Rock-Paper-Scissors Lattice Model


  • Nasir Ganikhodjaev
  • Pah Chin Hee International Islamic University Malaysia



Rock-Paper-Scissors game, quadratic stochastic operator, lattice model, Cayley tree


In this work, we introduce Rock-Paper-Scissors lattice model on Cayley tree of second order generated by Rock-Paper-Scissors game. In this strategic 2-player game, the rule is simple: rock beats scissors, scissors beat paper, and paper beats rock. A payoff matrix  of this game is a skew-symmetric. It is known that quadratic stochastic operator generated by this matrix is non-ergodic transformation. The Hamiltonian of Rock-Paper-Scissors Lattice Model is defined by this skew-symmetric payoff matrix . In this paper, we discuss a connection between three fields of research: evolutionary games, quadratic stochastic operators, and lattice models of statistical physics. We prove that a phase diagram of the Rock-Paper-Scissors model consists of translation-invariant and periodic Gibbs measure with period 3.


Akin, E., & Losert, V. (1984). Evolutionary dynamics of zero-sum game. Journal of Mathematical Biology, 20, 231–258.

Baxter, R. J. (1982). Exactly solvable models in statistical mechanics. London, New York: Academic Press.

Bernstein, S. N. (1924). The solution of a mathematical problem related to the theory of heredity. Uchn Zapiski NI Kaf Ukr Otd Mat, 1, 83–115.

Christensen, K., & Moloney, N. R. (2015). Complexity and Criticality. Imperial College Press.

Ganikhodjaev, N. N., Ganikhodjaev, R. N., & Jamilov, U. U. (2015). Quadratic stochastic operators and zero-sum game dynamics. Ergodic Theory and Dynamical Systems, 35, 1443–1473.

Ganikhodjaev, N. N., & Zanin, D. V. (2004). On a necessary condition for the ergodicity of quadratic operators defi ned on the two-dimensional simplex. Russian Mathematical Surveys, 59(3), 571–572.

Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. Berlin-New York: Walter de Gruyter.

Hofbauer, J., & Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press.

Hui, P. M., & Xu, C. (2018). When evolutionary games and the Ising model meet. Jurnal Fizik Malaysia, 39(2), 10030–10034.

Kesten, H. (1970). Quadratic transformations: A model for population growth II. Advances in Applied Probability, 2, 1–82.

Kindermann, R., & Snell, J. L. (1980). Markov random fields and their applications. Contemp. Math. v.1.

Liu, J., Xu, C., & Hui, P. M. (2017). Evolutionary games with self-questioning adaptive mechanism and the Ising model. EPL 119 68001.

Losert, V., & Akin, E. (1983). Dynamics of games and genes: Discrete versus continuous time. Journal of Mathematical Biology, 17(2), 241–251.

Minlos, R. A. (2000). Introduction to Mathematical Statistical Physics. University lecture series, AMS, v.19.

Potts, R. B. (1952). Some generalized order-disorder transformations. Proc. Cambridge Philos. Soc., 48, 106–109.

Ulam, S. (1960). A Collection of Mathematical Problems. New York: Interscience.

Zakharevich, M. I. (1978). On behavior of trajectories and the ergodic hypothesis for quadratic transformations of the simplex. Russian Mathematical Surveys, 33, 265–266.