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Research Article

Entropy and the value of information to investors

    Antonio Cabrales, Olivier Gossner, Roberto Serrano
    American Economic Review, 103: 360-377, 2013

    Abstract: Consider any investor who fears ruin when facing any set of investments that satisfy no-arbitrage. Before investing, he can purchase information about the state of nature in the form of an information structure. Given his prior, information structure α investment dominates information structure β if, whenever he is willing to buy β at some price, he is also willing to buy α at that price. We show that this informativeness ordering is complete and is represented by the decrease in entropy of his beliefs, regardless of his preferences, initial wealth, or investment problem. We also show that no prior-independent informativeness ordering based on similar premises exists.


    Reasoning-based introspection

      Olivier Gossner and Elias Tsakas
      Theory and Decision, 73: 513-523, 2012

      Abstract: We show that if an agent reasons according to standard inference rules, the axioms of truth and introspection extend from the set of non-epistemic propositions to the whole set of propositions. This implies that the usual axiomatization of the partitional possibility correspondence, which describes an agent who processes information rationally, is redundant.


      Performing best when it matters most: evidence from professional tennis

        Julio González-Días, Olivier Gossner and Brian Rogers
        Journal of Economic Behavior and Organizations, 84: 767-781, 2012

        Abstract: Stakes affect aggregate performance in a wide variety of settings. At the individual level, we define the critical ability as an agent’s ability to adapt performance to the importance of the situation. We identify individual critical abilities of professional tennis players, relying on point-level data from twelve years of the US Open tournament. We establish persistent heterogeneity in critical abilities. We find a significant statistical relationship between identified critical abilities and overall career success, which validates the identification procedure and suggests that response to pressure is a significant factor for success.


        Impermanent types and permanent reputations

          Olivier Gossner, Mehmet Ekmekci, Andrea Wilson
          Journal of Economic Theory, 147: 142-178, 2012

          Abstract: We study the impact of unobservable stochastic replacements for the long-run player in the classical reputation model with a long-run player and a series of short-run players. We provide explicit lower bounds on the Nash equilibrium payoffs of a long-run player, both ex-ante and following any positive probability history. Under general conditions on the convergence rates of the discount factor to one and of the rate of replacement to zero, both bounds converge to the Stackelberg payoff if the type space is sufficiently rich. These limiting conditions hold in particular if the game is played very frequently.


          When is the individually rational payoff in a repeated game equal to the minmax payoff?

            Olivier Gossner and Johannes Hörner
            Journal of Economic Theory, 145: 63-84, 2010


            Abstract: We study the relationship between a player’s lowest equilibrium payoff in a repeated game with imperfect monitoring and this player’s min max payoff in the corresponding one-shot game. We characterize the signal structures under which these two payoffs coincide for any payoff matrix. Under an identifiability assumption, we further show that, if the monitoring structure of an infinitely repeated game “nearly” satisfies this condition, then these two payoffs are approximately equal, independently of the discount factor. This provides conditions under which existing folk theorems exactly characterize the limiting payoff set.


            Informationally optimal correlation

              Olivier Gossner, Rida Laraki and Tristan Tomala
              Mathematical Programming series B, 116: 147-172, 2009


              Abstract: This papers studies an optimization problem under entropy constraints arising from repeated games with signals. We provide general properties of solutions and a full characterization of optimal solutions for 2 × 2 sets of actions. As an application we compute the min max values of some repeated games with signals.


              Information independence and common knowledge

                Olivier Gossner, Ehud Kalai, and Robert Weber
                Econometrica, 77: 1317–1328, 2009


                Abstract: In Bayesian environments with private information, as described by the types of Harsanyi, how can types of agents be (statistically) disassociated from each other and how are such disassociations reflected in the agents’ knowledge structure? Conditions studied are (i) subjective independence (the opponents’ types are independent conditional on one’s own) and (ii) type disassociation under common knowledge (the agents’ types are independent, conditional on some common-knowledge variable). Subjective independence is motivated by its implications in Bayesian games and in studies of equilibrium concepts. We find that a variable that disassociates types is more informative than any common-knowledge variable. With three or more agents, conditions (i) and (ii) are equivalent. They also imply that any variable which is common knowledge to two agents is common knowledge to all, and imply the existence of a unique common-knowledge variable that disassociates types, which is the one defined by Aumann.


                Entropy bounds on Bayesian learning

                  Olivier Gossner, Tristan Tomalar
                  Journal of Mathematical Economics, 44: 24-32, 2008


                  Abstract: An observer of a process (x_t) believes the process is governed by Q whereas the true law is P. We bound the expected average distance between P(x_t|x1,…,x_{t−1}) and Q(x_t|x_1,…,x_{t−1}) for t = 1,…,n by a function of the relative entropy between the marginals of P and Q on the n first realizations. We apply this bound to the cost of learning in sequential decision problems and to the merging of Q to P.


                  Secret correlation in repeated games with imperfect monitoring

                    Olivier Gossner, Tristan Tomala
                    Mathematics of Operations Research, 32: 413–424, 2007


                    Abstract: We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with imperfect monitoring. Our result relies on the optimal tradeoff for the team between optimization of stage payoffs and generation of signals for future correlation.


                    Optimal use of communicaiton resources

                      Olivier Gossner, Penélope Hernández, and Abraham Neyman
                      Econometrica, 74: 1603-1636, 2006


                      Abstract: We study a repeated game with asymmetric information about a dynamic state of nature. In the course of the game, the better-informed player can communicate some or all of his information to the other. Our model covers costly and/or bounded communication. We characterize the set of equilibrium payoffs and contrast these with the communication equilibrium payoffs, which by definition entail no communication costs.