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Unit-sphere games

Author

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  • Pingzhong Tang

    (Tsinghua University)

  • Hanrui Zhang

    (Tsinghua University)

Abstract

This paper introduces a class of games, called unit-sphere games, in which strategies are real vectors with unit 2-norms (or, on a unit-sphere). As a result, they should no longer be interpreted as probability distributions over actions, but rather be thought of as allocations of one unit of resource to actions and the payoff effect on each action is proportional to the square root of the amount of resource allocated to that action. The new definition generates a number of interesting consequences. We first characterize the sufficient and necessary condition under which a two-player unit-sphere game has a Nash equilibrium. The characterization reduces solving a unit-sphere game to finding all eigenvalues and eigenvectors of the product matrix of individual payoff matrices. For any unit-sphere game with non-negative payoff matrices, there always exists a unique Nash equilibrium; furthermore, the unique equilibrium is efficiently reachable via Cournot adjustment. In addition, we show that any equilibrium in positive unit-sphere games corresponds to approximate equilibria in the corresponding normal-form games. Analogous but weaker results are obtained in n-player unit-sphere games.

Suggested Citation

  • Pingzhong Tang & Hanrui Zhang, 2017. "Unit-sphere games," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 957-974, November.
    • Handle: RePEc:spr:jogath:v:46:y:2017:i:4:d:10.1007_s00182-017-0565-y
    DOI: 10.1007/s00182-017-0565-y
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