Immobile Hider Research Articles

Search for an Immobile Hider on a Binary Tree with Unreliable Locational Information

Search for an Immobile Hider on a Binary Tree with Unreliable Locational Information

Search strategies in the game with a team of pursuers and one evader on the plane

We consider a differential simple search game without phase constraints on the plane. Each pursuer from the team chooses an initial position and piecewise constant strategy. As an example, the game of simple search of immobile hider (evader) on the plane is considered, and the necessary conditions for detection are found. For cases with two and three pursuers, a comparison of the detection time of the evader was made and approximate information sets numerically constructed.

Search for an immobile hider on a stochastic network

Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry’s goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.

Search for an immobile Hider in a known subset of a network

A unit speed Searcher, constrained to start in a given closed set S, wishes to quickly find a point x known to be located in a given closed subset H of a metric network Q. This defines a game G=G(Q,H,S), where the payoff to the maximizing Hider is the time for the Searcher path to reach x. Lengths on Q are defined by a measure λ, which then defines distance as least length of connecting path. For trees Q, we find that the minimax search time (value V of G) is given by V=λ(H)−dH(S)/2, where dH(S) is what we call the ‘H-diameter of S’, and equals the usual diameter d(S) of S in the case H=Q. For the classical case of Gal where the S is a singleton and H=Q, our formula reduces to his result V=λ(Q). If S=H=Q, our formula gives Dagan and Gal's result V=λ(Q)−d(Q)/2. In all other cases, our result is new. Optimal searches consist of minimum length paths covering H which start and end at points of S, traversed equiprobably in either direction.

The search value of a set

We study search games in which the hider may hide in a finite number of locations. We assume that the cost of searching these locations does not depend on the order in which the locations are searched. From these assumptions we derive that the cost function is submodular, thus placing search games with an immobile hider in the context of coalitional games.

Optimal search and ambush for a hider who can escape the search region

Search games for a mobile or immobile hider traditionally have the hider permanently confined to a compact ‘search region’ making eventual capture inevitable. Hence the payoff can be taken as time until capture. However in many real life search problems it is possible for the hider to escape an area in which he was known to be located (e.g. Bin Laden from Tora Bora) or for a prey animal to escape a predator’s hunting territory. We model and solve such continuous time problems with escape where we take the probability of capture to be the searcher’s payoff.We assume the searcher, while cruise searching, can cover the search region at unit rate of area, for a given time horizon T known to the hider. The hider can stay still or choose any time to flee the region. To counter this, the searcher can also adopt an ambush mode which will capture a fleeing hider. The searcher wins the game if he either finds the hider while cruise searching or ambushes him while he is attempting to flee; the hider wins if he flees successfully (while the searcher is cruising) or has not been found by time T. The optimal searcher strategy involves decreasing the ambush probability over time, to a limit of zero. This surprising behaviour is opposite to that found recently by Alpern et al. (2011, 2013) in a predator-prey game with similar dynamics but without the possibility of the hider escaping. Our work also complements that of Zoroa et al. (2015) on searching for multiple prey and Gal and Casas (2014) for a combined model of search and pursuit.

A SEARCH GAME WITH UNKNOWN EXAMINATION COSTS AND TRAVEL TIMES

A search game on a finite and complete graph is studied between a (immobile) hider and a seeker. A strategy for the hider is to choose a node where he hides, except for a specified node. A strategy for the seeker is an ordering of nodes in which the seeker starts at the specified node, examines each node in that order until he finds the hider, traveling along edges. Associated with an examination of a node is the examination cost, and associated with a movement from a node to a node is a traveling cost. These costs are unknown for both players when they choose strategies. They know only probability laws of them. While the hider wishes to maximize the expectation of the sum of these costs which are required to find the hider, the seeker wishes to minimize it. The game is solved when the nodes are classified into two groups depending on the examination costs, and the edges are classified into three groups depending on both costs.

Find-and-Fetch Search on a Tree

We introduce a new type of search game called the “find-and-fetch” game F(Q, O). The Hider simply picks any point H in the network Q. The Searcher starts at time zero at a given point O of Q, moving at unit speed until he reaches H (finds the Hider). Then he returns at a given speed ρ along the shortest path back to O, arriving at time R, the payoff. This models the problem faced in many types of search, including search-and-rescue problems and foraging problems of animals (where food must be found and returned to the lair). When Q is a binary tree, we derive optimal probabilities for the Searcher to branch at the nodes. These probabilities give a positive bias towards searching longer branches first. We show that the minimax value of the return time R (the game value of F(Q, O)) is μ + D/ρ, where μ is the total length of Q and D is the mean distance from the root O to the leaves (terminal nodes) of Q, where the mean is taken with respect to what is known as the equal branch density distribution. As ρ goes to infinity, our problem reduces to the search game model where the payoff is simply the time to reach the Hider, and our results tend to those obtained by Gal [Gal, S. 1979. Search games with mobile and immobile hider. SIAM J. Control Optim. 17(1) 99–122] and Anderson and Gal [Anderson, E. J., S. Gal. 1990. Search in a maze. Probab. Engrg. Inform. Sci. 4(3) 311–318] for that model. We also apply our return time formula μ + D/ρ to determine the ideal location for the root (lair or rescue center) O, assuming it can be moved. In the traditional “find only” model, the location of O does not matter.

Network search games with immobile hider, without a designated searcher starting point

In the (zero-sum) search game Γ(G, x) proposed by Isaacs, the Hider picks a point H in the network G and the Searcher picks a unit speed path S(t) in G with S(0) = x. The payoff to the maximizing Hider is the time T = T(S, H) = min{t : S(t) = H} required for the Searcher to find the Hider. An extensive theory of such games has been developed in the literature. This paper considers the related games Γ(G), where the requirement S(0) = x is dropped, and the Searcher is allowed to choose his starting point. This game has been solved by Dagan and Gal for the important case where G is a tree, and by Alpern for trees with Eulerian networks attached. Here, we extend those results to a wider class of networks, employing theory initiated by Reijnierse and Potters and completed by Gal, for the fixed-start games Γ(G, x). Our results may be more easily interpreted as determining the best worst-case method of searching a network from an arbitrary starting point.

Searching for an Agent Who May OR May Not Want to be Found

There is an extensive theory regarding optimal continuous path search for a mobile or immobile “target.” The traditional theory assumes that the target is one of three types: (i) an object with a known distribution of paths, (ii) a mobile or immobile hider who wants to avoid or delay capture, or (iii) a rendezvouser who wants to find the searcher. This paper introduces a new type of search problem by assuming that aims of the target are not known to the searcher. The target may be either a type (iii) cooperator (with a known cooperation probability c) or a type (ii) evader. This formulation models search problems like that for a lost teenager who may be a “runaway,” or a lost intelligence agent who may be a defector. In any given search context, it produces a continuum of search problems τ(c), 0 ≤ c ≤ 1, linking a zero-sum search game (with c = 0) to a rendezvous problem (with c = 1). These models thus provide a theoretical bridge between two previously distinct parts of search theory, namely search games and rendezvous search.

On the optimality of a simple strategy for searching graphs

Consider a search game with an immobile hider in a graph. A Chinese postman tour is a closed trajectory which visits all the points of the graph and has minimal length. We show that encircling the Chinese postman tour in a random direction is an optimal search strategy if and only if the graph is weakly Eulerian (i.e it consists of several Eulerian curves connected in a tree-like structure).

Complexity of searching an immobile hider in a graph

We study the computational complexity of certain search-hide games on a graph. There are two players, called searcher and hider. The hider is immobile and hides in one of the nodes of the graph. The searcher selects a starting node and a search path of length at most k. His objective is to detect the hider, which he does with certainty if he visits the node chosen for hiding. Finding the optimal randomized strategies in this zero-sum game defines a fractional path covering problem and its dual, a fractional packing problem. If the length k of the search path is arbitrary, then the problem is NP-hard. The problem remains NP-hard if the searcher may freely revisit nodes that he has seen before. In that case, the searcher selects a connected subgraph of k nodes rather than a path of k nodes. If k is logarithmic in the number of nodes of the graph, then the problem can be solved in polynomial time. This is shown using a recent technique called color-coding due to Alon, Yuster and Zwick. The same results hold for edges instead of nodes, that is, if the hider hides in an edge and the searcher searches k edges on a path or on a connected subgraph.

A search game on the union of graphs with immobile hider

We consider a search game for an immobile hider on one arc of the union of n graphs joined at one or two points. We evaluate a lower bound on the value of a strategy for the hider on this union. When we have identical graphs, we give the conditions under which the value of the strategy for the hider on this union is greater than or equal to n times the value of this strategy on one graph. We also solve search games on graphs, consisting of an odd number of arcs, each of length one, joining two points. © 1995 John Wiley & Sons, Inc.

Search games with immobile hider

In this paper we discuss a type of search game with immobile hider on a graph. If the graph is weakly cyclic (i.e. no edge lies on two different cycles) we determine the value of the game and give an easy way to compute optimal strategies for the players. The value is half the length of a quasi-Euler path, i.e. a closed path of minimal length that visits all points of the network.

The search game on a network with immobile hider

AbstractIn this paper we discuss a search game on a network, Q, with two players, a searcher and a hider. The hider chooses a point on Q at which to hide, while the searcher starts at a given point on Q and moves with continuous trajectory subject to a maximal speed. This is a zero‐sum game with payoff given by the time elapsed until the searcher reaches the point occupied by the hider. The solution to this game is known for certain types of network. Here by formulating the problem as an infinite‐dimensional linear program, we derive an algorithm for its solution. We discuss the performance of the algorithm on three examples.

On a Discrete Search Problem on Three Arcs

A discrete search game with an immobile hider is posed and solved. The game is a discrete analogue of a known continuous game where the search takes place on three arcs which connect two points. A solution to the continuous game is obtained as a limiting case.

Search Games with Mobile and Immobile Hider

We consider search games in which the searcher moves along a continuous trajectory in a set Q until he captures the hider, where Q is either a network or a two (or more) dimensional region. We distinguish between two types of games; in the first type which is considered in the first part of the paper, the hider is immobile while in the second type of games which is considered in the rest of the paper, the hider is mobile. A complete solution is presented for some of the games, while for others only upper and lower bounds are given and some open problems associated with those games are presented for further research.