177,153 research outputs found
Computing optimal strategies for a cooperative hat game
We consider a `hat problem' in which each player has a randomly placed stack
of black and white hats on their heads, visible to the other player, but not
the wearer. Each player must guess a hat position on their head with the goal
of both players guessing a white hat. We address the question of finding the
optimal strategy, i.e., the one with the highest probability of winning, for
this game. We provide an overview of prior work on this question, and describe
several strategies that give the best known lower bound on the probability of
winning. Upper bounds are also considered here
An Algorithm for Probabilistic Alternating Simulation
In probabilistic game structures, probabilistic alternating simulation
(PA-simulation) relations preserve formulas defined in probabilistic
alternating-time temporal logic with respect to the behaviour of a subset of
players. We propose a partition based algorithm for computing the largest
PA-simulation, which is to our knowledge the first such algorithm that works in
polynomial time, by extending the generalised coarsest partition problem (GCPP)
in a game-based setting with mixed strategies. The algorithm has higher
complexities than those in the literature for non-probabilistic simulation and
probabilistic simulation without mixed actions, but slightly improves the
existing result for computing probabilistic simulation with respect to mixed
actions.Comment: We've fixed a problem in the SOFSEM'12 conference versio
Iterated Regret Minimization in Game Graphs
Iterated regret minimization has been introduced recently by J.Y. Halpern and
R. Pass in classical strategic games. For many games of interest, this new
solution concept provides solutions that are judged more reasonable than
solutions offered by traditional game concepts -- such as Nash equilibrium --.
Although computing iterated regret on explicit matrix game is conceptually and
computationally easy, nothing is known about computing the iterated regret on
games whose matrices are defined implicitly using game tree, game DAG or, more
generally game graphs. In this paper, we investigate iterated regret
minimization for infinite duration two-player quantitative non-zero sum games
played on graphs.
We consider reachability objectives that are not necessarily antagonist.
Edges are weighted by integers -- one for each player --, and the payoffs are
defined by the sum of the weights along the paths. Depending on the class of
graphs, we give either polynomial or pseudo-polynomial time algorithms to
compute a strategy that minimizes the regret for a fixed player. We finally
give algorithms to compute the strategies of the two players that minimize the
iterated regret for trees, and for graphs with strictly positive weights only.Comment: 19 pages. Bug in introductive example fixed
Mean-field games of optimal stopping: a relaxed solution approach
We consider the mean-field game where each agent determines the optimal time
to exit the game by solving an optimal stopping problem with reward function
depending on the density of the state processes of agents still present in the
game. We place ourselves in the framework of relaxed optimal stopping, which
amounts to looking for the optimal occupation measure of the stopper rather
than the optimal stopping time. This framework allows us to prove the existence
of the relaxed Nash equilibrium and the uniqueness of the associated value of
the representative agent under mild assumptions. Further, we prove a rigorous
relation between relaxed Nash equilibria and the notion of mixed solutions
introduced in earlier works on the subject, and provide a criterion, under
which the optimal strategies are pure strategies, that is, behave in a similar
way to stopping times. Finally, we present a numerical method for computing the
equilibrium in the case of potential games and show its convergence
Approximate Nash Equilibria via Sampling
We prove that in a normal form n-player game with m actions for each player,
there exists an approximate Nash equilibrium where each player randomizes
uniformly among a set of O(log(m) + log(n)) pure strategies. This result
induces an algorithm for computing an approximate Nash
equilibrium in games where the number of actions is polynomial in the number of
players (m=poly(n)), where is the size of the game (the input size).
In addition, we establish an inverse connection between the entropy of Nash
equilibria in the game, and the time it takes to find such an approximate Nash
equilibrium using the random sampling algorithm
- …