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Semidefinite Programming for Min-Max Problems and Games [article]

Rida Laraki
2009 arXiv   pre-print
Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice from global optimization suggests that very often few relaxations are needed for a good approximation  ...  polynomial game in randomized strategies and with compact basic semi-algebraic pure strategy sets.  ...  When the game is zero-sum and S i = [0, 1] for each player i = 1, 2, Parrilo [37] showed that finding an optimal solution is equivalent to solving a single semidefinite program.  ... 
arXiv:0810.3150v2 fatcat:qwenkmtwbfanfosr75fqgmfbi4

Semidefinite programming for min–max problems and games

R. Laraki, J. B. Lasserre
2010 Mathematical programming  
N -player games; Nash equlibria; min-max optimization problems; semidefinite programming. We would like to thank Bernhard von Stengel and the referees for their comments. The work of J.B.  ...  Lasserre was supported by the (French) ANR under grant NT05 − 3 − 41612. 1 Games with finitely many players where the set of pure actions of each player is finite. 2 Games with two players where the set  ...  When the game is zero-sum and S i = [0, 1] for each player i = 1, 2, Parrilo [37] showed that finding an optimal solution is equivalent to solving a single semidefinite program.  ... 
doi:10.1007/s10107-010-0353-y fatcat:skzmuaws35ahhctyxzsv7jxr5q

Polynomial stochastic games via sum of squares optimization

Parikshit Shah, Pablo A. Parrilo
2007 2007 46th IEEE Conference on Decision and Control  
It is shown that minimax equilibria and optimal strategies for such games may be obtained via semidefinite programming.  ...  We consider finite state two-player zero-sum stochastic games over an infinite time horizon with discounted rewards.  ...  Munther Dahleh for pointing out the linear programming solution to single controller finite stochastic games.  ... 
doi:10.1109/cdc.2007.4434492 dblp:conf/cdc/ShahP07 fatcat:ydv7ek334fcufaibucafgox77q

Polynomial stochastic games via sum of squares optimization [article]

Parikshit Shah, Pablo A. Parrilo
2008 arXiv   pre-print
It is shown that minimax equilibria and optimal strategies for such games may be obtained via semidefinite programming.  ...  We consider finite state two-player zero-sum stochastic games over an infinite time horizon with discounted rewards.  ...  By solving a primal-dual pair of semidefinite programs, we obtained minimax equilibria and optimal strategies for the players.  ... 
arXiv:0806.2469v1 fatcat:xtyz3p6ihjejjjjj526jdfefkm

Polynomial games and sum of squares optimization

Pablo A. Parrilo
2006 Proceedings of the 45th IEEE Conference on Decision and Control  
We show that the value of the game, and the corresponding optimal strategies, can be obtained by solving a single semidefinite programming problem.  ...  We study two-person zero-sum games, where the payoff function is a polynomial expression in the actions of the players. This class of games was introduced by Dresher, Karlin, and Shapley in 1950.  ...  Alexandre Megretski, for his help with the translation of [20] , and to Noah Stein, for bringing Karlin's quote in Section II-B to my attention.  ... 
doi:10.1109/cdc.2006.377261 dblp:conf/cdc/Parrilo06 fatcat:nyitmlx4l5ewbnw2rchk2o2g3m

Parallel Approximation of Non-interactive Zero-sum Quantum Games

Rahul Jain, John Watrous
2009 2009 24th Annual IEEE Conference on Computational Complexity  
This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to  ...  determine the players' payoffs.  ...  We therefore thank Rohit Khandekar for his implicit and indirect contribution to this work, and for allowing us to include it in this paper.  ... 
doi:10.1109/ccc.2009.26 dblp:conf/coco/JainW09 fatcat:qsani3o7tfc5bolt5tjlwrbn3e

Parallel approximation of non-interactive zero-sum quantum games [article]

Rahul Jain, John Watrous
2008 arXiv   pre-print
This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to  ...  determine the players' payoffs.  ...  We therefore thank Rohit Khandekar for his implicit and indirect contribution to this work, and for allowing us to include it in this paper.  ... 
arXiv:0808.2775v1 fatcat:mekzd65ykrfmhk7apdc2anpepa

Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming [article]

Xavier Allamigeon, Stéphane Gaubert, Ricardo D. Katz, Mateusz Skomra
2018 arXiv   pre-print
Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter.  ...  The geometric interpretation of the condition number relies in particular on a duality theorem for tropical semidefinite feasibility programs.  ...  Acknowledgement The second author thanks Vladimir Gurvich for enlightening discussions on the pumping algorithm of [GKK88] and its extension to stochastic games in [BEGM15] .  ... 
arXiv:1802.07712v1 fatcat:qypn46dt6vhdfdtxzqrtictitu

Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

Xavier Allamigeon, Stéphane Gaubert, Mateusz Skomra
2018 Journal of symbolic computation  
A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean.  ...  This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.  ...  Acknowledgments We thank the referees for their detailed comments. References  ... 
doi:10.1016/j.jsc.2017.07.002 fatcat:qvmjhlcyl5fb5pdqsgr3ya4dyi

Characterization and computation of correlated equilibria in infinite games

Noah D. Stein, Pablo A. Parrilo, Asuman Ozdaglar
2007 2007 46th IEEE Conference on Decision and Control  
Motivated by recent work on computing Nash equilibria in two-player zero-sum games with polynomial payoffs by semidefinite programming and in arbitrary polynomiallike games by discretization techniques  ...  Then we use these to construct algorithms for approximating correlated equilibria of polynomial games with arbitrary accuracy, including a sequence of semidefinite programming relaxation algorithms and  ...  Definition 2.1: A finite game consists of n < ∞ players, each of whom has a finite pure strategy set C i and a utility or payoff function u i : C → R, where C = Π n j=1 C j .  ... 
doi:10.1109/cdc.2007.4434890 dblp:conf/cdc/SteinPO07 fatcat:7zaxb4xhqzdx5prawpsfmlzrei

Computing correlated equilibria of polynomial games via adaptive discretization

Noah D. Stein, Asuman Ozdaglar, Pablo A. Parrilo
2008 2008 47th IEEE Conference on Decision and Control  
We construct a family of iterative discretization algorithms for computing sequences of finitely-supportedcorrelated equilibria of n-player games with polynomial utility functions.  ...  These algorithms can be implemented efficiently using semidefinite programming and sum of squares techniques.  ...  IMPLEMENTING THESE ALGORITHMS WITH SEMIDEFINITE PROGRAMS To implement these algorithms for polynomial games, we must be able to do two things.  ... 
doi:10.1109/cdc.2008.4739338 dblp:conf/cdc/SteinOP08 fatcat:3qvjifzk7rfbbn2zrqkl4h3ygu

Symmetric duality in quadratic programming and matrix game equivalence

D.S. Kim, K.A. Noh
2004 Applied Mathematics Letters  
In this note, we study equivalence between symmetric duality in quadratic programming and matrix games.  ...  In particular, we obtain two different zero-sum games (a symmetric game and a nonsymmetric game) whose Nash equilibria correspond to the solutions of pairs of quadratic primal-dual problems.  ...  Here A is an m x n matrix; C, a symmetric positive semidefinite n x n matrix; D, a symmetric positive semidefinite m x m matrix; b, an m x 1 vector; y, v, m x 1 vectors; p, an n x 1 vector; x, u, n x 1  ... 
doi:10.1016/j.aml.2004.02.003 fatcat:kkjat3cvfvafbbu2ozuefb2pwe

Optimal Symmetric Rendezvous Search on Three Locations

Richard Weber
2012 Mathematics of Operations Research  
In the symmetric rendezvous search game played on n locations two players are initially placed at two distinct locations.  ...  The game is played in discrete steps, at each of which each player can either stay where he is or move to a different location. The players share no common labelling of the locations.  ...  By pioneering the use of semidefinite programming as a method of addressing rendezvous search problems, Jimmy was the first in many years to obtain significantly improved lower bounds on the rendezvous  ... 
doi:10.1287/moor.1110.0528 fatcat:uqkh42xhangy3lgxlqyi43btv4

Page 626 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
The author studies the discrete problem of choosing two k-player teams from a pool of n players. Several protocols are proposed and illustrated with examples.  ...  Ramana, An exact duality theory for semidefinite programming and its complex- ity implications (129-162); Pascal Gahinet and Arkadi Nemirovski [A. S.  ... 

Correlated equilibria in continuous games: Characterization and computation

Noah D. Stein, Pablo A. Parrilo, Asuman Ozdaglar
2011 Games and Economic Behavior  
We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions  ...  We present several new characterizations of correlated equilibria in games with continuous utility functions.  ...  Silva for many discussions about ergodic theory, and in particular for the statement and proof of Corollary 4.18 using Birkhoff's ergodic theorem.  ... 
doi:10.1016/j.geb.2010.04.004 fatcat:f6c44rokovgdtnwr4rpod3u5qu
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