Complexity of Spatial Games

Authors Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, Jakub Svoboda

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Author Details

Krishnendu Chatterjee
  • Institute of Science and Technology Austria, Klosterneuburg, Austria
Rasmus Ibsen-Jensen
  • University of Liverpool, UK
Ismaël Jecker
  • University of Warsaw, Poland
Jakub Svoboda
  • Institute of Science and Technology Austria, Klosterneuburg, Austria

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Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Complexity of Spatial Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Spatial games form a widely-studied class of games from biology and physics modeling the evolution of social behavior. Formally, such a game is defined by a square (d by d) payoff matrix M and an undirected graph G. Each vertex of G represents an individual, that initially follows some strategy i ∈ {1,2,…,d}. In each round of the game, every individual plays the matrix game with each of its neighbors: An individual following strategy i meeting a neighbor following strategy j receives a payoff equal to the entry (i,j) of M. Then, each individual updates its strategy to its neighbors' strategy with the highest sum of payoffs, and the next round starts. The basic computational problems consist of reachability between configurations and the average frequency of a strategy. For general spatial games and graphs, these problems are in PSPACE. In this paper, we examine restricted setting: the game is a prisoner’s dilemma; and G is a subgraph of grid. We prove that basic computational problems for spatial games with prisoner’s dilemma on a subgraph of a grid are PSPACE-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • spatial games
  • computational complexity
  • prisoner’s dilemma
  • dynamical systems


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