6,318 research outputs found
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
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium
We present a direct reduction from k-player games to 2-player games that
preserves approximate Nash equilibrium. Previously, the computational
equivalence of computing approximate Nash equilibrium in k-player and 2-player
games was established via an indirect reduction. This included a sequence of
works defining the complexity class PPAD, identifying complete problems for
this class, showing that computing approximate Nash equilibrium for k-player
games is in PPAD, and reducing a PPAD-complete problem to computing approximate
Nash equilibrium for 2-player games. Our direct reduction makes no use of the
concept of PPAD, thus eliminating some of the difficulties involved in
following the known indirect reduction.Comment: 21 page
Mean-field games of optimal stopping: a relaxed solution approach
We consider the mean-field game where each agent determines the optimal time
to exit the game by solving an optimal stopping problem with reward function
depending on the density of the state processes of agents still present in the
game. We place ourselves in the framework of relaxed optimal stopping, which
amounts to looking for the optimal occupation measure of the stopper rather
than the optimal stopping time. This framework allows us to prove the existence
of the relaxed Nash equilibrium and the uniqueness of the associated value of
the representative agent under mild assumptions. Further, we prove a rigorous
relation between relaxed Nash equilibria and the notion of mixed solutions
introduced in earlier works on the subject, and provide a criterion, under
which the optimal strategies are pure strategies, that is, behave in a similar
way to stopping times. Finally, we present a numerical method for computing the
equilibrium in the case of potential games and show its convergence
Equilibria-based Probabilistic Model Checking for Concurrent Stochastic Games
Probabilistic model checking for stochastic games enables formal verification
of systems that comprise competing or collaborating entities operating in a
stochastic environment. Despite good progress in the area, existing approaches
focus on zero-sum goals and cannot reason about scenarios where entities are
endowed with different objectives. In this paper, we propose probabilistic
model checking techniques for concurrent stochastic games based on Nash
equilibria. We extend the temporal logic rPATL (probabilistic alternating-time
temporal logic with rewards) to allow reasoning about players with distinct
quantitative goals, which capture either the probability of an event occurring
or a reward measure. We present algorithms to synthesise strategies that are
subgame perfect social welfare optimal Nash equilibria, i.e., where there is no
incentive for any players to unilaterally change their strategy in any state of
the game, whilst the combined probabilities or rewards are maximised. We
implement our techniques in the PRISM-games tool and apply them to several case
studies, including network protocols and robot navigation, showing the benefits
compared to existing approaches
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