Ryusuke Hohzaki

An inspection game with multiple inspectees

Abstract This paper deals with a non-zero-sum inspection game with one inspector and several inspectees. Each inspectee country makes a decision from the standpoint of his national interest while the inspector distributes staff to inspectee facilities to detect illegal behavior. We propose a method to find a Nash equilibrium for the game, which could help the inspector make an effective plan about how to assign staff to individual suspect facilities in those countries. Avenhaus et al. studied this kind of two-inspectee model. This paper extends Avenhaus’ research to a multi-inspectee model taking account of optimal dispatching of inspection staff.

Search allocation game

This paper deals with a two-person zero-sum game called search allocation game (SAG), where a searcher allocates his searching resources in a search space to detect a target while the target takes a path running across the space to evade the searcher. We consider the discrete SAG and the continuous SAG defined on the discrete search space and the continuous one, respectively. In a general way, we prove an existence theorem of equilibrium points for both the SAGs and elucidate that an equilibrium of the continuous SAG is given by a convergence point of equilibria of the discrete SAG. After then we develop a method to solve a large size of the discrete problem with specific feasibility conditions. As one of numerical examples, we take so-called flaming datum search game, which is adequate to demonstrate the convergence theorem.

A search game when a search path is given

Abstract In this paper, we investigate a search game in discrete time and space. A searcher is given a search path in advance and his look on the path is determined by a randomized look strategy. A target selects a path from some options. The searcher gains a value on the detection of the target but expends search cost by the look. A pay-off function of the game for the searcher is the expected reward which is defined as the expected value minus the expected search cost. First, we show a recursive relation for the conditional optimal look strategy of the searcher given a target path. We prove its NP-completeness, though it looks simple, and clarify some characteristics of the solution. Then our original continuous game is converted to a matrix game. From these facts, we consider a relationship between the game and the one-sided optimizing problem and examine some examples.

Optimal strategy of route and look for the path constrained search problem with reward criterion

Abstract A target is moving among a finite number of cells K = {1,…, K} in discrete time T = {1,…, T} . Knowing probability of the targets path selection, a searcher is searching for the target with constraints that he can move from cell i to one of the adjacent cells I(i). The searcher detects the target with the probability pi on the look into cell i in which both the target and the searcher are there. He gains a value V(t) on the detection of the target at time t but expends cost c0(i, t) on the search in cell i at t. The searcher selects his route and determines whether he looks into his current cell on the route or not. In this paper, our purpose is to find an optimal strategy for the route and the look of the searcher, which maximizes the expected reward defined as the expected value minus the expected cost. The problem is formulated as an integer programming problem. For solving the problem, we use a branch and bound procedure with an upper bound estimation.

A Smuggling Game with Asymmetrical Information of Players

This paper deals with a smuggling game with multiple stages. Customs is allowed to patrol within the limited number of chances and obtain reward by the capture of a smuggler. The smuggler gets a reward depending on the amount of contraband he succeeds to ship in smuggling at each stage. The pay-off of the game is zero-sum. In almost all past studies, they adopt the alternative of smuggling or non-smuggling as the smugglers strategy. From the point of view of information, some researchers assumed that both players could observe their opponents behaviour at the past stage or a few assumed that both players had no information about their opponent. Other than these types of smuggling games with the symmetric information, we introduce the asymmetrical acquisition of information or the concept of perfect Bayesian equilibrium in the smuggling game for the first time.

An Attrition Game on an Acyclic Network

This paper deals with noncooperative games in which two players conflict on a network through an attrition phenomenon. The associated problem has a variety of applications, but we model the problem as a military conflict between an attacker and a defender on an acyclic network. The attacker marches from a starting node to a destination node, expecting to keep his initial members untouched during the march. The defender deploys his forces on arcs to intercept the attacker. If the attacker goes through an arc with deployed defenders, the attacker incurs casualties according to Lanchester’s linear law. In this paper, we discuss two games having the number of remaining attackers as the payoff and propose systems of linear programming formulations to derive their equilibrium points. One game is a two-person zero-sum (TPZS) one-shot game with no information and the other is a TPZS game with two stages separated by information acquisition about players’ opponents.

Optimal ambushing search for a moving target

Abstract This paper investigates a search problem for a moving target in which a searcher can anticipate the probabilities of routes selected by the target but does not have any time information about when the target transits the route. If the searcher had some time information, he could develop an efficient search plan by varying allocations of search effort based on time. Due to the lack of time information, the searcher must ambush the target by distributing search effort to places where the target is likely to pass. There are few papers that deal mathematically with this type of search problem with no time information. Employing the criterion of detection probability, we formulate the problem and obtain necessary and sufficient conditions for the optimal solution. By applying the conditions, we propose two methods for solving the problem. The convex programming problem can be easily solved numerically by some well-known methods, e.g. the gradient projection method or the multiplier method. By numerical comparison, it is verified that the proposed methods have the excellent performance in computational time. We also elucidate some properties of the optimal distribution of search effort by some numerical examples.

An Attrition Game on a Network Ruled by Lanchester's Square Law

We consider two-person zero-sum attrition games in which an attacker and a defender are in combat with each other on a network. The attacker marches from a starting node to a destination node, hoping that the initial members survive the march. The defender deploys his forces on arcs in order to intercept the attacker. If the attacker encounters the defender on an arc, the attacker incurs casualties according to Lanchester’s square law. We consider two models: a one-shot game in which the two players have no information about their opponents, and a two-stage game in which both players have some information about their opponents. For both games, the payoff is defined as the number of survivors for the attacker. The attacker’s strategy is to choose a path, and the defender’s is to deploy the defending forces on arcs. We propose a numerical algorithm, in which nonlinear programming is embedded, to derive the equilibrium of the game.

A patrol problem in a building by search theory

Art gallery problem has been extensively studied by computational geometry, where major issue was to find the minimum number of guards and their locations to watch inside an art gallery or a facility. In this paper, we are concerned with the dynamic and game-theoretic aspects of a security problem, where a thief tries to invade the gallery while watchmen try to prevent it. We consider the following problems: an invasion scheduling problem and an invasion route problem on thiefs side, a selection problem of patrol routes and a distribution problem of watching effort for the guards. We solve the first and the second problems by a dynamic programming formulation, and the third and the fourth problems by game theory and search theory. By the proposed methodology, we can evaluate the vulnerability of patrol routes and thus recommend better strategies for the security of a building or a facility.

A Nonzero-Sum Search Game with Two Competitive Searchers and a Target

In this paper, we deal with a nonzero-sum three-person noncooperative search game, where two searchers compete for detection of a target and the target tries to evade the searchers. We verify that there occurs cooperation between two searchers against the target in the game with a stationary target and for a special case of the game with a moving target. Using these characteristics, we can partially regard the three-person nonzero-sum game as an equivalent two-person zero-sum game with the detection probability of target as a payoff. For a general game with a moving target, however, there could be many Nash equilibria. We propose a numerical algorithm for a Nash equilibrium in the general case. The discussion on the nonzero-sum search game in this paper could help us to step forward to a cooperative search game, where a coalition of some searchers and the rest of searchers compete against each other for detection of target as the future work.