Steve Alpern

The Rendezvous Search Problem

The author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time. This time is called the rendezvous value of the region. It is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise. A general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players. These concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network. The discrete rendezvous problem on a cycle graph for players restricted to symmetric Markovian strategies is then solved. Finally, the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other.

Patrolling Games

A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games. The facility to be patrolled can be thought of as a network or graph Q of interconnected nodes (e.g., rooms, terminals), and the Attacker can choose to attack any node of Q within a given time T. He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus, the patrolling game is a win-lose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically either the Value of the game, or bounds on the Value, for various classes of graphs, and we discuss possible extensions and generalizations.

The search value of a network

Let Q be a connected network with a distinguished (starting) point qo, whose arc lengths sum to one. We associate with Q a “search value” V ( Q ) representing the expected time needed for a searcher, starting at qo and moving at unit speed, to find a moving hider. We assume neither sees the other until they meet. We demonstrate that the “figure-eight” network, consisting of two equal loops joined at a central starting point, has a search value not exceeding #. This contradicts a conjecture of Gal that the search value of any network is at least 1. In the other direction, we show that V ( Q ) S 6kD for a network with k edges and diameter D.

Rendezvous Search on the Line with More Than Two Players

Suppose n blind, speed one, players are placed by a random permutation onto the integers 1 to n , and each is pointed randomly to the right or left. What is the least expected time required form m ≤ n of them to meet together at a single point? If they must all use the same strategy we call this time the symmetric rendezvous value R n , m s ; otherwise the asymmetric value R n , m a . We show that R 3,2 a = 47/48, and that R n , n s is asymptotic to n /2. These results respectively extend those for two players given by Alpern and Gal (Alpern, S., S. Gal. 1995. Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33 1270–1276.) and Anderson and Essegaier (Anderson, E. J., S. Essegaier. 1995. Rendezvous search on the line with indistinguishable players. SIAM J. Control Optim. 33 1637–1642.).

Return times and conjugates of an antiperiodic transformation

Denote by G the group of all μ-preserving bijections of a Lebesgue probability space (X, Σ, μ) and by C the conjugacy class of an antiperiodic transformation σ in G. We present several new results concerning the denseness of C in G with respect to various topologies. One of these asserts that given any weakly mixing transformation τ in G and any F with μ(F) < 1, there is a transformation in C which agrees with τ a.e. on F.

Infiltration games on arbitrary graphs

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

A mixed-strategy minimax theorem without compactness

Minimax theorems for infinite games generally require that both players choose their pure strategies from compact sets and have semicontinuity requirements in both variables. This paper proves the following theorem. Let X be a compact Hausdorfl space and let

Asymmetric Rendezvous on the Line is a Double Linear Search Problem

(Y,A)

Ambush strategies in search games on graphs

be a measurable space. Let

Searching a Variable Speed Network

f:X \times Y