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Leadership (commitment) games

S. Leonardos \& C. Melolidakis. On the commitment value and commitment optimal strategies in bimatrix games, under revision, 2017. https://arxiv.org/abs/1612.08888.

Given a bimatrix game in normal form, the associated leadership games, one for each order of the players, are the games where one of the players commits to any of his (mixed) strategies and reveals his commitment (not its realization) to his opponent prior to the opponent’s strategy choice. Setting the foundations of contemporary game theory, von Neumann & Morgestern already used these games to obtain the value of two-person zero-sum games in their fundamental work, von Neumann & Morgenstern (1953). However, this optimization approach failed to generalize efficiently in non-zero sum games and it was Nash’s equilibrium approach that became the widely accepted generalization of the value in non-zero sum games.

Despite the vast literature and the various methods that have been applied in this setting, including recently the comprehensive work of von Stengel & Zamir (2010) who established a certain leadership advantage in bimatrix games, many questions remain open: Firstly, can we characterize the class of games at which the sequential play leads to an equilibrium of the simultaneous move game? Secondly, for which classes of bimatrix games may the leader not guarantee more than the safety level of his payoff matrix? Lastly, if the leader-follower approach does not lead to a unique value in non-zero sum games, is there still a some other solution concept that would generalize the optimality point of view? Variations of these questions, still constitute a main area of research for many contemporary authors and the relevant literature is continuously growing.

As the main topic of my PhD dissertation, I initiated my research by searching and studying a wide range of papers on this subject and realized quickly that there seemed to be a little space for something novel. All our initial ideas were rejected in view of papers that had satisfactorily developed them. To address this challenge, I invested time in becoming familiar with the existing results and the methods used in their derivation, in an effort to expand them. My focus was mainly on the comprehensive approach of von Stengel & Zamir (2010) that was closely related to our motivating questions.

Some first results were obtained by direct computations on the tractable case of 2×2 bimatrix games. Subsequently, we applied methods used in von Stengel & Zamir (2010) to derive a property that applies in bimatrix games of arbitrary dimension, namely that the strategies used in the equilibrium path of a pure subgame perfect equilibrium of a leadership game that is non-degenerate for the leader, constitute a pure strategy equilibrium of the simultaneous move game. This result corroborates the prevalent intuition that the full potential of the leader advantage is realized when he uses a mixed strategy, i.e. when he conceals (at least slightly) his true choice.

In an attempt to study the leadership game in a more specific context, we restricted attention to specific classes of bimatrix games. Along this line of research, we noted that the leader-follower approach of von Neumann and Morgenstern extends directly to the class of weakly unilateral competitive games, a result that is immediate when one has the right mathematical framework. In other classes of games, such as almost strictly competitive, pre-tight or even strategically zero-sum games (all of which include the class of zero-sum games) we established that the solutions of the leadership and simultaneous move game may not coincide, in view of several non-trivial counterexamples.

Still, many of our initial questions remain open and constitute a subject of ongoing research. Despite the difficulties, any advancements would have broad theoretical applications and would provide intuitions for problems that lie in the core of contemporary game theory. As a subject that addresses the fundamentals of game theory, it is appropriate to actively engage students that are interested in graduate studies in this area. On the one hand, it offers exposure to a wide range of conceptual questions and on the other hand, it constitutes a promising area for research for a student who will manage to view these established notions from a novel point of view or to draw connections with other mathematical objects.

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