# Cooperation in a Multi-Dimensional Local Interaction Model

*Alexander Tieman*,
*Harold Houba* () and
*Gerard van der Laan* ()

Game Theory and Information from EconWPA

**Abstract:**
We consider a local interaction model with a population on an h dimensional torus, in which in each round of play a random player gets a learning draw. This player plays a k+1 action stage game with players in his neighborhood, compares his own average payoff with the average payoff of the neighbors he played against and updates his action based on this comparison. Individuals use the update rule `Win Cooperate, Lose Defect', a multi-player variant of Tit-for-Tat. We prove that there are exactly k+1 stable states and that all of these can be reached with positive probability, for any dimension h of the torus. Furthermore, we prove that when k+1=2, both stable states will be reached with probability 1/2. For k+1>2 we provide some insight in the probability of reaching each of the stable states by presenting simulation results.

**Keywords:** Evolution; Local Interaction; Cooperation; Prisoner's Dilemma. (search for similar items in EconPapers)

**JEL-codes:** C78 (search for similar items in EconPapers)

**Date:** 1998-03-24

**Note:** Type of Document - dvi (compiled TeX); prepared on IBM PC - Scientific Workplace 2.5; to print on HP/PostScript; pages: 27 ; figures: included. Tinbergen Institute Discussion Paper

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