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     

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     

Technical note: Characterization of binomial semivalues through delegation games

Semivalues are allocation rules for cooperative games that assign to each player in a given game a weighted sum of his marginal contributions to all coalitions he belongs to, where the weighting coefficients depend only on the coalition size. Binomial semivalues are a special class of semivalues whose weighting coefficients are obtained by means of a unique parameter. In particular, the Banzhaf value is a binomial semivalue. In this article, we provide an axiomatic characterization for each binomial semivalue. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Interchange fee rate,merchant discount rate,and retail price in a credit card network: A game‐theoretic analysis

We consider two game‐theoretic settings to determine the optimal values of an issuer's interchange fee rate, an acquirer's merchant discount rate, and a merchant's retail price in a credit card network. In the first setting, we investigate a two‐stage game problem in which the issuer and the acquirer first negotiate the interchange fee rate, and the acquirer and the retailer then determine their merchant discount rate and retail price, respectively. In the second setting, motivated by the recent US bill “H.R. 2695,” we develop a three‐player cooperative game in which the issuer, the acquirer, and the merchant form a grand coalition and bargain over the interchange fee rate and the merchant discount rate. Following the cooperative game, the retailer makes its retail pricing decision. We derive both the Shapley value‐ and the nucleolus‐characterized, and globally‐optimal unique rates for the grand coalition. Comparing the two game settings, we find that the participation of the merchant in the negotiation process can result in the reduction of both rates. Moreover, the stability of the grand coalition in the cooperative game setting may require that the merchant should delegate the credit card business only to the issuer and the acquirer with sufficiently low operation costs. We also show that the grand coalition is more likely to be stable and the U.S. bill “H.R. 2695” is thus more effective, if the degree of division of labor in the credit card network is higher as the merchant, acquirer, and issuer are more specialized in the retailing, acquiring, and issuing operations, respectively. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

A cooperative game in search theory

Search theory originates from the military research efforts of WWII. Most researchers of that period modeled their search games in noncooperative games, where players are enemies or compete against each other. In this article, we deal with a cooperative search game, where multiple searchers behave cooperatively. First we describe several search problems and discuss the possibility of a coalition or cooperation among searchers. For the cooperative search game, we define a function named quasi‐characteristic function, which gives us a criterion similar to the so‐called characteristic function in the general coalition game with transferable utility. The search operation includes a kind of randomness with respect to whether the searchers can detect a target and get the value of the target. We also propose a methodology to divide the obtained target value among members of the coalition taking account of the randomness. As a concrete problem of the cooperative search game, we take the so‐called search allocation game, where searchers distribute their searching resources to detect a target in a cooperative way and the target moves in a search space to evade the searchers. Lastly, we discuss the core of the cooperative search allocation game. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

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.     

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.     

One-person games of economic survival

This paper discusses the one-person economic survival game model with a discrete probability distribution for the contribution to surplus variable. The general game model and strategies in these games are examined, and necessary conditions which an undominated, stationary strategy must satisfy are obtained. For a special class of these games a mathematical formulation of the value of the game is given, and examples and theorems which relate to undominated strategies in this class are presented. This paper, in some sense, is a sequel to a portion of a paper by Shubik and Thompson [7] which appeared in this journal.     

A note on values and multilinear extensions

An axiomatic formulation is given of a class of values for cooperative games. This class includes the Shapley value and the Banzhaf index, and is related to the multilinear extension of a game.     

Relaxed sequencing games have a nonempty core

We study sequencing situations with a fixed initial order and linear cost functions. Cost savings can be obtained by rearranging jobs. Next to finding an optimal order, an additional issue is formed by the division of these savings. Cooperative game theory studies this issue. A common assumption states that cooperation between players is restricted to groups that are connected according to the initial order. The value of disconnected groups is defined additively over their connected components. In this paper we allow players in disconnected coalitions to switch places as long as they do not hurt the players not in the coalition under consideration. The resulting games are called relaxed sequencing games. Although they have been studied before, no general results on stable profit divisions have been derived so far. In this paper we prove that relaxed sequencing games have a nonempty core, i.e., they all have stable profit divisions. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Multilinear extensions and the banzhaf value

It is shown that the Banzhaf value can be obtained by differentiating the multilinear extension of a game at the midpoint of the unit cube. This gives us a composition theorem for the value of compound games. As an example, the values of the electoral college and presidential election “games” are approximated by the method of extensions.     

Revisiting inspection game and inspector leadership through reaction networks

The inspection game is a two-player noncooperative game that models a situation where an inspector verifies whether the inspectee complies with the rules (on the assumption that the inspectee has the tendency to violate at least one of the rules). The usual approach in the analysis of this game seeks to find an optimal strategic inspection scheme for each of the two players yielding favorable payoffs. Recently, there have been some developments in the study of such games that use a mathematical structure known as reaction network involving a set of molecular species and the existing reactions among these species. In this paper, we use a reaction network to analyze the inspection game giving an alternative way of modeling the social situation. The molecular species play the role of the players' decision moves and their resulting gain or loss, while the reactions are the encounters of the decisions of the players which, as expected, yield payoffs. We reexamine the dynamics of the inspection game through the lens of reaction network theory and consider various situations that call for more detailed analyses such as equal or unequal reaction rates and inspection leadership. Conditions concerning reaction rates, initial population of decision species, benefits, and costs are determined in order to identify strategies that yield better payoffs both for the inspector and inspectee. These results illustrate practical insights rooted from the formulated simple game models.     

不同游戏方式下的暴力网络游戏对大学生攻击性的影响

目的:探讨不同游戏方式下的暴力网络游戏对不同攻击性特质大学生攻击性的影响。方法:采用攻击性特质问卷、词汇决策任务和竞争反应时任务对60名大学生进行问卷和实验研究。结果:在不同游戏方式下,被试表现出的攻击性认知和行为存在显著差异(F(2,52)=4.41,P<0.05;F(1,52)=50.06,P<0.001)。高、低攻击性特质者在游戏后的攻击性认知和行为上存在显著差异(F(1,52)=15.25,P<0.001;F(1,52)=82.35,P<0.001)。结论:不同游戏方式下的暴力网络游戏对大学生的攻击性存在不同的影响,相比竞争游戏和单人游戏,合作游戏是一种较好的游戏方式,可以在一定程度减少大学生,特别是高攻击性特质大学生的攻击性认知和行为。此外,暴力网络游戏对大学生攻击性认知和攻击性行为的影响存在一致性。     

具有策略偏好的模糊双矩阵对策模型及其应用

针对局中人对策略有偏好、支付值模糊的双矩阵对策给出了一种具体的求解方法,首先根据加权算子集结得到局中人的策略偏好向量、利用模糊数排序方式将模型转化为清晰的双矩阵对策,再利用粒子群优化算法求解.最后以雷达电子对抗为例,建立了具有策略偏好的模糊双矩阵对策的作战效能评估模型,研究结果对于双方资源分配,提高作战效能具有一定的军...     

Cost allocation rules for elastic single‐attribute situations

Many cooperative games, especially ones stemming from resource pooling in queueing or inventory systems, are based on situations in which each player is associated with a single attribute (a real number representing, say, a demand) and in which the cost to optimally serve any sum of attributes is described by an elastic function (which means that the per‐demand cost is non‐increasing in the total demand served). For this class of situations, we introduce and analyze several cost allocation rules: the proportional rule, the serial cost sharing rule, the benefit‐proportional rule, and various Shapley‐esque rules. We study their appeal with regard to fairness criteria such as coalitional rationality, benefit ordering, and relaxations thereof. After showing the impossibility of combining coalitional rationality and benefit ordering, we show for each of the cost allocation rules which fairness criteria it satisfies. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 271–286, 2017     

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.     

Multilinear extensions and quotients of simple games

Studied here is the problem of evaluating the formation of coalitions in cooperative games by considering its internal and external effects. Our analysis is based upon a formal theory of the quotient game and the use of multilinear extensions to compute the Shapley value and the coalitional value. When it applies to simple games, a series of interesting results is obtained. © 1996 John Wiley & Sons, Inc.     

Discrete search allocation game with false contacts

This paper deals with a two‐person zero‐sum game called a search allocation game, where a searcher and a target participate, taking account of false contacts. The searcher distributes his search effort in a search space in order to detect the target. On the other hand, the target moves to avoid the searcher. As a payoff of the game, we take the cumulative amount of search effort weighted by the target distribution, which can be derived as an approximation of the detection probability of the target. The searcher's strategy is a plan of distributing search effort and the target's is a movement represented by a path or transition probability across the search space. In the search, there are false contacts caused by environmental noises, signal processing noises, or real objects resembling true targets. If they happen, the searcher must take some time for their investigation, which interrupts the search for a while. There have been few researches dealing with search games with false contacts. In this paper, we formulate the game into a mathematical programming problem to obtain its equilibrium point. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Franchising cooperation through chance cross‐constrained games

This paper explores the role of franchising arrangements in the context of game theory. We assume a single franchisor and a single franchisee channel and address the impact of fixed lump‐sum fees, royalties, wholesale price, and retail price on the franchise contracts. We start with the chance cross‐constrained noncooperative situation where the franchisor, as the leader, first specifies his/her strategies. The franchisee, as the follower, then decides on his/her decision. We then relax the assumption of franchisee's inability to influence the franchisor's decisions and discuss cooperative situation between the franchisor and the franchisee. Nash's bargaining model is utilized to select the best Pareto‐efficient payment scheme for the franchisor and the franchisee to achieve their cooperation. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 669–685, 2000     

Systems defense games: Colonel blotto,command and control

The classical “Colonel Blotto” games of force allocation are generalized to include situations in which there are complementarities among the targets being defended. The complementarities are represented by means of a system “characteristic function,” and a valuation technique from the theory of cooperative games is seen to indicate the optimal allocations of defense and attack forces. Cost trade-offs between systems defense and alternative measures, such as the hardening of targets, are discussed, and a simple example is analyzed in order to indicate the potential of this approach.