Existence of Nash Equilibria on Integer Programming Games

Abstract

The recently defined class of integer programming games (IPG) models situations where multiple self-interested decision makers interact, with their strategy sets represented by a finite set of linear constraints together with integer requirements. Many real-world problems can suitably be fit in this class, and hence anticipating IPG outcomes is of crucial value for policy makers and regulators. Nash equilibria have been widely accepted as the solution concept of a game. Consequently, their computation provides a reasonable prediction of games outcome. In this paper, we start by showing the computational complexity of deciding the existence of a Nash equilibrium for an IPG. Then, using sufficient conditions for their existence, we develop two general algorithmic approaches that are guaranteed to approximate an equilibrium under mild conditions. We also showcase how our methodology can be changed to determine other equilibria definitions. The performance of our methods is analysed through computational experiments in a knapsack game, a competitive lot-sizing game and a kidney exchange game. To the best of our knowledge, this is the first time that equilibria computation methods for general integer programming games have been designed and computationally tested.

Copositive Duality for Discrete Markets and Games

“…and Cournot games with indivisible goods (Kostreva 1993). Most papers focus on algorithms for computing NE, and are limited to IPGs with only integer decisions (Köppe et al 2011, Sagratella 2016, Carvalho et al 2017. Theoretical results on IPGs include Mallick (2011) andSagratella (2016), which respectively provide NE existence conditions for two-person discrete games and 2-groups partitionable discrete games.…”

The Cut and Play Algorithm: Computing Nash Equilibria via Outer Approximations

“…Stein et al [57] prove that for all the MNE s in separable games -namely where players' payoffs take a sum-of-products form -there is a payoff equivalent MNE of finite support. Carvalho et al [9,12] focus on IPGs, where each player strategy set is defined by inequalities and integrality requirements. They point out that the problem of deciding if an IPG has an MNE -or even a PNE -is Σ p 2 -hard.…”

“…Games can broaden the modeling capabilities of MIP , and extend classical combinatorial and decision-making problems to multi-agent settings that can account for interactions among multiple decision-makers. For instance, bilevel programming [3,8,29,36,39,41]) and Integer Programming Games (IPGs) [9,11,19,24,33,40]. This recent research direction suggests that the joint endeavor between game theory and MIP can widen their theoretical understanding and practical impact.…”

ZERO: Playing Mathematical Programming Games

“…The Operations Research growing interest in game dynamics reflects a need to extend classical decision-making frameworks to multi-agent settings that can account for interactions among multiple decision-makers. The community devoted particular interest -to name a few -to Bilevel Programming (e.g., Hu and Ralph (2007), DeNegre and Ralphs (2009), Labbé and Violin (2013), Caprara et al (2016), Fischetti et al (2017), Basu et al (2019), Kleinert et al (2021)) and its application in electricity markets and network pricing problems (Feijoo and Das 2015, Labbé et al 1998, Brotcorne et al 2001, Equilibrium Problems with Equilibrium Constraints (Luo et al 1996, Carvalho et al 2019, and more recently to Integer Programming Games (Carvalho et al 2018, Cronert and Minner 2021, Guo et al 2021, Köppe et al 2011, Dragotto and Scatamacchia 2021. On the one hand, such empowering modeling capabilities unquestionably offer a tempting opportunity for extending the domain of influence of operations research.…”

“…Additionally, they can help embed socially beneficial outcomes by enlightening the nature of interaction among possibly adversarial decision-makers. For instance, Carvalho et al (2019) provides insights on the role of a carbon tax in competitive international energy markets, Carvalho et al (2017) proves that the most rational outcome in their cross-border kidney exchange maximizes the social welfare (e.g., the sum of the objectives of all players). On the other hand, multi-agent models are as helpful as one can efficiently compute equilibria (or equivalent solution paradigms), thus highlighting the importance of theoretical and practical contributions in equilibria computation.…”