On a noisy-silent-versus-silent duel with equal accuracy functions

This paper deals with the noisy-silent-versus-silent duel with equal accuracy functions. Player I has a gun with two bullets and player II has a gun with one bullet. The first bullet of player I is noisy, the second bullet of player I is silent, and the bullet of player II is silent. Each player can fire their bullets at any time in [0, 1] aiming at his opponent. The accuracy function ist for both players. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is –1. The optimal strategies and the value of the game are obtained. Although optimal strategies in past works concerning games of timing does not depend on the firing moments of the players, the optimal strategy obtained for player II depends explicitly on the firing moment of player I's noisy bullet.     

Two-noisy-versus-one-silent duel with equal accuracy functions

This paper deals with the two-noisy-versus-one-silent duel which is still open, as pointed out by Styszyński (Ref. 1). Player I has a noisy gun with two bullets, and player II has a silent gun with one bullet. Each player fires his bullets aiming at his opponent at any time in [0, 1]. The accuracy function (the probability that one player hits his opponent if he fires at timet) isp(t)=t for each player. If player I hits player II, without being hit himself before, the payoff of the duel is +1; if player I is hit by player II, without hitting player II before, the payoff is taken to be ?1. In this paper, we determine the optimal strategies and the value of the game. The strategy for player II depends explicitly on the firing moment of player I's first shot.     

On a duel with time lag and equal accuracy functions

This paper deals with a duel with time lag that has the following structure: Each of two players I and II has a gun with one bullet and he can fire his bullet at any time in [0, 1], aiming at this opponent. The gun of player I is silent and the gun of player II is noisy with time lagt (i.e., if player II fires at timex, then player I knows it at timex+t). They both have equal accuracy functions. Furthermore, if player I hits player II without being hit himself before, the payoff is +1; if player I is hit by player II without hitting player II before, the payoff is –1; if they hit each other at the same time or both survive, the payoff is 0.This paper gives the optimal strategy for each player, the game value, and some examples.     

A silent nversus 1 bullet duel with retreat after the shots

This paper is concerned with silent duel in which the first player has nbullets and the second one-one bullet. The accuracy functions are the same. It is assumed that each player removes to the back after firing all his bullets. The situation when players have different speeds are considered as well as that in which the speeds are the same. In both situations the optimal strategies are determined and the value of the game is found.     

On a duel with time lag and arbitrary accuracy functions

This paper deals with a two-person zero-sum game called duel with the following structure: Each of two players I and II has a gun with one bullet and he can fire his bullet at any time in [0, 1], aiming at his opponent. If I or II fires at timex, he hits his opponent with probabilityp (x) orq(x), respectively. The gun of I is silent, and hence, II does not know whether his opponent has fired or not, and the gun of II is noisy with time lagt, wheret is a positive constant,i.e., if II fires at timex then I knows it at timex +t. Further, if I hits II without being hit himself before, the payoff is 1; if I is hit by II without hitting II before, the payoff is ?1; if they hit each other at the same time or both survive, the payoff is 0. This paper gives optimal strategy for each player and the value of the game.     

Silent-noisy marksmanship contest with random termination

We consider a marksmanship contest in which Player I has one silent bullet, whereas Player II has one noisy bullet, the first contestant to hit his target wins, and the contest is to be terminated at a random timeT with cdfH(t). The model is a silent-noisy version of our previous paper (Ref. 8), and an extension of silent-noisy duel to nonzero-sum games of timing under an uncertain environment. It is shown that the uncertainty on the termination of the contest has influence on the equilibrium strategies and the equilibrium values, but the silent player has no advantages over the noisy one, in such a nonzero-sum model.The author thanks Professor M. Sakaguchi, Osaka University, who contributed to the research on mathematical decision-making problems and expresses appreciation for his continuous encouragement and guidance. The author also thanks Professor G. Kimeldorf, The University of Texas at Dallas, who invited the author to his university. Finally, the author expresses appreciation to Professors K. Sugahara and W. Fukui, Himeji Institute of Technology, for their encouragement and support.     

Teraoka-type two-person nonzero-sum silent duel

The paper discusses a silent nonzero-sum duel between two players each of whom has a single bullet. The duel is terminated at a random time in [0, 1] given by a cumulative distribution function. It is shown that the game has a unique Nash equilibrium under a wide range of possible payoff values for simultaneous firing. This contrasts with a very similar game considered by Teraoka for which there are many Nash equilibria.This work was carried out while the second author was visiting the University of Southampton on a Postdoctoral Fellowship of The Royal Society of London.     

The simultaneous truel

The simultaneous truel is a three-person game which is a generalization of the simple duel. The players' positions are fixed and their firing is simultaneous. Each player's only decision is which of his opponents will be his target. The (simultaneous) firing continues until there is at most one survivor or until all survivors have fired a specified number of times. Each player is assumed to be concerned only with his own survival; he is indifferent to the fate of his opponents. These games (parametrized by the maximum possible number of shots by each player) are examined for equilibrium points. It is found that, in many cases, the truel has a unique equilibrium point at which the player who is the poorest marksman has the greatest chance of survival.     

Linear differential games with delayed and noisy information

This paper deals with a linear-quadratic-Gaussian zero-sum game in which one player has delayed and noisy information and the other has perfect information. Assuming that the player with perfect information can deduce his opponent's state estimate, the optimal closed-loop control laws are derived. Then, it is shown that the separation theorem is satisfied for the player with imperfect information and his optimal state estimate is given by a delay-differential equation.     

On Simple MIX game

We consider zero-sum game which is called Simple MIX game. Each of two players (I and II) draws a number (x andy respectively) according to a uniform distribution on [0, 1]. After observing his number each player can then choose to offer or not offer to exchange his number for the other player's number. Conditions for an exchange are the following: 1) both players must offer for a trade to occur with certainty; 2) if only one player offers, a trade occurs with probabilityp. A player's payoff is equal to 1, 0 or — 1 if the value of the number which he finally gets is greater, equal or less than the number of his opponent. In the present paper we shall investigate Simple MIX game in which both of the players can obtain additional information about the opponent's number. Besides, we consider two-stage variant of this game.     

Existence of value in differential games with terminal cost function

A two-person, zero-sum differential game with general type phase constraints and terminal (not fixed) cost function is investigated. Player II (possessing complete information) can choose any strategy in the Varaiya-Lin sense, while his opponent (having incomplete information) can select any lower II-strategy introduced by Friedman (Ref. 1). The existence of a value and an optimal player II's strategy is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. The present paper largely extends the results of Ref. 2.     

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inR ⁿ. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.Many thanks are due to the referees for their detailed comments.     

A generalization of Ruckle's results for an ambush game

This article considers a two-person zero-sum game in which a traveller, player I chooses a point in the closed unit interval [0,1] so as to minimise the probability of being ambushed by two obstacles placed there by his adversary, player II. The payoff to player I equals 1 if the point chosen by him is not in either of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, and Lee.     

Random Walk Analysis in Antagonistic Stochastic Games

Abstract

This article deals with two “antagonistic random processes” that are intended to model classes of completely noncooperative games occurring in economics, engineering, natural sciences, and warfare. In terms of game theory, these processes can represent two players with opposite interests. The actions of the players are manifested by a series of strikes of random magnitudes imposed onto the opposite side and rendered at random times. Each of the assaults is aimed to inflict damage to vital areas. In contrast with some strictly antagonistic games where a game ends with one single successful hit, in the current setting, each side (player) can endure multiple strikes before perishing. Each player has a fixed cumulative threshold of tolerance which represents how much damage he can endure before succumbing. Each player will try to defeat the adversary at his earliest opportunity, and the time when one of them collapses is referred to as the “ruin time”. We predict the ruin time of each player, and the cumulative status of all related components for each player at ruin time. The actions of each player are formalized by a marked point process representing (an economic) status of each opponent at any given moment of time. Their marks are assumed to be weakly monotone, which means that each opposite side accumulates damages, but does not have the ability to recover. We render a time-sensitive analysis of a bivariate continuous time parameter process representing the status of each player at any given time and at the ruin time and obtain explicit formulas for related functionals.     

On the existence of good stationary strategies for nonleavable stochastic games

This paper discusses the problem regarding the existence of optimal or nearly optimal stationary strategies for a player engaged in a nonleavable stochastic game. It is known that, for these games, player I need not have an -optimal stationary strategy even when the state space of the game is finite. On the contrary, we show that uniformly -optimal stationary strategies are available to player II for nonleavable stochastic games with finite state space. Our methods will also yield sufficient conditions for the existence of optimal and -optimal stationary strategies for player II for games with countably infinite state space. With the purpose of introducing and explaining the main results of the paper, special consideration is given to a particular class of nonleavable games whose utility is equal to the indicator of a subset of the state space of the game.     

基于特征曲线和曲面匹配的弹痕自动比对方案

提出了消除测量误差的两套方案:基于特征曲线提取的方案和基于曲面匹配的方案.建立了中心重合法、近似法矢重合法与多点预定位法相结合的曲面匹配模型;讨论并给出了用于弹痕比对的六个差异度衡量指标;建立了弹痕及弹头相似度模型;给出了方案的有效性评价体系;最后,对一组实际的弹痕数据,进行有效性验证,取得了较高的正确率.     

On a generalization of Kaplansky's game

In this paper, games of the following general kind are studied: Two players move alternately by selecting unselected integer coordinate points in the plane. On each move, the first player selects exactly r points and the second player selects exactly one point. The first player wins if he can select p points on a line having none of his opponent's points before his opponent selects q points on a line having none of his own. If this latter eventuality occurs first, the second player wins. It is shown that if p ? c(r)q, then the second player can always win.     

The tug-of-war without noise and the infinity Laplacian in a wedge

Consider the ending time of the tug-of-war without noise in a wedge. There is a critical angle for finiteness of its expectation when player I maximizes the distance to the boundary and player II minimizes the distance. There is also a critical angle such that for smaller angles, player II can find a strategy where the expected ending time is finite, regardless of player I’s strategy. For larger angles, for each strategy of player II, player I can find a strategy making the expected ending time infinite. Using connections with the inhomogeneous infinity Laplacian, we bound this critical angle.     

Sequencing games with repeated players

Two classes of one machine sequencing situations are considered in which each job corresponds to exactly one player but a player may have more than one job to be processed, so called RP(repeated player) sequencing situations. In max-RP sequencing situations it is assumed that each player’s cost function is linear with respect to the maximum completion time of his jobs, whereas in min-RP sequencing situations the cost functions are linear with respect to the minimum completion times. For both classes, following explicit procedures to go from the initial processing order to an optimal order for the coalition of all players, equal gain splitting rules are defined. It is shown that these rules lead to core elements of the associated RP sequencing games. Moreover, it is seen that min-RP sequencing games are convex. We thank two referees for their valuable suggestions for improvement. Financial support for P. Calleja has been given by the Ministerio de Educación y Ciencia and FEDER under grant SEJ2005-02443/ECON, and by the Generalitat de Catalunya through a BE grant from AGAUR and grant 2005SGR00984.     

On some games of timing of mixed type

The paper considers a zero-sum, two-person game of timing on [0, 1] in which Players 1 and 2 behave as in the so-called discrete-fire duel and non-discrete fire duel, respectively. Player 1 is in possession of one action. Accuracy functions of the players are continuous and nondecreasing, from [0, 1] onto [0, 1]. The game is analysed in three versions and the form of optimal strategies for the players is found.