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Collusive game solutions via optimization

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Abstract

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported.

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Authors and Affiliations

  1. Department of Economics, The Johns Hopkins University, Baltimore, Maryland, 21218-2682, USA

    J.E. Harrington

  2. Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, Maryland, 21218-2682, USA

    B.F. Hobbs

  3. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, 12180-3590, USA

    J.S. Pang

  4. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland, 21218-2682, USA

    A. Liu

  5. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland, 21218-2682, USA

    G. Roch

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  1. J.E. Harrington
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  2. B.F. Hobbs
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  3. J.S. Pang
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  4. A. Liu
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  5. G. Roch
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Correspondence to J.S. Pang.

Additional information

It is with great honor that we dedicate this paper to Professor Terry Rockafellar on the occasion of his 70th birthday. Our work provides another example showing how Terry's fundamental contributions to convex and variational analysis have impacted the computational solution of applied game problems.

This author's research was partially supported by the National Science Foundation under grant ECS-0080577.

This author's research was partially supported by the National Science Foundation under grant CCR-0098013.

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Harrington, J., Hobbs, B., Pang, J. et al. Collusive game solutions via optimization. Math. Program. 104, 407–435 (2005). https://doi.org/10.1007/s10107-005-0622-3

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  • DOI: https://doi.org/10.1007/s10107-005-0622-3

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