42 research outputs found
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
We show that 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. Extending our result for the Harsanyi-Selten
process, we show that several other homotopy-based algorithms for finding
equilibria of games are also 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 strengthening of the result in
[Savani and von Stengel]. These results show that our techniques can be widely
applied and suggest that the PSPACE-completeness of implementing homotopy
methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc
Geometry and equilibria in bimatrix games
This thesis studies the application of geometric concepts and methods in the analysis
of strategic-form games, in particular bimatrix games. Our focus is on three
geometric concepts: the index, geometric algorithms for the computation of Nash
equilibria, and polytopes.
The contribution of this thesis consists of three parts. First, we present an algorithm
for the computation of the index in degenerate bimatrix games. For this, we define
a new concept, the “lex-index” of an extreme equilibrium, which is an extension of
the standard index. The index of an equilibrium component is easily computable
as the sum of the lex-indices of all extreme equilibria of that component.
Second, we give several new results on the linear tracing procedure, and its bimatrix
game implementation, the van den Elzen-Talman (ET) algorithm. We compare
the ET algorithm to two other algorithms: On the one hand, we show that the
Lemke-Howson algorithm, the classic method for equilibrium computation in bimatrix
games, and the ET algorithm differ substantially. On the other hand, we
prove that the ET algorithm, or more generally, the linear tracing procedure, is a
special case of the global Newton method, a geometric algorithm for the computation
of equilibria in strategic-form games. As the main result of this part of the
thesis, we show that there is a generic class of bimatrix games in which an equilibrium
of positive index is not traceable by the ET algorithm. This result answers an
open question regarding sustainability.
The last part of this thesis studies the index in symmetric games. We use a construction
of polytopes to prove a new result on the symmetric index: A symmetric
equilibrium has symmetric index +1 if and only if it is “potentially unique”, in the
sense that there is an extended symmetric game, with additional strategies for the
players, where the given symmetric equilibrium is unique
07471 Abstracts Collection -- Equilibrium Computation
From 18 to 23 November 2007, the Dagstuhl Seminar 07471 ``Equilibrium Computation\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
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
An Empirical Study of Finding Approximate Equilibria in Bimatrix Games
While there have been a number of studies about the efficacy of methods to
find exact Nash equilibria in bimatrix games, there has been little empirical
work on finding approximate Nash equilibria. Here we provide such a study that
compares a number of approximation methods and exact methods. In particular, we
explore the trade-off between the quality of approximate equilibrium and the
required running time to find one. We found that the existing library GAMUT,
which has been the de facto standard that has been used to test exact methods,
is insufficient as a test bed for approximation methods since many of its games
have pure equilibria or other easy-to-find good approximate equilibria. We
extend the breadth and depth of our study by including new interesting families
of bimatrix games, and studying bimatrix games upto size .
Finally, we provide new close-to-worst-case examples for the best-performing
algorithms for finding approximate Nash equilibria
Uniqueness of Stationary Equilibrium Payoffs in Coalitional Bargaining
We study a model of sequential bargaining in which, in each period before an agreement is reached, the proposer’s identity (and whether there is a proposer) are randomly determined; the proposer suggests a division of a pie of size one; 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. The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique. This generalizes Eraslan (2002) insofar as: (a) there are no restrictions on the structure of sets of winning coalitions; (b) different proposers may have different sets of winning coalitions; (c) there may be a positive probability that no proposer is selected.