s ′ , θ ′ ) = 1 / . Consider an arbitrary extended act f ∗ . If the payoff of f ∗ on ( s, θ ) increasesby ǫ > , its evaluation increases by ǫ. When θ is observed, the DM’s posterior over { s, s ′ } is (1 / , / f ∗ | θ increases by ǫ ifthe payoff of f ∗ on ( s, θ ) increases by ǫ .The key observation from the above example is that the DM becomes more sensitiveto payoff changes on ( s, θ ) after signal θ is observed, since the realization of θ rules out event S × (Θ \ θ ), which makes payoff changes on ( s, θ ) more likely to occur. This argument doesnot rely on whether the information is ambiguous or not. Based on this observation, weintroduce axiom ISU. Definition 2.
An updating rule satisfies axiom ISU if for any extended evaluation function V , any θ ∈ Θ V and any f ∗ , g ∗ ∈ F ∗ with g ∗ ( s, θ ) > f ∗ ( s, θ ) and g ∗ = ( S × Θ) \ ( s,θ ) f ∗ , V ( f ∗ ) = V θ ( f ∗ | θ ) implies V ( g ∗ ) ≤ V θ ( g ∗ | θ ) . Since V ( f ∗ ) = V θ ( f ∗ | θ ) and the increase of payoff on ( s, θ ) affects the DM more after θ is observed, we have V ( g ∗ ) ≤ V θ ( g ∗ | θ ). However, the next example shows that axiom ISUis violated by both FB and ML. Example 2.
Let S = { s, s ′ } and Θ = { θ, θ ′ } . The DM’s prior set over S × Θ is asimple set P , which consists of all convex combinations of p and p . p and p are shown inTable 2. Table 2 p θ θ ′ s / / s ′ / / p θ θ ′ s / / s ′ / / f ∗ θ θ ′ s s ′ g ∗ θ θ ′ s / s ′ V be the DM’s extended evaluation function, represented by P . Consider extended14cts f ∗ and g ∗ , as shown in Table 2. We can verify that V ( f ∗ ) = E { p ,p } ( f ∗ ) = E p ( f ∗ ) = 9 / , and V ( g ∗ ) = E { p ,p } ( g ∗ ) = E p ( g ∗ ) = 5 / . Let signal θ be observed.First, assume that the DM updates according to FB. The DM’s posterior set over S consists of all convex combinations of p | θ = (1 / , /
4) and p | θ = (8 / , / f ∗ | θ as E { p | θ,p | θ } ( f ∗ | θ ) and evaluates g ∗ | θ as E { p | θ,p | θ } ( g ∗ | θ ) . We have E { p | θ,p | θ } ( f ∗ | θ ) = E p | θ ( f ∗ | θ ) = 9 / , and E { p | θ,p | θ } ( g ∗ | θ ) = E p | θ ( g ∗ | θ ) = 39 / . Next, assume that the DM follows ML. The DM selects the prior that maximizesthe probability of S × θ, which is p . Thus, his posterior set is { p | θ } . Again, his ex-postevaluations of f ∗ | θ and g ∗ | θ are given by E p | θ ( f ∗ | θ ) = 9 / E p | θ ( g ∗ | θ ) = 39 / . Since E p ( f ∗ ) = E p | θ ( f ∗ | θ ) and E p ( g ∗ ) > E p | θ ( g ∗ | θ ) , both ML and FB violate axiomISU. In contrast, CML satisfies axiom ISU, which is implied by the following proposition. Proposition 1.
Let V ∈ V , θ ∈ Θ V and V θ = Γ( V, θ ) , where Γ is CML. For any extendedacts f ∗ and g ∗ and any ( s, θ ) ∈ S × Θ , if g ∗ ( s, θ ) > f ∗ ( s, θ ) and g ∗ = ( S × Θ) \ ( s,θ ) f ∗ , then V ( g ∗ ) − V ( f ∗ ) ≤ V θ ( g ∗ | θ ) − V θ ( f ∗ | θ ) . Proposition 1 says that CML satisfies a stronger version of axiom ISU. If a DM updatesaccording to CML, he is always more sensitive to payoff changes on ( s, θ ) after θ is observed,no matter whether his ex-ante utility level equals the ex-post level or not.15 Characterization
In this section, we provide an axiomatic foundation for CML. We provide three axiomsto fully characterize CML, two of which are also satisfied by FB. The only axiom that isviolated by FB is axiom ISU. After that, we consider the situation in which a DM’s extendedevaluation function is not observable. Instead, we observe the DM’s prior belief over the statespace and posterior belief after observing each signal. We provide a sufficient and necessarycondition under which the DM’s belief profile can be rationalized by CML, i.e., there existsa set of interpretations of signals such that CML updating leads to the desired posterior ofthe DM under each signal.
First, we define some notations. For any two simple set of priors P and P ′ over S × Θ andany E ⊆ S × Θ , P and P ′ are said to agree on E , denoted by P ≈ E P ′ , if(1) for any p ∈ P , there exists p ′ ∈ P ′ such that for any E ′ ⊆ E, p ( E ′ ) = p ′ ( E ′ ) , and(2) for any p ′ ∈ P ′ , there exists p ∈ P such that for any E ′ ⊆ E, p ′ ( E ′ ) = p ( E ′ ).By the definition, P ≈ E P ′ means that the two prior sets induce exactly the same set ofdistributions over E . For any two extended evaluation functions V and V ′ and any E ⊆ S × Θ, V and V ′ are said to agree on E , denoted by V ≈ E V ′ , if for any extended act f ∗ and anypayoff x , V ( f ∗ [ E ] x ) = V ′ ( f ∗ [ E ] x ). Obviously, V and V ′ agree on E if and only if P and P ′ agree on E , where P and P ′ represent V and V ′ respectively.For any evaluation function U and any S ′ ⊆ S , U is said to be strictly increasing on S ′ if for any f ∈ F and any x, y ∈ K with x < y , U ( x [ S ′ ] f ) < U ( y [ S ′ ] f ). If U is representedby Q , we can show that U is strictly increasing on S ′ if and only if min µ ∈Q µ ( S ′ ) >
0. Weproceed to state the axioms.
Independence of Irrelevant Signals (Axiom IIS).
For any { V, W } ⊆ V and any θ ∈ Θ V , if V ≈ S × θ W , then V θ = W θ . V be represented by P and W represented by P ′ . Note that V ≈ S × θ W indicatesthat P and P ′ agree on S × θ. Hence, axiom IIS says that the DM’s posterior belief afterobserving signal θ only depends on the details of the DM’s priors on θ and is irrelevant withthe details of other signals. Obviously, this axiom is satisfied by FB since the FB posteriorset consists of each prior’s conditional distribution on S × θ when θ is observed. Ratio Consistency (Axiom RC).
For any { V, W } ⊆ V , any θ ∈ Θ V ∩ Θ W and any { s, s ′ } ⊆ S with s = s ′ , if V θ and W θ are strictly increasing on { s, s ′ } , and V ≈ { s,s ′ }× Θ W ,then for any { y, z, w } ⊆ K , V θ ( y [ s ] z [ s ′ ] w ) = w if and only if W θ ( y [ s ] z [ s ′ ] w ) = w. Again, let V be represented by P and W represented by P ′ . The primitive conditionsof axiom RC indicate that P and P ′ agree on { s, s ′ } × Θ. Axiom RC says that how theDM trades off payoffs on s and s ′ after observing signal θ is completely determined by thedetails of the DM’s priors on the two states. More concretely, let V θ be represented by Q θ and W θ represented by Q ′ θ . Since V θ and W θ are strictly increasing on { s, s ′ } , each posteriorin Q θ ∪ Q ′ θ assigns a positive probability to { s, s ′ } . The condition that V θ ( y [ s ] z [ s ′ ] w ) = w ifand only if W θ ( y [ s ] z [ s ′ ] w ) = w is equivalent to thatmax µ ∈Q θ µ ( s ) µ ( s ′ ) = max µ ′ ∈Q ′ θ µ ′ ( s ) µ ′ ( s ′ ) , and min µ ∈Q θ µ ( s ) µ ( s ′ ) = min µ ′ ∈Q ′ θ µ ′ ( s ) µ ′ ( s ′ ) , where the maximization and minimization are allowed to be positive infinite as thedenominator can be equal to zero. We note that FB also satisfies this axiom: with FB,the ratios of the two states’ posterior probabilities are completely determined by their ex-ante probability ratios on the realized signal. The last axiom is axiom ISU. increased Sensitivity after Updating (Axiom ISU).
For any V ∈ V , any θ ∈ Θ V and any { f ∗ , g ∗ } ⊆ F ∗ satisfying f ∗ ( s, θ ) < g ∗ ( s, θ ) and f ∗ = ( S × Θ) \ ( s,θ ) g ∗ , V ( f ∗ ) = V θ ( f ∗ | θ ) We prove that FB satisfies axiom RC in Appendix B. V ( g ∗ ) ≤ V θ ( g ∗ | θ ).The three axioms fully characterize CML. Theorem 1.
An updating rule is CML if and only if it satisfies axioms IIS, RC and ISU.
We sketch the proof of the sufficiency part of the theorem. Consider an extendedevaluation function V and let it be represented by P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ). Recall that µ is the DM’s prior over the state space, and { c t ( ·| s ) s ∈ S } t ∈ T are the DM’s interpretations ofsignals. Consider some θ ∈ Θ V and let V θ be represented by Q θ ⊆ ∆( S ).The most important step is to show that min µ ∗ ∈Q θ µ ∗ ( s ) ≥ µ ( s ) if µ ( s ) > t ∈ T c t ( θ | s ) = 1. To see this, consider an extended act f ∗ satisfying that(1) f ∗ ( s, θ ) = 0 , (2) f ∗ ( s, θ ′ ) = 1 for each θ ′ = θ and(3) f ∗ (ˆ s, ˆ θ ) = 0 for each (ˆ s, ˆ θ ) ∈ ( S \ s ) × Θ. The P -evaluation of f ∗ , i.e., V ( f ∗ ), is 0 since max t ∈ T c t ( θ | s ) = 1, which indicates that theminimal probability of s × (Θ \ θ ) is zero. For ǫ ∈ (0 , ǫ increase of payoff on ( s, θ )increases the P -evaluation of f ∗ by µ ( s ) ǫ . When θ is observed, the act f ∗ | θ yields payoff 0constantly. Hence, the P -evaluation of f ∗ equals to the Q θ -evaluation of f ∗ | θ . Note that an ǫ increase of payoff on ( s, θ ) for f ∗ increases the Q θ -evaluation of f ∗ | θ by min µ ∗ ∈Q θ µ ∗ ( s ) ǫ .Axiom ISU implies that min µ ∗ ∈Q θ µ ∗ ( s ) ≥ µ ( s ) . Given the above observation, the rest of the proof can be illustrated through thefollowing simple example, in which there is no ambiguity. General cases can be shownsimilarly.Let S = { s, s ′ , s ′′ } and Θ = { θ, θ ′ } ∪ { θ i } + ∞ i =1 . Consider P = ( µ, { c ( ·| ˆ s ) ˆ s ∈ S } ), where µ = (1 / , / , /
3) and c ( θ | s ) = c ( θ ′ | s ) = c ( θ | s ′ ) = c ( θ ′ | s ′ ) = c ( θ | s ′′ ) = c ( θ ′ | s ′′ ) = 1 / . When Here, we assume that both 0 and 1 are in the payoff space K . 0 and 1 can be replaced by any payoffs x and y satisfying x < y . is observed, we want to show that for each one of the DM’s posteriors, the probability ratiobetween states s and s ′ is equal to µ ( s ) c ( θ | s ) µ ( s ′ ) c ( θ | s ′ ) . (3)Consider the extended evaluation function V ′ , represented by P ′ = ( µ ′ , { c ′ ( ·| ˆ s ) ˆ s ∈ S } )where µ ′ ( s ) = µ ( s ) c ( θ | s ) = 1 / , µ ′ ( s ′ ) = µ ( s ′ )( θ | s ′ ) = 1 / , µ ′ ( s ′′ ) = 2 / ,c ′ ( θ | s ) = c ′ ( θ | s ′ ) = 1 , and c ′ ( θ | s ′′ ) = 1 / . We can verify that
P ≈ S × θ P ′ . By axiom IIS, V θ = V ′ θ . Let V ′ θ be represented by Q ′ θ . Wehave Q ′ θ = Q θ .Consider another extended evaluation function V ′′ , represented by P ′′ = ( µ ′′ , { c ′′ ( ·| ˆ s ) ˆ s ∈ S } )where µ ′′ (ˆ s ) = µ ′ (ˆ s ) , ∀ ˆ s ∈ S, and c ′′ ( θ | s ) = c ′′ ( θ | s ′ ) = c ′′ ( θ | s ′′ ) = 1 . Obviously, P ′ ≈ { s,s ′ }× Θ P ′′ . Note that for each ˆ s ∈ S , c ′′ ( θ | ˆ s ) = 1. Hence, if V ′′ θ is representedby Q ′′ θ , we have min µ ∗ ∈Q ′′ θ µ ∗ (ˆ s ) ≥ µ ′′ (ˆ s ) for all ˆ s ∈ S . This indicates that Q ′′ θ = { µ ′′ } . Since P ′ ≈ { s,s ′ }× Θ P ′′ , by axiom RC, we know that for each µ ∗ ∈ Q ′ θ ( Q θ ), the ratio between µ ∗ ( s )and µ ∗ ( s ′ ) is equal to the ratio between µ ′′ ( s ) and µ ′′ ( s ′ ), which is the desired ratio (3).For completeness, we show that any two out of the three axioms are not sufficient foran updating rule to be CML in Appendix B. In this section, we study the situation in which the DM’s extended evaluation function cannotbe observed. Let the DM’s ex-ante evaluation function be U . U describes the DM’s choicesover acts before any information. Let U be represented by µ ∈ ∆( S ) . We assume throughout19his section that µ ∈ ∆ o ( S ) . The DM’s choices over acts under signal θ is characterizedby the ex-post evaluation function U θ . Let { U θ } θ ∈ Θ ∗ be the DM’s set of ex-post evaluationfunctions, where Θ ∗ is a finite set of signals. U θ is represented by µ θ ∈ ∆( S ) . Given theprofile ( µ, { µ θ } θ ∈ Θ ∗ ) , we want to provide a sufficient and necessary condition under which theex-post beliefs of the DM are updated according to CML under some set of interpretationsof signals. Definition 3. ( µ, { µ θ } θ ∈ Θ ∗ ) is CML rationalizable if there exists a finite set of interpretations { c t ( ·| s ) s ∈ S } t ∈ T over Θ ∗ such that for each θ ∈ Θ ∗ , X s ′ ∈ S µ ( s ′ ) max t ∈ T c t ( θ | s ′ ) > , (4) and µ θ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈ T c t ( θ | s ′ ) , ∀ s ∈ S. (5)Condition (4) requires that all observed signals are not unexpected. Condition (5) saysthat the DM updates according to CML. The following example shows that not all beliefprofiles are CML rationalizable. Example 3.
Let S = { s, s ′ , s ′′ } and Θ ∗ = { θ, θ ′ } . Suppose that the DM’s priorover S is µ = (1 / , / , / µ θ = (1 / , / , /
3) under signal θ and µ θ ′ = (1 / , / , /
3) under signal θ ′ . We argue that ( µ, { µ θ , µ θ ′ } ) is not CML rationalizable.The reason is that the posterior probabilities of state s are too small compared to theposterior probabilities of state s ′ . When signal θ is realized, the conditional likelihoods ratioused for updating between s and s ′ is 1 : 3. Since the conditional maximum likelihood of θ on s ′ is at most 1, the conditional maximum likelihood of θ on s is at most 1 / . By a similarargument, the conditional maximum likelihood of θ ′ on s is at most 1 / . Obviously, this is If µ ( s ) = 0, we can simply delete state s and define the state space to be S \{ s } . If any posterior of theDM assigns positive probability to s , we can directly reject the hypothesis that the DM follows CML. Again, if U θ is represented by a non-singleton set Q θ , we can reject the hypothesis that the DM followsCML. θ and θ ′ on s is strictly less than one.The next theorem completely characterizes the set of CML rationalizable belief profiles.The characterization exactly rules out the situation in Example 3: conditional on any state s , the summation of the probabilities of all signals cannot be strictly less than one. Theorem 2. ( µ, { µ θ } θ ∈ Θ ∗ ) is CML rationalizable if and only if for each s ∈ S, X θ ∈ Θ µ θ ( s ) µ ( s ) max s ′ ∈ S ′ µ θ ( s ′ ) µ ( s ′ ) ! − ≥ . In this section, we give three applications of CML. First, we show that CML is divisible, i.e.,CML updating is unaffected by the order independent signals arrive. Second, we apply CMLto study how DMs update with signals of unknown accuracy. We show that CML predictsunder-reaction to such signals and illustrate how DMs learn with such signals. Finally,we apply CML to study an information design problem where a designer can introduceambiguous information structures to affect the action taken by an agent. We show that ifthe agent follows CML, the designer can almost achieve the maximal payoff.
An important question in the literature of belief updating is whether DMs’ beliefs are affectedby the order independent signals arrive. An updating rule is divisible, or path-independent,if the posterior set of the DM is unaffected by the order of independent signals. Cripps(2018) gives the representations of divisible rules when there is no ambiguity. In this section,we show that CML satisfies divisibility.For simplicity, let S , Θ and Θ ′ be finite. Θ and Θ ′ are two different sets of signals. Letthe DM’s prior over S be µ . Let the DM’s set of interpretations over Θ be { c t ( ·| s ) s ∈ S } t ∈ T ′ be { c t ′ ( ·| s ) s ∈ S } t ′ ∈ T ′ . The two signal sets Θ and Θ ′ areindependent: the DM’s set of interpretations over Θ × Θ ′ is given by { c ( t,t ′ ) ( ·| s ) s ∈ S } ( t,t ′ ) ∈ T × T ′ where for each ( t, t ′ ) ∈ T × T ′ , c ( t,t ′ ) ( θ, θ ′ | s ) = c t ( θ | s ) · c t ′ ( θ ′ | s ) for each θ ∈ Θ and θ ′ ∈ Θ ′ . Fix the two independent signal sets. We argue that CML updating is path-independent.First, assume that signals θ and θ ′ arrive simultaneously. The DM’s posterior µ θ,θ ′ is givenby µ θ,θ ′ ( s ) = µ ( s ) max ( t,t ′ ) ∈ T × T ′ c ( t,t ′ ) ( θ, θ ′ | s ) P s ′ ∈ S µ ( s ′ ) max ( t,t ′ ) ∈ T × T ′ c ( t,t ′ ) ( θ, θ ′ | s ′ ) , ∀ s ∈ S. Since c ( t,t ′ ) ( θ, θ ′ | s ) = c t ( θ | s ) · c t ′ ( θ ′ | s ), we have µ θ,θ ′ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) max t ′ ∈ T ′ c t ′ ( θ ′ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈ T c t ( θ | s ′ ) max t ′ ∈ T ′ c t ′ ( θ ′ | s ′ ) , ∀ s ∈ S. Based on the updating formula, the conditional likelihoods used for updating is themultiplication of the conditional maximum likelihoods of the two signals on the each state.Next, consider the case where signal θ is observed first. The DM updates his prior µ to µ θ : µ θ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈ T c t ( θ | s ′ ) , ∀ s ∈ S. Then, he observes signal θ ′ and updates µ θ to ( µ θ ) θ ′ :( µ θ ) θ ′ ( s ) = µ θ ( s ) max t ′ ∈ T ′ c t ′ ( θ ′ | s ) P s ′ ∈ S µ θ ( s ′ ) max t ′ ∈ T ′ c t ′ ( θ ′ | s ′ ) , ∀ s ∈ S. Obviously, µ θ,θ ′ and ( µ θ ) θ ′ are the same. Similarly, if signal θ ′ arrives before θ , the DM’sposterior remains to be µ θ,θ ′ . Hence, CML is divisible. SO19 and L19 study how DMs react to information of ambiguous accuracy. Some of theirexperimental findings are inconsistent with FB and ML under the max-min expected utilityframework. For example, the experimental evidence of SO19 suggests that ambiguity averseDMs may not dilate their prior when processing ambiguous information. In addition to the22on-dilation property, we show that CML can accommodate another experimental findingby L19: under-reaction to ambiguous information. We then discuss how DMs learn in thelong run with CML when information has ambiguous accuracy.We introduce the framework. Let S = { s , s } be the state space and Θ = { θ , θ } thesignal space. There are two possible levels of accuracy of the signals: H and L . We requirethat H, L ∈ (0 , H > L and H + L ≥ . The DM’s set of interpretations of signals aregiven by { c H , c L } where c H ( θ | s ) = c H ( θ | s ) = H and c L ( θ | s ) = c L ( θ | s ) = L. H + L ≥ s i is weakly increased after θ i is observed for each i ∈ { , } .Otherwise, we can swap the labels of the two signals.The first observation is that when H = 1 − L , the DM will maintain his prior over S upon receiving any signal with CML. Since given any signal θ i , the conditional maximumlikelihood of θ i on s i is H and the conditional maximum likelihood of θ i on s j ( j = i ) is 1 − L .The two conditional likelihoods cancel out. This coincides with the experimental finding ofSO19 that ambiguous averse DMs do not adjust their value of bets after such signals.Next, we introduce the definition of under-reaction to unambiguous information byL19. Consider acts f and g satisfying that f ( s ) = g ( s ) = 1 and f ( s ) = g ( s ) = 0 . Consider another conditional distribution c ( ·| s ) s ∈ S where c ( θ | s ) = c ( θ | s ) = H + L . A DM under-reacts to ambiguous information if when signal θ is observed, his ex-postevaluation of f under the interpretations { c H , c L } is strictly lower than that under { c } , andhis ex-post evaluation of g under the interpretations { c H , c L } is strictly higher than thatunder { c } . To interpret, observing signal θ is good news for f and bad news for g . Hence,after observing θ , the DM increases his evaluation of f and decreases his evaluation of g . If he under-reacts to ambiguous information, he increases less for his evaluation of f and decreases less for his evaluation of g when the interpretation of signals has multiplepossibilities. 23ML predicts under-reaction to ambiguous information when H + L > . To see this,let the DM’s prior over { s , s } be µ = ( α, − α ) , where α ∈ (0 , . Assume that signal θ isobserved. Given the set of interpretations { c H , c L } , the DM’s CML posterior µ θ satisfies µ θ ( s ) = αHαH + (1 − α )(1 − L ) ,µ θ ( s ) = (1 − α )(1 − L ) αH + (1 − α )(1 − L ) . Given the unambiguous interpretation { c } , the DM’s CML posterior µ ∗ θ satisfies µ ∗ θ ( s ) = α H + L α H + L + (1 − α )(1 − H + L ) ,µ ∗ θ ( s ) = (1 − α )(1 − H + L ) α H + L + (1 − α )(1 − H + L ) . To see that the DM always under-reacts to ambiguous information, note that the twoconditional likelihoods ratios satisfy H − L − ( H + L ) / − ( H + L ) / H − L − H + L (1 − L )(2 − H − L )= ( H − L )(1 − H − L )(1 − L )(2 − H − L ) < . Hence, the DM updates more when the information is unambiguous. The same argumentapplies for signal θ .We end this section by discussing the implications of CML on learning. Consider thesituation in which the DM’s set of interpretations of signals consists of c H and c L . Assumethat H + L >
1. Suppose that the true signal generating process is given by some c ∗ where c ∗ ( θ | s ) = c ∗ ( θ | s ) = λ ∈ (0 , . Fix the true state of the world. At each period, a signalfrom { θ , θ } is generated according to c ∗ independently. CML has simple predictions onlearning in such an environment.First, note that the DM always updates according to the conditional likelihood ratio H − L . Hence, the DM updates his belief as if the accuracy of signals is HH +1 − L > /
2. If c ∗ is24eakly consistent with the DM’s perception of the information, i.e., λ > / , the DM willfinally learn the true state of the world. In contrast, if λ < / , the DM will finally learnthe wrong state of the world. Hence, even with ambiguous information, as long as the DM’sinterpretations of signals are biased towards the true signal generating process, he will finallylearn the true state if he updates according to CML. The problem of information design is first studied by Kamenica and Gentzkow (2011), inwhich information structures are unambiguous. Beauchêne, Li, and Li (2019) study the casewhere the information designer can use ambiguous information structures, and the agentfollows FB. In this section, we consider ambiguous information structures and assume thatthe agent follows CML.Let a finite set S be the state space. A designer and an agent have a common prior µ ∈ ∆ o ( S ) . Let a finite set A be the set of all actions. The designer can design informationstructures for the agent. After observing a signal, the agent chooses an action. The utilitiesof the designer and the agent depend on both the true state and the action taken by theagent. Let v : S × A → R be the designer’s utility function. Let u : S × A → R be theagent’s utility function.Since the agent follows CML, the posterior set of the agent is always a singleton. Forany µ ∗ ∈ ∆( S ) , let BR ( µ ∗ ) be the set of actions that maximize the expected utility of theagent under µ ∗ , i.e., BR ( µ ∗ ) = { a ∈ A : X s ∈ S µ ∗ ( s ) u ( s, a ) ≥ X s ∈ S µ ∗ ( s ) u ( s, a ′ ) , ∀ a ′ ∈ A } . Let A ∗ ⊆ A be the set of all optimal actions of the agent, i.e., A ∗ = [ µ ∗ ∈ ∆( S ) BR ( µ ∗ ) . We impose the following assumption on A ∗ . ssumption 1. For each a ∈ A ∗ , there exists µ ∗ ∈ ∆ o ( S ) such that a ∈ BR ( µ ∗ ) . The assumption says that the agent’s optimal actions can be achieved by posteriors in theinterior of ∆( S ) . Note that this assumption is generically true.We next define ambiguous information structures. An ambiguous information structureis a tuple (Θ , { c t ( ·| s ) s ∈ S } t ∈ T ) where Θ is a finite set of signals and { c t ( ·| s ) s ∈ S } t ∈ T ⊆ (∆(Θ)) S is a finite set of signal generating systems. We require that for each θ ∈ Θ , max t ∈ T c t ( θ | s ) > s ∈ S . Let µ θ be the CML posterior of µ under this information structure whensignal θ is observed, i.e., µ θ satisfies condition (2).Consider an ambiguous information structure (Θ , { c t ( ·| s ) s ∈ S } t ∈ T ) and a set of actions { a θ } θ ∈ Θ , we say that G = (Θ , { c t ( ·| s ) s ∈ S } t ∈ T , { a θ } θ ∈ Θ ) is implementable if1. For each θ ∈ Θ , a θ ∈ BR ( µ θ ) .
2. For any t, t ′ ∈ T, X s ∈ S X θ ∈ Θ µ ( s ) c t ( θ | s ) v ( s, a θ ) = X s ∈ S X θ ∈ Θ µ ( s ) c t ′ ( θ | s ) v ( s, a θ ) . Condition 1 says that a θ is the agent’s optimal action when observing signal θ . Condition 2ensures the commitment power of the designer: she has no incentive to pick any particularsignal generating system since under each signal generation system, her expected payoff is thesame. If G = (Θ , { c t ( ·| s ) s ∈ S } t ∈ T , { a θ } θ ∈ Θ ) is implementable, let v ∗ ( G ) denote the designer’sexpected payoff under G, i.e., v ∗ ( G ) = X s ∈ S X θ ∈ Θ µ ( s ) c t ( θ | s ) v ( s, a θ ) , ∀ t ∈ T. Ideally, the designer’s maximal payoff is given by X s ∈ S (cid:18) µ ( s ) max a ∈ A ∗ v ( s, a ) (cid:19) . That is, under each state of the world, the agent takes the designer’s optimal action fromthe feasible action set A ∗ . The next theorem states that the maximal payoff of the designercan be almost achieved with suitable information structures.26 heorem 3. Suppose that assumption 1 holds. For any ǫ > , there exists an ambiguousinformation structure (Θ , { c t ( ·| s ) s ∈ S } t ∈ T ) and a set of actions { a θ } θ ∈ Θ such that G =(Θ , { c t ( ·| s ) s ∈ S } t ∈ T , { a θ } θ ∈ Θ ) is implementable and v ∗ ( G ) > X s ∈ S (cid:18) µ ( s ) max a ∈ A ∗ v ( s, a ) (cid:19) − ǫ. We give a simple example to illustrate the idea of Theorem 3. Let S = { s , s } and A ∗ = { a , a } . a is the designer’s optimal action on state s . a is the designer’s optimalaction on s . The common prior of the designer and the agent is µ = (1 / , / a if his posterior is µ = (1 / , / a if hisposterior is µ = (2 / , / { θ j,l } j ∈{ ,..., } ,l ∈{ , } and 20000 signal generating systems { c j,l } j ∈{ ,..., } ,l ∈{ , } . Our targetis that for l ∈ { , } , when signal θ j,l is realized, the agent’s CML posterior is µ l . Thus, theagent chooses action a l at signal θ j,l for any j . Moreover, we want the agent’s posterior atsignal θ j,l to be induced by c j,l for each j and each l . We illustrate the construction of c j, for some j as an example. Let c j, ( θ j, | s ) = 0 . , and c j, ( θ j, | s ) = 0 . . The two conditional probabilities indeed induce the posterior µ = (1 / , /
3) given the prior µ = (1 / , / c j, ( θ j ′ , | s ) = 1 − . − , ∀ j ′ = j,c j, ( θ j ∗ , | s ) = 1 − . , ∀ j ∗ ,c j, ( θ j ∗ , | s ) = 0 , ∀ j ∗ , and c j, ( θ j ′ , | s ) = 0 , ∀ j ′ = j. With c j, , conditional on s , the agent chooses action a with probability 1. Condition on s , the agent chooses action a with probability 0 .
98. Other signal generating systems can27e constructed similarly. We need many signals to ensure the conditional probability of θ j,l to be very small under c j ′ ,l ′ if ( j, l ) = ( j ′ , l ′ ) so that only c j,l plays a role for controlling theagent’s posterior at signal θ j,l . Note that the only payoff loss under c j, is when ( s , θ j, ) isrealized. However, we can let the payoff loss be smaller by adding more signals and shrinkingthe conditional probability c j, ( θ j, | s ).We note that the optimal payoff is usually not achievable. For example, assume that a ∈ A ∗ is the the designer’s unique optimal action at state s and is not an optimal actionof the designer at any other state. Assume further that the agent chooses a only if theprobability of state s ′ ( s ′ = s ) is greater than half. In this case, the designer must incura payoff loss in order to induce action a , where the payoff loss comes from the situation inwhich a is chosen at state s ′ . A natural extension of our framework is to allow the DM to have multiple priors over the statespace. However, we show that axiom ISU will be violated by any updating rule. The onlyassumption we impose on the updating rule is that if ( s, θ ) has zero prior probability, then s has zero posterior probability after signal θ is observed. Consider the following example. Example 4.
Let S = { s , s , s , s } and θ = { θ , θ } . The DM’s prior set over S × Θis P , consisting of all convex combinations of p and p , as shown in Table 3. Note that P is not simple since it does not induce a unique prior over S .28 able 3 p θ θ s . s . s . s . p θ θ s . s . s . s . θ be observed. Since only ( s , θ ) , ( s , θ ) , ( s , θ ) and ( s , θ ) have non-zero priorprobabilities, the DM’s posterior set is given by some Q ⊆ ∆( { s , s } ). Assume without lossof generality that the payoff space is R . Consider two extended acts f ∗ and g ∗ : f ∗ ( s , θ ) = 0 , f ∗ ( s , θ ) = 1 , f ∗ ( s , θ ) = 10 , f ∗ ( s , θ ) = x,g ∗ ( s , θ ) = 1 , g ∗ ( s , θ ) = 0 , g ∗ ( s , θ ) = − , g ∗ ( s , θ ) = y, where x and y will be determined later on. The P -evaluations of f ∗ and g ∗ are given by E { p ,p } ( f ∗ ) = E p ( f ∗ ) = 1 . . x, E { p ,p } ( g ∗ ) = E p ( g ∗ ) = − . . y. After θ is observed, the ex-post evaluations are given by E Q ( f ∗ | θ ) = min µ ∗ ∈Q µ ∗ ( s ) , E Q ( g ∗ | θ ) = min µ ∗ ∈Q µ ∗ ( s ) . Let ǫ > f ∗ on ( s , θ ) by ǫ increases its P -evaluation by p ( s , θ ) ǫ and increases the Q -evaluation of f ∗ | θ by min µ ∗ ∈Q µ ∗ ( s ) ǫ .Similarly, increasing the payoff of g ∗ on ( s , θ ) by ǫ increases its P -evaluation by p ( s , θ ) ǫ and increases the Q -evaluation of g ∗ | θ by min µ ∗ ∈Q µ ∗ ( s ) ǫ . We can pick x and y such that E p ( f ∗ ) = E Q ( f ∗ | θ ) , E p ( g ∗ ) = E Q ( g ∗ | θ ) .
29y axiom ISU, we conclude thatmin µ ∗ ∈ Q µ ∗ ( s ) ≥ p ( s , θ ) = 0 . , and min µ ∗ ∈ Q µ ∗ ( s ) ≥ p ( s , θ ) = 0 . , which is impossible.Example 4 shows that axiom ISU is generally violated under the max-min expectedutility framework. This gives a justification for our framework where ambiguity only comesfrom signals, in which ISU assumption works. In this section, we compare CML with the proxy rule by Gul and Pesendorfer (2018). Tostart with, we introduce the proxy rule. For simplicity, assume that S × Θ is finite. Proxyrule works for totally monotone capacities. Totally monotone capacities have the followingequivalent multi-prior characterization. For any nonempty E ⊆ S × Θ , let P E be the set ofall probability distributions that have support E , i.e., P E = { p ∈ ∆( S × Θ) : p ( E ) = 1 } . Atotally monotone capacity has a multiple-prior representation given by X E ⊆ S × Θ: E = ∅ α E P E , (6)where α E ∈ [0 ,
1] for each nonempty E ⊆ S × Θ and P E ⊆ S × Θ: E = ∅ α E = 1 . Let | E | denote thecardinality of event E . With the proxy rule, when θ is observed, the DM’s set of posteriorsis given by X E ⊆ S × Θ: E = ∅ α E · | E ∩ ( S × θ ) | · | E | − P E ′ ⊆ S × Θ: E ′ = ∅ ( α E ′ · | E ′ ∩ ( S × θ ) | · | E ′ | − ) P E ∩ ( S × θ ) . A key property of the proxy rule is that “not all news is bad news”: given an informationstructure and an extended act f ∗ , the DM’s ex-post evaluation of f ∗ | θ should be weaklyhigher than his evaluation of f ∗ under some signal θ . The following example illustrates thatCML violates “not all news is bad news”. 30 xample 5. Let S = { s, s ′ } and Θ = { θ, θ ′ } . The DM’s prior set is P , which is simpleand consists of all convex combinations of p , p , p and p , as shown in Table 4. Table 4 p θ θ ′ s /
20 1 / s ′ / / p θ θ ′ s /
20 9 / s ′ /
10 2 / p θ θ ′ s /
20 1 / s ′ /
10 2 / p θ θ ′ s /
20 9 / s ′ / / P admits a totally monotone capacity since it satisfies condition (6): P = 120 P { ( s,θ ) } + 120 P { ( s,θ ′ ) } + 25 P { ( s,θ ) , ( s,θ ′ ) } + 110 P { ( s ′ ,θ ) } + 110 P { ( s ′ ,θ ′ ) } + 310 P { ( s ′ ,θ ) , ( s ′ ,θ ′ ) } . Consider an extended act f ∗ : f ∗ ( s, θ ) = f ∗ ( s, θ ′ ) = 0 and f ∗ ( s ′ , θ ) = f ∗ ( s ′ , θ ′ ) = 1. The DM’sprior over S is µ = (1 / , /
2) according to P . When θ is realized, the DM’s CML posteriorover S is ( , ). When θ ′ is realized, the DM’s CML posterior is ( , ). Obviously, theDM lowers his evaluation of f ∗ after each signal. As a result, CML violates “not all news isbad news”. The next example shows that the proxy rule violates axiom ISU. Example 6.
Let S = { s, s ′ } and Θ = { θ, θ ′ } . The DM’s prior set is P , which is simpleand consists of all convex combinations of p and p . p and p are shown in Table 5. Table 5 p θ θ ′ s / / s ′ / p θ θ ′ s / / s ′ / P admits a totally monotone capacity since P = 13 P { ( s,θ ) } + 16 P { ( s,θ ′ ) } + 12 P { ( s ′ ,θ ) , ( s ′ ,θ ′ ) } . { s, s ′ } is µ = (1 / , /
2) according to P . With the proxy rule, when θ is observed, the DM’s posterior over S is µ ∗ = (4 / , / f ∗ : f ∗ ( s, θ ) = f ( s ′ , θ ′ ) = 1 , f ∗ ( s ′ , θ ) = 0 and f ∗ ( s, θ ′ ) = 10 / . We can verify that the evaluation E P ( f ∗ ) is 4 /
7. With the proxy rule, the ex-post evaluation of f ∗ | θ is given by E µ ∗ ( f ∗ | θ ),which is again equal to 4 / . Consider another extended act g ∗ where g ∗ ( s, θ ) = g ( s ′ , θ ′ ) = 1 ,g ∗ ( s ′ , θ ) = 1 / g ∗ ( s, θ ′ ) = 10 / . g ∗ differs from f ∗ only at ( s ′ , θ ), where g ∗ yields a higherpayoff. We have E P ( g ∗ ) = 23 / > E µ ∗ ( g ∗ | θ ) = 11 /
14. Hence, axiom ISU is violated by theproxy rule.
In this paper, we axiomatize a new updating rule, CML, for updating ambiguous information.Different from existing rules, CML satisfies and can be characterized by axiom ISU. Weshow that CML satisfies divisibility, accommodates recent experimental findings and hassimple predictions on learning. When an agent updates according to CML, we show thatan information designer can benefit from introducing ambiguous information and almostachieves the maximal payoff.We propose two streams of future works. First, axiom ISU can be tested in the lab.Testing axiom ISU is straightforward once we collect DMs’ ex-ante and ex-post evaluationfunctions. Second, we can investigate whether axiom ISU is compatible with other theoreticalframeworks of ambiguity, e.g., the dual-self expected utility framework. See Chandrasekher, Frick, Iijima, and Le Yaouanq (2020) for the dual-self expected utility theory. Appendix
Proof of Proposition 1.
Let V ∈ V be represented by P . Since P is simple, we can write P as ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ). Consider f ∗ and g ∗ satisfying the conditions stated in the proposition.We have V ( g ∗ ) − V ( f ∗ ) = E P ( g ∗ ) − E P ( f ∗ )= E P ( g ∗ ) − E p ∗ ( f ∗ ) ≤ E p ∗ ( g ∗ ) − E p ∗ ( f ∗ )= p ∗ ( s, θ )( g ∗ ( s, θ ) − f ∗ ( s, θ )) ≤ max p ∈P p ( s, θ )( g ∗ ( s, θ ) − f ∗ ( s, θ ))= µ ( s ) max t ∈T c t ( θ | s )( g ∗ ( s, θ ) − f ∗ ( s, θ )) , where p ∗ ∈ P minimizes the expectation of f ∗ . Other inequalities are obvious. Let µ θ bethe CML posterior when signal θ is observed. For V θ ( g ∗ | θ ) and V θ ( f ∗ | θ ), we have V θ ( g ∗ | θ ) − V θ ( f ∗ | θ ) = µ θ ( s )( g ∗ ( s, θ ) − f ∗ ( s, θ ))= µ ( s ) max t ∈T c t ( θ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈T c t ( θ | s ′ ) ( g ∗ ( s, θ ) − f ∗ ( s, θ )) ≥ µ ( s ) max t ∈T c t ( θ | s )( g ∗ ( s, θ ) − f ∗ ( s, θ )) . The last inequality holds since P s ′ ∈ S µ ( s ′ ) max t ∈T c t ( θ | s ′ ) ≤ P s ′ ∈ S µ ( s ′ ) = 1 . The propositionis thus shown.
Proof of Theorem 1.
Let V ∈ V be represented by P . It is obvious that θ ∈ Θ V if and onlyif there exists p ∈ P such that p ( s, θ ) > s . For any nonempty, convex and closedset of probability distributions Q ⊆ ∆( S ) and any S ′ ⊆ S , if min µ ∈Q µ ( S ′ ) >
0, let Q| S ′ bethe set of conditional probabilities of Q on S ′ . That is, µ S ′ ∈ Q| S ′ if and only if µ S ′ is theBayesian posterior of some µ ∈ Q on S ′ . Since min µ ∈Q µ ( S ′ ) > Q| S ′ is nonempty, convex33nd closed. For any evaluation function U that is represented by Q , U is strictly increasingon S ′ ⊆ S if and only if min µ ∈Q µ ( S ′ ) >
0. Before proceeding, we first prove a lemma.
Lemma 1.
For any { s, s ′ } ⊆ S and any two evaluation functions U and U ′ , representedby Q and Q ′ respectively, if min µ ∈Q µ ( { s, s ′ } ) > and min µ ′ ∈Q ′ µ ′ ( { s , s } ) > , then thefollowing two conditions are equivalent.1. For any x, y, z ∈ K , U ( x [ s ] y [ s ′ ] z ) = z if and only if U ′ ( x [ s ] y [ s ′ ] z ) = z .2. Q|{ s, s ′ } = Q ′ |{ s, s ′ } . Proof of Lemma 1.
Assume that condition 1 holds. We prove condition 2. Assume to thecontrary that
Q|{ s, s ′ } 6 = Q ′ |{ s, s ′ } . There exist x, y ∈ K such thatmin µ { s,s ′} ∈Q|{ s,s ′ } (cid:16) µ { s,s ′ } ( s ) x + µ { s,s ′ } ( s ′ ) y (cid:17) = z > min µ ′{ s,s ′} ∈Q ′ |{ s,s ′ } (cid:16) µ ′{ s,s ′ } ( s ) x + µ ′{ s,s ′ } ( s ′ ) y (cid:17) . This implies that U ( x [ s ] y [ s ′ ] z ) = z and U ′ ( x [ s ] y [ s ′ ] z ) < z , which is a contradiction.Inversely, assume that condition 2 holds. We only need to show that U ( x [ s ] y [ s ′ ] z ) = z (7)if and only if min µ { s,s ′} ∈Q|{ s,s ′ } (cid:16) µ { s,s ′ } ( s ) x + µ { s,s ′ } ( s ′ ) y (cid:17) = z. (8)Then, condition 2 implies condition 1.Suppose that condition (7) holds. There exists µ ∗ ∈ Q such that µ ∗ ( s ) x + µ ∗ ( s ′ ) y + (1 − µ ∗ ( { s, s ′ } )) z = z . Thus, µ ∗ ( s ) x + µ ∗ ( s ′ ) y = ( µ ∗ ( s ) + µ ∗ ( s ′ )) z . Since min µ ∈Q µ ( { s, s ′ } ) > µ ∗ ( s ) + µ ∗ ( s ′ ) > µ ∗ ( s ) µ ∗ ( s ) + µ ∗ ( s ′ ) x + µ ∗ ( s ′ ) µ ∗ ( s ) + µ ∗ ( s ′ ) y = z. Note that µ ∗{ s,s ′ } = ( µ ∗ ( s ) µ ∗ ( s )+ µ ∗ ( s ′ ) , µ ∗ ( s ′ ) µ ∗ ( s )+ µ ∗ ( s ′ ) ) ∈ Q|{ s, s ′ } . Hence, we havemin µ { s,s ′} ∈Q|{ s,s ′ } (cid:16) µ { s,s ′ } ( s ) x + µ { s,s ′ } ( s ′ ) y (cid:17) ≤ z.
34f the above inequality holds strictly, by the assumption that min µ ∈Q µ ( { s, s ′ } ) >
0, we know U ( x [ s ] y [ s ′ ] z ) < z , which is a contradiction. Hence, condition (8) must hold. Showing thatcondition (8) implies condition (7) is similar. (Necessity.) For axiom IIS, consider
V, W ∈ V . Let V be represented by P and W represented by P ′ . If V ( f ∗ [ S × θ ] x ) = W ( f ∗ [ S × θ ] x ) for all f ∗ ∈ F ∗ and x ∈ K , thenwe have P ≈ S × θ P ′ . Therefore, θ ∈ Θ V implies that θ ∈ Θ W . Moreover, for each s ∈ S, max p ∈P p ( s, θ ) = max p ′ ∈P ′ p ′ ( s, θ ) . By the formula of CML posterior (1), we have V θ = W θ . For axiom RC, consider
V, W ∈ V . Let V be represented by P and W representedby P ′ . If V ( f ∗ [ { s, s ′ } × Θ] x ) = W ( f ∗ [ { s, s ′ } × Θ] x ) for all f ∗ ∈ F ∗ and x ∈ K , then P ≈ { s,s ′ }× Θ P ′ . For any θ ∈ Θ V ∩ Θ W , we knowmax p ∈P p ( s, θ ) = max p ′ ∈P ′ p ′ ( s, θ ) , and max p ∈P p ( s ′ , θ ) = max p ′ ∈P ′ p ′ ( s ′ , θ ) . (9)Let V θ be represented by Q θ = { µ θ } and W θ represented by Q ′ θ = { µ ′ θ } . Since V θ and W θ are strictly increasing on { s, s ′ } , we have µ θ ( { s, s ′ } ) > µ ′ θ ( { s, s ′ } ) >
0. By the CMLposterior formula (1) and condition (9), we have Q θ |{ s, s ′ } = Q ′ θ |{ s, s ′ } . By Lemma 1, axiomRC holds. Axiom ISU is shown by Proposition 1. (Sufficiency.) Assume that axioms IIS,RC and ISU all hold. Through out the proofof the sufficiency part, assume the payoff space to be R . This is without loss of generalitysince for any payoff y not in K , we can pick x in the interior of K and take the convexcombination αx + (1 − α ) y of x and y such that αx + (1 − α ) y is in K . Lemma 2.
Let V ∈ V be represented by P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ) and V θ represented by Q θ ,where θ ∈ Θ V . If X s ∈ S µ ( s ) max t ∈ T c t ( θ | s ) = 1 , (10) then Q θ = { µ } . roof. Fix some s ∈ S with µ ( s ) >
0. By condition (10), max t ∈ T c t ( θ | s ) = 1 . Consider anextended act f ∗ satisfying that f ∗ ( s, θ ) = 0 ,f ∗ ( s, θ ′ ) = 1 , ∀ θ ′ ∈ Θ \ θ,f ∗ ( s ′ , θ ′ ) = 0 , ∀ ( s ′ , θ ′ ) ∈ ( S × Θ) \ ( s × Θ) . For such an extended act, we have V ( f ∗ ) = µ ( s )(1 − max t ∈ T c t ( θ | s )) = 0 = V θ ( f ∗ | θ ) . Consider an extended act g ∗ satisfying that g ∗ ( s, θ ) = 12 ,g ∗ ( s, θ ′ ) = 1 , ∀ θ ′ ∈ Θ \ θ,g ∗ ( s ′ , θ ′ ) = 0 , ∀ ( s ′ , θ ′ ) ∈ ( S × Θ) \ ( s × Θ) . We have V ( g ∗ ) = 12 µ ( s ) , and V θ ( g ∗ | θ ) = 12 min µ ∗ ∈Q θ µ ∗ ( s ) . For extended acts f ∗ and g ∗ , axiom ISU implies thatmin µ ∗ ∈Q θ µ ∗ ( s ) ≥ µ ( s ) . Since this holds for each s ∈ S with µ ( s ) >
0, we have for each µ ∗ ∈ Q θ and each s ∈ S , µ ∗ ( s ) = µ ( s ) . That is, Q θ = { µ } . Lemma 3.
Let V ∈ V be represented by P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ) and V θ represented by Q θ , where θ ∈ Θ V . If µ ( s ) max t ∈ T c t ( θ | s ) > , then for any s ∈ S that is different from s andany µ ∗ ∈ Q θ , it holds that µ ∗ ( s ) > and µ ∗ ( s ) µ ∗ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) µ ( s ) max t ∈ T c t ( θ | s ) . roof. Fix s and s that satisfy the conditions stated in the lemma. Pick s from S \{ s , s } . Consider V ′ ∈ V , represented by P ′ = ( µ ′ , { b t ( ·| s ) s ∈ S } t ∈ T ), where ( µ ′ , { b t ( ·| s ) s ∈ S } t ∈ T ) satisfiesthe following conditions:(1) µ ′ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) and µ ′ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ).(2) µ ′ ( s ) = µ ( s ) + µ ( s ) + µ ( s ) − µ ′ ( s ) − µ ′ ( s ).(3) µ ′ (ˆ s ) = µ (ˆ s ) for each ˆ s ∈ S \{ s , s , s } .(4) b t ( θ | s ) = c t ( θ | s ) (cid:16) max ˆ t ∈ T c ˆ t ( θ | s ) (cid:17) − for all t ∈ T. (5) b t ( θ | s ) = c t ( θ | s ) (cid:16) max ˆ t ∈ T c ˆ t ( θ | s ) (cid:17) − for all t ∈ T if µ ′ ( s ) > b t ( θ | s ) = 0 for all t ∈ T if µ ( s ) = 0.(7) b t ( θ | s ) = µ ( s ) µ ′ ( s ) c t ( θ | s ) for all t ∈ T if µ ( s ) > . (8) b t ( θ | ˆ s ) = c t ( θ | ˆ s ) for all ˆ s ∈ S \{ s , s , s } and all t ∈ T .With the above conditions, we can verify that for each t ∈ T and each s ∈ S , µ ( s ) c t ( θ | s ) = µ ′ ( s ) b t ( θ | s ). Therefore, P ≈ S × θ P ′ . By axiom IIS, we know V θ = V ′ θ . Let V ′ θ be represented by Q ′ θ . We have Q θ = Q ′ θ . By a similar argument as in Lemma 2, we canshow that µ ∗ ( s ) ≥ µ ′ ( s ) > µ ∗ ∈ Q ′ θ > V ′′ ∈ V , represented by P ′′ = ( µ ′′ , { a t ( ·| s ) s ∈ S } t ∈ T ), where ( µ ′′ , { a t ( ·| s ) s ∈ S } t ∈ T )satisfies the following conditions:(1) µ ′′ ( s ) = µ ′ ( s ) for all s ∈ S .(2) a t ( θ ′ | s ) = b t ( θ ′ | s ) and a t ( θ ′ | s ) = b t ( θ ′ | s ) for all θ ′ ∈ Θ and all t ∈ T .(3) a t ( θ | ˆ s ) = 1 for all ˆ s ∈ S \{ s , s } and all t ∈ T .Let V ′′ θ be represented by Q ′′ θ . With the above three conditions, we have X s ∈ S µ ′′ ( s ) max t ∈ T a t ( θ | s ) = 1 . By Lemma 2, we have Q ′′ θ = { µ ′′ } = { µ ′ } . We can verify that µ ′ ( s ) b t ( θ ′ | s ) = µ ′′ ( s ) a t ( θ ′ | s )for each s ∈ { s , s } , each θ ′ ∈ Θ and each t ∈ T . Hence, we have P ′ ≈ { s ,s }× Θ P ′′ . Since µ ∗ ( s ) > µ ∗ ∈ Q ′ θ and Q ′′ θ = { µ ′ } satisfies µ ′ ( s ) > V ′ θ and V ′′ θ are both strictly37ncreasing on { s , s } . By axiom RC and Lemma 1, we know that for each µ ∗ ∈ Q ′ θ ,µ ∗ ( s ) µ ∗ ( s ) = µ ′′ ( s ) µ ′′ ( s ) = µ ′ ( s ) µ ′ ( s ) = µ ( s ) max t ∈ T c t ( θ | s ) µ ( s ) max t ∈ T c t ( θ | s ) . Since Q θ = Q ′ θ , for each µ ∗ ∈ Q θ , the above condition holds. This finishes the proof of thelemma.By Lemma 3, if µ (ˆ s ) max t ∈ T c t ( θ | ˆ s ) > s ∈ S , then for each µ ∗ ∈ Q θ , µ ∗ (ˆ s ) > µ ∗ ( s ) µ ∗ (ˆ s ) = µ ( s ) max t ∈ T c t ( θ | s ) µ (ˆ s ) max t ∈ T c t ( θ | ˆ s )for each s ∈ S. Hence, for each µ ∗ ∈ Q θ and each s ∈ S , µ ∗ ( s ) = µ ∗ ( s ) P s ′ ∈ S µ ∗ ( s ′ ) = µ ∗ ( s ) µ ∗ (ˆ s ) P s ′ ∈ S µ ∗ ( s ′ ) µ ∗ (ˆ s ) = µ ( s ) max t ∈ T c t ( θ | s ) P s ′ ∈ S µ ( s ′ ) max t ∈ T c t ( θ | s ′ ) . We are done.
Proof of Theorem 2.
For necessity, suppose that { ( c t ( ·| s )) s ∈ S } t ∈ T is the DM’s set of inter-pretations of signals that satisfies conditions (4) and (5). For each s ∈ S and each θ ∈ Θ ∗ , µ θ ( s ) µ ( s ) max s ′ ∈ S ′ µ θ ( s ′ ) µ ( s ′ ) ! − = max t ∈ T c t ( θ | s )max s ′ ∈ S (max t ∈ T c t ( θ | s ′ )) ≥ max t ∈ T c t ( θ | s ) . Therefore, X θ ∈ Θ ∗ µ θ ( s ) µ ( s ) max s ′ ∈ S ′ µ θ ( s ′ ) µ ( s ′ ) ! − ≥ X θ ∈ Θ ∗ (cid:18) max t ∈ T c t ( θ | s ) (cid:19) ≥ . This shows the necessity part.For sufficiency, suppose that the belief profile ( µ, { µ θ } θ ∈ Θ ∗ ) satisfies that for each s ∈ S, X θ ∈ Θ ∗ µ θ ( s ) µ ( s ) max s ′ ∈ S ′ µ θ ( s ′ ) µ ( s ′ ) ! − ≥ . (11)We construct the finite set of interpretations. Define { c θ ( ·| s ) s ∈ S } θ ∈ Θ ∗ , where each interpre-tation c θ is labeled by a signal θ ∈ Θ ∗ . Let S θ := arg max s ∈ S ( µ θ ( s ) /µ ( s )) . For each s ∈ S and θ ∈ Θ ∗ , let c θ ( θ | s ) = µ θ ( s ) µ ( s ) max s ′ ∈ S ′ µ θ ( s ′ ) µ ( s ′ ) ! − . c θ ( θ | s ) ≤ s ∈ S and c θ ( θ | s ) = 1 for each s ∈ S θ . By condition (11), wehave X θ ∈ Θ ∗ c θ ( θ | s ) ≥ s ∈ S . Hence, we can find non-negative numbers { c θ ( θ ′ | s ) } θ ′ ∈ Θ ∗ \ θ for each θ ∈ Θ ∗ and each s ∈ S such that X θ ′ ∈ Θ ∗ c θ ( θ ′ | s ) = 1 , and c θ ( θ ′ | s ) ≤ c θ ′ ( θ ′ | s ) . Since { c θ ′ ( θ | s ) } θ ′ ∈ Θ ∗ is maximized at θ ′ = θ for each s and each θ , we have the desiredposterior µ θ for each θ ∈ Θ ∗ . Proof of Theorem 3.
If there exists a ∗ ∈ T s ∈ S arg max a ∈ A ∗ v ( s, a ), let a s = a ∗ for each s ∈ S. Otherwise, for each s ∈ S, pick a s ∈ arg max a ∈ A ∗ v ( s, a ) . Pick µ s ∈ ∆ o ( S ) such that a s ∈BR ( µ s ) . { µ s } s ∈ S are the target posteriors. For each µ s , pick a vector ( λ ss ′ ) s ′ ∈ S ∈ (0 , S suchthat µ ( s ′ ) λ ss ′ µ ( s ′′ ) λ ss ′′ = µ s ( s ′ ) µ s ( s ′′ )for each s ′ , s ′′ ∈ S. Fix the vector ( λ ss ′ ) s ′ ∈ S for each s ∈ S. For any { r s } s ∈ S ⊆ (0 , , let n ( { r s } s ∈ S ) be an integer satisfying thatmax s ′ ∈ S,s ′′ ∈ S − r s ′ λ s ′ s ′′ n ( { r s } s ∈ S ) − < min s ′ ∈ S,s ′′ ∈ S r s ′ λ s ′ s ′′ . (12)For the given ǫ >
0, pick { r s } s ∈ S ⊆ (0 ,
1) such that ∀ s ∈ S, X s ′ ∈ S µ ( s ′ )( r s λ ss ′ v ( s ′ , a s ) + (1 − r s λ ss ′ ) v ( s ′ , a s ′ )) = l ∗ , (13)where l ∗ satisfies that l ∗ > X s ∈ S (cid:18) µ ( s ) max a ∈ A ∗ v ( s, a ) (cid:19) − ǫ. (14)39n the case a s = a ∗ , we have l ∗ = X s ∈ S (cid:18) µ ( s ) max a ∈ A ∗ v ( s, a ) (cid:19) , which implies that { r s } s ∈ S exist. For the case where T s ∈ S arg max a ∈ A ∗ v ( s, a ) = ∅ , since as r s converges to zero, X s ′ ∈ S µ ( s ′ )( r s λ ss ′ v ( s ′ , a s ) + (1 − r s λ ss ′ ) v ( s ′ , a s ′ ))is strictly lower than but converges to X s ′ ∈ S µ ( s ′ ) v ( s ′ , a s ′ ) = X s ∈ S (cid:18) µ ( s ) max a ∈ A ∗ v ( s, a ) (cid:19) , the numbers { r s } s ∈ S that satisfy the desired conditions exist. Fix the numbers { r s } s ∈ S .We proceed to construct the information structure. Let N = n ( { r s } s ∈ S ). Let S = { s , ..., s m } . Let Θ = { θ j,l } j ∈{ ,...,m } ,l ∈{ ,...,N } . Let there be m · N signal generating systems.Each signal generating system is denoted by c j,l for some j ∈ { , ..., m } and l ∈ { , ..., N } . For each c j,l , let c j,l ( θ j,l | s ) = r s j λ s j s , ∀ s ∈ S,c j,l ( θ j ′ ,l ′ | s ) = 1 − r s j λ s j s N − , if s = s j ′ and l = l ′ , and c j,l ( θ j ′ ,l ′ | s ) = 0 otherwise.By condition (12), for each θ j,l , its conditional maximum probability on each s ∈ S is achieveduniquely by c j,l and equal to r s j λ s j s . Hence, the CML posterior at signal θ j,l is µ s j .Consider G = (Θ , { c j,l ( ·| s ) s ∈ S } j ∈{ ,...,m } ,l ∈{ ,...,N } , { a θ j,l } θ j,l ∈ Θ ) , where a θ j,l = a s j . Weargue G satisfies the desired conditions of the theorem. Note that a θ j,l is indeed the optimalaction of the agent given the CML posterior µ s j . For each signal generating system c j,l , conditional on state s j , action a s j is taken by the agent with probability 1. Conditional onstate s = s j , action a s is taken by the agent with probability 1 − − r sj λ sjs N − , and action a s j istaken with probability − r sj λ sjs N − . By conditions (13) and (14), we know that G satisfies theconditions in the statement of the theorem. 40 .2 Appendix B: Other Proofs In this section, we show that any two of the three axioms are not sufficient for CML.
Axiom IIS and Axiom RC.
FB satisfies both axiom IIS and axiom RC and violatesaxiom ISU. Let V ∈ V be represented by P . If θ ∈ Θ V , the ex-post evaluation V θ specified byFB is represented by the closure of P| θ , denoted by cl ( P| θ ). Obviously, FB satisfies axiomIIS. We show that FB satisfies axiom RC. Let V ∈ V and W ∈ V be represented by P and P ′ respectively. Let θ ∈ Θ V ∩ Θ W . With FB, the two evaluation functions V θ and W θ are represented by cl ( P| θ ) and cl ( P ′ | θ ) respectively. Fix two distinct states s and s ′ . To show axiom RC, we need to show that P ≈ { s,s ′ }× Θ P ′ , inf µ ∈P| θ ( { s, s ′ } ) > µ ′ ∈P ′ | θ ( { s, s ′ } ) > cl ( P| θ ) |{ s, s ′ } = cl ( P ′ | θ ) |{ s, s ′ } . Then, by Lemma 1, we aredone. Note that min µ { s,s ′} ∈ cl ( P| θ ) |{ s,s ′ } µ { s,s ′ } ( s ) µ { s,s ′ } ( s ′ ) = inf p ∈P : p ( s,θ )+ p ( s ′ ,θ ) > p ( s, θ ) p ( s ′ , θ ) , and min µ ′{ s,s ′} ∈ cl ( P ′ | θ ) |{ s,s ′ } µ ′{ s,s ′ } ( s ) µ ′{ s,s ′ } ( s ′ ) = inf p ′ ∈P ′ : p ′ ( s,θ )+ p ′ ( s ′ ,θ ) > p ′ ( s, θ ) p ′ ( s ′ , θ ) . Since
P ≈ { s,s ′ }× Θ P ′ , we havemin µ { s,s ′} ∈ cl ( P| θ ) |{ s,s ′ } µ { s,s ′ } ( s ) µ { s,s ′ } ( s ′ ) = min µ ′{ s,s ′} ∈ cl ( P ′ | θ ) |{ s,s ′ } µ ′{ s,s ′ } ( s ) µ ′{ s,s ′ } ( s ′ ) . Similarly, we have max µ { s,s ′} ∈ cl ( P| θ ) |{ s,s ′ } µ { s,s ′ } ( s ) µ { s,s ′ } ( s ′ ) = max µ ′{ s,s ′} ∈ cl ( P ′ | θ ) |{ s,s ′ } µ ′{ s,s ′ } ( s ) µ ′{ s,s ′ } ( s ′ ) . This implies that cl ( P| θ ) |{ s, s ′ } = cl ( P ′ | θ ) |{ s, s ′ } . We are done. Axiom IIS and Axiom ISU.
Consider the following updating rule. For any V ∈ V that is represented by P and any θ ∈ Θ V , the evaluation function V θ is represented by Q θ = X s ∈ S max p ∈P p ( s, θ ) ! µ θ + − X s ∈ S max p ∈P p ( s, θ ) !! cl ( P| θ ) , µ θ is the CML posterior. That is, the DM’s posterior set is a convex combination ofthe CML posterior set and the FB posterior set, where weights are given by X s ∈ S max p ∈P p ( s, θ ) and 1 − X s ∈ S max p ∈P p ( s, θ ) ! . This updating rule obviously satisfies axiom IIS. For axiom ISU, note that for any µ ∗ ∈ Q θ ,µ ∗ ( s ) ≥ X s ∈ S max p ∈P p ( s, θ ) ! µ θ ( s ) = max p ∈P p ( s, θ ) . This indicates that the ex-post sensitivity is at least max p ∈P p ( s, θ ). Therefore, axiom ISUis satisfied. Axiom RC and Axiom ISU.
Consider the following updating rule. For any V ∈ V that is represented by P = ( µ, { c t ( ·| s ) s ∈ S } t ∈ T ) and any θ ∈ Θ V , the evaluation function V θ is represented by µ λθ for some λ ∈ (0 ,
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