Elchanan Ben-Porath

Rationality, Nash Equilibrium and Backwards Induction in Perfect-Information Games

We say that a player is certain of an event A if she assigns probability 1 to A. There is common certainty (CC) of A if the event A occurred, each player is certain of A, each player is certain that every other player is certain of A, and so forth. It is shown that in a generic perfect-information game the set of outcomes that are consistent with common certainty of rationality (CCR) at the beginning of the game coincides with the set of outcomes that survive one deletion of weakly dominated strategies and then iterative deletion of strongly dominated strategies. Thus, the backward induction outcome is not the only outcome that is consistent with CCR. In particular, cooperation in Rosenthals (1981) centipede game, and fighting in Seltens (1978) chainstore game are consistent with CCR at the beginning of the game. Next, it is shown that, if in addition to CCR, there is CC that each player assigns a positive probability to the true strategies and beliefs of the other players, and if there is CC of the support of the beliefs of each player, then the outcome of the game is a Nash equilibrium outcome.

Signaling future actions and the potential for sacrifice

We consider extensions of games where some players have the option of signaling future actions by incurring costs. The main result is that in a class of games, if one player can incur costs, then forwards induction selects her most preferred outcome. Surprisingly, the player does not have to incur any costs to achieve this—the option alone suffices. However, when all players can incur costs, one players attempt to signal a future action is vulnerable to a counter-signal by the opponent. This vulnerability to counter-signaling distinguishes signaling future actions from signaling types.

Cheap talk in games with incomplete information

Abstract The paper studies Bayesian games which are extended by adding pre-play communication. Let Γ be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q , that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of Γ , provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in Γ such that for each type t i of each player i the expected payoff of t i in q is larger than the expected payoff of t i in s .

The complexity of computing a best response automaton in repeated games with mixed strategies

Abstract I examine the complexity of computing a best response automaton in a two-person repeated game when there is uncertainty about the automaton selected by the other player. There are two versions of this problem: (1) Finding a best response automaton; (2) Deciding if a given automaton is a best response. It is shown that these problems are “difficult” (i.e., not polynomial). Next it is shown that if the size of the support of the distribution of the other players automata is known in advance (for example, when it is known that the other player is restricted to automata of a certain size), the problems are polynomial. The problems are polynomial even if there is uncertainty in the game that will be played (i.e., even if it is a repeated game with incomplete information).

Communication in repeated games with costly monitoring

Abstract We study repeated games with discounting where perfect monitoring is possible, but costly. It is shown that if players can make public announcements, then every payoff vector which is an interior point in the set of feasible and individually rational payoffs can be implemented in a sequential equilibrium of the repeated game when the discount factor is high enough. Thus, efficiency can be approximated even when the cost of monitoring is high, provided that the discount factor is high enough.

Implementation with Partial Provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agentsʼ preferences or beliefs and agents need little information about each othersʼ preferences. Our results are robust to evidence forgery at any strictly positive cost.

Common Knowledge of Rationality and Market Clearing in Economies with Asymmetric Information

Consider an exchange economy with asymmetric information. What is the set of outcomes that are consistent with common knowledge of rationality and market clearing? To address this question we define an epistemic model for the economy that provides a complete description not only of the beliefs of each agent on the relationship between states of nature and prices but also of the whole system of interactive beliefs. The main result, theorem 1, provides a characterization of outcomes that are consistent with common knowledge of rationality and market clearing (henceforth, CKRMC outcomes) in terms of a solution notion - Ex - Post Rationalizability - that is defined directly in terms of the parameters that define the economy. We then apply theorem 1 to characterize the set of CKRMC outcomes in a general class of economies with two commodities. CKRMC manifests several intuitive properties that stand in contrast to the full revelation property of Rational Expectations Equilibrium: In particular, we obtain that for a robust class of economies: (1) there is a continuum of prices that are consistent with CKRMC in every state of nature, and hence these prices do not reveal the true state, (2) the range of CKRMC outcomes is monotonically decreasing as agents become more informed about the economic fundamentals, and (3) trade is consistent with common knowledge of rationality and market clearing even when there is common knowledge that there are no mutual gains from trade.