Constraint Games: Framework and Local Search Solver

Abstract

Game theory is a highly successful paradigm for strategic decision making between multiple agents having conflicting objectives. Since a few years, games have been studied in a computational perspective, raising new issues like complexity of equilibria or succinctness of representation. Indeed, the main representation for general games is still a n-dimensional matrix of exponential size called normal form. In this paper, we introduce the framework of Constraint Games to model strategic interaction between players. A Constraint Game is composed of a set of variables shared by all the players. Among these variables, each player owns a set of decision variables she can control and a Constraint Optimization Problem defining her preferences. Since the preferences of a player depend on the decisions taken by the other players, each player may try to improve her position by choosing an assignment that optimizes her preferences. Pure Nash equilibria are situations in which no player may improve her preferences unilaterally. Constraint Games are thus a generic tool to model general games and can be exponentially more succinct than their normal form. We show the practical utility of the framework by modelling a few realistic problems and we propose an algorithm based on tabu search to compute pure Nash equilibria in Constraint games that outperforms the algorithms based on normal form. In addition, Constraint Games raise some interesting research issues that deserve further attention.

On a Conjecture for Dynamic Priority Queues and Nash Equilibrium for Quality of Service Sensitive Markets

“…A related model of optimal pricing with pre-emptive scheduling has been recently analyzed in [10]. On the other hand, the study of equilibrium behaviour of customers and servers in queueing system has long history and researchers have developed a broad literature in this field by now (see [3,6,12,13,14,15,23] and references there in). In above literature, often customers (or customer classes) are considered as player whereas we consider market as a player by aggregating all customers into a market segment.…”

“…Strategy sets of one player depends on the other and hence they are coupled as in constraint games (See [23]). Service provider's strategy β is constrained by its dependence on strategy of secondary class customers, λ s , (say, constraint (3)).…”

Resource Graph Games: A Compact Representation for Games with Structured Strategy Spaces

“…A related line of work aims to compactly represent games' decisions and utilities using either Boolean logic (Harrenstein et al 2001;Dunne and Wooldridge 2012) or constraint satisfaction problems (Nguyen, Lallouet, and Bordeaux 2013;Nguyen and Lallouet 2014). Each player in these formulations can control multiple decision variables, and as a result these formulations can model games with structured strategy spaces.…”

A Complete Solver for Constraint Games

“…This key issue has been adressed by several types of compact representations. Some are based on some assumptions on the interactions between players, like graphical games [18] or action-graph games [17] while other are language-based, like Boolean Games [14,9,4] or Constraint Games [22]. We focus our interest here in Constraint Games for which utilities are expressed by Constraint Satisfaction Problems (CSP) or Constraint Optimization Problems (COP).…”

“…For Boolean Games, there are techniques limited to specific categories of games like games with acyclic interaction graph [2] or using specific bargaining techniques [10]. For Constraint Games, only a solver based on local search has been proposed [22]. Not surprisingly, large games can be solved but with no guarantee of finding an equilibrium or prove the absence of equilibrium.…”