Uncertain Information Structures and Backward Induction

In everyday economic interactions, it is not clear whether sequential choices are visible or not to other participants: agents might be deluded about opponents'capacity to acquire,interpret or keep track of data, or might simply unexpectedly forget what they previously observed (but not chose). Following this idea, this paper drops the assumption that the information structure of extensive-form games is commonly known; that is, it introduces uncertainty into players' capacity to observe each others' past choices. Using this approach, our main result provides the following epistemic characterisation: if players (i) are rational,(ii) have strong belief in both opponents' rationality and opponents' capacity to observe others' choices, and (iii) have common belief in both opponents' future rationality and op-ponents' future capacity to observe others' choices, then the backward induction outcome obtains. Consequently, we do not require perfect information, and players observing each others' choices is often irrelevant from a strategic point of view. The analysis extends {from generic games with perfect information to games with not necessarily perfect information{the work by Battigalli and Siniscalchi (2002) and Perea (2014), who provide different sufficient epistemic conditions for the backward induction outcome.Financial support from the Spanish Ministry of Science and Technology (grant ECO2009-11213) is acknowledged

Approximate knowledge of rationality and correlated equilibria

We extend Aumann's [3] theorem deriving correlated equilibria as a consequence of common priors and common knowledge of rationality by explicitly allowing for non-rational behavior. We replace the assumption of common knowledge of rationality with a substantially weaker notion, joint p-belief of rationality, where agents believe the other agents are rational with probabilities p = (pi)i2I or more. We show that behavior in this case constitutes a constrained correlated equilibrium of a doubled game satisfying certain p-belief constraints and characterize the topological structure of the resulting set of p-rational outcomes. We establish continuity in the parameters p and show that, for p su ciently close to one, the p-rational outcomes are close to the correlated equilibria and, with high probability, supported on strategies that survive the iterated elimination of strictly dominated strategies. Finally, we extend Aumann and Dreze's [4] theorem on rational expectations of interim types to the broader p-rational belief systems, and also discuss the case of non-common priors.Spanish Ministry of Science and Technology (Grants SEJ2007-64340 and ECO2011-28965) Spanish Ministry of Science and Technology (Grant ECO2009-11213

Replicator Dynamics and Evolutionary Stable Strategies in Heterogeneous Games

We generalise and extend the work of Iñarra and Laruelle (2011) by studying two person symmetric evolutionary games with two strategies, a heterogenous population with two possible types of individuals and incomplete information. Comparing such games with their classic homogeneous version vith complete information found in the literature, we show that for the class of anti-coordination games the only evolutionarily stable strategy vanishes. Instead, we find infinite neutrally stable strategies. We also model the evolutionary process using two different replicator dynamics setups, each with a different inheritance rule, and we show that both lead to the same results with respect to stability.

Uncertain Information Structures and Backward Induction

In everyday economic interactions, it is not clear whether sequential choices are visible or not to other participants: agents might be deluded about opponents'capacity to acquire,interpret or keep track of data, or might simply unexpectedly forget what they previously observed (but not chose). Following this idea, this paper drops the assumption that the information structure of extensive-form games is commonly known; that is, it introduces uncertainty into players' capacity to observe each others' past choices. Using this approach, our main result provides the following epistemic characterisation: if players (i) are rational,(ii) have strong belief in both opponents' rationality and opponents' capacity to observe others' choices, and (iii) have common belief in both opponents' future rationality and op-ponents' future capacity to observe others' choices, then the backward induction outcome obtains. Consequently, we do not require perfect information, and players observing each others' choices is often irrelevant from a strategic point of view. The analysis extends {from generic games with perfect information to games with not necessarily perfect information{the work by Battigalli and Siniscalchi (2002) and Perea (2014), who provide different sufficient epistemic conditions for the backward induction outcome.Financial support from the Spanish Ministry of Science and Technology (grant ECO2009-11213) is acknowledged

Games with perceptions

We assume that 2 x 2 matrix games are publicly known and that players perceive a dichotomous characteristic on their opponents which defines two types for each player. In turn, each type has beliefs concerning her opponent's types, and payoffs are assumed to be type-independent. We analyze whether the mere possibility of different types playing different strategies generates discriminatory equilibria. Given a specific information structure we find that in equilibrium a player discriminates between her types if and only if her opponent does so. We also find that for dominant solvable 2x2 games no discriminatory equilibrium exists, while under different conditions of concordance between players' beliefs discrimination appears for coordination and for competitive games. A complete characterization of the set of Bayesian equilibria is provided.Spanish Ministerio de Ciencia e Innovación under project ECO2009-11213, co-funded by ERDF. Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07

Replicator Dynamics and Evolutionary Stable Strategies in Heterogeneous Games

We generalise and extend the work of Iñarra and Laruelle (2011) by studying two person symmetric evolutionary games with two strategies, a heterogenous population with two possible types of individuals and incomplete information. Comparing such games with their classic homogeneous version vith complete information found in the literature, we show that for the class of anti-coordination games the only evolutionarily stable strategy vanishes. Instead, we find infinite neutrally stable strategies. We also model the evolutionary process using two different replicator dynamics setups, each with a different inheritance rule, and we show that both lead to the same results with respect to stability.This research is supported by the Spanish Ministerio de Ciencia e Innovación under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07