A search game on a hypergraph with booby traps

Abstract

A set of n boxes, located on the vertices of a hypergraph G, contain known but different rewards. A Searcher opens all the boxes in some hyperedge of G with the objective of collecting the maximum possible total reward. Some of the boxes, however, are booby trapped. If the Searcher opens a booby trapped box, the search ends and she loses all her collected rewards. We assume the number k of booby traps is known, and we model the problem as a zero-sum game between the maximizing Searcher and a minimizing Hider, where the Hider chooses k boxes to booby trap and the Searcher opens all the boxes in some hyperedge. The payoff is the total reward collected by the Searcher. This model could reflect a military operation in which a drone gathers intelligence from guarded locations, and a booby trapped box being opened corresponds to the drone being destroyed or incapacitated. It could also model a machine scheduling problem, in which rewards are obtained from successfully processing jobs but the machine may crash. We solve the game when G is a 1-uniform hypergraph (the hyperedges are all singletons), so the Searcher can open just 1 box. When G is the complete hypergraph (containing all possible hyperedges), we solve the game in a few cases: (1) same reward in each box, (2) $k=1$, and (3) $n=4$ and $k=2$. The solutions to these few cases indicate that a general simple, closed form solution to the game appears unlikely.

Introduction

We consider the following game between a Hider and a Searcher. There is a set $[n]\equiv \{1,\dots ,n\}$ of boxes, with box i containing a reward of $r_i\ge 0$, for $i\in [n]$. We also make a standing assumption that, without loss of generality, $r_1\ge \cdots \ge r_n$. The boxes are identified with the vertices of a hypergraph G. The Hider sets booby traps in k of the boxes, where $1\le k\le n-1$, so his strategy set is ${[n]}^{(k)}\equiv \{H\subset [n]:|H|=k\}$. The Searcher chooses a subset $S\subset [n]$ of boxes to search, where S is the hyperedge of a hypergraph G with vertices V and hyperedges $E\subset 2^V$.

If the Hider plays H and the Searcher plays S, the payoff $R(S,H)$ is given by

$$R(S,H)=\{\begin{array}{cc}r(S),\hfill & \text{if }H\cap S=\varnothing ,\hfill \\ 0,\hfill & \text{otherwise,}\hfill \end{array}$$

where $r(S)\equiv {\sum }_{i\in S}{r}_i$ is the sum of the rewards in S. In other words, the Searcher keeps the sum of all the rewards in the boxes she opens unless one or more of them is booby trapped, in which case, she gets nothing. If the Searcher uses a mixed strategy p (that is, a probability distribution over subsets $S\subset [n]$) and the Hider uses a mixed strategy q (a probability distribution over subsets $H\in {[n]}^{(k)}$), we write the expected payoff as $R(p,q)$. We also write $R(p,H)$ and $R(S,q)$ if one player uses a pure strategy while the other player uses a mixed strategy.

This game could be an appropriate model for a military scenario in which a drone is used to gather intelligence at several locations, and $r_i$ is the expected value of the intelligence gathered at location i. A known number k of the locations are guarded, and flying the drone near these locations would result in its incapacitation. Alternatively, the Searcher may be collecting rewards in the form of stolen weapons or drugs from locations at which capture is possible, or the Searcher could be a burglar stealing valuable possessions from houses in a neighborhood, some of which are monitored by security cameras. The graph structure could correspond to geographical constraints. The case of the complete hypergraph, where $E=2^V$, corresponds to no constraints on the Searcher's choice of subset. The case where E is 1-uniform, so that every hyperedge consists of a single vertex, corresponds to the Searcher being limited to searching only one location. If E is 2-uniform, so that G is a graph, the Searcher must choose locations corresponding to the endpoints of an edge of the graph.

The game could also model a scheduling problem in which there are n jobs with utilities $r_i$ which are obtained from a successful execution of job i. For example, jobs may correspond to computer programs. A total of k of the programs are bugged, and each bug will crash the machine so that all data is lost. The objective is to find a subset of jobs to run that maximizes the worst-case expected utility, assuming Nature chooses which k jobs are bugged.

This work lies in the field of search games, as discussed in [2], [3], and [5]. Search games involving objects hidden in boxes have previously been considered in [7] and [8]. In these works, the objective of the Searcher is to minimize a total cost of finding a given number of hidden objects. [1] consider a machine scheduling problem in which rewards are collected from processing jobs and the machine may crash, similarly to our problem. But in their setting, each job will independently cause the machine to fail with a given probability.

Since this is a zero-sum game, it could be solved by standard linear programming methods, but this approach would be inefficient for large k, or if the hypergraph has a large number of hyperedges. In this work, we concentrate on two special cases of the game, with the aim of finding concise, closed-form solutions. We first solve the case where G is a 1-uniform hypergraph in Section 2. In Section 3, we consider the complete hypergraph, and solve the game for three special cases: (1) same reward in each box, (2) $k=1$, and (3) $n=4$, $k=2$. We also give some general bounds, and make a conjecture on the form of the optimal solution. Finally, we offer concluding remarks in Section 4.