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On the value of commitment

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Abstract

In game theory, it is well known that being able to commit to a strategy before other players move can be beneficial. In this paper, we analyze how much benefit a player can derive from commitment in various types of games, in a quantitative sense that is similar to concepts such as the value of mediation and the price of anarchy. Specifically, we introduce and study the value of pure commitment (the benefit of committing to a pure strategy), the value of mixed commitment (the benefit of committing to a mixed strategy), and the mixed versus pure commitment ratio (how much can be gained by committing to a mixed strategy rather than a pure one). In addition to theoretical results about how large these values are in the extreme case in various classes of games, we also give average-case results based on randomly drawn normal-form games.

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Notes

  1. As is common in this literature, the algorithm presented in this paper assumes that ties are broken in the leader’s favor. The same algorithm appears in a GEB-2010 paper by von Stengel and Zamir [1] to compute the highest possible payoff for the leader; they also give a linear-programming algorithm to compute the lowest possible payoff for the leader, and point out that these payoffs must be the same in generic bimatrix games.

  2. They assume that every subset of a valid schedule is also a valid schedule.

  3. Due to numerical issues, our MIP-based Nash solver computes a higher NE utility for player 1 in some traveler’s dilemma games. This happens because there is an epsilon-NE with extremely small epsilon in which each player randomizes over several strategies. We think this also says something interesting about this game, namely that extremely small deviations from rationality are sufficient to completely change the game-theoretic prediction and thereby the point this game is intended to make.

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Acknowledgments

This work has been supported by NSF Awards IIS-0812113, IIS-0953756, CCF-1101659, ARO Grants W911NF-09-1-0459, W911NF-11-1-0332, W911NF-12-1-0550, and an Alfred P. Sloan fellowship. We would like to thank Kamesh Munagala for his suggestion to consider games that are close to constant-sum.

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  1. Duke University, Durham, NC, USA

    Joshua Letchford, Dmytro Korzhyk & Vincent Conitzer

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  1. Joshua Letchford
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  2. Dmytro Korzhyk
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  3. Vincent Conitzer
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Correspondence to Joshua Letchford.

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Letchford, J., Korzhyk, D. & Conitzer, V. On the value of commitment. Auton Agent Multi-Agent Syst 28, 986–1016 (2014). https://doi.org/10.1007/s10458-013-9246-9

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