Common belief of weak-dominance rationality in strategic-form games: A qualitative analysis (original) (raw)

Qualitative Analysis of Common Belief of Rationality in Strategic-Form Games

SSRN Electronic Journal

In this paper we study common belief in rationality in strategic-form games with ordinal utilities, employing a model of qualitative beliefs. We characterize the three main solution concepts for such games, viz., Iterated Deletion of Strictly Dominated Strategies (IDSDS), Iterated Deletion of Börgers-dominated Strategies (IDBS) and Iterated Deletion of Inferior Strategy Profiles (IDIP), by means of gradually restrictive properties imposed on the models of qualitative beliefs. As a corollary, we prove that IDIP refines IDBS, which refines IDSDS.

Dominance rationality: A unified approach

Games and Economic Behavior, 2019

There are four types of dominance depending on whether domination is strict or weak and whether the dominating strategy is pure or mixed. Letting d vary over these four types of dominance, we say that a player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows. For weak dominance by a mixed strategy, Stalnaker (1994) introduced a process of iterative maximal elimination of certain profiles that we call here flaws. We define here, analogously, d-flaws for each type of dominance d, and show that for each d, iterative elimination of d-flaws is order independent. We then show that the characterization of common knowledge of d-dominance rationality is the same for each d. A strategy profile can be played when d-dominance rationality is commonly known if and only if it survives an iterative elimination of d-flaws.

A Syntactic Approach to Rationality in Games with Ordinal Payoffs

2008

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided.

Iterated Admissibility Through Forcing in Strategic Belief Models

Journal of Logic, Language and Information, 2020

Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving m +1 rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality (R∞AR), might not be satisfied by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy R∞AR. One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen's technique of forcing. Keywords Iterated admissibility • Possibility models • Forcing 1 Introduction Non-Cooperative Game Theory is concerned with the interactions of self-interested agents in structured environments. While its main elements were introduced by von B Fernando Tohmé

Forcing Iterated Admissibility in Strategic Belief Models

Iterated admissibility (IA) can be seen as exhibiting a minimal criterion of rationality in games. In order to make this intuition more precise, the epistemic characterization of this game-theoretic solution has been actively investigated in recent times: it has been shown that strategies surviving m+1 rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality (RmAR) in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, RinftyARR\infty ARRinftyAR, might not be satisfied by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we introduce a weaker notion of completeness which is nonetheless sufficient to characterize IA in a highly general way as the class of strategies that indeed satisfy RinftyARR\infty ARRinftyAR. The key methodological innovation involves defining a new notion of generic types and employing ...

A syntactic approach to rationality in games

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided.

Weakly Dominated Strategies : A Mystery Cracked

2013

An informal argument shows that common knowledge of rationality implies the iterative elimination of strongly dominated strategies. Rationality here means that players do not play strategies that are strongly dominated relative to their knowledge. We formalize and prove this claim. When by rationality we mean that players do not play strategies that are weakly dominated relative to their knowledge, then common knowledge of rationality does not imply iterative elimination of weakly dominated strategies. We show that it does imply an iterative elimination of flaws of weakly dominated strategies. The iterative elimination of flaws of strongly dominated strategies coincides with the iterative elimination of strongly dominated strategies.

StrategicRequirementswithIndierence: Single Peaked versus SinglePlateaued Preferences1

We concentrate on the problem of the provision of one pure public good whenever agents that form the society have either single-plateaued preferences or single-peaked preferences over the set of alternatives. We are interested in comparing the relationships between di¤erent nonmanipulability notions under these two domains. On the single-peaked domain, under strategy-proofness, non-bossyness is equivalent to convex range. Thus, minmax rules are the only strategy-proof non-bossy rules. On the single-plateaued domain, only constant rules are non-bossy or Maskin monotonic; but strategy-proofness and weak non-bossy are equivalent to strict Maskin monotonicity. Moreover, strategy-proofness and plateau-invariant guarantee convexity of the range. JEL Classi…cation Number: D71.