Journal of Mathematical Biology

, Volume 71, Issue 6–7, pp 1551–1574 | Cite as

A study of the dynamics of multi-player games on small networks using territorial interactions

  • Mark BroomEmail author
  • Charlotte Lafaye
  • Karan Pattni
  • Jan Rychtář


Recently, the study of structured populations using models of evolutionary processes on graphs has begun to incorporate a more general type of interaction between individuals, allowing multi-player games to be played among the population. In this paper, we develop a birth-death dynamics for use in such models and consider the evolution of populations for special cases of very small graphs where we can easily identify all of the population states and carry out exact analyses. To do so, we study two multi-player games, a Hawk–Dove game and a public goods game. Our focus is on finding the fixation probability of an individual from one type, cooperator or defector in the case of the public goods game, within a population of the other type. We compare this value for both games on several graphs under different parameter values and assumptions, and identify some interesting general features of our model. In particular there is a very close relationship between the fixation probability and the mean temperature, with high temperatures helping fitter individuals and punishing unfit ones and so enhancing selection, whereas low temperatures give a levelling effect which suppresses selection.


Evolutionary graph theory Structured populations Evolution  Game theory Territory 

Mathematics Subject Classification

91A22 91A43 60J10 



The research was supported by Simons Foundation grant 245400 to JR, and a City of London Corporation grant for KP.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Mark Broom
    • 1
    Email author
  • Charlotte Lafaye
    • 2
  • Karan Pattni
    • 1
  • Jan Rychtář
    • 3
  1. 1.Department of MathematicsCity University LondonLondonUK
  2. 2.École Centrale ParisGrande Voie des VignesChâtenay-MalabryFrance
  3. 3.Department of Mathematics and StatisticsThe University of North Carolina at GreensboroGreensboroUSA

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