New results on rendezvous search on the interval

In a rendezvous search problem, two players are placed in a network and must try to meet each other in the least possible expected time. We look at rendezvous search on a discrete interval in which the players are initially placed using independent draws (usually assumed to be from the same distribution). Some optimal solutions are known if this distribution is uniform, and also for certain other special types of distribution. In this article, we present two new results. First, we characterize the complete set of solutions for the uniform case, showing that all optimal strategies must have two specific properties (namely, of being swept and strictly geodesic). Second, we relate search strategies on the interval to proper binary trees, and use this correspondence to derive a recurrence relation for solutions to the symmetric rendezvous problem for any initial distribution. This relation allows us to solve any such problem computationally by dynamic programming. Finally, some ideas for future research are discussed. © Wiley Periodicals, Inc. Naval Research Logistics 60: 454–467, 2013     

Solving partially observable agent‐intruder games with an application to border security problems

In this paper, we introduce partially observable agent‐intruder games (POAIGs). These games model dynamic search games on graphs between security forces (an agent) and an intruder given possible (border) entry points and high value assets that require protection. The agent faces situations with dynamically changing, partially observable information about the state of the intruder and vice versa. The agent may place sensors at selected locations, while the intruder may recruit partners to observe the agent's movement. We formulate the problem as a two‐person zero‐sum game, and develop efficient algorithms to compute each player's optimal strategy. The solution to the game will help the agent choose sensor locations and design patrol routes that can handle imperfect information. First, we prove the existence of ?‐optimal strategies for POAIGs with an infinite time horizon. Second, we introduce a Bayesian approximation algorithm to identify these ?‐optimal strategies using belief functions that incorporate the imperfect information that becomes available during the game. For the solutions of large POAIGs with a finite time horizon, we use a solution method common to extensive form games, namely, the sequence form representation. To illustrate the POAIGs, we present several examples and numerical results.