Approximate solutions for expanding search games on general networks

Abstract

We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point H on a network Q moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the "expanding search" paradigm recently introduced by the authors. Here the regions S (t) that have been searched by time t are increasing from the starting point and have total length t. Roughly speaking the search follows a sequence of arcs a i such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where H is hidden to take a long time to find) for the cases where Q is a tree or is 2-arc-connected. This paper solves the game on some additional families of networks. However the main contribution is to give strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). We identify cases where our strategies are in fact optimal.

Competitive search in a network

“…Expanding search on a network was introduced in (Alpern & Lidbetter, 2013), with the focus on the Bayesian problem of minimizing the expected search time against a known Hider distribution. In a followup paper (Alpern & Lidbetter, 2019), the same authors studied expanding search on general networks and gave two strategy classes that have expected search times that are within a factor close to 1 of the value of the game. Both these works apply to the unnormalized search time.…”

Games for the optimal deployment of security forces

“…Introduction the network: the intruder and the agent. The intruder tries to optimize the value of the system, for example by (1) computing the shortest path between a source node and a sink node [14,36,59]; (2) maximizing the amount of flow through the network [18,76,113]; (3) maximizing the probability of completing a route [29,92,93,101]. The agent attempts to intercept the intruder before the goal is achieved.…”

“…Consider a network with 3 nodes. There are two players with routes R (1) = {{1}, {2}} and R (2) = {{2}, {3}}. The payoff functions for player 1 and 2 are given by:…”

“…where p (1) is the probability that player 1 selects node 1 and 1 − p (1) is the probability that he selects node 2. Similar, p (2) is the probability that player 2 selects node 3 and 1 − p (2) is the probability that he selects node 2.…”

“…We solve this network using Lagrange multipliers. We assume that µ 1 > λ (1) , µ 3 > λ (2) and µ 2 > λ (1) + λ (2) .…”