Proper Consistency

Proper consistency is defined by the properties that each player takes all opponent strategies into account (is cautious) and deems one opponent strategy to be infinitely more likely than another if the opponent prefers the one to the other (respects preferences). When there is common certain belief of proper consistency, a most preferred strategy is properly rationalizable. Any strategy used with positive probability in a proper equilibrium is properly rationalizable. Only strategies that lead to the backward induction outcome is properly rationalizable in the strategic form of a generic perfect information game. Proper rationalizability can be used to test the robustness of inductive procedures.

Rationalizability and Minimal Complexity in Dynamic Games

This paper presents a formal epistemic framework for dynamic games in which players, during the course of the game, may revise their beliefs about the opponents'' utility functions. We impose three key conditions upon the players'' beliefs: (a) throughout the game, every move by the opponent should be interpreted as being part of a rational strategy, (b) the belief about the opponents'' relative ranking of two strategies should not be revised unless one is certain that the opponent has decided not to choose one of these strategies, and (c) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Types that, throughout the game, respect common belief about these three events, are called persistently rationalizable for the profile u of utility functions. It is shown that persistent rationalizability implies the backward induction procedure in generic games with perfect information. We next focus on persistently rationalizable types for u that hold a theory about the opponents of ``minimal complexity'''', resulting in the concept of minimal rationalizability. For two-player simultaneous move games, minimal rationalizability is equivalent to the concept of Nash equilibrium strategy. In every outside option game, as defined by van Damme (1989), minimal rationalizability uniquely selects the forward induction outcome.microeconomics ;

An Epistemic Characterisation of Extensive Form Rationalisability

We use an extensive form, universal type space to provide the following epistemic characterisation of extensive form rationalisability. Say that player i strongly believes event E if i is certain of E conditional on each of her information sets consistent with E. Our main contribution is to show that a strategy profile s is extensive form rationalisable if and only if there is a state in which s is played and (0) everybody is rational, (1) everybody strongly believes (0), (2) everybody strongly believes (0) & (1), (3) everybody strongly believes (0) & (1) & (2), .... This result also allows us to provide sufficient epistemic conditions for the backward induction outcome and to relate extensive form rationalisability and conditional common certainty of rationality.Rationalisability, Extensive Form Games

Refinements of Rationalizability for Normal-Form Games

In normal-form games, rationalizability (Bernheim [3], Pearce [11]) on its own fails to exclude some very implausible strategy choices. Three main refinements of rationalizability have been proposed in the literature: cautious, perfect, and proper rationalizability. Nevertheless, some of these refinements also fail to eliminate unreasonable outcomes and suffer from several drawbacks. Therefore, we introduce the trembling-hand rationalizability concept, where the players' actions have to be best responses also against perturbed conjectures. We also propose another refinement: weakly perfect rationalizability, where players' actions that are not best responses are only played with a very small probability. We show the relationship between perfect rationalizability and weakly perfect rationalizability as well as the relationship between proper rationalizability and weakly perfect rationalizability : weakly perfect rationalizability is a weaker refinement than both perfect and proper rationalizability. Moreover, in two-player games it holds that weakly perfect rationalizability is a weaker refinement than trembling-hand rationalizability. The other relationships between the various refinements are illustrated by means of examples. For the relationship between any other two refinements we give examples showing that the remaining set of strategies corresponding to the first redinement can be either smaller or larger than the one corresponding to the second refinement.rationalizability;refinements

Prudent Rationalizability in Generalized Extensive-Form Games

We define an extensive-form analogue of iterated admissibility, called Prudent Rationalizability (PR). In each round of the procedure, for each information set of a player a surviving strategy of hers is required to be rational vis-a-vis a belief system with a full-support belief on the opponents' previously surviving strategies that reach that information set. Somewhat surprisingly, prudent rationalizable strategies may not refine the set of Extensive-Form Rationalizable (EFR) strategies (Pearce 1984). However, we prove that the paths induced by PR strategy-profiles (weakly) refine the set of paths induced by EFR strategies. PR applies also to generalized extensive-form games which model mutual unawareness of actions (Heifetz, Meier and Schipper, 2011a). We demonstrate the applicability of PR in the analysis of verifiable communication, and show that it yields the same, full information unraveling prediction as does the unique sequential equilibrium singled out by Milgrom and Roberts (1986); yet, we also show that under unawareness full unraveling might fail.Prudent rationalizability, caution, extensive-form rationalizability, extensive-form games, unawareness, verifiable communication

EPISTEMIC FOUNDATIONS OF SOLUTION CONCEPTS IN GAME THEORY: AN INTRODUCTION

We give an introduction to the literature on the epistemic foundations of solution concepts in game theory. Only normal-form games are considered. The solution concepts analyzed are rationalizability, strong rationalizability, correlated equilibrium and Nash equilibrium. The analysis is carried out locally in terms of properties of the belief hierarchies. Several examples are used throughout to illustrate definitions and concepts.

Dynamic Unawareness and Rationalizable Behavior

We define generalized extensive-form games which allow for mutual unawareness of actions. We extend Pearce's (1984) notion of extensive-form (correlated) rationalizability to this setting, explore its properties and prove existence. We define also a new variant of this solution concept, prudent rationalizability, which refines the set of outcomes induced by extensive-form rationalizable strategies. Finally, we define the normal form of a generalized extensive-form game, and characterize in it extensive-form rationalizability by iterative conditional dominance.Unawareness, extensive-form games, extensive-form rationalizability, prudent rationalizability, iterative conditional dominance