scholarly journals Pure Nash Equilibria: Hard and Easy Games

2005 ◽  
Vol 24 ◽  
pp. 357-406 ◽  
Author(s):  
G. Gottlob ◽  
G. Greco ◽  
F. Scarcello
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Equilibrium Problems ◽  
Small Neighborhood ◽  
Constraint Satisfaction Problems ◽  
Pure Nash Equilibrium ◽  
Bounded Treewidth ◽  
Dependency Structure ◽  
Strong Nash Equilibrium ◽  
Restrictive Settings

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems.

scholarly journals Convex generalized Nash equilibrium problems and polynomial optimization

2021 ◽  
Author(s):  
Jiawang Nie ◽  
Xindong Tang
Keyword(s):  
Nash Equilibrium ◽  
Numerical Experiments ◽  
Nash Equilibria ◽  
Equilibrium Problems ◽  
Polynomial Optimization ◽  
Generalized Nash Equilibrium ◽  
Semidefinite Relaxations ◽  
The Moment ◽  
Generalized Nash Equilibria ◽  
Generalized Nash Equilibrium Problems

AbstractThis paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing generalized Nash equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.

THE PRICE OF ANARCHY FOR RESTRICTED PARALLEL LINKS

2006 ◽  
Vol 16 (01) ◽  
pp. 117-131 ◽  
Author(s):  
MARTIN GAIRING ◽  
THOMAS LÜCKING ◽  
MARIOS MAVRONICOLAS ◽  
BURKHARD MONIEN
Keyword(s):  
Nash Equilibrium ◽  
System Performance ◽  
Nash Equilibria ◽  
Price Of Anarchy ◽  
Pure Nash Equilibrium ◽  
Worst Case ◽  
Comprehensive Collection ◽  
Parallel Links ◽  
Special Case

In the model of restricted parallel links, n users must be routed on m parallel links under the restriction that the link for each user be chosen from a certain set of allowed links for the user. In a (pure) Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links. The Price of Anarchy is a widely adopted measure of the worst-case loss (relative to optimum) in system performance (maximum latency) incurred in a Nash equilibrium. In this work, we present a comprehensive collection of bounds on Price of Anarchy for the model of restricted parallel links and for the special case of pure Nash equilibria. Specifically, we prove: • For the case of identical users and identical links, the Price of Anarchy is [Formula: see text]. • For the case of identical users, the Price of Anarchy is [Formula: see text]. • For the case of identical links, the Price of Anarchy is [Formula: see text], which is asymptotically tight. • For the most general case of arbitrary users and related links, the Price of Anarchy is at least m – 1 and less than m. The shown bounds reveal the dependence of the Price of Anarchy on n and m for all possible assumptions on users and links.

scholarly journals Pure Nash Equilibria in Online Fair Division

2017 ◽  
Author(s):  
Martin Aleksandrov ◽  
Toby Walsh
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Specific Property ◽  
Fair Division ◽  
Pareto Efficiency ◽  
Pure Nash Equilibrium ◽  
Group Strategy ◽  
Is Strategy ◽  
Strategy Proof ◽  
Online Mechanism

We consider a fair division setting in which items arrive one by one and are allocated to agents via two existing mechanisms: LIKE and BALANCED LIKE. The LIKE mechanism is strategy-proof whereas the BALANCED LIKE mechanism is not. Whilst LIKE is strategy-proof, we show that it is not group strategy-proof. Indeed, our first main result is that no online mechanism is group strategy-proof. We then focus on pure Nash equilibria of these two mechanisms. Our second main result is that computing a pure Nash equilibrium is tractable for LIKE and intractable for BALANCED LIKE. Our third main result is that there could be multiple such profiles and counting them is also intractable even when we restrict our attention to equilibria with a specific property (e.g. envy-freeness, Pareto efficiency).

scholarly journals Pure Nash Equilibria in Resource Graph Games

2021 ◽  
Vol 72 ◽  
Author(s):  
Tobias Harks ◽  
Max Klimm ◽  
Jannik Matuschke
Keyword(s):  
Nash Equilibrium ◽  
Arbitrary Function ◽  
Nash Equilibria ◽  
Mutual Influence ◽  
Pure Nash Equilibrium ◽  
Strategy Space ◽  
Graph Games ◽  
Finite Set ◽  
Load Vector ◽  
The Cost

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium. For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

A Theory of Patronized Goods: Inefficient and Efficient Equilibria

2011 ◽  
pp. 65-87 ◽  
Author(s):  
A. Rubinstein
Keyword(s):  
Nash Equilibrium ◽  
Social Policy ◽  
Nash Equilibria ◽  
Institutional Environment ◽  
Social Motivation ◽  
Pareto Optimal ◽  
Market Failures ◽  
Equilibrium Conditions ◽  
Policy Issues

The article considers some aspects of the patronized goods theory with respect to efficient and inefficient equilibria. The author analyzes specific features of patronized goods as well as their connection with market failures, and conjectures that they are related to the emergence of Pareto-inefficient Nash equilibria. The key problem is the analysis of the opportunities for transforming inefficient Nash equilibrium into Pareto-optimal Nash equilibrium for patronized goods by modifying the institutional environment. The paper analyzes social motivation for institutional modernization and equilibrium conditions in the generalized Wicksell-Lindahl model for patronized goods. The author also considers some applications of patronized goods theory to social policy issues.

scholarly journals Expressiveness and Nash Equilibrium in Iterated Boolean Games

2021 ◽  
Vol 22 (2) ◽  
pp. 1-38
Author(s):  
Julian Gutierrez ◽  
Paul Harrenstein ◽  
Giuseppe Perelli ◽  
Michael Wooldridge
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Infinite Sequence ◽  
Multi Agent Systems ◽  
Temporal Logics ◽  
Agent Systems ◽  
Temporal Properties ◽  
Multi Agent ◽  
Game Theoretic ◽  
Boolean Games

We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent  has a goal  , represented using (a fragment of) Linear Temporal Logic ( ) . The goal  captures agent  ’s preferences, in the sense that the models of  represent system behaviours that would satisfy  . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of  ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

An optimization method for Nash equilibrium problems

2017 ◽  
Vol 20 (6-7) ◽  
pp. 1465-1469
Author(s):  
Jian Hou ◽  
Qing Chang ◽  
Zong-Chuan Wen
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Problems ◽  
Optimization Method

scholarly journals A note on supervised classification and Nash-equilibrium problems

2017 ◽  
Vol 51 (2) ◽  
pp. 329-341
Author(s):  
Nicolas Couellan
Keyword(s):  
Nash Equilibrium ◽  
Support Vector Machines ◽  
Supervised Classification ◽  
Dual Space ◽  
Equilibrium Problems ◽  
Support Vector ◽  
Generalized Nash Equilibrium ◽  
Class Separation ◽  
Vector Machines ◽  
Generalized Nash Equilibrium Problems

In this note, we investigate connections between supervised classification and (Generalized) Nash equilibrium problems (NEP & GNEP). For the specific case of support vector machines (SVM), we exploit the geometric properties of class separation in the dual space to formulate a non-cooperative game. NEP and Generalized NEP formulations are proposed for both binary and multi-class SVM problems.

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