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A mixed duel under arbitrary motion and uncertain existence of the shot

Stanisław Trybuła (1993)

Applicationes Mathematicae

The purpose of the paper is to solve a mixed duel in which the numbers of shots given to the players are independent 0-1-valued random variables. The players know their distributions as well as the accuracy function P, the same for both players. It is assumed that the players can move as they like and that the maximal speed of the first player is greater than that of the second player. It is shown that the game has a value, and a pair of optimal strategies is found.

A survey on topological games and their applications in analysis.

Jiling Cao, Warren B. Moors (2006)

RACSAM

In this survey article we shall summarise some of the recent progress that has occurred in the study of topological games as well as their applications to abstract analysis. The topics given here do not necessarily represent the most important problems from the area of topological games, but rather, they represent a selection of problems that are of interest to the authors.

Discrepancy games.

Alon, Noga, Krivelevich, Michael, Spencer, Joel, Szabó, Tibor (2005)

The Electronic Journal of Combinatorics [electronic only]

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle...

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

Edge-disjoint odd cycles in graphs with small chromatic number

Claude Berge, Bruce Reed (1999)

Annales de l'institut Fourier

For a simple graph, we consider the minimum number of edges which block all the odd cycles and the maximum number of odd cycles which are pairwise edge-disjoint. When these two coefficients are equal, interesting consequences appear. Similar problems (but interchanging “vertex-disjoint odd cycles” and “edge-disjoint odd cycles”) have been considered in a paper by Berge and Fouquet.

Game saturation of intersecting families

Balázs Patkós, Máté Vizer (2014)

Open Mathematics

We consider the following combinatorial game: two players, Fast and Slow, claim k-element subsets of [n] = 1, 2, …, n alternately, one at each turn, so that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed k-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone...

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