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The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
[article]

2011
*
arXiv
*
pre-print

We show that

arXiv:1006.5352v2
fatcat:tnukvlwxgrf5zokw47hdxic6ri
*the*widely used*homotopy**method*for solving fixpoint problems, as well as*the*Harsanyi-Selten*equilibrium**selection*process for games, are PSPACE-complete to implement. ... A further application*of*our techniques yields*the*result that it is PSPACE-complete to compute any*of**the*equilibria that could be found via*the*classical*Lemke*-*Howson*algorithm, a*complexity*-theoretic ... This paper is a*complexity*-theoretic critique*of**the*tracing procedure*and**the**homotopy**method*: we show that finding*the**solutions*they prescribe requires*the*power*of*PSPACE. ...##
###
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

2011
*
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
*

We show that

doi:10.1109/focs.2011.26
dblp:conf/focs/GoldbergPS11
fatcat:fc7o54gmfrbxlpxz73tqzmumci
*the*widely used*homotopy**method*for solving fixpoint problems, as well as*the*Harsanyi-Selten*equilibrium**selection*process for games, are PSPACEcomplete to implement. ... A further application*of*our techniques yields*the*result that it is PSPACE-complete to compute any*of**the*equilibria that could be found via*the*classical*Lemke*-*Howson*algorithm, a*complexity*-theoretic ... Papadimitriou was supported by an NSF grant*and*by a Google grant. ...##
###
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

2013
*
ACM Transactions on Economics and Computation
*

We show that

doi:10.1145/2465769.2465774
fatcat:uqaj6yrr7vabnj6s2w6ieijtem
*the*widely used*homotopy**method*for solving fixpoint problems, as well as*the*Harsanyi-Selten*equilibrium**selection*process for games, are PSPACEcomplete to implement. ... A further application*of*our techniques yields*the*result that it is PSPACE-complete to compute any*of**the*equilibria that could be found via*the*classical*Lemke*-*Howson*algorithm, a*complexity*-theoretic ... Papadimitriou was supported by an NSF grant*and*by a Google grant. ...##
###
How Do You Like Your Equilibrium Selection Problems? Hard, or Very Hard?
[chapter]

2010
*
Lecture Notes in Computer Science
*

In

doi:10.1007/978-3-642-16170-4_2
fatcat:v5lgj7akdfhehgnwf5o6n4ur24
*the*talk I will give an overview*of*this topic,*and*a summary*of*recent progress showing that*the*equilibria that are found by*the**Lemke*-*Howson*algorithm, as well as related*homotopy**methods*, are PSPACE-complete ... Thus we show that there are no short cuts to*the**Lemke*-*Howson**solutions*, subject only to*the*hardness*of*PSPACE. I mention some open problems. ...*Homotopy**methods**The*survey paper [8] discusses*homotopy**methods*in detail, including*the**Lemke*-*Howson*algorithm, which falls within this framework. ...##
###
A Survey of PPAD-Completeness for Computing Nash Equilibria
[article]

2011
*
arXiv
*
pre-print

*The*most well-known such problem is that

*of*computing a Nash

*equilibrium*

*of*a game. Other examples include

*the*search for market equilibria,

*and*envy-free allocations in

*the*context

*of*cake-cutting. ... I also give an overview

*of*some recent work that uses these ideas to show PSPACE-completeness for

*the*computation

*of*specific equilibria found by

*homotopy*

*methods*. ... Related algorithms include

*the*linear tracing procedure

*and*

*homotopy*

*methods*[39, 41, 42, 43, 44] discussed below: in these papers

*the*motivation is

*equilibrium*

*selection*-in cases where multiple equilibria ...

##
###
Structure Learning for Approximate Solution of Many-Player Games

2020
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

We experimentally demonstrate

doi:10.1609/aaai.v34i02.5586
fatcat:in26jvhfpjcwpgvrmm4lchjphi
*the*efficacy*of*both*methods*in reaching quality*solutions**and*uncovering hidden structure, on both perfectly*and*approximately structured game instances. ... Our first algorithm, K-Roles, exploits symmetry by learning a role assignment for players*of**the*game through unsupervised learning (clustering)*methods*. ... Acknowledgment This work was supported in part by*the*US Army Research Office under MURI W911NF-18-1-0208. ...##
###
Multiagent systems: algorithmic, game-theoretic, and logical foundations

2009
*
ChoiceReviews
*

a sample Nash

doi:10.5860/choice.46-5662
fatcat:pr2pmv7k2bad3pp5bxgogecgnq
*equilibrium*91 4.2.2 An LCP formulation*and**the**Lemke*-*Howson*algorithm 93 4.2.3 Searching*the*space*of*supports 101 4.2.4 Beyond sample*equilibrium*computation 104 4.3 Computing Nash equilibria ...*Solution*Concepts*of*Normal-Form Games 89 4.1 Computing Nash equilibria*of*two-player, zero-sum games 89 4.2 Computing Nash equilibria*of*two-player, general-sum games 91 4.2.1*Complexity**of*computing ... We will call*the*first type*of*belief "mere belief" (or sometimes "belief" for short)*and**the*second type*of*belief "certainty." ...##
###
Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra
[article]

2006
*
arXiv
*
pre-print

We then explain how to find all

arXiv:math/0612462v1
fatcat:2lng5b4majabfa5jt2qyehczfu
*the**complex*roots*of**the*polynomial systems for arbitrary generic games, by polyhedral*homotopy*continuation starting from*the**solutions*to*the*specially constructed games ... We describe*the*use*of*Groebner bases to solve these polynomial systems*and*to learn geometric information about how*the**solution*set varies with*the*payoff functions. ... We would like to express our gratitude to*the*following for generously taking*the*time to personally discuss with us*the*use*of*their software packages: Andrew McLennan*and*Ted Turocy (Gambit [18] ), ...##
###
Finding all Nash equilibria of a finite game using polynomial algebra

2009
*
Economic Theory
*

We then explain how to find all

doi:10.1007/s00199-009-0447-z
fatcat:tr6n7jjz5vgwleky56t2mxgeky
*the**complex*roots*of**the*polynomial systems for arbitrary generic games, by polyhedral*homotopy*continuation starting from*the**solutions*to*the*specially constructed games ... We describe*the*use*of*Gröbner bases to solve these polynomial systems*and*to learn geometric information about how*the**solution*set varies with*the*payoff functions. ... Readers may be familiar with*the**homotopy*continuation*method*under*the*guise*of*"tracing procedures", such as*the*techniques*of**Lemke*-*Howson*or Govindan*and*Wilson. ...##
###
Computing Nash Equilibria in Bimatrix Games: GPU-Based Parallel Support Enumeration

2014
*
IEEE Transactions on Parallel and Distributed Systems
*

*The*algorithm achieves significant speedups relative to

*the*OpenMP-based parallel implementation

*of*

*the*support enumeration

*method*running on conventional multicore machines. ... Computing Nash equilibria is a very important problem in strategic analysis

*of*markets, conflicts

*and*resource allocation. ... ACKNOWLEDGMENT This research was supported in part by NSF grants DGE-0654014

*and*CNS-1116787. ...

##
###
Computing Nash equilibria in bimatrix games: GPU-based parallel support enumeration

2012
*
2012 IEEE 31st International Performance Computing and Communications Conference (IPCCC)
*

*The*algorithm achieves significant speedups relative to

*the*OpenMP-based parallel implementation

*of*

*the*support enumeration

*method*running on conventional multicore machines. ... Computing Nash equilibria is a very important problem in strategic analysis

*of*markets, conflicts

*and*resource allocation. ... ACKNOWLEDGMENT This research was supported in part by NSF grants DGE-0654014

*and*CNS-1116787. ...

##
###
Uniqueness of Stationary Equilibrium Payoffs in Coalitional Bargaining

2011
*
Social Science Research Network
*

*The*theory

*of*

*the*fixed point index is used to show that stationary

*equilibrium*expected payoffs

*of*this coalitional bargaining game are unique. ... each other agent either approves or rejects

*the*proposal,

*and*

*the*proposal is implemented if

*the*set

*of*approving agents is a winning coalition for

*the*proposer. ... based on linear pivoting such as

*the*simplex algorithm for linear programming,

*the*

*Lemke*-

*Howson*algorithm (

*Lemke*

*and*

*Howson*(1964) ) for two player games, or

*the*

*Lemke*(1965) algorithm for linear complementary ...

##
###
Uniqueness of stationary equilibrium payoffs in coalitional bargaining

2013
*
Journal of Economic Theory
*

*The*theory

*of*

*the*fixed point index is used to show that stationary

*equilibrium*expected payoffs

*of*this coalitional bargaining game are unique. ... each other agent either approves or rejects

*the*proposal,

*and*

*the*proposal is implemented if

*the*set

*of*approving agents is a winning coalition for

*the*proposer. ... based on linear pivoting such as

*the*simplex algorithm for linear programming,

*the*

*Lemke*-

*Howson*algorithm (

*Lemke*

*and*

*Howson*(1964) ) for two player games, or

*the*

*Lemke*(1965) algorithm for linear complementary ...

##
###
Equilibria, fixed points, and complexity classes

2009
*
Computer Science Review
*

A B S T R A C T Many models from a variety

doi:10.1016/j.cosrev.2009.03.004
fatcat:lhr6iqyodfgond7vjrochf44la
*of*areas involve*the*computation*of*an*equilibrium*or fixed point*of*some kind. ... There are certain common computational principles underlying different types*of*equilibria, which are captured by*the**complexity*classes PLS, PPAD,*and*FIXP. ... Newton-based,*and**homotopy**methods*(some*of*these assume differentiability*and*use also*the*derivatives*of**the*function). ...##
###
Toward Large-Scale Agent Guidance in an Urban Taxi Service
[article]

2012
*
arXiv
*
pre-print

Nash

arXiv:1210.4849v1
fatcat:iu5vs2sdlvhpjklatgt5b6d5d4
*equilibrium*policy is proposed as*the**solution*to*the*game, where no driver has*the*incentive to singly deviate from it. ... Empirical result shows that both formulations improve*the*efficiency*of**the*service significantly. ...*The*most well-known*of*such*methods*is*the**Lemke*-*Howson*algorithm. See (Herings*and*Peeters, 2009) for a recent survey,*and*(Goldberg et al., 2011) for a discussion on*the**complexity**of*such*methods*. ...
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