2,209 research outputs found
A Potential Reduction Algorithm for Two-person Zero-sum Mean Payoff Stochastic Games
We suggest a new algorithm for two-person zero-sum undiscounted stochastic
games focusing on stationary strategies. Given a positive real , let
us call a stochastic game -ergodic, if its values from any two
initial positions differ by at most . The proposed new algorithm
outputs for every in finite time either a pair of stationary
strategies for the two players guaranteeing that the values from any initial
positions are within an -range, or identifies two initial positions
and and corresponding stationary strategies for the players proving
that the game values starting from and are at least
apart. In particular, the above result shows that if a stochastic game is
-ergodic, then there are stationary strategies for the players
proving -ergodicity. This result strengthens and provides a
constructive version of an existential result by Vrieze (1980) claiming that if
a stochastic game is -ergodic, then there are -optimal stationary
strategies for every . The suggested algorithm is based on a
potential transformation technique that changes the range of local values at
all positions without changing the normal form of the game
A potential reduction algorithm for two-person zero-sum mean payoff stochastic games
We suggest a new algorithm for two-person zero-sum undiscounted
stochastic games focusing on stationary strategies. Given a positive real
, let us call a stochastic game -ergodic, if its values from any two initial
positions dier by at most . The proposed new algorithm outputs for
every > 0 in nite time either a pair of stationary strategies for the two
players guaranteeing that the values from any initial positions are within
an -range, or identies two initial positions u and v and corresponding
stationary strategies for the players proving that the game values starting
from u and v are at least =24 apart. In particular, the above result
shows that if a stochastic game is -ergodic, then there are stationary
strategies for the players proving 24-ergodicity. This result strengthens
and provides a constructive version of an existential result by Vrieze (1980)
claiming that if a stochastic game is 0-ergodic, then there are -optimal
stationary strategies for every > 0. The suggested algorithm is based
on a potential transformation technique that changes the range of local
values at all positions without changing the normal form of the game
A potential reduction algorithm for two-person zero-sum mean payoff stochastic games
We suggest a new algorithm for two-person zero-sum undiscounted
stochastic games focusing on stationary strategies. Given a positive real
, let us call a stochastic game -ergodic, if its values from any two initial
positions dier by at most . The proposed new algorithm outputs for
every > 0 in nite time either a pair of stationary strategies for the two
players guaranteeing that the values from any initial positions are within
an -range, or identies two initial positions u and v and corresponding
stationary strategies for the players proving that the game values starting
from u and v are at least =24 apart. In particular, the above result
shows that if a stochastic game is -ergodic, then there are stationary
strategies for the players proving 24-ergodicity. This result strengthens
and provides a constructive version of an existential result by Vrieze (1980)
claiming that if a stochastic game is 0-ergodic, then there are -optimal
stationary strategies for every > 0. The suggested algorithm is based
on a potential transformation technique that changes the range of local
values at all positions without changing the normal form of the game
A potential reduction algorithm for two-person zero-sum mean payoff stochastic games
We suggest a new algorithm for two-person zero-sum undiscounted
stochastic games focusing on stationary strategies. Given a positive real
, let us call a stochastic game -ergodic, if its values from any two initial
positions dier by at most . The proposed new algorithm outputs for
every > 0 in nite time either a pair of stationary strategies for the two
players guaranteeing that the values from any initial positions are within
an -range, or identies two initial positions u and v and corresponding
stationary strategies for the players proving that the game values starting
from u and v are at least =24 apart. In particular, the above result
shows that if a stochastic game is -ergodic, then there are stationary
strategies for the players proving 24-ergodicity. This result strengthens
and provides a constructive version of an existential result by Vrieze (1980)
claiming that if a stochastic game is 0-ergodic, then there are -optimal
stationary strategies for every > 0. The suggested algorithm is based
on a potential transformation technique that changes the range of local
values at all positions without changing the normal form of the game
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
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
A pseudo-polynomial algorithm for mean payoff stochastic games with perfect information and few random positions
We consider two-person zero-sum stochastic mean payoff games with perfect information,
or BWR-games, given by a digraph G = (V;E), with local rewards r : E Z, and three
types of positions: black VB, white VW, and random VR forming a partition of V . It is a long-
standing open question whether a polynomial time algorithm for BWR-games exists, or not,
even when |VR| = 0. In fact, a pseudo-polynomial algorithm for BWR-games would already
imply their polynomial solvability. In this paper, we show that BWR-games with a constant
number of random positions can be solved in pseudo-polynomial time. More precisely, in any
BWR-game with |VR| = O(1), a saddle point in uniformly optimal pure stationary strategies
can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward, and the
common denominator of the transition probabilities
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