Epistemic conditions for rationalizability

In this paper I present conditions, not involving common knowledge of rationality, that lead to (correlated) rationalizability. The basic observation is that, if the actual world belongs to a set of states where the set Z of action profiles is played, everyone is rational and it is mutual knowledge that the action profiles played are in Z, then the actions played at the actual world are rationalizable actions. Alternatively, if at the actual world the support of the conjecture of player i is Dⁱ, there is mutual knowledge of: (i) the game being played, (ii) that the players are rational, and (iii) that for every i the support of the conjecture of player i is contained in Dⁱ, then every strategy in the support of the conjectures is rationalizable. The results do not require common knowledge of anything and are valid for games with any number of players.     

On p-rationalizability and approximate common certainty of rationality

This paper shows that common p-belief of rationality implies p-rationalizability for games with compact strategy sets. We also establish the Bayesian foundation for the perfect p-rationalizability for finite games. The p-rationalizability is then used to analyze the robustness of rationalizable sets. For any game with compact strategy sets, we show that the rationalizable set is robust, i.e., the strategies characterized by common p-belief of rationality are close to the rationalizable set when p→1.     

An indistinguishability result on rationalizability under general preferences

In this paper, we show that, in the class of games where each player??s strategy space is compact Hausdorff and each player??s payoff function is continuous and ??concave-like,?? rationalizability in a variety of general preference models yields the unique set of outcomes of iterated strict dominance. The result implies that rationalizable strategic behavior in these preference models is observationally indistinguishable from that in the subjective expected utility model, in this class of games. Our indistinguishability result can be applied not only to mixed extensions of finite games, but also to other important applications in economics, for example, the Cournot?Coligopoly model.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

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.     

Informational properties of the Nash solutions of two stochastic nonzero-sum games

Within the framework of stochastic two-person nonzero-sum games, we deal with two commonly used models in engineering and economics—namely, the LQG (Linear-Quadratic-Gaussian) and the duopoly problems. We investigate how variations in information available to either player affect the equilibrium Nash strategies for these two models, whose existence and uniqueness have been proven in the paper. We show that for the LQG model better information for either player results in lower average Nash costs for both players; whereas for the duopoly model better information for one player helps him alone to achieve a higher average Nash profit, and it hurts the other player in the sense that his average Nash profit decreases. We further relate these properties of the Nash solutions for these two games to some of the distinct features of zero-sum games and team problems.     

Implementation without incentive compatibility: Two stories with partially informed planners

We consider two stories, Typhoon by Conrad and Traveler's Dilemma by Basu, 1994, Basu, 2007, as implementation problems under incomplete information without incentive compatibility, but where the planner has some private information regarding the state. If social choice functions do not satisfy incentive compatibility, full implementation is unattainable via existing approaches. For each story, we construct a direct mechanism that relies on the planner's private information. We provide a sufficient condition on players' beliefs regarding the state under which every player has a unique rationalizable action, namely, telling the truth. Thus, in these stories, the planner's information can bypass the lack of incentive compatibility.     

Disappearing private reputations in long-run relationships

For games of public reputation with uncertainty over types and imperfect public monitoring, Cripps et al. [Imperfect monitoring and impermanent reputations, Econometrica 72 (2004) 407-432] showed that an informed player facing short-lived uninformed opponents cannot maintain a permanent reputation for playing a strategy that is not part of an equilibrium of the game without uncertainty over types. This paper extends that result to games in which the uninformed player is long-lived and has private beliefs, so that the informed player's reputation is private. The rate at which reputations disappear is uniform across equilibria and reputations also disappear in sufficiently long discounted finitely repeated games.     

An experimental analysis of ultimatum bargaining

There are many experimental studies of bargaining behavior, but suprisingly enough nearly no attempt has been made to investigate the so-called ultimatum bargaining behavior experimentally. The special property of ultimatum bargaining games is that on every stage of the bargaining process only one player has to decide and that before the last stage the set of outcomes is already restricted to only two results. To make the ultimatum aspect obvious we concentrated on situations with two players and two stages. In the ‘easy games’ a given amount c has to be distributed among the two players, whereas in the ‘complicated games’ the players have to allocate a bundle of black and white chips with different values for both players. We performed two main experiments for easy games as well as for complicated games. By a special experiment it was investigated how the demands of subjects as player 1 are related to their acceptance decisions as player 2.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

Bayesian and consistent assessments

In a Bayesian assessment, beliefs are computed from the strategy profile applying Bayes rule at positive probability information sets. A consistent assessment is the limit point of a sequence of completely mixed Bayesian assessments. We characterize the set of extensive forms for which the sets of Bayesian and consistent assessments coincide. As an illustration of the results, we characterize consistency in some multi-period games with simultaneous actions.     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

Correlated quantal responses and equilibrium selection

This paper considers incomplete information games with payoffs subject to correlated random disturbances. It explains the connection between the uniqueness of quantal response equilibria, where large noise is required, and the uniqueness of equilibria in global games, where small noise is required.     

Fictitious play in ‘one-against-all’ multi-player games

Summary. A compound game is an (n + 1) player game based on n two-person subgames. In each of these subgames player 0 plays against one of the other players. Player 0 is regulated, so that he must choose the same strategy in all n subgames. We show that every fictitious play process approaches the set of equilibria in compound games for which all subgames are either zero-sum games, potential games, or games. Received: July 18, 1997; revised version: December 4, 1998     

Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283-314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games. Journal of Economic Literature Classification Numbers: C72, D72.     

Commitment and observability in a contracting environment

This paper examines leader-follower games where a leader must purchase an essential input from a price-setting supplier in order to take an action. We show that equilibrium outcomes when the followers perfectly observe the leaders' actions cannot be approximated by mixed equilibrium outcomes of the game where followers imperfectly observe the leaders' actions, i.e. they are not accessible. Accessibility fails since in a pure strategy equilibrium, a supplier makes positive profits; however in an equilibrium where a leader randomizes, supplier profits must be zero. Our result follows from a generalized indifference principle that mixed strategies must satisfy in economic environments. While supplier profits cannot be approximated, player action profiles are accessible. Our results also apply to games with costly observation.     

On Sustaining Cooperation without Public Observations

This paper examines a dynamic game in which each player only observes a private and imperfect signal on the actions played. Our main result is that in a repeated prisoner's dilemma where defections are irreversible (at least for a long enough period of time), patient enough players may achieve almost efficient outcomes. Dealing with models of imperfect private monitoring is difficult because (i) continuation games are games of incomplete information, hence they do not have the same structure as the original game. In particular, continuation equilibria are correlated equilibria. (ii) Players are typically uncertain about their opponents' past observations and actions, and they use their entire own private history to learn about these actions. As a result equilibrium strategies are in general nontrivial and increasingly complex functions of past observations. We bypass these difficulties by looking at correlated equilibria of the original game and find correlated equilibria in which the decision problem faced by each player remains the same over time. Journal of Economic Literature Classification Numbers: C72.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.