Modified fictitious play

We describe a modification of Brown's fictitious play method for solving matrix (zero-sum two-person) games and apply it to both symmetric and general games. If the original game is not symmetric, the basic idea is to transform the given matrix game into an equivalent symmetric game (a game with a skew-symmetric matrix) and use the solution properties of symmetric games (the game value is zero and both players have the same optimal strategies). The fictitious play method is then applied to the enlarged skew-symmetric matrix with a modification that calls for the periodic restarting of the process. At restart, both players' strategies are made equal based on the following considerations: Select the maximizing or minimizing player's strategy that has a game value closest to zero. We show for both symmetric and general games, and for problems of varying sizes, that the modified fictitious play (MFP) procedure approximates the value of the game and optimal strategies in a greatly reduced number of iterations and in less computational time when compared to Brown's regular fictitious play (RFP) method. For example, for a randomly generated 50% dense skew-symmetric 100 × 100 matrix (symmetric game), with coefficients |a_ij| ≤ 100, it took RFP 2,652,227 iterations to reach a gap of 0.03118 between the lower and upper bounds for the game value in 70.71 s, whereas it took MFP 50,000 iterations to reach a gap of 0.03116 in 1.70 s. Improved results were also obtained for general games in which the MFP solves a much larger equivalent symmetric game. © 1996 John Wiley & Sons, Inc.     

Finding saddle points on polyhedra: Solving certain continuous minimax problems

This article reviews procedures for computing saddle points of certain continuous concave-convex functions defined on polyhedra and investigates how certain parameters and payoff functions influence equilibrium solutions. The discussion centers on two widely studied applications: missile defense and market-share attraction games. In both settings, each player allocates a limited resource, called effort, among a finite number of alternatives. Equilibrium solutions to these two-person games are particularly easy to compute under a proportional effectiveness hypothesis, either in closed form or in a finite number of steps. One of the more interesting qualitative properties we establish is the identification of conditions under which the maximizing player can ignore the values of the alternatives in determining allocation decisions. © 1996 John Wiley & Sons, Inc.     

Cumulative search-evasion games

Cumulative search-evasion games (CSEGs) are two-person zero-sum search-evasion games where play proceeds throughout some specified period without interim feedback to either of the two players. Each player moves according to a preselected plan. If (X_t, Y_t,) are the positions of the two players at time t, then the game's payoff is the sum over t from 1 to T of A(X_t, Y_t, t). Additionally, all paths must be “connected.” That is, the finite set of positions available for a player in any time period depends on the position selected by that player in the previous time period. One player attempts to select a mixed strategy over the feasible T-time period paths to maximize the expected payoff. The other minimizes. Two solution procedures are given. One uses the Brown-Robinson method of fictitious play and the other linear programming. An example problem is solved using both procedures.     

Counterexamples to optimal permutation schedules for certain flow shop problems

It is well known that a minimal makespan permutation sequence exists for the n × 3 flow shop problem and for the n × m flow shop problem with no inprocess waiting when processing times for both types of problems are positive. It is shown in this paper that when the assumption of positive processing times is relaxed to include nonnegative processing times, optimality of permutation schedules cannot be guaranteed.     

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.     

Reliability of a 2‐dimensional k‐within‐consecutive‐r × s‐out‐of‐m × n:F system

A 2‐dimensional rectangular (cylindrical) k‐within‐consecutive‐r × s‐out‐of‐m × n:F system is the rectangular (cylindrical) m × n‐system if the system fails whenever k components in a r × s‐submatrix fail. This paper proposes a recursive algorithm for the reliability of the 2‐dimensional k‐within‐consecutive‐r × s‐out‐m × n:F system, in the rectangular case and the cylindrical case. This algorithm requires min ( O (mk^r(n?s)), O (nk^s(m?r))), and O (mk^rn) computing time in the rectangular case and the cylindrical case, respectively. The proposed algorithm will be demonstrated and some numerical examples will be shown. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 625–637, 2001.     

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r,s)‐out‐of‐(m,n):F system

A 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system consists of m × n components, and fails if and only if k or more components fail in an r × s submatrix. This system can be treated as a reliability model for TFT liquid crystal displays, wireless communication networks, etc. Although an effective method has been developed for evaluating the exact system reliability of small or medium‐sized systems, that method needs extremely high computing time and memory capacity when applied to larger systems. Therefore, developing upper and lower bounds and accurate approximations for system reliability is useful for large systems. In this paper, first, we propose new upper and lower bounds for the reliability of a 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system. Secondly, we propose two limit theorems for that system. With these theorems we can obtain accurate approximations for system reliabilities when the system is large and component reliabilities are close to one. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A family of games on 1,∞)2 with payoff a function of y/x

A two-parameter class of games on 1,∞)² is studied. The games may be regarded as analogs of Silverman games, having continuous payoff function in place of a step function of y/x. This change is motivated by a desire to move toward a model for competitive situations where the penalty for overspending increases with the amount of overspending. There are some similarities to games with bell-shaped kernel. For most of the region considered in the plane of the two parameters there are solutions of finite type, which are obtained explicitly. There are, however, pockets in this plane where no optimal strategies have been found and possibly where none of finite type exist.     

Exploiting self‐canceling demand point aggregation error for some planar rectilinear median location problems

When solving location problems in practice it is quite common to aggregate demand points into centroids. Solving a location problem with aggregated demand data is computationally easier, but the aggregation process introduces error. We develop theory and algorithms for certain types of centroid aggregations for rectilinear 1‐median problems. The objective is to construct an aggregation that minimizes the maximum aggregation error. We focus on row‐column aggregations, and make use of aggregation results for 1‐median problems on the line to do aggregation for 1‐median problems in the plane. The aggregations developed for the 1‐median problem are then used to construct approximate n‐median problems. We test the theory computationally on n‐median problems (n ≥ 1) using both randomly generated, as well as real, data. Every error measure we consider can be well approximated by some power function in the number of aggregate demand points. Each such function exhibits decreasing returns to scale. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 614–637, 2003.     

Characterization of the nucleolus for a class of n-person games

An inductive procedure is given for finding the nucleolus of an n-person game in which all coalitions with less than n-1 players are totally defeated. It is shown that, for such a game, one of three things may occur: (a) all players receive the same amount; (b) each player receives his quota, plus a certain constant (which may be positive, nerative, or zero); (c) the weakest player receives one half his quota, and the other players divide the remaining profit according to the nucleolus of a similar (n-1)-person game. It is also shown that the nucleolus of such a game yields directly the nucleolus of each derived game. An example is worked out in detail.     

Mathematical aspects of the 3 × n job-shop sequencing problem

The paper discusses mathematical properties of the well-known Bellman-Johnson 3 × n sequencing problem. Optimal rules for some special cases are developed. For the case min B_i ≥ maxA_j we find an optimal sequence of the 2 × n problem for machines B and C and move one item to the front of the sequence to minimize (7); when min B_i ≥ max C_j we solve a 2 × n problem for machines A and B and move one item to the end of the optimal sequence so as to minimize (9). There is also given a sufficient optimality condition for a solution obtained by Johnson's approximate method. This explains why this method so often produces an optimal solution.     

Adaptive competitive decision in repeated play of a matrix game with uncertain entries

This study is concerned with a game model involving repeated play of a matrix game with unknown entries; it is a two-person, zero-sum, infinite game of perfect recall. The entries of the matrix ((pij)) are selected according to a joint probability distribution known by both players and this unknown matrix is played repeatedly. If the pure strategy pair (i, j) is employed on day k, k = 1, 2, …, the maximizing player receives a discounted income of β^{k - 1} X_ij, where β is a constant, 0 ≤ β ? 1, and X_ij assumes the value one with probability p_ij or the value zero with probability 1 - p_ij. After each trial, the players are informed of the triple (i, j, X_ij) and retain this knowledge. The payoff to the maximizing player is the expected total discounted income. It is shown that a solution exists, the value being characterized as the unique solution of a functional equation and optimal strategies consisting of locally optimal play in an auxiliary matrix determined by the past history. A definition of an ?-learning strategy pair is formulated and a theorem obtained exhibiting ?-optimal strategies which are ?-learning. The asymptotic behavior of the value is obtained as the discount tends to one.     

On the number of games played in a best of (2n – 1) series

Let p(⩾0.5) denote the probability that team A beats B in a single game. The series continues until either A or B wins n games. Assuming that these games are independent replications, we study some features of the distribution of X_n, the number of games played in the series. It is shown that X_n is unimodal, has an IFRA distribution, and is stochastically decreasing in p. Close approximations to its mode, mean, and variance are given. Finally, it is shown that the maximum-likelihood estimator of p based on X_n is unique.     

New results on a Ruckle problem in discrete games of ambush

This article deals with a two‐person zero‐sum game in which player I chooses in integer interval [1, N] two integer intervals consisting of p and q points where p + q < N, and player II chooses an integer point in [1, N]. The payoff to player I equals 1 if the point chosen by player II is at least in one of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, Lee, Garnaev, and Zoroa, Zoroa and Fernández‐Sáez. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 98–106, 2001     

Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations

A Markovian arrival process of order n, MAP(n), is typically described by two n × n transition rate matrices in terms of rate parameters. While it is straightforward and intuitive, the Markovian representation is redundant since the minimal number of parameters is n² for non‐redundant MAP(n). It is well known that the redundancy complicates exact moment fittings. In this article, we present a minimal and unique Laplace‐Stieltjes transform (LST) representations for MAP(n)s. Even though the LST coefficients vector itself is not a minimal representation, we show that the joint LST of stationary intervals can be represented with the minimum number of parameters. We also propose another minimal representation for MAP(3)s based on coefficients of the characteristic polynomial equations of the two transition rate matrices. An exact moment fitting procedure is presented for MAP(3)s based on two proposed minimal representations. We also discuss how MAP(3)/G/1 departure process can be approximated as a MAP(3). A simple tandem queueing network example is presented to show that the MAP(3) performs better than the MAP(2) in queueing approximations especially under moderate traffic intensities. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 549–561, 2016     

The advantage of deeper pockets in Silverman's game on intervals

Silverman's game on (1, B) × (1, B) was analyzed by R. J. Evans, who showed that optimal strategies exist (and found them) only on a set of measure zero in the parameter plane. We examine the corresponding game on (1, B) × (1, D) with D > B, and show that optimal strategies exist in about half the parameter plane. Optimal strategies and game value are obtained explicitly. © 1995 John Wiley & Sons, Inc.     

Localization of optimal strategies in certain games

Assume the payoffs of a matrix game are concave in the index of the maximizing player. That player is shown to have an optimal strategy which uses at most two consecutive pure strategies, identifiable through approximate solution of a related continuous game. Generalizations are given, and the results are applied to a motivating hidden-target model due to Shapley. © 1994 John Wiley & Sons, Inc.     

Solving partially observable agent‐intruder games with an application to border security problems

In this paper, we introduce partially observable agent‐intruder games (POAIGs). These games model dynamic search games on graphs between security forces (an agent) and an intruder given possible (border) entry points and high value assets that require protection. The agent faces situations with dynamically changing, partially observable information about the state of the intruder and vice versa. The agent may place sensors at selected locations, while the intruder may recruit partners to observe the agent's movement. We formulate the problem as a two‐person zero‐sum game, and develop efficient algorithms to compute each player's optimal strategy. The solution to the game will help the agent choose sensor locations and design patrol routes that can handle imperfect information. First, we prove the existence of ?‐optimal strategies for POAIGs with an infinite time horizon. Second, we introduce a Bayesian approximation algorithm to identify these ?‐optimal strategies using belief functions that incorporate the imperfect information that becomes available during the game. For the solutions of large POAIGs with a finite time horizon, we use a solution method common to extensive form games, namely, the sequence form representation. To illustrate the POAIGs, we present several examples and numerical results.     

A strategic model for state compliance verification

The theory of directed graphs and noncooperative games is applied to the problem of verification of State compliance to international treaties on arms control, disarmament and nonproliferation of weapons of mass destruction. Hypothetical treaty violations are formulated in terms of illegal acquisition paths for the accumulation of clandestine weapons, weapons‐grade materials or some other military capability. The paths constitute the illegal strategies of a sovereign State in a two‐person inspection game played against a multi‐ or international Inspectorate charged with compliance verification. The effectiveness of existing or postulated verification measures is quantified in terms of the Inspectorate's expected utility at Nash equilibrium. A prototype software implementation of the methodology and a case study are presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 260–271, 2016     

Application of disjunctive programming to the linear complementarity problem

In this article we address the question of developing deep cuts for disjunctive programs using rectilinear distance measures. The method is applied to linear complementarity problems where the matrix M need not be copositive plus. Some modifications that are needed as a computational expediency are discussed. The computation results for matrix M of size up to 30 × 30 are discussed.