A Classical Search Game in Discrete Locations

TL;DR

This paper analyzes a strategic search game where a searcher and hider choose locations with probabilistic detection, proving the existence of optimal strategies and providing algorithms for their computation.

Contribution

It establishes the existence of optimal strategies for both players and introduces an algorithm to compute these strategies in the discrete search game.

Findings

Optimal strategies exist for both players.

The hider's strategy involves randomizing over all locations.

An algorithm is developed to compute the optimal strategies.

Abstract

Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among $n$ discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location $i$ takes $t_i$ time units and detects the hider -- if hidden there -- independently with probability $q_i$, for $i=1,\ldots,n$. The hider aims to maximize the expected time until detection, while the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, the hider's optimal mixed strategy hides in each location with a nonzero probability, and the searcher's optimal mixed strategy can be constructed with up to $n$ simple search sequences. We develop an algorithm to compute an optimal strategy for each player, and compare the optimal hiding strategy with the simple hiding strategy which gives the searcher no location preference at the beginning of the search.