359,235 research outputs found
Solving parity games: Explicit vs symbolic
In this paper we provide a broad investigation of the symbolic approach for solving Parity Games. Specifically, we implement in a fresh tool, called, four symbolic algorithms to solve Parity Games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been
Positional games on random graphs
We introduce and study Maker/Breaker-type positional games on random graphs.
Our main concern is to determine the threshold probability for the
existence of Maker's strategy to claim a member of in the unbiased game
played on the edges of random graph , for various target families
of winning sets. More generally, for each probability above this threshold we
study the smallest bias such that Maker wins the biased game. We
investigate these functions for a number of basic games, like the connectivity
game, the perfect matching game, the clique game and the Hamiltonian cycle
game
On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games
We study the convergence time of the best response dynamics in
player-specific singleton congestion games. It is well known that this dynamics
can cycle, although from every state a short sequence of best responses to a
Nash equilibrium exists. Thus, the random best response dynamics, which selects
the next player to play a best response uniformly at random, terminates in a
Nash equilibrium with probability one. In this paper, we are interested in the
expected number of best responses until the random best response dynamics
terminates.
As a first step towards this goal, we consider games in which each player can
choose between only two resources. These games have a natural representation as
(multi-)graphs by identifying nodes with resources and edges with players. For
the class of games that can be represented as trees, we show that the
best-response dynamics cannot cycle and that it terminates after O(n^2) steps
where n denotes the number of resources. For the class of games represented as
cycles, we show that the best response dynamics can cycle. However, we also
show that the random best response dynamics terminates after O(n^2) steps in
expectation.
Additionally, we conjecture that in general player-specific singleton
congestion games there exists no polynomial upper bound on the expected number
of steps until the random best response dynamics terminates. We support our
conjecture by presenting a family of games for which simulations indicate a
super-polynomial convergence time
When Can Limited Randomness Be Used in Repeated Games?
The central result of classical game theory states that every finite normal
form game has a Nash equilibrium, provided that players are allowed to use
randomized (mixed) strategies. However, in practice, humans are known to be bad
at generating random-like sequences, and true random bits may be unavailable.
Even if the players have access to enough random bits for a single instance of
the game their randomness might be insufficient if the game is played many
times.
In this work, we ask whether randomness is necessary for equilibria to exist
in finitely repeated games. We show that for a large class of games containing
arbitrary two-player zero-sum games, approximate Nash equilibria of the
-stage repeated version of the game exist if and only if both players have
random bits. In contrast, we show that there exists a class of
games for which no equilibrium exists in pure strategies, yet the -stage
repeated version of the game has an exact Nash equilibrium in which each player
uses only a constant number of random bits.
When the players are assumed to be computationally bounded, if cryptographic
pseudorandom generators (or, equivalently, one-way functions) exist, then the
players can base their strategies on "random-like" sequences derived from only
a small number of truly random bits. We show that, in contrast, in repeated
two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then
random bits remain necessary for equilibria to exist
How to Play Unique Games against a Semi-Random Adversary
In this paper, we study the average case complexity of the Unique Games
problem. We propose a natural semi-random model, in which a unique game
instance is generated in several steps. First an adversary selects a completely
satisfiable instance of Unique Games, then she chooses an epsilon-fraction of
all edges, and finally replaces ("corrupts") the constraints corresponding to
these edges with new constraints. If all steps are adversarial, the adversary
can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard
as in the worst case. In our semi-random model, one of the steps is random, and
all other steps are adversarial. We show that known algorithms for unique games
(in particular, all algorithms that use the standard SDP relaxation) fail to
solve semi-random instances of Unique Games.
We present an algorithm that with high probability finds a solution
satisfying a (1-delta) fraction of all constraints in semi-random instances (we
require that the average degree of the graph is Omega(log k). To this end, we
consider a new non-standard SDP program for Unique Games, which is not a
relaxation for the problem, and show how to analyze it. We present a new
rounding scheme that simultaneously uses SDP and LP solutions, which we believe
is of independent interest.
Our result holds only for epsilon less than some absolute constant. We prove
that if epsilon > 1/2, then the problem is hard in one of the models, the
result assumes the 2-to-2 conjecture.
Finally, we study semi-random instances of Unique Games that are at most
(1-epsilon) satisfiable. We present an algorithm that with high probability,
distinguishes between the case when the instance is a semi-random instance and
the case when the instance is an (arbitrary) (1-delta) satisfiable instance if
epsilon > c delta
Strong games played on random graphs
In a strong game played on the edge set of a graph G there are two players,
Red and Blue, alternating turns in claiming previously unclaimed edges of G
(with Red playing first). The winner is the first one to claim all the edges of
some target structure (such as a clique, a perfect matching, a Hamilton cycle,
etc.). It is well known that Red can always ensure at least a draw in any
strong game, but finding explicit winning strategies is a difficult and a quite
rare task. We consider strong games played on the edge set of a random graph G
~ G(n,p) on n vertices. We prove, for sufficiently large and a fixed
constant 0 < p < 1, that Red can w.h.p win the perfect matching game on a
random graph G ~ G(n,p)
Connectivity and equilibrium in random games
We study how the structure of the interaction graph of a game affects the
existence of pure Nash equilibria. In particular, for a fixed interaction
graph, we are interested in whether there are pure Nash equilibria arising when
random utility tables are assigned to the players. We provide conditions for
the structure of the graph under which equilibria are likely to exist and
complementary conditions which make the existence of equilibria highly
unlikely. Our results have immediate implications for many deterministic graphs
and generalize known results for random games on the complete graph. In
particular, our results imply that the probability that bounded degree graphs
have pure Nash equilibria is exponentially small in the size of the graph and
yield a simple algorithm that finds small nonexistence certificates for a large
family of graphs. Then we show that in any strongly connected graph of n
vertices with expansion the distribution of the number
of equilibria approaches the Poisson distribution with parameter 1,
asymptotically as .Comment: Published in at http://dx.doi.org/10.1214/10-AAP715 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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