27,685 research outputs found
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
Positional Determinacy of Games with Infinitely Many Priorities
We study two-player games of infinite duration that are played on finite or
infinite game graphs. A winning strategy for such a game is positional if it
only depends on the current position, and not on the history of the play. A
game is positionally determined if, from each position, one of the two players
has a positional winning strategy.
The theory of such games is well studied for winning conditions that are
defined in terms of a mapping that assigns to each position a priority from a
finite set. Specifically, in Muller games the winner of a play is determined by
the set of those priorities that have been seen infinitely often; an important
special case are parity games where the least (or greatest) priority occurring
infinitely often determines the winner. It is well-known that parity games are
positionally determined whereas Muller games are determined via finite-memory
strategies.
In this paper, we extend this theory to the case of games with infinitely
many priorities. Such games arise in several application areas, for instance in
pushdown games with winning conditions depending on stack contents.
For parity games there are several generalisations to the case of infinitely
many priorities. While max-parity games over omega or min-parity games over
larger ordinals than omega require strategies with infinite memory, we can
prove that min-parity games with priorities in omega are positionally
determined. Indeed, it turns out that the min-parity condition over omega is
the only infinitary Muller condition that guarantees positional determinacy on
all game graphs
Positional Games and QBF: The Corrective Encoding
Positional games are a mathematical class of two-player games comprising
Tic-tac-toe and its generalizations. We propose a novel encoding of these games
into Quantified Boolean Formulas (QBF) such that a game instance admits a
winning strategy for first player if and only if the corresponding formula is
true. Our approach improves over previous QBF encodings of games in multiple
ways. First, it is generic and lets us encode other positional games, such as
Hex. Second, structural properties of positional games together with a careful
treatment of illegal moves let us generate more compact instances that can be
solved faster by state-of-the-art QBF solvers. We establish the latter fact
through extensive experiments. Finally, the compactness of our new encoding
makes it feasible to translate realistic game problems. We identify a few such
problems of historical significance and put them forward to the QBF community
as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on
Theory and Applications of Satisfiability Testing (SAT2020
Efficient winning strategies in random-turn Maker-Breaker games
We consider random-turn positional games, introduced by Peres, Schramm,
Sheffield and Wilson in 2007. A -random-turn positional game is a two-player
game, played the same as an ordinary positional game, except that instead of
alternating turns, a coin is being tossed before each turn to decide the
identity of the next player to move (the probability of Player I to move is
). We analyze the random-turn version of several classical Maker-Breaker
games such as the game Box (introduced by Chv\'atal and Erd\H os in 1987), the
Hamilton cycle game and the -vertex-connectivity game (both played on the
edge set of ). For each of these games we provide each of the players with
a (randomized) efficient strategy which typically ensures his win in the
asymptotic order of the minimum value of for which he typically wins the
game, assuming optimal strategies of both players.Comment: 20 page
Fast strategies in biased Maker--Breaker games
We study the biased Maker--Breaker positional games, played on the
edge set of the complete graph on vertices, . Given Breaker's bias
, possibly depending on , we determine the bounds for the minimal number
of moves, depending on , in which Maker can win in each of the two standard
graph games, the Perfect Matching game and the Hamilton Cycle game
Games where you can play optimally without any memory
International audienceReactive systems are often modelled as two person antagonistic games where one player represents the system while his adversary represents the environment. Undoubtedly, the most popular games in this context are parity games and their cousins (Rabin, Streett and Muller games). Recently however also games with other types of payments, like discounted or mean-payoff , previously used only in economic context, entered into the area of system modelling and verification. The most outstanding property of parity, mean-payoff and discounted games is the existence of optimal positional (memoryless) strategies for both players. This observation raises two questions: (1) can we characterise the family of payoff mappings for which there always exist optimal positional strategies for both players and (2) are there other payoff mappings with practical or theoretical interest and admitting optimal positional strategies. This paper provides a complete answer to the first question by presenting a simple necessary and sufficient condition on payoff mapping guaranteeing the existence of optimal positional strategies. As a corollary to this result we show the following remarkable property of payoff mappings: if both players have optimal positional strategies when playing solitary one-player games then also they have optimal positional strategies for two-player games
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
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