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Convergence of the Iterated Prisoner's Dilemma Game

Published online by Cambridge University Press:  25 April 2002

MARTIN DYER ,
LESLIE ANN GOLDBERG ,
CATHERINE GREENHILL ,
GABRIEL ISTRATE  and
MARK JERRUM
MARTIN DYER
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, United Kingdom (e-mail: dyer@comp.leeds.ac.uk)
LESLIE ANN GOLDBERG
Affiliation:
Department of Computer Science, University of Warwick, Coventry CV4 7AL, United Kingdom (e-mail: leslie@dcs.warwick.ac.uk)
CATHERINE GREENHILL
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3502, Australia (e-mail: csg@ms.unimelb.edu.au)
GABRIEL ISTRATE
Affiliation:
Center for Nonlinear Science and CIC-3 Division, Los Alamos National Laboratory, Mail Stop B258, Los Alamos, NM 87545, USA (e-mail: istrate@lanl.gov)
MARK JERRUM
Affiliation:
School of Computer Science, University of Edinburgh, King's Building, Edinburgh EH9 3JZ, United Kingdom (e-mail: mrj@dcs.ed.ac.uk)
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Abstract

We consider a stochastic process based on the iterated prisoner's dilemma game. During the game, each of n players has a state, either cooperate or defect. The players are connected by an ‘interaction graph’. During each step of the process, an edge of the graph is chosen uniformly at random and the states of the players connected by the edge are modified according to the Pavlov strategy. The process converges to a unique absorbing state in which all players cooperate. We prove two conjectures of Kittock: the convergence rate is exponential in n when the interaction graph is a complete graph, and it is polynomial in n when the interaction graph is a cycle. In fact, we show that the rate is O(n log n) in the latter case.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 2 , March 2002 , pp. 135 - 147
Copyright
2002 Cambridge University Press

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Footnotes

This work was supported in part by the EPSRC Research Grant ‘Sharper Analysis of Randomised Algorithms: a Computational Approach’ and by the ESPRIT Projects RAND-APX and ALCOM-FT.