Pairwise interact-and-imitate dynamics

This paper introduces and studies a class of evolutionary dynamics—pairwise interact-and-imitate dynamics (PIID)—in which agents are matched in pairs, engage in a symmetric game, and imitate the opponent with a probability that depends on the difference in their payoffs. We provide a condition on the underlying game, named supremacy, and show that the population state in which all agents play the supreme strategy is globally asymptotically stable. We extend the framework to allow for payoff uncertainty, and check the robustness of our results to the introduction of some heterogeneity in the revision protocol followed by agents. Finally, we show that PIID can allow the survival of strictly dominated strategies, leads to the emergence of inefficient conventions in social dilemmas, and makes assortment ineffective in promoting cooperation.

Pairwise Interact-and-Imitate Dynamics

This paper introduces and studies a class of evolutionary dynamics --- Pairwise Interact-and-Imitate Dynamics (PIID) --- in which agents are matched in pairs, engage in a symmetric game, and imitate the opponent with a probability that depends on the difference in their payoffs. We provide a condition on the underlying game, named supremacy, and show that the population state in which all agents play the supreme strategy is globally asymptotically stable. We extend the framework to allow for payoff uncertainty, and check the robustness of our results to the introduction of some heterogeneity in the revision protocol followed by agents. Finally, we show that PIID can allow the survival of strictly dominated strategies, leads to the emergence of inefficient conventions in social dilemmas, and makes assortment ineffective in promoting cooperation.

Optimal guessing in ‘Guess Who’

Are you Richard? Are you Anne? We look at the strategic problem in the children’s guessing game Guess Who, which is a form of zero-sum symmetric game with perfect information. We discuss some preliminary strategic insights and formally derive an optimal strategy and win-probabilities for the game. We discuss the first-mover advantage in the game and other strategic aspects coming out of the optimal strategy. While the paper is based on the popular children’s game, our analysis generalises the actual game by allowing any initial game state with an arbitrarily large number of starting characters. With the aid of these mathematical results you can now comprehensively thrash your young children and be a terrible parent!

Applications to Game Theory

This chapter explores several applications of quantal response equilibrium (QRE) to specific games in order to illustrate and expand on the wide range of game-theoretic principles and phenomena associated with QRE that have been highlighted in the previous chapters. The first application considered belongs to the class of continuous games. With a continuum of decisions, QRE predicts a choice distribution that is not merely a (possibly asymmetric) spread to each side of a Nash equilibrium, since “feedback effects” from deviations by one player alter others' expected payoff profiles, which would induce further changes. The second application is a symmetric game with binary actions where players have continuously distributed private information about an unknown state of the world that affects both players' payoffs. The remainder of the chapter looks at three applications to extensive-form games, all of which are games of incomplete information.