Rationalizability is a key concept in game theory that helps predict player behavior. It assumes players are rational and believe others are too, leading to the elimination of strategies that are never best responses.
This concept builds on the idea of dominant strategies by considering players' beliefs about others' actions. It provides a broader framework for understanding strategic decision-making in games with multiple equilibria or strategic uncertainty.
Rationalizability and Best Response
Rationalizability and Best Response Strategies
- Rationalizability assumes players are rational and believe others are rational, eliminating strategies that are never best responses
- Best response strategy maximizes a player's payoff given their beliefs about the strategies of other players
- Iterated best response involves players repeatedly choosing best response strategies based on their beliefs about others' strategies
- Rationalizable strategies are those that survive iterated elimination of strategies that are never best responses for any beliefs about others' strategies
Applications and Examples
- In the Prisoner's Dilemma, cooperating is not rationalizable because defecting is always a best response regardless of beliefs about the other player's strategy
- In the Battle of the Sexes game, both (Opera, Opera) and (Football, Football) are rationalizable outcomes as they involve best responses to the believed strategy of the other player
- Rationalizability can help predict behavior in games like auctions where bidders choose strategies based on beliefs about others' valuations and bidding strategies
Common Knowledge and Strategic Uncertainty
Common Knowledge of Rationality
- Common knowledge of rationality means all players are rational, know all players are rational, know that all players know all players are rational, and so on
- Assumes an infinite hierarchy of beliefs about rationality that allows players to reason about others' strategies based on common knowledge
- Lack of common knowledge can lead to outcomes that differ from equilibrium predictions (Centipede game)
Strategic Uncertainty and Epistemic Game Theory
- Strategic uncertainty refers to players' lack of knowledge about others' strategies or beliefs
- Epistemic game theory models players' reasoning and beliefs to analyze games with strategic uncertainty
- Focuses on belief hierarchies, where players have beliefs about others' strategies, beliefs about others' beliefs, and so on
- Relaxing common knowledge assumptions can explain deviations from equilibrium play and learning in repeated games
Relationship to Nash Equilibrium
Comparing Rationalizability and Nash Equilibrium
- Nash equilibrium is a stronger solution concept than rationalizability, as it requires strategies to be best responses to each other
- Every Nash equilibrium strategy profile is rationalizable, but not every rationalizable strategy profile is a Nash equilibrium
- In some games (Matching Pennies), the set of rationalizable strategies is larger than the set of Nash equilibrium strategies
- Rationalizability may be more appropriate when there is strategic uncertainty or lack of common knowledge, while Nash equilibrium assumes complete information
Examples and Applications
- In the Stag Hunt game, both (Stag, Stag) and (Hare, Hare) are rationalizable and Nash equilibria, but (Stag, Stag) is payoff-dominant while (Hare, Hare) is risk-dominant
- In the Traveler's Dilemma, the unique Nash equilibrium is for both players to choose the lowest possible number, but higher numbers are rationalizable and often observed in experiments
- Comparing rationalizability and Nash equilibrium can provide insights into the role of strategic uncertainty, coordination, and equilibrium selection in games