Infiltration games on arbitrary graphs

Abstract

The purpose of this note is to extend the work of Lalley [L] and Auger [A] on “infiltration games” on certain classes of graphs to an arbitrary graph by a simple application of Menger’s Theorem [H, Theorem 5.91. Infiltration games, as proposed by Gal [G, Sect. 4.61 and generalized by Auger [A], are determined by specifying two vertices a and b of a connected graph G, and a “capture probability” 1 2. In the infiltration game r= f(G, a, b, 2) the Infiltrator chooses a path p = (a =p(O), p(l), ,,,, p(T) = b) from a to b in G of any length, and the Guard chooses a map g: { 1,2, . ..} + V{a, b), where V denotes the vertex set of G. The interpretation is that at time t the Infiltrator and Guard are at the vertices p(t) and g(t), and the Guard will capture the Inliltrato’i with probability 1 A if they coincide. In the zero-sum game r the payoff I7 is the probability that the Infiltrator (maximizer) safely reaches the target vertex b, Z7(p, g) = A*, where q is the number of times p and g coincide. In some versions a time bound n is put on the length T of Infiltrator paths p, and the game is denoted f(G, a, b, 1, n). Related “searchlight” games are discussed in [BB, OPl, OP2, R]. Recently Auger [A] has extended the work of Lalley [L] by giving a complete solution, for arbitrary time bounds n, of the infiltration games on the graphs L = Lk(mI, m2, . . . . mk) consisting of two vertices a and b joined by k vertex disjoint paths with m,, . . . . mk (total m = C: mi) interior vertices. The value of the game T(L, a, b, A, n) is given by