# Schedule - Parallel Session 4 - Updating and Ambiguity 1

Engineering F1.10 (note the change of room from F105-106 to F1.10) - 11:00 - 12:30#### Epistemic Foundation of Equilibria under Ambiguity

Adam Dominiak and Juergen Eichberger

**Adam Dominiak**

#### Communicating with an Ambiguity Averse Receiver

Tigran Melkonyan; Surajeet Chakravarty

#### Abstract

Cheap talk plays a key role in most market and non-market interactions. Professional advice, announcements by government agencies, marketing campaigns, bargaining, and delegation of decision rights and communication in organizations are just few examples of situations with asymmetric information and possibility of sending costless messages. We analyze games where a sender costlessly communicates with a receiver who takes an action on behalf of the sender. The receiver is not only unaware of the type of the sender and is also ignorant about the distribution of the type the sender may be. There has been considerable amount of work on strategic communication recently (Sobel, 2011) most of which has mainly used the Crawford and Sobel (1982) as a model of communication. We develop our analysis in the same spirit of Crawford and Sobel (1982) (CS from now on). In CS the sender communicates a one-dimensional signal to the receiver and the sender has a uniform bias where the sender’s preferred action is always to the right of the receiver. In comparison to CS we consider the case when the receiver has ambiguity-averse preferences. The sender therefore knows this and will try and send a message to the receiver in order to convince him to take an action according to his own preferences. We characterize the equilibria of such a game. We show that communication can be informative, and can also be influential. While there is no fully separating equilibrium, we characterize the partial separating equilibria and the pooling equilibrium. And we find like CS, there is an equilibrium where for any message the receiver takes the same action. Finally, we show that as ambiguity increases, the equilibria, which are affected, are the ones where the sender chooses to partially mix and the receiver responds by also partially mixing. But as ambiguity increases the message of the sender becomes more precise. The second contribution we intend to make is in the analysis of authority within organizations. Dessein (2002) uses a CS setup to demonstrate that delegation of decision rights to the agent is preferred to the communication mechanism if the principal’s and agent’s preferences do not differ too much relative to principal’s uncertainty. Our results shed light on how this trade-off between the two authority structures is affected by ambiguity. We show that communication may become more informative as ambiguity increases. The intuition behind this seemingly counterintuitive finding is that the agent will communicate a more informative signal since he will anticipate that increased ambiguity will cause the principal to select an action, which will be further away from his preferred action. To counter this he will send a more informative signal.

**Surajeet Chakravarty**

#### Predicting the Unpredictable: A Theory of Learning Under Unawareness

Idione Meneghel; Rabee Tourky

#### Abstract

We provide a model of statistical inference in a setting where unawareness matters. In this setting, a decision maker forms an assessment regarding some alternatives available, but before making this assessment he can collect information about the environment. More specifically, we assume that the decision maker views data as being generated by an underlying stochastic process that satisfies a condition we denote conditional exchangeability. That is, there is a sequence of random variables (X_t) that represents the realization of repeated trials of an experiment. The sequence (X_t) is said to be conditionally exchangeable if, for every n, (X_1, … , X_n, X_n+1) is distributed as (X_1, … , X_n, X_n+2). Conditional exchangeability is related to the notion of exchangeability: exchangeable random variables are obviously conditionally exchangeable. There are, however, sequences of random variables that are conditionally exchangeable but not exchangeable. In particular, conditionally exchangeable sequences of random variables may fail to be stationary. At any point t in time, the decision maker makes an assessment regarding bets whose payoffs depend on these realizations. However, the decision maker’s level of awareness restricts his perception of the realized state of the world: he can only partially observe the realizations of the sequence of random variables (X_t). As information trickles, the decision maker discovers new states. Consequently his awareness level increases and his conceivable state space expands. Moreover, as new states are discovered, probability mass is shifted from old, non-null events to the events just created. When facing the choice of how much probability mass to shift, due to the lack of familiarity with the new events, the decision maker updates his assessment by taking into consideration the largest set of probability measures that is consistent with his previous assessment. As a result, newly learned events are initially seen as ambiguous. As evidence accumulates, the ambiguity associated with those events gradually resolves and the assessment made by the decision maker converges to the true conditional probability of those events. Our contribution can thus be summarized by three core features: 1. We provide a model of learning under unawareness that, similarly to Epstein and Schneider [2007], accommodates ambiguous beliefs. 2. We explicitly model the process of inductive reasoning implied by the dynamics of growing awareness described in Karni and Vierø [2013]. 3. Because ambiguity emerges endogenously, we provide a foundation for the unanimity rule preference representation axiomatized in Bewley [2002] and Gilboa, Macheroni, Marinacci, and Schmeidler [2010].

**Idione Meneghel**

#### Dynamically Consistent Beliefs Must Be Decomposable

Daniel Schoch

#### Abstract

It has been argued that, as long as both dynamic consistency and consequentialism are imposed in sufficient generality and the valuation is independent of the form of the decision tree, the decision maker’s beliefs are represented by additive measures. In particular, valuations of Savageian acts are linear and beliefs are updated by the Bayesian rule, which amounts to Subjective Expected Utility (SEU). However, these studies are either within the framework of Choquet Expected Utility (CEU) (Eichberger and Kelsey IER 1996, Sarin and Wakker JRU 1998), or presume a variant of Savage’s P4 axiom, which implies beliefs to be ordinally representable by probabilities (Machina and Schmeidler Econometrica 1992, Epstein and Le Breton JET 1993). Moreover, under a certain condition on null events, dynamic consistency implies the Sure Thing Principle, excluding both Allais and Ellsberg paradoxa. In this paper we present a (weakly) dynamically consistent and consequentialist decision model in a Savagian framework. We drop the condition on null events, leave the CEU framework, and weaken Savage’s P4, and impose dynamic consistency. One example, Maximal Possibilistic Utility (MPU), allows representation of risk aversion in the sense of Allais’s Paradox. It induces Bayesian belief update and excludes Ellsberg Paradox. Beliefs are represented by possibility fuzzy measures (capacities). MPU has a preference represententation similar to a Sugeno Integral, but with cardinal utilities on a ratio scale. SEU and MPU are among the many cardinal decision models, which satisfy the following three axioms: Weak dynamic consistency says that all ex ante best plans remain best plans ex post. Consequentialism claims independence of conditional preferences on counterfactual/bygone events. The third axiom is on belief update; it states that conditionalization on an event ordinally preserves belief degrees of subevents. In general, belief is shown to be represented by a capacity, which is decomposable (“additive”) over a continuous t-conorm, which again turns out to be representable as an ordinal sum of pseudo-additions. When restricted to binary acts, the first two axioms translate to similar conditions for belief update. All three conditions for belief update are altogether equivalent to a single axiom expressing a kind of backward induction consistency and implies the absence of Ellsberg Paradoxa.

**Daniel Schoch**