SCIENCE CHINA Information Sciences, Volume 62, Issue 1: 012207(2019) https://doi.org/10.1007/s11432-017-9411-0

Symmetry-based decomposition of finite games

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  • ReceivedOct 28, 2017
  • AcceptedMar 5, 2018
  • PublishedDec 19, 2018


The symmetry-based decompositions of finite games are investigated.First, the vector space of finite games is decomposed into a symmetric subspace and an orthogonal complement of the symmetric subspace. The bases of the symmetric subspace and those of its orthogonal complement are presented.Second, the potential-based orthogonal decompositions of two-player symmetric/antisymmetric games are presented. The bases and dimensions of all dual decomposed subspaces are revealed.Finally, some properties of these decomposed subspaces are obtained.


This work was supported by National Natural Science Foundation of China (Grant Nos. 61473099, 61273013, 61333001).


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  • Table 1   Payoff bi-matrix of rock-paper-scissors
    Rock PaperScissor
    Rock 0, 0 $-1$, 11, $-1$
    Paper 1, $-1$ 0, 0$-1$,$~1$
    Scissor $-1$,$~1$ 1, $-1$0, 0
  • Table 2   Payoff bi-matrix of matching pennies
    Heads Tails
    Heads 1, $-1$ $-1$, 1
    Tails $-1$,$~1$ 1, $-1$

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