Applying a one-shot and infinite repeated inspection game to materials management

An inspection game models a conflict situation between an inspector and an inspectee. The mathematical analysis aims to determine optimal behaviour of the inspectee. In this paper, the controller’s (inspector) job is to audit a planner’s (inspectee) decision with regard to its optimality and to submit a report to the company’s top managers. Based on Fandel and Trockel (Int J Prod Econ 133:256–261, 2011) this inspection game is discussed here as an infinite two-person game under the aspect of costs minimization. The results show that a higher costs deviation because of a competitive behaviour of the strategic players will decrease their payoffs, so that the optimal cooperative behaviour will finally lead to a better solution for the company with respect to cost minimization as normally expected.     

Avoiding non-optimal management decisions by applying a three-person inspection game

An inspection game models a conflict situation between an inspector and an inspectee. The mathematical analysis aims to generate optimal behavior of the inspectee under the assumption that an undesirable action of the inspectee could otherwise be carried out strategically. In this paper the controller’s (inspector’s) particular job is to audit a manager’s (inspectee’s) decision and to submit a report to the company’s top managers for examination. Thus, a conflict as regards the choice of behavioral actions of the manager, the controller and the top management impends. Based on Fandel and Trockel (2011a) this modified inspection game is discussed here for the first time as a three-person game in the context of a manager’s faulty decision that will unnecessarily add to the company’s costs and that the top management understandably wishes to minimize. We will first examine the conditions under which a Nash equilibrium occurs in this three-person game in which poor management, poor monitoring and poor revision coincide. We will then examine the effects that the penalties and bonuses exert on the Nash equilibrium solution. We will find that penalties and bonuses can neutralize each other in their effects on the improved decision making by the manager and the controller.     

An inspection game with multiple inspectees

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.     

A polynomial-time method to compute all Nash equilibria solutions of a general two-person inspection game

We consider a two-person nonzero-sum simultaneous inspection game that takes place at multiple sites. The inspector has a limited inspection resource. She needs to decide which sites to inspect, and with how much effort, while adhering also to local restrictions on the permitted inspections levels at the sites. The inspectee has several employees who work on his behalf. He needs to decide how to distribute them across the sites, and how they should act there. Computation of Nash equilibria is challenging for this sort of games. Still, we develop a linear-time algorithm that determines all Nash equilibria solutions of the game, and provide explicit (easily computable) expressions for all possible Nash equilibria. We then derive some managerial insights by applying the algorithm to several examples, and examining the Nash equilibria, including an outcome that an increase in the inspection resource may induce the inspectee to cooperate more at sites without increasing the inspection levels at them.     

Timely inspection and deterrence

Verification of compliance to formal agreements requires the performance of routine inspections for detection of illegal behavior. Deterrence of violations is generally taken to be the primary objective of any verification regime. If the inspections are carried out in such a way that the gain the inspectee may expect from illegal behavior is smaller than that for behaving legally, then the inspections may indeed be said to have deterred violation. In this paper inspection problems characterized additionally by a `critical time' are considered for the first time in this context. The critical time is the maximum time interval within which illegal activity must be detected in order to meet the objectives of the agreement. Critically time-dependent inspection problems are modeled as two-person non-cooperative games between inspector and inspectee. They extend over a reference time interval in which the inspector performs by agreement precisely k inspections and in which the inspectee behaves illegally at most once. The inspections are assumed to incur statistical errors of the first and second kind, the former being detrimental to both protagonists. Variants for which the inspectee may or may not be in a position to take advantage of information gained during the reference time interval are considered. Equilibria are investigated and conditions for the existence of deterring inspection strategies are determined.     

The relaxed game chromatic number of outerplanar graphs

The (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004     

Some exploration of the game SET®

Since the game SET® was first introduced to the public in 1993, it has stimulated some interesting studies. While the game itself is rather straightforward, a plethora of decent mathematical questions lies beneath the surface. It is perhaps because the game ties in so closely with such an underlying mathematical term that its implications can be seen in many major fields of mathematics. This note introduces a generalized version of the game SET® in which there are n ^m cards, m???4,?n???3,?m???n. Each card has m features and each feature has n options. A precise formula is presented for calculating the total number of SETs in a general deck.     

基于多方参与的“碳核查”业务权利寻租行为监管及控制研究

针对“碳核查”业务权利寻租问题,考虑政府、控排企业及第三方核查企业对其治理的影响,运用演化博弈理论分析各参与主体在“碳核查”业务权利寻租问题治理过程中的决策行为。研究结果表明：碳核查”业务权利寻租行为的发生受各主体不同决策行为的共同作用,如何约束控排企业的行为是解决“碳核查”业务权利寻租问题的关键;中低强度的复查抽检率无法有效约束控排企业的寻租行为,而高复查抽检率则可有效约束控排企业的寻租行为;因寻租成本不同,控排企业及第三方核查企业对政府惩罚力度的敏感性不同;中高强度的复查率及惩罚力度,有助于降低控排企业及第三方核查企业权力寻租行为的发生。     

Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework

We study optimal reinsurance in the framework of stochastic Stackelberg differential game, in which an insurer and a reinsurer are the two players, and more specifically are considered as the follower and the leader of the Stackelberg game, respectively. An optimal reinsurance policy is determined by the Stackelberg equilibrium of the game, consisting of an optimal reinsurance strategy chosen by the insurer and an optimal reinsurance premium strategy by the reinsurer. Both the insurer and the reinsurer aim to maximize their respective mean–variance cost functionals. To overcome the time-inconsistency issue in the game, we formulate the optimization problem of each player as an embedded game and solve it via a corresponding extended Hamilton–Jacobi–Bellman equation. It is found that the Stackelberg equilibrium can be achieved by the pair of a variance reinsurance premium principle and a proportional reinsurance treaty, or that of an expected value reinsurance premium principle and an excess-of-loss reinsurance treaty. Moreover, the former optimal reinsurance policy is determined by a unique, model-free Stackelberg equilibrium; the latter one, though exists, may be non-unique and model-dependent, and depend on the tail behavior of the claim-size distribution to be more specific. Our numerical analysis provides further support for necessity of integrating the insurer and the reinsurer into a unified framework. In this regard, the stochastic Stackelberg differential reinsurance game proposed in this paper is a good candidate to achieve this goal.     

Robust non-zero-sum investment and reinsurance game with default risk

This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, who are concerned about the potential model ambiguity and aim to seek the robust optimal reinsurance and investment strategies. The ambiguity-averse insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks; and can invest in a financial market consisting of one risk-free asset, one risky asset and one defaultable corporate bond. The objective of each insurer is to maximize the expected exponential utility of his terminal surplus relative to that of his competitor under the worst-case scenario of the alternative measures. Applying the techniques of stochastic dynamic programming, we derive the robust Nash equilibrium reinsurance and investment policies explicitly and present the corresponding verification theorem. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.     

多重因素约束下的网格检查对策问题研究

基于物品数量及每列容量等限制因素,构造局中人的可行策略集合;考虑隐藏成本,处罚规则与检查成功概率等因素,构造相应的支付函数,建立多重因素约束下的网格检查对策模型.根据矩阵对策性质,将对策论问题转化为非线性整数规划问题,利用H(o|¨)lder不等式获得实数条件下的规划问题的解,然后转化为整数解,得到特定条件下的模型的对策值及局中人的最优混合策略.最后,给出一个实例,说明上述模型的实用性及方法的有效性.     

Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R₀) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R₀ and holds great promise for lowering the number of children and youth who addict to online game in Thailand.     

A game-theoretic equivalence to the Hahn-Banach theorem

Let Γ_X() = X, A (X), υ be a cooperative von Neumann game with side payments, where X is a nonempty set of arbitrary cardinality, A(X) the Boolean ring generated from P(X) with the operations Δ and ∩ for addition and multiplication, respectively, such that S² =S for all S ε A (X), and with ;() = 0. The Shapley-Bondareva-Schmeidler Theorem, which states that a game of the form Γ_X() = X, A (X), is weak if and only if the core of Γ_X(),ζ(Γ_X()), is normal, may be regarded as the fundamental theorem for weak cooperative games with side-payments. In this paper we use an ultrapower construction on the reals, , to summarize a common mathematical theme employed in various constructions used to establish the Shapley-Bondareva-Schmeidler Theorem in the literature (Dalbaen, 1974; Kannai, 1969; Schmeidler, 1967, 1972). This common mathematical theme is that the space L, comprised of finite, real linear combinations of the collection of functions, {χ_a : a ε A (X)}, possesses a certain extension property that is intimately related to the Hahn-Banach Theorem of functional analysis. A close inspection of the extension property reveals that the Shapley-Bondareva-Schmeidler Theorem is in fact equivalent to the Hahn-Banach Theorem.     

John von Neumann, the Mathematician

Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner would be John von Neumann. Reasons are seen, for instance, in the title of the excellent biography [M] by Macrae: John von Neumann. The Scientific Genius who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Indeed, he was a fundamental figure not only in designing modern computers but also in defining their place in society and envisioning their potential. His minimax theorem, the first theorem of game theory, and later his equilibrium model of economy, essentially inaugurated the new science of mathematical economics. He played an important role in the development of the atomic bomb. However, behind all these, he was a brilliant mathematician. My goal here is to concentrate on his development and achievements as a mathematician and the evolution of his mathematical interests.     

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework.The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly.The claim process of the insurer is governed by a Brownian motion with a drift.A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle.Both the insurer and the reinsurer are assumed to invest in a risky asset,which is distinct for each other and driven by a constant elasticity of variance model.The optimal decision is formulated on a weighted sum of the insurer’s and the reinsurer’s surplus processes.Upon a verification theorem,which is established with a formal proof for a more general problem,explicit solutions are obtained for the proposed investment-reinsurance model.Moreover,numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.     

桑蚕成品卵微粒子病抽样检验样本容量的确定

蚕业生产上对桑蚕种成品卵微粒子病控制得很严,要求将总体病卵率为0.15%的蚕种以98.5%的概率将其判为不合格。在此标准下,运用概率统计方法,给出了抽样检验时样本最少抽检量为2800粒。并在控制把合格蚕种判为不合格蚕种要求下,给出了抽样检验时确定样本容量的数学模型。     

Weighted multiple majority games with unions: Generating functions and applications to the European Union

An a priori system of unions or coalition structure is a partition of a finite set of players into disjoint coalitions which have made a prior commitment to cooperate in playing a game. This paper provides “ready-to-apply” procedures based on generating functions that are easily implementable to compute coalitional power indices in weighted multiple majority games. As an application of the proposed procedures, we calculate and compare coalitional power indices under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the Council of the European Union.     

A single-shot game of multi-period inspection

This paper deals with an inspection game of Customs and a smuggler during some days. Customs has two options of patrolling or not. The smuggler can take two strategies of shipping its cargo of contraband or not. Two players have several opportunities to take an action during a limited number of days but they may discard some of the opportunities. When the smuggling coincides with the patrol, there occurs one of three events: the capture of the smuggler by Customs, a success of the smuggling and nothing new. If the smuggler is captured or no time remains to complete the game, the game ends. There have been many studies on the inspection game so far by the multi-stage game model, where both players at a stage know players’ strategies taken at the previous stage. In this paper, we consider a two-person zero-sum single-shot game, where the game proceeds through multiple periods but both players do not know any strategies taken by their opponents on the process of the game. We apply dynamic programming to the game to exhaust all equilibrium points on a strategy space of player. We also clarify the characteristics of optimal strategies of players by some numerical examples.     

Risk and quality control in a supply chain: competitive and collaborative approaches

This paper provides a quantitative and comparative economic and risk approach to strategic quality control in a supply chain, consisting of one supplier and one producer, using a random payoff game. Such a game is first solved in a risk-neutral framework by assuming that both parties are competing with each other. We show in this case that there may be an interior solution to the inspection game. A similar analysis under a collaborative framework is shown to be trivial and not practical, with a solution to the inspection game being an ‘all or nothing’ solution to one or both the parties involved. For these reasons, the sampling random payoff game is transformed into a Neyman–Pearson risk constraints game, where the parties minimize the expected costs subject to a set of Neyman–Pearson risk (type I and type II) constraints. In this case, the number of potential equilibria can be large. A number of such solutions are developed and a practical (convex) approach is suggested by providing an interior (partial sampling) solution for the collaborative case. Numerical examples are developed to demonstrate the procedure used. Thus, unlike theoretical approaches to the solution of strategic quality control random payoff games, the approach we construct is both practical and consistent with the statistical risk Neyman–Pearson approach.     

Automorphisms and weighted values

The notion of automorphism is an essential tool to capture the meaning of any mathematical structure. We apply this idea to cooperative games and obtain two interesting characterizations of the automorphisms of such a game: the one, in the complete case, as the permutations of players which preserve the (classical) Shapley value; the other, for the general case, as the permutations preservingall weighted Shapley values.Carreras' research supported by project UPV 036321-HA186/92 of the Basque Country University. Owen's research supported by the National Science Foundation, Grant DMS-9116416. This collaboration was made possible by the Polytechnic University of Catalonia, which sponsored Owen's visit to Terrassa.