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.     

A price leadership method for solving the inspector's non-constant-sum game

An inspector's game is a non-constant-sum two-person game in which one player has promised to perform a certain duty and the other player is allowed to inspect and verify occasionally that the duty has indeed been performed. A solution to a variant of such a game is given in this paper, based on the assumption that the inspector can announce his mixed strategy in advance, if he so wishes, whereas the other player, who has already given his promise, cannot threaten by explicitly saying that he will not keep his word.     

Estimation of strategies in a markov game

In this paper a two-person Markov game, in discrete time, and with perfect state information, is considered from the point of view of a single player (player A) only. It is assumed that A's opponent (player B) uses the same strategy every time the game is played. It is shown that A can obtain a consistent estimate of B's strategy on the basis of his past experience of playing the game with B. Two methods of deriving such an estimate are given. Further, it is shown that using one of these estimates A can construct a strategy for himself which is asymptotically optimal. A simple example of a game in which the above method may be useful is given.     

A transportation problem with competition

The classic transportation problem can be generalized with many carriers and one owner. From the formulation the competition in sense of game theory naturally appears. Here we present and solve this problem using a generalized n-person game. Besides the same composition properties about solutions and regarding zones, related results are considered. Finally, the problem in which there is a modification of the set of destinations assigned to the carriers is also studied.     

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     

Interchange fees for bank ATM networks

Banks have found it advantageous to connect their Automated Teller Machines (ATMs) in networks so that customers of one bank may use the ATMs of any bank in the network. When this occurs, an interchange fee is paid by the customer's bank to the one that owns the ATM. These have been set by historic interbank negotiation. The paper investigates how a model based on n-player game theory concepts of Shapley value and nucleolus could be used as an alternative way of setting such fees. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 407–417, 1998     

On some stocxastic tactical antisubmarine games

Mathematical models of tactical problems in Hntisubmarine Warfare (ASW) are developed. Specifically, a game of pursuit between a hunter-killer force. player 1, and a possible submarine, player 2 is considered. The game consists of a sequence of moves and terminates when player 2 is tcaught or evades player 1. When the players move they observe the actual tactical configuration of the forces (state) and each player choosa-s a tactical plan from a finite collection. This joint choice of tactical plans determines an immediate payoff and a transition probability distribution over the states. Hence an expected payoff function is defined, Formally this game is a Terminating Stochastic Game (TSG). Shapley demonstrated the existence of a value and optimal strategies (solution), An iterative technique to approximate the solution to within desired accuracy is proposed. Each iteration of the technique is obtained by solving a set of linear programs. To introduce more realism into the game several variations of the TSG are also considered. One variation is a finite TSG and linear programming techniques are employed to find the solution.     

A duel complicated by false targets and uncertainty as to opponent type

An attacker, being one of two types, initiates an attack at some time in the interval [-T, 0]. The a priori probabilities of each type are known. As time elapses the defender encounters false targets which occur according to a known Poisson process and which can be properly classified with known probability. The detection and classification probabilities for each type attacker are given. If the defender responds with a weapon at the time of attack, he survives with a probability which depends on the number of weapons in his possession and on attacker type. If he does not respond, his survival probability is smaller. These probabilities are known, as well as the current number of weapons in the defender's possession. They decrease as the number of weapons decreases. The payoff is the defender's survival probability. An iterative system of first-order differential equations is derived whose unique solution V₁(t),V₂(t),…,V_k(t) is shown to be the value of the game at time t, when the defender has 1, 2,…, k,… weapons, respectively. The optimal strategies are determined. Limiting results are obtained as t→-∞, while the ratio of the number of weapons to the expected number of false targets remaining is held constant.     

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     

Continuous accumulation games on discrete locations

In an accumulation game, a HIDER attempts to accumulate a certain number of objects or a certain quantity of material before a certain time, and a SEEKER attempts to prevent this. In a continuous accumulation game the HIDER can pile material either at locations $1, 2, …, n, or over a region in space. The HIDER will win (payoff 1) it if accumulates N units of material before a given time, and the goal of the SEEKER will win (payoff 0) otherwise. We assume the HIDER can place continuous material such as fuel at discrete locations i = 1, 2, …, n, and the game is played in discrete time. At each time k > 0 the HIDER acquires h units of material and can distribute it among all of the locations. At the same time, k, the SEEKER can search a certain number s < n of the locations, and will confiscate (or destroy) all material found. After explicitly describing what we mean by a continuous accumulation game on discrete locations, we prove a theorem that gives a condition under which the HIDER can always win by using a uniform distribution at each stage of the game. When this condition does not hold, special cases and examples show that the resulting game becomes complicated even when played only for a single stage. We reduce the single stage game to an optimization problem, and also obtain some partial results on its solution. We also consider accumulation games where the locations are arranged in either a circle or in a line segment and the SEEKER must search a series of adjacent locations. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 60–77, 2002; DOI 10.1002/nav.1048     

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     

Asymptotic joint normality of an increasing number of multivariate order statistics and associated cell frequencies

Take n independent identically distributed (IID) observations from a continuous r-variate population, and choose some order statistics from each of the r variates. These order statistics are used to construct a grid in r-dimensional space. Under certain conditions, it is shown that as n increases we can choose an increasing number of order statistics in such a way that the asymptotic joint distribution of the chosen order statistics and of the frequencies of sample points falling in the cells of the grid can be assumed to be a normal distribution. An application to testing independence of random variables is given.     

Game theoretic analysis of maximum cooperative purchasing situations

This article introduces maximum cooperative purchasing (MCP)‐situations, a new class of cooperative purchasing situations. Next, an explicit alternative mathematical characterization of the nucleolus of cooperative games is provided. The allocation of possible cost savings in MCP‐situations, in which the unit price depends on the largest order quantity within a group of players, is analyzed by defining corresponding cooperative MCP‐games. We show that a decreasing unit price is a sufficient condition for a nonempty core: there is a set of marginal vectors that belong to the core. The nucleolus of an MCP‐game can be derived in polynomial time from one of these marginal vectors. To show this result, we use the new mathematical characterization for the nucleolus for cooperative games. Using the decomposition of an MCP‐game into unanimity games, we find an explicit expression for the Shapley value. Finally, the behavior of the solution concepts is compared numerically. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 607–624, 2013     

Analytic solution for the nucleolus of a three‐player cooperative game

The nucleolus solution for cooperative games in characteristic function form is usually computed numerically by solving a sequence of linear programing (LP) problems, or by solving a single, but very large‐scale, LP problem. This article proposes an algebraic method to compute the nucleolus solution analytically (i.e., in closed‐form) for a three‐player cooperative game in characteristic function form. We first consider cooperative games with empty core and derive a formula to compute the nucleolus solution. Next, we examine cooperative games with nonempty core and calculate the nucleolus solution analytically for five possible cases arising from the relationship among the value functions of different coalitions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

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     

Assembly of systems having maximum reliability

The first problem considered in this paper is concerned with the assembly of independent components into parallel systems so as to maximize the expected number of systems that perform satisfactorily. Associated with each component is a probability of it performing successfully. It is shown that an optimal assembly is obtained if the reliability of each assembled system can be made equal. If such equality is not attainable, then bounds are given so that the maximum expected number of systems that perform satisfactorily will lie within these stated bounds; the bounds being a function of an arbitrarily chosen assembly. An improvement algorithm is also presented. A second problem treated is concerned with the optimal design of a system. Instead of assembling given units, there is an opportunity to “control” their quality, i.e., the manufacturer is able to fix the probability, p, of a unit performing successfully. However, his resources, are limited so that a constraint is imposed on these probabilities. For (1) series systems, (2) parallel systems, and (3) k out of n systems, results are obtained for finding the optimal p's which maximize the reliability of a single system, and which maximize the expected number of systems that perform satisfactorily out of a total assembly of J systems.     

Algorithms for minimizing total cost,bottleneck time and bottleneck shipment in transportation problems

We consider the transportation problem of determining nonnegative shipments from a set of m warehouses with given availabilities to a set of n markets with given requirements. Three objectives are defined for each solution: (i) total cost, TC, (ii) bottleneck time, BT (i.e., maximum transportation time for a positive shipment), and (iii) bottleneck shipment, SB (i.e., total shipment over routes with bottleneck time). An algorithm is given for determining all efficient (pareto-optimal or nondominated) (TC, BT) solution pairs. The special case of this algorithm when all the unit cost coefficients are zero is shown to be the same as the algorithms for minimizing BT. provided by Szwarc and Hammer. This algorithm for minimizing BT is shown to be computationally superior. Transportation or assignment problems with m=n=100 average about a second on the UNIVAC 1108 computer (FORTRAN V)) to the threshold algorithm for minimizing BT. The algorithm is then extended to provide not only all the efficient (TC, BT) solution pairs but also, for each such BT, all the efficient (TC, SB) solution pairs. The algorithms are based on the cost operator theory of parametric programming for the transportation problem developed by the authors.     

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 with a protector

A classic problem in Search Theory is one in which a searcher allocates resources to the points of the integer interval [1, n] in an attempt to find an object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We model the situation as a two‐person non‐zero‐sum game so that we can take into account the fact that using resources can be costly. It is shown that this game has a unique Nash equilibrium when the searcher's probability of finding an object located at point i is of the form (1 − exp (−λ_ix_i)) exp (−μ_iy_i) when the searcher and protector allocate resources x_i and y_i respectively to point i. An algorithm to find this Nash equilibrium is given. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47:85–96, 2000     

Capacity games for partially complementary products under multivariate random demands

In this study, we consider n firms, each of which produces and sells a different product. The n firms face a common demand stream which requests all their products as a complete set. In addition to the common demand stream, each firm also faces a dedicated demand stream which requires only its own product. The common and dedicated demands are uncertain and follow a general, joint, continuous distribution. Before the demands are realized, each firm needs to determine its capacity or production quantity to maximize its own expected profit. We formulate the problem as a noncooperative game. The sales price per unit for the common demand could be higher or lower than the unit price for the dedicated demand, which affects the firm's inventory rationing policy. Hence, the outcome of the game varies. All of the prices are first assumed to be exogenous. We characterize Nash equilibrium(s) of the game. At the end of the article, we also provide some results for the endogenous pricing. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 59: 146–159, 2012