158 research outputs found
A Continuation Method for Nash Equilibria in Structured Games
Structured game representations have recently attracted interest as models
for multi-agent artificial intelligence scenarios, with rational behavior most
commonly characterized by Nash equilibria. This paper presents efficient, exact
algorithms for computing Nash equilibria in structured game representations,
including both graphical games and multi-agent influence diagrams (MAIDs). The
algorithms are derived from a continuation method for normal-form and
extensive-form games due to Govindan and Wilson; they follow a trajectory
through a space of perturbed games and their equilibria, exploiting game
structure through fast computation of the Jacobian of the payoff function. They
are theoretically guaranteed to find at least one equilibrium of the game, and
may find more. Our approach provides the first efficient algorithm for
computing exact equilibria in graphical games with arbitrary topology, and the
first algorithm to exploit fine-grained structural properties of MAIDs.
Experimental results are presented demonstrating the effectiveness of the
algorithms and comparing them to predecessors. The running time of the
graphical game algorithm is similar to, and often better than, the running time
of previous approximate algorithms. The algorithm for MAIDs can effectively
solve games that are much larger than those solvable by previous methods
Natural Strategic Ability
International audienc
Nash equilibria, gale strings, and perfect matchings
This thesis concerns the problem 2-NASH of ļ¬nding a Nash equilibrium of a bimatrix
game, for the special class of so-called āhard-to-solveā bimatrix games. The term āhardto-solveā relates to the exponential running time of the famous and often used Lemkeā
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 deļ¬ned by the āGale evenness conditionā that characterise the vertices of these
polytopes.
We deļ¬ne the combinatorial problem āAnother completely labeled Gale stringā whose
solutions deļ¬ne the Nash equilibria of any game deļ¬ned by cyclic polytopes, including
the games where the LemkeāHowson algorithm takes exponential time. We show that
āAnother completely labeled Gale stringā is solvable in polynomial time by a reduction to
the āPerfect matchingā problem in Euler graphs. We adapt the LemkeāHowson 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
LemkeāHowson 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 ļ¬nding 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 difļ¬culties 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
Randomized approximation algorithms : facility location, phylogenetic networks, Nash equilibria
Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions
New Perspectives on Games and Interaction
This volume is a collection of papers presented at the 2007 colloquium on new perspectives on games and interaction at the Royal Dutch Academy of Sciences in Amsterdam. The purpose of the colloquium was to clarify the uses of the concepts of game theory, and to identify promising new directions. This important collection testifies to the growing importance of game theory as a tool to capture the concepts of strategy, interaction, argumentation, communication, cooperation and competition. Also, it provides evidence for the richness of game theory and for its impressive and growing application
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