More on the search for an infiltrator

We have asymptotically solved a discrete search game on an array of n ordered cells with two players: infiltrator (hider) and searcher, when the probability of survival approaches 1. The infiltrator wishes to reach the last cell in finite time, and the searcher has to defend that cell. When the players occupy the same cell, the searcher captures the infiltrator with probability 1 ? z. The payoff to the hider is the probability that the hider reaches the last cell without getting captured. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 1–14, 2002; DOI 10.1002/nav.1047     

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017     

The solution to an open problem for a caching game

In a caching game introduced by Alpern et al. (Alpern et al., Lecture notes in computer science (2010) 220–233) a Hider who can dig to a total fixed depth normalized to 1 buries a fixed number of objects among n discrete locations. A Searcher who can dig to a total depth of h searches the locations with the aim of finding all of the hidden objects. If he does so, he wins, otherwise the Hider wins. This zero‐sum game is complicated to analyze even for small values of its parameters, and for the case of 2 hidden objects has been completely solved only when the game is played in up to 3 locations. For some values of h the solution of the game with 2 objects hidden in 4 locations is known, but the solution in the remaining cases was an open question recently highlighted by Fokkink et al. (Fokkink et al., Search theory: A game theoretic perspective (2014) 85–104). Here we solve the remaining cases of the game with 2 objects hidden in 4 locations. We also give some more general results for the game, in particular using a geometrical argument to show that when there are 2 objects hidden in n locations and n→∞, the value of the game is asymptotically equal to h/n for h≥n/2. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 23–31, 2016     

An inspection game: Taking account of fulfillment probabilities of players' aims

This paper deals with an inspection game of customs and a smuggler. The customs can take two options of assigning a patrol or not. The smuggler has two strategies of shipping its cargo of contraband or not. Two players have several opportunities to take actions during a limited number of days. When both players do, there are some possibilities that the customs captures the smuggler and, simultaneously, the smuggler possibly makes a success of the smuggling. If the smuggler is captured or there remain no days for playing the game, the game ends. In this paper, we formulate the problem into a multi‐stage two‐person zero‐sum stochastic game and investigate some characteristics of the equilibrium solution, some of which are given in a closed form in a special case. There have been some studies so far on the inspection game. However, some consider the case that the smuggler has only one opportunity of smuggling or the perfect‐capture case that the customs can certainly arrest the smuggler on patrol, and others think of a recursive game without the probabilities of fulfilling the players' purposes. In this paper, we consider the inspection game taking account of the fulfillment probabilities of the players' aims. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Information security: Designing a stochastic‐network for throughput and reliability

Todas information and communication network requires a design that is secure to tampering. Traditional performance measures of reliability and throughput must be supplemented with measures of security. Recognition of an adversary who can inflict damage leads toward a game‐theoretic model. Through such a formulation, guidelines for network designs and improvements are derived. We opt for a design that is most robust to withstand both natural degradation and adversarial attacks. Extensive computational experience with such a model suggests that a Nash‐equilibrium design exists that can withstand the worst possible damage. Most important, the equilibrium is value‐free in that it is stable irrespective of the unit costs associated with reliability vs. capacity improvement and how one wishes to trade between throughput and reliability. This finding helps to pinpoint the most critical components in network design. From a policy standpoint, the model also allows the monetary value of information‐security to be imputed. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Exact algorithms and bounds for the dynamic assignment interdiction problem

We study a multi‐stage dynamic assignment interdiction (DAI) game in which two agents, a user and an attacker, compete in the underlying bipartite assignment graph. The user wishes to assign a set of tasks at the minimum cost, and the attacker seeks to interdict a subset of arcs to maximize the user's objective. The user assigns exactly one task per stage, and the assignment costs and interdiction impacts vary across stages. Before any stage commences in the game, the attacker can interdict arcs subject to a cardinality constraint. An interdicted arc can still be used by the user, but at an increased assignment cost. The goal is to find an optimal sequence of assignments, coupled with the attacker's optimal interdiction strategy. We prove that this problem is strongly NP‐hard, even when the attacker can interdict only one arc. We propose an exact exponential‐state dynamic‐programming algorithm for this problem as well as lower and upper bounds on the optimal objective function value. Our bounds are based on classical interdiction and robust optimization models, and on variations of the DAI game. We examine the efficiency of our algorithms and the quality of our bounds on a set of randomly generated instances. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 373–387, 2017     

Decentralized admission control of a queueing system: A game‐theoretic model

Consider a distributed system where many gatekeepers share a single server. Customers arrive at each gatekeeper according to independent Poisson processes with different rates. Upon arrival of a new customer, the gatekeeper has to decide whether to admit the customer by sending it to the server, or to block it. Blocking costs nothing. The gatekeeper receives a reward after a customer completes the service, and incurs a cost if an admitted customer finds a busy server and therefore has to leave the system. Assuming an exponential service distribution, we formulate the problem as an n‐person non‐zero‐sum game in which each gatekeeper is interested in maximizing its own long‐run average reward. The key result is that each gatekeeper's optimal policy is that of a threshold type regardless what other gatekeepers do. We then derive Nash equilibria and discuss interesting insights. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 702–718, 2003.     

A search game taking account of attributes of searching resources

This article deals with a two‐person zero‐sum game called a search allocation game (SAG), in which a searcher and a target participate as players. The searcher distributes his searching resources in a search space to detect the target. The effect of resources lasts a certain period of time and extends to some areas at a distance from the resources' dropped points. On the other hand, the target moves around in the search space to evade the searcher. In the history of search games, there has been little research covering the durability and reachability of searching resources. This article proposes two linear programming formulations to solve the SAG with durable and reachable resources, and at the same time provide an optimal strategy of distributing searching resources for the searcher and an optimal moving strategy for the target. Using examples, we will analyze the influences of two attributes of resources on optimal strategies. © 2007 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Timely exposure of a secret project: Which activities to monitor?

A defender wants to detect as quickly as possible whether some attacker is secretly conducting a project that could harm the defender. Security services, for example, need to expose a terrorist plot in time to prevent it. The attacker, in turn, schedules his activities so as to remain undiscovered as long as possible. One pressing question for the defender is: which of the project's activities to focus intelligence efforts on? We model the situation as a zero‐sum game, establish that a late‐start schedule defines a dominant attacker strategy, and describe a dynamic program that yields a Nash equilibrium for the zero‐sum game. Through an innovative use of cooperative game theory, we measure the harm reduction thanks to each activity's intelligence effort, obtain insight into what makes intelligence effort more effective, and show how to identify opportunities for further harm reduction. We use a detailed example of a nuclear weapons development project to demonstrate how a careful trade‐off between time and ease of detection can reduce the harm significantly.     

Technical note: Find a hidden “treasure”

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two‐sided allocation non‐zero‐sum game on an integer interval [1,n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i ∈ [1,n] is characterized by a detection parameter λ_i (μ_i) for Player 1 (Player 2) such that p_i(1 ? exp(?λ_ix_i)) (p_i(1 ? exp(?μ_iy_i))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x_i (y_i) applied at point i where p_i ∈ (0,1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player 1 (Player 2) is 1 if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q₁ and Player 2 gets q₂ where q₁ + q₂ ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Probabilistic resource pooling games

We study a setting with a single type of resource and with several players, each associated with a single resource (of this type). Unavailability of these resources comes unexpectedly and with player‐specific costs. Players can cooperate by reallocating the available resources to the ones that need the resources most and let those who suffer the least absorb all the costs. We address the cost savings allocation problem with concepts of cooperative game theory. In particular, we formulate a probabilistic resource pooling game and study them on various properties. We show that these games are not necessarily convex, do have non‐empty cores, and are totally balanced. The latter two are shown via an interesting relationship with Böhm‐Bawerk horse market games. Next, we present an intuitive class of allocation rules for which the resulting allocations are core members and study an allocation rule within this class of allocation rules with an appealing fairness property. Finally, we show that our results can be applied to a spare parts pooling situation.     

Optimal patrol to uncover threats in time when detection is imperfect

Consider a patrol problem, where a patroller traverses a graph through edges to detect potential attacks at nodes. An attack takes a random amount of time to complete. The patroller takes one time unit to move to and inspect an adjacent node, and will detect an ongoing attack with some probability. If an attack completes before it is detected, a cost is incurred. The attack time distribution, the cost due to a successful attack, and the detection probability all depend on the attack node. The patroller seeks a patrol policy that minimizes the expected cost incurred when, and if, an attack eventually happens. We consider two cases. A random attacker chooses where to attack according to predetermined probabilities, while a strategic attacker chooses where to attack to incur the maximal expected cost. In each case, computing the optimal solution, although possible, quickly becomes intractable for problems of practical sizes. Our main contribution is to develop efficient index policies—based on Lagrangian relaxation methodology, and also on approximate dynamic programming—which typically achieve within 1% of optimality with computation time orders of magnitude less than what is required to compute the optimal policy for problems of practical sizes. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 557–576, 2014     

A multi‐objective model for bank ATM networks

In this paper we present an application of the core solution concepts for multi‐objective games to a bank ATM network model. In these games, the worth of a coalition is given by a subset of vectors of the k‐dimensional space rather than by a scalar. The paper investigates how an ATM network model based on multi‐objective cooperative game theory could be used as an alternative way of setting interchange fees paid by the customer's bank to the one that owns the ATM. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Network disconnection problems in a centralized network

We consider three network disconnection problems in a centralized network where a source node provides service to the other nodes, called demand nodes. In network disconnection problems, each demand node gets a certain benefit when connected to a source node and a network attacker destroys edges to prevent demand nodes from achieving benefits. As destroying edges incurs expenses, an attacker considers the following three different strategies. The first is to maximize the sum of benefits of the disconnected nodes while keeping the total edge destruction cost no more than a given budget. The second is to minimize the total destruction cost needed to make a certain amount of benefits not accomplished. The last is to minimize the ratio of the total destruction cost to the benefits not accomplished. In this paper, we develop exact algorithms to solve the above three problems. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A three‐stage procurement optimization problem under uncertainty

This article examines a problem faced by a firm procuring a material input or good from a set of suppliers. The cost to procure the material from any given supplier is concave in the amount ordered from the supplier, up to a supplier‐specific capacity limit. This NP‐hard problem is further complicated by the observation that capacities are often uncertain in practice, due for instance to production shortages at the suppliers, or competition from other firms. We accommodate this uncertainty in a worst‐case (robust) fashion by modeling an adversarial entity (which we call the “follower”) with a limited procurement budget. The follower reduces supplier capacity to maximize the minimum cost required for our firm to procure its required goods. To guard against uncertainty, the firm can “protect” any supplier at a cost (e.g., by signing a contract with the supplier that guarantees supply availability, or investing in machine upgrades that guarantee the supplier's ability to produce goods at a desired level), ensuring that the anticipated capacity of that supplier will indeed be available. The problem we consider is thus a three‐stage game in which the firm first chooses which suppliers' capacities to protect, the follower acts next to reduce capacity from unprotected suppliers, and the firm then satisfies its demand using the remaining capacity. We formulate a three‐stage mixed‐integer program that is well‐suited to decomposition techniques and develop an effective cutting‐plane algorithm for its solution. The corresponding algorithmic approach solves a sequence of scaled and relaxed problem instances, which enables solving problems having much larger data values when compared to standard techniques. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

New results on a stochastic duel game with each force consisting of heterogeneous units

Two forces engage in a duel, with each force initially consisting of several heterogeneous units. Each unit can be assigned to fire at any opposing unit, but the kill rate depends on the assignment. As the duel proceeds, each force—knowing which units are still alive in real time—decides dynamically how to assign its fire, in order to maximize the probability of wiping out the opposing force before getting wiped out. It has been shown in the literature that an optimal pure strategy exists for this two‐person zero‐sum game, but computing the optimal strategy remained cumbersome because of the game's huge payoff matrix. This article gives an iterative algorithm to compute the optimal strategy without having to enumerate the entire payoff matrix, and offers some insights into the special case, where one force has only one unit. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 56–65, 2014     

Flow shop scheduling with earliness,tardiness, and intermediate inventory holding costs

We consider the problem of scheduling customer orders in a flow shop with the objective of minimizing the sum of tardiness, earliness (finished goods inventory holding), and intermediate (work‐in‐process) inventory holding costs. We formulate this problem as an integer program, and based on approximate solutions to two different, but closely related, Dantzig‐Wolfe reformulations, we develop heuristics to minimize the total cost. We exploit the duality between Dantzig‐Wolfe reformulation and Lagrangian relaxation to enhance our heuristics. This combined approach enables us to develop two different lower bounds on the optimal integer solution, together with intuitive approaches for obtaining near‐optimal feasible integer solutions. To the best of our knowledge, this is the first paper that applies column generation to a scheduling problem with different types of strongly ????‐hard pricing problems which are solved heuristically. The computational study demonstrates that our algorithms have a significant speed advantage over alternate methods, yield good lower bounds, and generate near‐optimal feasible integer solutions for problem instances with many machines and a realistically large number of jobs. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Inventory pooling games for expensive,low‐demand spare parts

We consider several independent decision makers who stock expensive, low‐demand spare parts for their high‐tech machines. They can collaborate by full pooling of their inventories via free transshipments. We examine the stability of such pooling arrangements, and we address the issue of fairly distributing the collective holding and downtime costs over the participants, by applying concepts from cooperative game theory. We consider two settings: one where each party maintains a predetermined stocking level and one where base stock levels are optimized. For the setting with fixed stocking levels, we unravel the possibly conflicting effects of implementing a full pooling arrangement and study these effects separately to establish intuitive conditions for existence of a stable cost allocation. For the setting with optimized stocking levels, we provide a simple proportional rule that accomplishes a population monotonic allocation scheme if downtime costs are symmetric among participants. Although our whole analysis is motivated by spare parts applications, all results are also applicable to other pooled resource systems of which the steady‐state behavior is equivalent to that of an Erlang loss system. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Base‐stock policies in capacitated assembly systems: Convexity properties

We study an assembly system with a single finished product managed using an echelon base‐stock or order‐up‐to policy. Some or all operations have capacity constraints. Excess demand is either backordered in every period or lost in every period. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base‐stock levels and capacity levels. When the holding costs are also included in the objective function, we show that the cost function can be written as a sum of a convex function and a concave function. Throughout the article, we discuss algorithmic implications of our results for making optimal inventory and capacity decisions in such systems.© 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

To catch an intruder: Part A—uncluttered scenario

We analyze an interdiction scenario where an interceptor attempts to catch an intruder as the intruder moves through the area of interest. A motivating example is the detection and interdiction of drug smuggling vessels in the Eastern Pacific and Caribbean. We study two models in this article. The first considers a nonstrategic target that moves through the area without taking evasive action to avoid the interdictor. We determine the optimal location the interceptor should position itself to best respond when a target arrives. The second model analyzes the strategic interaction between the interceptor and intruder using a Blotto approach. The intruder chooses a route to travel on and the interceptor chooses a route to patrol. We model the interaction as a two‐player game with a bilinear payoff function. We compute the optimal strategy for both players and examine several extensions. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 29–40, 2017