A basic searchlight game

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS:Vol. 71, No, 1, OCTOBER I991

A Basic Searchlight Game V, J. BASTON 1 AND F, A. BOSTOCK 2

Communicated by M. Pachter

Abstract. A searchlight game is a two-person zero-sum dynamic game of the pursuit-evasion type in which at least one of the two players has a searchlight. A searchlight can be flashed a given number of times within a fixed time period and the objective is to catch the opponent in the region illuminated by the flash. Olsder and Papavassilopoulos instituted the study of these games and, in this paper, we supplement their results, obtaining a closed formula for the value and optimal strategies for the players in their basic game.

Key Words. Two-person games, zero-sum games, dynamic games, searchlight games.

1. Introduction

Searchlight games are a class of two-person zero-sum dynamic games o f the pursuit-evasion type and have aroused considerable interest in recent years. In a searchlight game, at least one of the two players has a searchlight which can be flashed a certain number of times within a given time period. A flash of the searchlight illuminates a region of known shape and the objective of a player with a searchlight is to catch his opponent in thisregion at the time of the flash. However, when a player flashes his searchlight, he automatically discloses his position to his opponent wherever his opponent is. Thus, if both players have a searchlight and a player does not catch his opponent in the region illuminated by the flash, his position can be more vulnerable as he has provided his opponent with potentially valuable information; such information is called dynamic since it depends on the actions of the players. ~Iecturer, Faculty of Mathematical Studies, University of Southampton, Southampton, England. 2Lecturer. Faculty o f Mathematical Studies, University of Southampton, Southampton, England.

47 0022-3239/91/1000-0047506.50/0 ~" 1991 PlenumPublishingCorporation

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JOTA: VOL. 71, NO. 1, OCTOBER 1991

As is pointed out in Ref. l, Section 5.4, it is conceptually very complicated to define mixed a n d / o r behavior strategies for games which proceed continuously in time. It is therefore not surprising that a number of investigations into searchlight games assume that play takes place in discrete time and that the players are constrained to move in a finite state space; i.e., the players move in a network with a finite number of nodes. In particular, Olsder and Papavassilopoulos (Refs. 2 and 3) have considered the case where the nodes are positioned on the circumference of a circle. Other authors have analyzed games which are essentially of searchlight type. For instance, Baston and Bostock (Ref. 4) investigated the (continuous) problem of a helicopter with k bombs trying to destroy a submarine in a narrow channel, but it could equally well have been formulated as a searchlight game. Similar comments apply to other games considered by Baston and Bostock (Ref. 5), Lee (Refs. 6 and 7), and Bernhard, Colomb, and Papavassilopoulos (Ref. 8). The main purpose of this paper is to supplement the results of Olsder and Papavassilopoulos. In Ref. 9, they show that the optimal strategies of what they term the basic game can be found via a certain matrix game. By adopting a different approach, we obtain explicit optimal strategies for the players in this basic game and a closed formula for its value. The particularly simple nature of our results enable us to deduce the values of some of the other games they consider.

2. Basic Game As its name implies, the basic game is a building block for many other searchlight games. It is a zero-sum game played on n equally spaced points on the circumference of a circle by two players called Pursuer and Evader. We will usually think of the points as being labelled 0, 1 , . . . , n - 1 in clockwise order (so that n - 1 is adjacent to 0) but, for convenience of notation, we will sometimes denote the point i by a number j satisfying j = i(mod n). Thus, for example, we often use - 1 to represent the point n - 1. Only Pursuer has a searchlight and he can flash it just once; the flash illuminates Pursuer's position together with the two points adjacent to it. Play terminates when the flash is used or at a fixed time T (known to both players), whichever is the shorter time. Play is assumed to take place in discrete time and, at the initial instant of time t = 0, Pursuer is at the point 0 while Evader is at the point d; this information is again known to both players. If d = 0, 1, or n - 1, the game is trivial since Pursuer can achieve his objective by flashing immediately. Thus we may assume n >~4 and, taking advantage of symmetry, we may further assume that 2~d~ I n / 2 ] - 1, Pursuer can obtain an expectation of at least 2/(4m + 1) when n = 6m + 2 and d = 3s + 1, s = 1. . . . . m.

JOTA: VOL. 71, NO. 1, OCTOBER 1991

Proof.

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It is easy to check that Pursuer can choose a point at random

from 2m

-- s

~=~ { 3 ( s - r e + u ) - 1, 3(s-re+u)} s--I

u ~o { 3 ( s + m - v ) - 1, 3 ( s + m - v ) } w {3s+ 3m +2}, and flash his searchlight there at time 3m. Notice that every point other than 3s+ 3m + 2 is caught with probability 2/(4m + 1). Since Evader cannot get to 3s + 3rn + 2 at time 3m when he starts at 3s + 1, the result now follows. []

4. Evader Optimal Strategies The optimal strategies for Evader are not so straightforward as those for Pursuer, and we need to introduce some notation before we can describe them. After a time [ n / 2 ] - 2 , Evader can get to any of the points d [n/21+2+w, for w = 0 , 1. . . . . 2 [ n / 2 ] - 4 , and the Evader strategies that we give involve choosing a point from a particular subset of these points. Thus, throughout this and the following section, we shall employ the following notation. Let

A=[(2[n/2l-d)/3J,

B=[d/3],

& = { d - [ n / 2 1 + 2 + 3 u : u = O , 1. . . . , A - I}, & = { d + [ n / 2 ] - 2 - 3 v : v = 0 , 1. . . . . B - 1}, and

S=S, u $2 • {in/21}.

(1)

Note that S~ and $2 are disjoint. For 0 ~