On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games

Stefanos Leonardos () and Costis Melolidakis ()
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Stefanos Leonardos: Department of Mathematics, National & Kapodistrian University of Athens, Panepistimioupolis GR — 157 84, Athens, Greece
Costis Melolidakis: Department of Mathematics, National & Kapodistrian University of Athens, Panepistimioupolis GR — 157 84, Athens, Greece

International Game Theory Review (IGTR), 2018, vol. 20, issue 03, 1-28

Abstract: 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 nondegenerate 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 are 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 2 × 2 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.

Keywords: Bimatrix game; Nash equilibrium; subgame perfect; commitment optimal; commitment value; weakly unilaterally competitive games; pure strategy equilibrium; commitment advantageous games (search for similar items in EconPapers)
JEL-codes: C72 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0219198918400017

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