1,143,628 research outputs found
Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds
Dull, weak and nested solitaire games are important classes of parity games,
capturing, among others, alternation-free mu-calculus and ECTL* model checking
problems. These classes can be solved in polynomial time using dedicated
algorithms. We investigate the complexity of Zielonka's Recursive algorithm for
solving these special games, showing that the algorithm runs in O(d (n + m)) on
weak games, and, somewhat surprisingly, that it requires exponential time to
solve dull games and (nested) solitaire games. For the latter classes, we
provide a family of games G, allowing us to establish a lower bound of 2^(n/3).
We show that an optimisation of Zielonka's algorithm permits solving games from
all three classes in polynomial time. Moreover, we show that there is a family
of (non-special) games M that permits us to establish a lower bound of 2^(n/3),
improving on the previous lower bound for the algorithm.Comment: In Proceedings GandALF 2013, arXiv:1307.416
Convergence to Equilibrium of Logit Dynamics for Strategic Games
We present the first general bounds on the mixing time of the Markov chain
associated to the logit dynamics for wide classes of strategic games. The logit
dynamics with inverse noise beta describes the behavior of a complex system
whose individual components act selfishly and keep responding according to some
partial ("noisy") knowledge of the system, where the capacity of the agent to
know the system and compute her best move is measured by the inverse of the
parameter beta.
In particular, we prove nearly tight bounds for potential games and games
with dominant strategies. Our results show that, for potential games, the
mixing time is upper and lower bounded by an exponential in the inverse of the
noise and in the maximum potential difference. Instead, for games with dominant
strategies, the mixing time cannot grow arbitrarily with the inverse of the
noise.
Finally, we refine our analysis for a subclass of potential games called
graphical coordination games, a class of games that have been previously
studied in Physics and, more recently, in Computer Science in the context of
diffusion of new technologies. We give evidence that the mixing time of the
logit dynamics for these games strongly depends on the structure of the
underlying graph. We prove that the mixing time of the logit dynamics for these
games can be upper bounded by a function that is exponential in the cutwidth of
the underlying graph and in the inverse of noise. Moreover, we consider two
specific and popular network topologies, the clique and the ring. For games
played on a clique we prove an almost matching lower bound on the mixing time
of the logit dynamics that is exponential in the inverse of the noise and in
the maximum potential difference, while for games played on a ring we prove
that the time of convergence of the logit dynamics to its stationary
distribution is significantly shorter
Subgame-Perfect Equilibria in Stochastic Timing Games
We introduce a notion of subgames for stochastic timing games and the related
notion of subgame-perfect equilibrium in possibly mixed strategies. While a
good notion of subgame-perfect equilibrium for continuous-time games is not
available in general, we argue that our model is the appropriate version for
timing games. We show that the notion coincides with the usual one for
discrete-time games. Many timing games in continuous time have only equilibria
in mixed strategies -- in particular preemption games, which often occur in the
strategic real option literature. We provide a sound foundation for some
workhorse equilibria of that literature, which has been lacking as we show. We
obtain a general constructive existence result for subgame-perfect equilibria
in preemption games and illustrate our findings by several explicit
applications.Comment: 27 pages, 1 figur
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