Abstract
We aim to investigate a new class of games, where each player’s set of strategies is a union of polyhedra. These are called integer programming games. To motivate our work, we describe some practical examples suitable to be modeled under this paradigm. We analyze the problem of determining whether or not a Nash equilibria exists for an integer programming game, and demonstrate that it is complete for the second level of the polynomial hierarchy.
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Notes
- 1.
The equilibrium conditions (2.3) only reflect a player p deviation to strategy in \(X^p\) and not in \(\varDelta ^p\), because a strategy in \(\varDelta ^p\) is a convex combination of strategies in \(X^p\), and thus cannot lead to a better payoff than one in \(X^p\).
- 2.
In specific, a player’s payoff is a summation over all pure profiles of strategies, where each term is the product of the associated binary variables and the associated payoff.
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Acknowledgements
Part of this work was performed while the first author was in the Faculty of Sciences University of Porto and INESC TEC. The first author thanks the support of Institute for data valorisation (IVADO), the Portuguese Foundation for Science and Technology (FCT) through a PhD grant number SFRH/BD/79201/2011 and the ERDF European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project POCI-01-0145-FEDER-006961, and National Funds through the FCT (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013. We thank the referees for comments and questions that helped clarifying the presentation.
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Carvalho, M., Lodi, A., Pedroso, J.P. (2018). Existence of Nash Equilibria on Integer Programming Games. In: Vaz, A., Almeida, J., Oliveira, J., Pinto, A. (eds) Operational Research. APDIO 2017. Springer Proceedings in Mathematics & Statistics, vol 223. Springer, Cham. https://doi.org/10.1007/978-3-319-71583-4_2
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