Computing All Solutions of Nash Equilibrium Problems with Discrete Strategy Sets

Sagratella, Simone

doi:10.1137/15m1052445

Cited by 36 publications

(34 citation statements)

References 30 publications

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“…Generalized Nash Equilibrium Problem (GNEP) is another important modeling tool in multi-agent contexts. GNEPs, that, unlike SBPs, are problems in which all agents act at the same level, have been extensively studied in the literature and many methods have been proposed for their solutions in the last decades, see, e.g., [15,18,19,21,31,32]. For further details, we refer the interested reader to [17].…”

Section: Introductionmentioning

confidence: 99%

A Bridge Between Bilevel Programs and Nash Games

Lampariello

Sagratella

2017

J Optim Theory Appl

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We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our "uneven" horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our "uneven" horizontal model, in some sense, lies between the vertical bilevel model and a "pure" horizontal game.

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Section: Introductionmentioning

confidence: 99%

A Bridge Between Bilevel Programs and Nash Games

Lampariello

Sagratella

2017

J Optim Theory Appl

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“…This is accomplished in two steps. First (see subsection 5.1), by referring to [16, Theorem 3.2 and Remark 3.2], we replace the pure hierarchical problem (5) by a Generalized Nash Equilibrium Problem (GNEP for short, see e.g., [1,10,11,12,14,22,23,24,25]) with two players, which is parametric in x. Then (see subsection 5.2), we deal with the resulting problem by replacing the lower level game with its Karush-Kuhn-Tucker (KKT) conditions, thus obtaining a corresponding mathematical program with complementarity constraints.…”

Section: How To Address (Spbp ε )mentioning

confidence: 99%

The Standard Pessimistic Bilevel Problem

Lampariello¹,

Sagratella²,

Stein³

2019

SIAM J. Optim.

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Pessimistic bilevel optimization problems, as optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allow us to approximate such finite infima arbitrarily well by minimal values of a sequence of solvable single-level problems. To this end, as already done for optimistic problems, for the first time in the literature we introduce the standard version of the pessimistic bilevel problem. For its algorithmic treatment, we reformulate it as a standard optimistic bilevel program with a two follower Nash game in the lower level. The latter lower level game, in turn, is replaced by its Karush-Kuhn-Tucker conditions, resulting in a single-level mathematical program with complementarity constraints. We prove that the perturbed pessimistic bilevel problem, its standard version, the two follower game as well as the mathematical program with complementarity constraints are equivalent with respect to their global minimal points. We study the more intricate connections between their local minimal points in detail. As an illustration, we numerically solve a regulator problem from economics for different values of the perturbation parameters.

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“…Such good results are strongly related to the possibility of reformulating the GNEP as a quasi-variational inequality, and, then, as a constrained system of equations. As described in [30], these reformulations cannot be used whenever, in the GNEP, there are discrete variables. Therefore, it is very difficult to obtain similar local error bound results in our mixed-integer framework.…”

Section: Remarkmentioning

confidence: 99%

“…But, in spite of the fact that there are many applications in which some or all the variables of the players must be assumed to be integers, all the above-mentioned methods for GNEPs work only if the variables of all the players are continuous. Very recently, some methods have been proposed that can deal with integer variables [30,31], but they are designed only for standard Nash equilibrium problems (NEPs), that are GNEPs in which the feasible region of any player is independent on the other players' variables.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for generalized potential games with mixed-integer variables

Sagratella

2017

Comput Optim Appl

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We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixedinteger variables, i.e., games in which some variable are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches. Keywords Generalized Nash equilibrium problem • Generalized potential game • Mixed-integer nonlinear problem • Parametric optimization The work of the author has been partially supported by Avvio alla Ricerca 2015 Sapienza University of Rome, under grant 488.

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Computing All Solutions of Nash Equilibrium Problems with Discrete Strategy Sets

Cited by 36 publications

References 30 publications

A Bridge Between Bilevel Programs and Nash Games

A Bridge Between Bilevel Programs and Nash Games

The Standard Pessimistic Bilevel Problem

Algorithms for generalized potential games with mixed-integer variables

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