On the commitment value and commitment optimal strategies in bimatrix games

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Department of Mathematics
On the commitment value and commitment optimal strategies in bimatrix games
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Given a bimatrix game, the associated leadership or commitment games are defined as the games at
which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower,
chooses his strategy after being informed of the irrevocable commitment of the leader (but not
of its realization in case it is mixed). Based on a result by Von Stengel and Zamir [2010], the notions
of commitment value and commitment optimal strategies for each player are discussed as a possible
solution concept. It is shown that in non-degenerate bimatrix games (a) pure commitment optimal
strategies together with the follower’s best response constitute Nash equilibria, and (b) strategies that
participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies,
provided they are not matrix game optimal. For various classes of bimatrix games that generalize
zero sum games, the relationship between the maximin value of the leader’s payoff matrix, the Nash
equilibrium payoff and the commitment optimal value is discussed. For the Traveler’s Dilemma, the
commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable
as a solution than the unique Nash equilibrium. Finally, the relationship between commitment
optimal strategies and Nash equilibria in 2x2 bimatrix games is thoroughly examined and in addition,
necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership
game than at any Nash equilibrium of the simultaneous move game are provided.
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Stefanos Leonardos
Costis Melolidakis
Game Theory, Bimatrix Games, Nash Equilibrium, Commitment Value, Commitment Advantageous Games, Weakly Unilaterally Competitive Games
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Project information:
Research funding Program ARISTEIA II: "Optimization of stochastic systems under partial information and applications" of NSRF.
Stefanos Leonardos acknowledges a scholarship of the Alexander S. Onassis Public Benefit Foundation.
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