91 research outputs found
Computing the Equilibria of Bimatrix Games using Dominance Heuristics
We propose a formulation of a general-sum bimatrix game as a bipartite
directed graph with the objective of establishing a correspondence between the
set of the relevant structures of the graph (in particular elementary cycles)
and the set of the Nash equilibria of the game. We show that finding the set of
elementary cycles of the graph permits the computation of the set of
equilibria. For games whose graphs have a sparse adjacency matrix, this serves
as a good heuristic for computing the set of equilibria. The heuristic also
allows the discarding of sections of the support space that do not yield any
equilibrium, thus serving as a useful pre-processing step for algorithms that
compute the equilibria through support enumeration
Entropy and typical properties of Nash equilibria in two-player games
We use techniques from the statistical mechanics of disordered systems to
analyse the properties of Nash equilibria of bimatrix games with large random
payoff matrices. By means of an annealed bound, we calculate their number and
analyse the properties of typical Nash equilibria, which are exponentially
dominant in number. We find that a randomly chosen equilibrium realizes almost
always equal payoffs to either player. This value and the fraction of
strategies played at an equilibrium point are calculated as a function of the
correlation between the two payoff matrices. The picture is complemented by the
calculation of the properties of Nash equilibria in pure strategies.Comment: 6 pages, was "Self averaging of Nash equilibria in two player games",
main section rewritten, some new results, for additional information see
http://itp.nat.uni-magdeburg.de/~jberg/games.htm
On the computation of stable sets for bimatrix games
In this paper it is shown how to compute stable sets, defined by Mertens (1989), inthe context of bimatrix games only using linear optimization techniques.combinatorics;
Complementary vertices and adjacency testing in polytopes
Our main theoretical result is that, if a simple polytope has a pair of
complementary vertices (i.e., two vertices with no facets in common), then it
has at least two such pairs, which can be chosen to be disjoint. Using this
result, we improve adjacency testing for vertices in both simple and non-simple
polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq
0} and a list of its V vertices, we describe an O(n) test to identify whether
any two given vertices are adjacent. For simple polytopes this test is perfect;
for non-simple polytopes it may be indeterminate, and instead acts as a filter
to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2)
precomputation, which is acceptable in settings such as all-pairs adjacency
testing. These results improve upon the more general O(nV) combinatorial and
O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal
version, which strengthens and extends the results in Section 2 (see p1 of
the paper for details
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