Fictitious play property of the Nash demand game

We show that the Nash demand game has the fictitious play property. We also show that almost every fictitious play process and its associated belief path converge to a pure-strategy Nash equilibrium in the Nash demand game.     

Learning in games with strategic complementarities revisited

Fictitious play is a classical learning process for games, and games with strategic complementarities are an important class including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond games with a unique equilibrium, global convergence has only been claimed for games with diminishing returns [V. Krishna, Learning in games with strategic complementarities, HBS Working Paper 92-073, Harvard University, 1992]. This result remained unpublished, and it relies on a specific tie-breaking rule. Here we prove an extension of it by showing that the ordinal version of strategic complementarities suffices. The proof does not rely on tie-breaking rules and provides some intuition for the result.     

Knightian games and robustness to ambiguity

This paper introduces a notion of robustness to ambiguous beliefs for Bayesian Nash equilibria. An equilibrium is robust if the corresponding strategies remain approximately optimal for a class of games with ambiguous beliefs that results from an appropriately defined perturbation of the belief structure of the original non-ambiguous belief game. The robustness definition is based on a novel definition of equilibrium for games with ambiguous beliefs that requires equilibrium strategies to be approximate best responses for all measures that define a player's belief. Conditions are derived under which robustness is characterized by a newly defined strategic continuity property, which can be verified without reference to perturbations and corresponding ambiguous belief games.     

Learning in games with unstable equilibria

We propose a new concept for the analysis of games, the TASP, which gives a precise prediction about non-equilibrium play in games whose Nash equilibria are mixed and are unstable under fictitious play-like learning. We show that, when players learn using weighted stochastic fictitious play and so place greater weight on recent experience, the time average of play often converges in these “unstable” games, even while mixed strategies and beliefs continue to cycle. This time average, the TASP, is related to the cycle identified by Shapley [L.S. Shapley, Some topics in two person games, in: M. Dresher, et al. (Eds.), Advances in Game Theory, Princeton University Press, Princeton, 1964]. The TASP can be close to or quite distinct from Nash equilibrium.     

Coordination failure in repeated games with private monitoring

Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2×2$2\times 2$ coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals.     

Perturbed communication games with honest senders and naive receivers

This paper studies communication games in which the sender is possibly honest (tells the truth) and the receiver is possibly naive (follows messages as if truthful). The characterization of message-monotone equilibria in the perturbed games explain several important aspects of strategic communication including sender exaggeration, receiver skepticism and message clustering. Surprisingly, the strategic receiver may respond to more aggressive claims with more moderate actions. In the limit as the probabilities of the non-strategic players approach zero, (i) the limit equilibrium corresponds to a most-informative equilibrium of the limit (Crawford-Sobel) game; (ii) only the top messages are sent.     

Between-game rule learning in dissimilar symmetric normal-form games

Rule learning posits that decision makers, rather than choosing over actions, choose over behavioral rules with different levels of sophistication. Rules are reinforced over time based on their historically observed payoffs in a given game. Past works on rule learning have shown that when playing a single game over a number of rounds, players can learn to form sophisticated beliefs about others. Here we are interested in learning that occurs between games where the set of actions is not directly comparable from one game to the next. We study a sequence of ten thrice-played dissimilar games. Using experimental data, we find that our rule learning model captures the ability of players to learn to reason across games. However, this learning appears different from within-game rule learning as previously documented. The main adjustment in sophistication occurs by switching from non-belief-based strategies to belief-based strategies. The sophistication of the beliefs themselves increases only slightly over time.     

Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

Efficiency results in N player games with imperfect private monitoring

We demonstrate that efficiency is achievable in a certain class of N player repeated games with private, almost perfect monitoring. Our equilibrium requires only one period memory and can be implemented by two state automata. Furthermore, we show that this efficiency result holds with any degree of accuracy of monitoring if private signals are hemiindependent. Whereas most existing research focuses on two player cases or only a special example of N player games, our results are applicable to a wide range of N player games of economic relevance, such as trading goods games and price-setting oligopolies.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Adaptive learning in extensive form games and sequential equilibrium

Summary. This paper studies adaptive learning in extensive form games and provides conditions for convergence points of adaptive learning to be sequential equilibria. Precisely, we present a set of conditions on learning sequences such that an assessment is a sequential equilibrium if and only if there is a learning sequence fulfilling the conditions, which leads to the assessment. Received: November 5, 1996; revised version: May 28, 1997     

Testing the TASP: An experimental investigation of learning in games with unstable equilibria

We report experiments studying mixed strategy Nash equilibria that are theoretically stable or unstable under learning. The Time Average Shapley Polygon (TASP) predicts behavior in the unstable case. We study two versions of Rock-Paper-Scissors that include a fourth strategy, Dumb. The unique Nash equilibrium is identical in the two games, but the predicted frequency of Dumb is much higher in the game where the NE is stable. Consistent with TASP, the observed frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment. However, Dumb is played too frequently in all treatments.     

Time-consistent policies

In many cases the optimal open-loop policy to influence agents who solve dynamic problems is time inconsistent. We show how to construct a time-consistent open-loop policy rule. We also consider an additional restriction under which the time-consistent open-loop policy is stationary. We use examples to illustrate the properties of these tax rules.     

Long persuasion games

This paper characterizes geometrically the sets of all Nash and perfect Bayesian equilibrium payoffs achievable with unmediated communication in persuasion games, i.e., games with an informed expert and an uninformed decisionmaker in which the expert's information is certifiable. The first equilibrium characterization is provided for unilateral persuasion games, and the second for multistage, bilateral persuasion games. As in Aumann and Hart [R.J. Aumann, S. Hart, Long cheap talk, Econometrica 71 (6) (2003) 1619-1660], we use the concepts of diconvexification and dimartingale. A leading example illustrates both geometric characterizations and shows how the expert, whatever his type, can increase his equilibrium payoff compared to all equilibria of the unilateral persuasion game by delaying information certification.     

When mandatory disclosure hurts: Expert advice and conflicting interests

We study the quality of advice that an informed and biased expert gives to an uninformed decision maker. We compare two scenarios: mandatory disclosure of the bias and nondisclosure, where information about the bias can only be revealed through cheap-talk. We find that in many scenarios nondisclosure allows for higher welfare for both parties. Hiding the bias allows for more precise communication for the more biased type and, if different types are biased in different directions, may allow for the same for the less biased type. We identify contexts where equilibrium revelation allows but mandatory disclosure prevents meaningful communication.     

On the convergence of reinforcement learning

This paper examines the convergence of payoffs and strategies in Erev and Roth's model of reinforcement learning. When all players use this rule it eliminates iteratively dominated strategies and in two-person constant-sum games average payoffs converge to the value of the game. Strategies converge in constant-sum games with unique equilibria if they are pure or if they are mixed and the game is 2×2. The long-run behaviour of the learning rule is governed by equations related to Maynard Smith's version of the replicator dynamic. Properties of the learning rule against general opponents are also studied.     

Contagion and efficiency

We consider a population of agents, either finite or countably infinite, located on an arbitrary network. Agents interact directly only with their immediate neighbors, but are able to observe the behavior of (some) other agents beyond their interaction neighborhood, and learn from that behavior by imitating successful actions. If interactions are not “too global” but information is fluid enough, we show that the efficient action is the only one which can spread contagiously to the whole population from an initially small, finite subgroup. This result holds even in the presence of an alternative, -dominant action.     

Noise-independent selection in multidimensional global games

This paper examines many-player many-action global games with multidimensional state parameters. It establishes that the notion of noise-independent selection introduced by Frankel, Morris and Pauzner [D. Frankel, S. Morris, A. Pauzner, Equilibrium selection in global games with strategic complementarities, J. Econ. Theory 108 (2003) 1–44] for one-dimensional global games is robust when the setting is extended to the one proposed by Carlsson and Van Damme [H. Carlsson, E. Van Damme, Global games and Equilibrium selection, Econometrica 61 (1993) 989–1018]. More precisely, our main result states that if an action profile of some complete information game is noise-independently selected in one-dimensional global games, then it is also noise-independently selected in all multidimensional global games.     

R&D games in a Cournot duopoly with isoelastic demand functions: A comment

In a recent paper, Tramontana (Economic Modelling, 27; 350-357, 2010) investigates the stability properties of a Cournot Duopoly game when the demand function is isoelastic. In this note, we show that for some well known applications of two-stage Cournot games (D. Aspremont and Jacquemin, American Economic Review, 78, 1122-1137, 1988) an isoelastic demand function can guarantee both the existence and the uniqueness of a Nash Equilibrium even in cases where existence is not obtained with linear demands.