Copositive Duality for Discrete Markets and Games

Guo, Cheng; Bodur, Merve; Taylor, Joshua A.

doi:10.48550/arxiv.2101.05379

Cited by 4 publications

(4 citation statements)

References 38 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Carvalho¹,

Dragotto²,

Lodi³

et al. 2021

Preprint

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The concept of Nash equilibrium enlightens the structure of rational behavior in multi-agent settings. However, the concept is as helpful as one may compute it efficiently. We introduce the Cut-and-Play, an algorithm to compute Nash equilibria for a class of non-cooperative simultaneous games where each player's objective is linear in their variables and bilinear in the other players' variables. Using the rich theory of integer programming, we alternate between constructing (i.) increasingly tighter outer approximations of the convex hull of each player's feasible set -by using branching and cutting plane methods -and (ii.) increasingly better inner approximations of these hulls -by finding extreme points and rays of the convex hulls. In particular, when these convex hulls are polyhedra, we prove the correctness of our algorithm and leverage the mixed integer programming technology to compute equilibria for a large class of games. Further, we integrate existing cutting plane families inside the algorithm, significantly speeding up equilibria computation. We showcase a set of extensive computational results for Integer Programming Games and simultaneous games among bilevel leaders. In both cases, our framework outperforms the state-of-the-art in computing time and solution quality.

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Section: Introductionmentioning

confidence: 99%

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

Carvalho¹,

Dragotto²,

Lodi³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…AGT attracted significant attention from the computer science and optimization community in the last two decades. Several recent works [10,18,21] considered Integer Programming Games (IPGs), namely games where the agents solve (parametrized) integer programs. In this work, we study a class of simultaneous and non-cooperative IPGs among n players (agents) as in Definition 1, where every player has m integer variables.…”

Section: Introductionmentioning

confidence: 99%

The ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

Dragotto¹,

Scatamacchia²

2021

Preprint

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In Algorithmic Game Theory (AGT ), designing efficient algorithms to compute Nash equilibria poses considerable challenges. We make progress in the field and shed new light on the intersection between Algorithmic Game Theory and Integer Programming. We introduce ZERO Regrets, a general cutting plane algorithm to compute, enumerate, and select Pure Nash Equilibria (PNE s) in Integer Programming Games, a class of simultaneous and non-cooperative games. We present a theoretical foundation for our algorithmic reasoning and provide a polyhedral characterization of the convex hull of the Pure Nash Equilibria. We introduce the concept of equilibrium inequality, and devise an equilibrium separation oracle to separate non-equilibrium strategies from PNE s. We test ZERO Regrets on two paradigmatic classes of games: the Knapsack Game and the Network Formation Game, a well-studied game in AGT . Our algorithm successfully solves relevant instances of both games and shows promising applications for equilibria selection.

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“…Of course, Theorem 3 holds more generally. While it is difficult to establish conditions under which strong duality holds for general nonconvex problems (although see [16] for the notion of "copositive duality" for mixed-binary quadratic programs), Theorem 3 establishes that if we solve the max social welfare problem and its dual, we will arrive at a minimum disequilibrium solution that may have value in certain situations or as an approximate equilibrium.…”

mentioning

confidence: 99%

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Harwood¹,

Trespalacios²,

Papageorgiou³

et al. 2021

Preprint

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This paper introduces the concept of optimization equilibrium as an equivalently versatile definition of a generalized Nash equilibrium for multi-agent non-cooperative games. Through this modified definition of equilibrium, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and social welfare maximization. Significantly, this is all in a general setting without the assumption of convexity. Along the way, we introduce the idea of minimum disequilibrium as a solution concept that reduces to traditional equilibrium when equilibrium exists. The connections with bilevel programming and related semi-infinite programming permit us to adapt global optimization methods for those classes of problems, such as constraint generation or cutting plane methods, to the problem of finding a minimum disequilibrium solution. We show that this method works, both theoretically and with a numerical example, even when the agents are modeled by mixed-integer programs.

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Copositive Duality for Discrete Markets and Games

Cited by 4 publications

References 38 publications

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

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

The ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

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