Optimal Behavior of Contrite Tit-for-Tat Under Infinitesimal Rate of Error

Date of Award

2003

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

McDonald, Patrick

Keywords

Game Theory, Mathematics, Prisoner's Dilemma

Area of Concentration

Mathematics

Abstract

The repeated play, with possibility of error, of the Prisoner's Dilemma is studied through the pure Nash equilibria for sets of strategies described by finite state transducers. Payoffs for the repeated game are defined by a limit-of-means or discounting approach. Tit-for-Tat is given as an example of a Nash equilibrium for the discounting payoff and the limit-of-means payoff in the error-free case. Toward an analysis of Nash equilibrium strategies under error, a Markov chain M� with associated stationary distribution �� is defined over the states of two given finite state transducers, with transition probabilities as a function of the error-rate parameter �. The long-run behavior of two finite state transducers under infinitesimal error is given by lim�-0 (��). This distribution is analyzed by the method of stochastic stability as given by Peyton Young and applied to a theory of equilibrium selection in convention games. The results of Peyton Young are discussed and sharpened slightly. A new payoff, limit-of-means under infinitesimal error, is defined as a weighted sum over the possible payoffs given by ��-0. Tit-for-Tat is shown to have poor performance under infinitesimal error, as ��?0 gives non-zero probability to states producing defection. The self-correcting strategy Contrite Tit-for-Tat is shown to be an efficient Nash equilibrium for the set of finite-state transducer strategies with payoff given by limit-of-means under infinitesimal error. Specifically, any other finite-state transducer played against Contrite Tit-for-Tat either produces a lower payoff for that player or is, against Contrite Tit-for-Tat, equivalent to it under noise.

Rights

This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.

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