2,224 research outputs found
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
The Value 1 Problem Under Finite-memory Strategies for Concurrent Mean-payoff Games
We consider concurrent mean-payoff games, a very well-studied class of
two-player (player 1 vs player 2) zero-sum games on finite-state graphs where
every transition is assigned a reward between 0 and 1, and the payoff function
is the long-run average of the rewards. The value is the maximal expected
payoff that player 1 can guarantee against all strategies of player 2. We
consider the computation of the set of states with value 1 under finite-memory
strategies for player 1, and our main results for the problem are as follows:
(1) we present a polynomial-time algorithm; (2) we show that whenever there is
a finite-memory strategy, there is a stationary strategy that does not need
memory at all; and (3) we present an optimal bound (which is double
exponential) on the patience of stationary strategies (where patience of a
distribution is the inverse of the smallest positive probability and represents
a complexity measure of a stationary strategy)
Robust Draws in Balanced Knockout Tournaments
Balanced knockout tournaments are ubiquitous in sports competitions and are
also used in decision-making and elections. The traditional computational
question, that asks to compute a draw (optimal draw) that maximizes the winning
probability for a distinguished player, has received a lot of attention.
Previous works consider the problem where the pairwise winning probabilities
are known precisely, while we study how robust is the winning probability with
respect to small errors in the pairwise winning probabilities. First, we
present several illuminating examples to establish: (a)~there exist
deterministic tournaments (where the pairwise winning probabilities are~0 or~1)
where one optimal draw is much more robust than the other; and (b)~in general,
there exist tournaments with slightly suboptimal draws that are more robust
than all the optimal draws. The above examples motivate the study of the
computational problem of robust draws that guarantee a specified winning
probability. Second, we present a polynomial-time algorithm for approximating
the robustness of a draw for sufficiently small errors in pairwise winning
probabilities, and obtain that the stated computational problem is NP-complete.
We also show that two natural cases of deterministic tournaments where the
optimal draw could be computed in polynomial time also admit polynomial-time
algorithms to compute robust optimal draws
The Big Match in Small Space
In this paper we study how to play (stochastic) games optimally using little
space. We focus on repeated games with absorbing states, a type of two-player,
zero-sum concurrent mean-payoff games. The prototypical example of these games
is the well known Big Match of Gillete (1957). These games may not allow
optimal strategies but they always have {\epsilon}-optimal strategies. In this
paper we design {\epsilon}-optimal strategies for Player 1 in these games that
use only O(log log T ) space. Furthermore, we construct strategies for Player 1
that use space s(T), for an arbitrary small unbounded non-decreasing function
s, and which guarantee an {\epsilon}-optimal value for Player 1 in the limit
superior sense. The previously known strategies use space {\Omega}(logT) and it
was known that no strategy can use constant space if it is {\epsilon}-optimal
even in the limit superior sense. We also give a complementary lower bound.
Furthermore, we also show that no Markov strategy, even extended with finite
memory, can ensure value greater than 0 in the Big Match, answering a question
posed by Abraham Neyman
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