A payoff matrix converted from A and B where player 1 has two possible actions V and W and player 2 has actions X, Y and Z

In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrices - matrix describing the payoffs of player 1 and matrix describing the payoffs of player 2.[1]

Player 1 is often called the "row player" and player 2 the "column player". If player 1 has possible actions and player 2 has possible actions, then each of the two matrices has rows by columns. When the row player selects the -th action and the column player selects the -th action, the payoff to the row player is and the payoff to the column player is .

The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector of length such that: . Similarly, a mixed strategy for the column player is a non-negative vector of length such that: . When the players play mixed strategies with vectors and , the expected payoff of the row player is: and of the column player: .

Nash equilibrium in bimatrix games

Every bimatrix game has a Nash equilibrium in (possibly) mixed strategies. Finding such a Nash equilibrium is a special case of the Linear complementarity problem and can be done in finite time by the Lemke–Howson algorithm.[1]

There is a reduction from the problem of finding a Nash equilibrium in a bimatrix game to the problem of finding a competitive equilibrium in an economy with Leontief utilities.[2]

A zero-sum game is a special case of a bimatrix game in which .

References

  1. 1 2 Chandrasekaran, R. "Bimatrix games" (PDF). Retrieved 17 December 2015.
  2. Codenotti, Bruno; Saberi, Amin; Varadarajan, Kasturi; Ye, Yinyu (2006). "Leontief economies encode nonzero sum two-player games". Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06. p. 659. doi:10.1145/1109557.1109629. ISBN 0898716055.
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