10171 Abstracts Collection -- Equilibrium Computation

From April 25 to April 30, 2010, the Dagstuhl Seminar 10171 ``Equilibrium Computation\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterise the vertices of these polytopes. We define the combinatorial problem “Another completely labeled Gale string” whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful

Fractional solutions for capacitated NTU-games, with applications to stable matchings

Abstract. In this paper we investigate some new applications of Scarf’s Lemma. First, we introduce the notion of fractional core for NTU-games, which is always nonempty by the Lemma. Stable allocation is a general solution concept for games where both the players and their possible cooperations can have capacities. We show that the problem of finding a stable allocation, given a finitely generated NTU-game with capacities, is always solvable by a variant of Scarf’s Lemma. Then we describe the interpretation of these results for matching games. Finally we consider an even more general setting where players ’ contributions in a joint activity may be different. We show that a stable allocation can be found by the Scarf algorithm in this case as well, and we demonstrate the usage of this method for the hospitals resident problem with couples. This problem is relevant in many practical applications, such as NRM

Carrizozo News, 09-25-1908

https://digitalrepository.unm.edu/carrizozo_news/1014/thumbnail.jp

White Oaks Eagle, 03-12-1896

https://digitalrepository.unm.edu/wo_eagle_news/1032/thumbnail.jp

January 19, 1929

The Breeze is the student newspaper of James Madison University in Harrisonburg, Virginia

Santa Fe New Mexican, 04-25-1898

https://digitalrepository.unm.edu/sfnm_news/6907/thumbnail.jp

Lincoln County Leader, 02-09-1889

https://digitalrepository.unm.edu/lcl_news/1328/thumbnail.jp