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.     

Closed-loop Stackelberg solution to a multistage linear-quadratic game

A new solution of a two-person, nonzero-sum Stackelberg game, with linear dynamics, quadratic performance criteion, and closed-loop information available to both players, is presented. This solution is applicable to all problems where the leader is able to influence the objective function of the follower, and this function is strictly convex with respect to the control variable handled by the follower. The resulting equilibrium strategies adapt to the possible nonoptimal behavior of players at some stages of the game. The strategy of the leader has a simple interpretation of a threat formulated by the leader toward the follower and, if necessary, carried out one stage after the follower has played inconsistently with the leader's wishes.     

Multimodel design of a Nash strategy

For interconnected systems with slow and fast modes, we analyze a multimodel design of a Nash equilibrium strategy. Decision makers use different models of the same system to obtain a composite strategy that approximates the exact Nash strategy. It is discussed in what sense this approximation holds.This work was supported by the Department of Energy, Electrical Energy Systems Division, under Contract No. EX-76-C-01-2088. The author wishes to thank Professor P. V. Kokotovic for his helpful suggestions in the course of this work.     

Differential games results for military expenditures of unequal antagonists in the third world

In developing countries, resources are scarce and military expenditure can hinder development. Thus, allocation of resources between defense and civilian sectors is an important decision. Also, in developing countries, most defense spending is related to regional conflicts where one frequently encounters unequal participants (e.g., India—Pakistan, Iran—Iraq, China—Vietnam). In such situations, the bigger power aspires to be the regional leader, and allocates resources to the defense sector accordingly.

A model incorporating the two above mentioned stylized facts has been examined by Deger and Sen [1] in two different ways. They first look at the problem from a single country's point of view, which takes the other's reaction function as given, and show that in the long run, military expenditures and armament stock levels will exhibit stability. Next, they formulate the problem as a noncooperative differential game in which each country assumes knowledge of the other country's welfare function, and conjecture that stability will once again be exhibited. The present paper attempts to verify that conjecture using a simplifying assumption which is necessary to be able to qualitatively solve the problem. This simplifying assumption does not change the results for the optimal control models, however, the conjecture is not valid for the simplified differential game model.     

On linear stochastic differential games with average cost criterions

In this paper, we consider scalar linear stochastic differential games with average cost criterions. We solve the dynamic programming equations for these games and give the synthesis of saddle-point and Nash equilibrium solutions.The authors wish to thank A. Ichikawa for providing the initial impetus and helpful advice.     

On Nash equilibria inN-person games over reflexive Banach spaces

In this note, an extended version of the Ky Fan inequality is obtained for reflexive Banach spaces. As an application of the result, it is proved that Nash equilibrium points exist under weaker conditions than those of Williams (Ref. 1).     

Some comments on a theorem of Hardy and Littlewood

In this note, we reconstruct a proof of a classical result due to Hardy and Littlewood. While this result has played an important role in the modern theories of Markov decision processes and stochastic games, it is not that easy to find its proof in the literature in the format in which it has been applied. Furthermore, we supply either examples or complete citations for the other related cases which are not covered by the Hardy-Littlewood theorem.     

Compromise values in cooperative game theory

The aim of this paper is to give a survey on several well-known compromise values in cooperative game theory and its applications. Special attention is paid to the τ-value for TU-games, the Raiffa-Kalai-Smorodinsky solution for bargaining problems, and the compromise value for NTU-games.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.