## Convergence of the Iterated Prisoner's Dilemma Game 1

 MARTIN DYER a1, LESLIE ANN GOLDBERG a2, CATHERINE GREENHILL a3 2 , GABRIEL ISTRATE a4 and MARK JERRUM a5a1 School of Computing, University of Leeds, Leeds LS2 9JT, United Kingdom (e-mail: dyer@comp.leeds.ac.uk)a2 Department of Computer Science, University of Warwick, Coventry CV4 7AL, United Kingdom (e-mail: leslie@dcs.warwick.ac.uk)a3 Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3502, Australia (e-mail: csg@ms.unimelb.edu.au)a4 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)a5 School of Computer Science, University of Edinburgh, King's Building, Edinburgh EH9 3JZ, United Kingdom (e-mail: mrj@dcs.ed.ac.uk)

## 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.

(Received October 7 2000)
(Revised May 25 2001)

### Footnotes

1 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.

2 Supported by an Australian Research Council Postdoctoral Fellowship.