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 the noisy-silent versus silent-noisy duel with equal accuracy functions

This paper deals with the noisy-silent versus silent-noisy duel with equal accuracy functions. Each of player I and player II has a gun with two bullets and he can fire his bullets at any time in [0, 1] aiming at his opponent. The first bullet of player I and the second bullet of player II are noisy, and the second bullet of player I and the first bullet of player II are silent. It is assumed that both players have equal accuracy functions. 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 value of the game and the optimal strategies are obtained. We find that the firing time of the silent bullet by player II's optimal strategy depends directly on the firing time of player I's noisy bullet.     

Repeated Games with Bonuses

This paper deals with 2-player zero-sum repeated games in which player 1 receives a bonus at stage t if he repeats the action he played at stage t−1. We investigate the optimality of simple strategies for player 1. A simple strategy for player 1 consists of playing the same mixed action at every stage, irrespective of past play. Furthermore, for games in which player 1 has a simple optimal strategy, we characterize the set of stationary optimal strategies for player 2.     

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.     

Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

The combinatorics of the colliding bullets

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let (V_i)_{1 ≤ i ≤ n} be an i.i.d. family of random variables with uniform distribution on [0,1]; shoot n bullets one after another at times 1,2,…,n, where the ith bullet has speed V_i. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalization of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.     

Axiomatic characterizations under players nullification

Many axiomatic characterizations of values for cooperative games invoke axioms which evaluate the consequences of removing an arbitrary player. Balanced contributions (Myerson, 1980) and balanced cycle contributions (Kamijo and Kongo, 2010) are two well-known examples of such axioms. We revisit these characterizations by nullifying a player instead of deleting her/him from a game. The nullification (Béal et al., 2014a) of a player is obtained by transforming a game into a new one in which this player is a null player, i.e. the worth of the coalitions containing this player is now identical to that of the same coalition without this player. The degree with which our results are close to the original results in the literature is connected to the fact that the targeted value satisfies the null player out axiom (Derks and Haller, 1999). We also revisit the potential approach (Hart and Mas-Colell, 1989) similarly.     

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 unifying model for matrix-based pairing situations

We present a unifying framework for transferable utility coalitional games that are derived from a non-negative matrix in which every entry represents the value obtained by combining the corresponding row and column. We assume that every row and every column is associated with a player, and that every player is associated with at most one row and at most one column. The instances arising from this framework are called pairing games, and they encompass assignment games and permutation games as two polar cases. We show that the core of a pairing game is always non-empty by proving that the set of pairing games coincides with the set of permutation games. Then we exploit the wide range of situations comprised in our framework to investigate the relationship between pairing games that have different player sets, but are defined by the same underlying matrix. We show that the core and the set of extreme core allocations are immune to the merging of a row player with a column player. Moreover, the core is also immune to the reverse manipulation, i.e., to the splitting of a player into a row player and a column player. Other common solution concepts fail to be either merging-proof or splitting-proof in general.     

An axiomatization of the Shapley value using a fairness property

In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games. Revised August 2001     

A multistage game with incomplete information requiring an infinite memory

This paper describes a zero-sum, discrete, multistage, time-lag game in which, for one player, there is no integerk such that an optimal strategy, for a new move during play, can always be determined as a function of the pastk state positions; that is, the player requires an infinite memory. The game is a pursuit-evasion game with the payoff to the maximizing player being the time to capture.This paper is the result of work carried out at the University of Adelaide, Adelaide, Australia, under an Australian Commonwealth Postgraduate Award.The author should like to thank the referee for his valued suggestions.     

On the NP-completeness of finding an optimal strategy in games with common payoffs

Consider a very simple class of (finite) games: after an initial move by nature, each player makes one move. Moreover, the players have common interests: at each node, all the players get the same payoff. We show that the problem of determining whether there exists a joint strategy where each player has an expected payoff of at least r is NP-complete as a function of the number of nodes in the extensive-form representation of the game. Received January 2001/Final version May 1, 2001     

The Sprague–Grundy function of the real game Euclid

The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternately, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. Unfortunately, the name Euclid has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a,b) with a<b are precisely the same as those for Cole and Davie’s game. Nevertheless, the Sprague–Grundy functions are not the same for the two games. We give an explicit formula for the Sprague–Grundy function for the original game of Euclid and we explain how the Sprague–Grundy functions of the two games are related.     

Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.     

On convex body chasing

A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setF _i is specified, and the player has to move to a point inF _i. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceF _i the cost to the player is within a constant (multiplicative) factor of the off-line cost (i.e., the least possible cost when allF _i are known in advance). We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces. The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line/off-line problems such as thek-server problem; we make some remarks on these more general problems.Joel Friedman wishes to acknowledge the National Science Foundation for supporting this research in part under a PYI Grant, CCR-8858788, and IBM for an IBM faculty development award. Nathan Linial's work was supported by a grant from the Binational Science Foundation Israel/US and from the Israel Academy of Sciences.     

The Guessing Secrets problem: a probabilistic approach

We introduce a probabilistic variant of the Guessing Secrets problem proposed by Chung et al. in [Electron. J. Combin. 8 (2001) R13]. In our variation, a player tries to discover the identity of a set S of n unknown secrets drawn by a second player, from a space Ω of N secrets. The first player tries to learn as much as possible about the elements of S by asking binary questions. For each question asked, the second player randomly chooses one of the n secrets of S that he uses in supplying the answer, which in any case must be truthful. We define a simple probabilistic guessing algorithm that allows us to guess all secrets of S with probability one. We show that the expected number of questions needed to guess all secrets is 2n²log₂N and the expected time complexity of the algorithm is . We also propose a generalization of this probabilistic guessing secrets problem, and provide some similar results for this generalization.     

One class of simultaneous pursuit games

Let A and B be given convex closed bounded nonempty subsets in a Hilbert space H; let the first player choose points in the set A and let the second one do those in the set B. We understand the payoff function as the mean value of the distance between these points. The goal of the first player is to minimize the mean value, while that of the second player is to maximize it. We study the structure of optimal mixed strategies and calculate the game value.     

Analyzing n-player impartial games

Combinatorial game theory is the study of two player perfect information games. While work has been done in the past on expanding this field to include n-player games we present a unique method which guarantees a single winner. Specifically our goal is to derive a function which, given an n-player game, is able to determine the winning player (assuming all n players play optimally). Once this is accomplished we use this function in analyzing a certain family of three player subtraction games along with a complete analysis of three player, three row Chomp. Furthermore we make use of our new function in producing alternative proofs to various well known two player Chomp games. Finally the paper presents a possible method of analyzing a two player game where one of the players plays a completely random game. As it turns out this slight twist to the rules of combinatorial game theory produces rather interesting results and is certainly worth the time to study further.