4,118 research outputs found
Recursive Concurrent Stochastic Games
We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent
analysis of recursive simple stochastic games to a concurrent setting where the
two players choose moves simultaneously and independently at each state. For
multi-exit games, our earlier work already showed undecidability for basic
questions like termination, thus we focus on the important case of single-exit
RCSGs (1-RCSGs).
We first characterize the value of a 1-RCSG termination game as the least
fixed point solution of a system of nonlinear minimax functional equations, and
use it to show PSPACE decidability for the quantitative termination problem. We
then give a strategy improvement technique, which we use to show that player 1
(maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM)
strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM
strategies. Thus, such games are r-SM-determined. These results mirror and
generalize in a strong sense the randomized memoryless determinacy results for
finite stochastic games, and extend the classic Hoffman-Karp strategy
improvement approach from the finite to an infinite state setting. The proofs
in our infinite-state setting are very different however, relying on subtle
analytic properties of certain power series that arise from studying 1-RCSGs.
We show that our upper bounds, even for qualitative (probability 1)
termination, can not be improved, even to NP, without a major breakthrough, by
giving two reductions: first a P-time reduction from the long-standing
square-root sum problem to the quantitative termination decision problem for
finite concurrent stochastic games, and then a P-time reduction from the latter
problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure
Exact Algorithms for Solving Stochastic Games
Shapley's discounted stochastic games, Everett's recursive games and
Gillette's undiscounted stochastic games are classical models of game theory
describing two-player zero-sum games of potentially infinite duration. We
describe algorithms for exactly solving these games
Qualitative Analysis of Concurrent Mean-payoff Games
We consider concurrent games played by two-players on a finite-state graph,
where in every round the players simultaneously choose a move, and the current
state along with the joint moves determine the successor state. We study a
fundamental objective, namely, mean-payoff objective, where a reward is
associated to each transition, and the goal of player 1 is to maximize the
long-run average of the rewards, and the objective of player 2 is strictly the
opposite. The path constraint for player 1 could be qualitative, i.e., the
mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative,
i.e., a given threshold between the minimal and maximal reward. We consider the
computation of the almost-sure (resp. positive) winning sets, where player 1
can ensure that the path constraint is satisfied with probability 1 (resp.
positive probability). Our main results for qualitative path constraints are as
follows: (1) we establish qualitative determinacy results that show that for
every state either player 1 has a strategy to ensure almost-sure (resp.
positive) winning against all player-2 strategies, or player 2 has a spoiling
strategy to falsify almost-sure (resp. positive) winning against all player-1
strategies; (2) we present optimal strategy complexity results that precisely
characterize the classes of strategies required for almost-sure and positive
winning for both players; and (3) we present quadratic time algorithms to
compute the almost-sure and the positive winning sets, matching the best known
bound of algorithms for much simpler problems (such as reachability
objectives). For quantitative constraints we show that a polynomial time
solution for the almost-sure or the positive winning set would imply a solution
to a long-standing open problem (the value problem for turn-based deterministic
mean-payoff games) that is not known to be solvable in polynomial time
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
The Complexity of Nash Equilibria in Limit-Average Games
We study the computational complexity of Nash equilibria in concurrent games
with limit-average objectives. In particular, we prove that the existence of a
Nash equilibrium in randomised strategies is undecidable, while the existence
of a Nash equilibrium in pure strategies is decidable, even if we put a
constraint on the payoff of the equilibrium. Our undecidability result holds
even for a restricted class of concurrent games, where nonzero rewards occur
only on terminal states. Moreover, we show that the constrained existence
problem is undecidable not only for concurrent games but for turn-based games
with the same restriction on rewards. Finally, we prove that the constrained
existence problem for Nash equilibria in (pure or randomised) stationary
strategies is decidable and analyse its complexity.Comment: 34 page
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