Email Game Research Articles

Uncertain rationality, depth of reasoning and robustness in games with incomplete information

Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and Yildiz's discontinuity. We weaken ICR to ICR λ , where λ is a sequence whose term λ n is the probability players attach to ( n − 1)th‐order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when λ n is above an appropriate threshold for all n, but fails when λ n converges to 0. That is, if players' confidence in mutual rationality persists at high orders, the discontinuity persists, but if confidence vanishes at high orders, the discontinuity vanishes.

False Consensus in Games: Embedding Level-k Models into Games of Incomplete Information

Level-k models have recently gained popularity as a framework for strategic behavior with bounded rationality. These models make two assumptions: agents of a higher level have richer beliefs, and can also perform more computations. I develop a model that incorporates Level-k models into games of incomplete information, by assuming that players’ beliefs exhibit the false consensus effect, which is the empirical observation that people think others share their beliefs, what allows players to believe others have the same level. A player’s level determines how complex her beliefs and I consider two solution concepts: Interim Correlated Rationalizability (ICR, with which level-0 behavior becomes endogenous), and k-th Best Response, which is the usual concept for Level-k games. My main result is that in this model the product topology and the uniform strategic topology coincide, what implies that players with similar beliefs behave similarly. Therefore, unlike for general type spaces, predictions will be robust to small specification errors. I show that in the Email game of Rubinstein (1989), when players receive few messages they never attack according to ICR; however, when they receive enough messages, they behave as if there was complete information. Other applications show that the model is consistent with experimental evidence of trading in “no-trade” contexts, and recovering multiplicity of equilibria in global games.

Cooperative and Competitive Reasoning: From Games to Revolutions

I develop a game theoretic model where players use two different reasoning processes in strategic situations: cooperative and competitive. Players always consider cooperating at first: if they believe others will cooperate with enough probability, they will do so; otherwise they behave competitively. The model generalizes Level-k and team reasoning models, and provides a unified explanation for several important phenomena. In Rubinstein’s Email game, players coordinate successfully upon receiving enough messages. In 2×2 games of complete information, the solution concept lies between Pareto dominance and risk-dominance. In coordination games, the model explains several experimental facts that cannot be accounted for by global games, especially the fact that people coordinate more with public rather than private information. I provide a generalization of the model that relaxes the epistemic requirements for cooperative behavior, and apply it to analyze revolutions.

Depth of reasoning and higher order beliefs

As demonstrated by the email game of Rubinstein (1989), the predictions of the standard equilibrium models of game theory are sensitive to assumptions about the fine details of the higher order beliefs. This paper shows that models of bounded depth of reasoning based on level-k thinking or cognitive hierarchy make predictions that are independent of the tail assumptions on the higher order beliefs. The framework developed here provides a language that makes it possible to identify general conditions on depth of reasoning, instead of committing to a particular model such as level-k thinking or cognitive hierarchy.

The robust selection of rationalizability

We propose a notion of selecting rationalizable actions by perturbing players' higher-order beliefs, which we call robust selection. Similarly to WY selection [28], robust selection generalizes the idea behind the equilibrium selection in the email game [27] and the global game [3]. In contrast to WY selection, however, we require selection to be robust to misspecifications of payoffs. Robust selection is a strong notion in the sense that, among types with multiple rationalizable actions, “almost all” selections are fragile; but it is also a weak notion in the sense that any strictly rationalizable action can be robustly selected. We show that robust selection is fully characterized by the curb collection, a notion that generalizes the curb set in [2]. We also use the curb collection to characterize critical types [12] in any fixed finite game.

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.

The e-mail game phenomenon

The e-mail game in Rubinstein (1989) shows that types with arbitrarily close higher-order beliefs may differ substantially in strategic behaviors. We define a notion called strategic discontinuity in arbitrary incomplete-information scenarios to generalize this e-mail game phenomenon. We show that almost all types involved in economic analysis — types in finite or common-prior models — display strategic discontinuity in simple games.

Depth of Reasoning and Higher Order Beliefs

As demonstrated by the email game of Rubinstein (1989), the predictions of the standard equilibrium models of game theory are sensitive to assumptions about the fine details of the higher order beliefs. This paper shows that models of bounded depth of reasoning based on level-k thinking or cognitive hierarchy make predictions that are independent of the tail assumptions on the higher order beliefs. In addition to this finding, the tools developed in this paper offer a new direction for the analysis of models of bounded depth of reasoning and their applications to various economic settings.

Do Conventions Need to Be Common Knowledge?

Do conventions need to be common knowledge in order to work? David Lewis builds this requirement into his definition of a convention. This paper explores the extent to which his approach finds support in the game theory literature. The knowledge formalism developed by Robert Aumann and others militates against Lewis’s approach, because it shows that it is almost impossible for something to become common knowledge in a large society. On the other hand, Ariel Rubinstein’s Email Game suggests that coordinated action is no less hard for rational players without a common knowledge requirement. But an unnecessary simplifying assumption in the Email Game turns out to be doing all the work, and the current paper concludes that common knowledge is better excluded from a definition of the conventions that we use to regulate our daily lives.

THE E-MAIL GAME REVISITED — MODELING ROUGH INDUCTIVE REASONING

I study the robustness of Rubinstein's (1989) E-Mail Game results by varying the information that players can utilize. The article follows one of Morris' (2002) reactions to the E-Mail game "that one should try to come up with a model of boundedly rational behavior that delivers predictions that are insensitive to whether there is common knowledge or a large number of levels of knowledge". Players in my model are presumed to use 'rough inductive reasoning' because they cannot utilize exact information. The information structure in the E-Mail game is generalized and the conditions are characterized under which Rubinstein's results hold. I find that rough inductive reasoning generates a payoff dominant equilibrium where the expected payoffs change continuously (instead of discretely) in the probability of "faulty" communication.

Efficiency and equilibrium in the electronic mail game; the general case

In the Email Game (Amer. Econom. Rev. 79 (1989) 385) noisy information channels may prevent risky-efficient coordination, even when the game is almost common knowledge. In this paper, we show that this is the case whenever message failure probabilities are not sufficiently different. Quite intuitively, the extent of the difference is governed by the game payoffs, and in particular by the mixed Nash Equilibrium strategy of one of the two games to be played. This is because, conditionally to having observed one's type, a player's beliefs on the opponent's choices are governed by the reliability of communication channels.

Coordination in an Email Game without ``Almost Common Knowledge''

The paper presents a variation of the EMAIL Game, originally proposed by Rubinstein (American Economic Review, 1989), in which coordination of the more rewarding-risky joint course of actions is shown to obtain, even when the relevant game is, at most, ``mutual knowledge.'' In the example proposed, a mediator is introduced in such a way that two individuals are symmetrically informed, rather than asymmetrically as in Rubinstein, about the game chosen by nature. As long as the message failure probability is sufficiently low, with the upper bound being a function of the game payoffs, conditional beliefs in the opponent's actions can allow players to choose a more rewarding-risky action. The result suggests that, for efficient coordination to obtain, the length of interactive knowledge on the game, possibly up to ``almost common knowledge,'' does not seem to be a major conceptual issue and that emphasis should be focused instead on the communication protocol and an appropriate relationship between the reliability of communication channels and the payoffs at stake.

Efficiency and Equilibrium in The Electronic Mail Game; The General Case

In the Email Game (Rubinstein, AER 89) noisy information channels may prevent efficient coordination, even when the game is almost common knowledge. In the paper we show that this is the case whenever message failure probabilities are not sufficiently different. Quite intuitively, the extent of the difference is governed by the game payoffs, and in particular by the purely mixed Nash equilibrium strategy of one of the two coordination games to be played. This is because, conditional to having observed oneis type, a playeris beliefs on the opponentis choices are governed by the reliability of communication channels.

Coordination in an Email Game Without Mutual Knowledge

The paper presents a version of the EMAIL Game, originally proposed by Rubinstein (AER,1989), in which efficient coordination is shown to obtain even when the relevant coordination game is not mutual knowledge. In the model investigated a mediator is introduced in such a way that the two individuals are symmetrically informed on the game chosen by nature, rather than asymmetrically as in Rubinstein. As long as the message failure probability is sufficiently low, with the upper bound being a function of the game payoffs, conditional beliefs on the opponent's actions can allow players to coordinate on the more rewarding-risky choice.