A game theoretic approach to option valuation under Markovian regime-switching models

In this paper, we consider a game theoretic approach to option valuation under Markovian regime-switching models, namely, a Markovian regime-switching geometric Brownian motion (GBM) and a Markovian regime-switching jump-diffusion model. In particular, we consider a stochastic differential game with two players, namely, the representative agent and the market. The representative agent has a power utility function and the market is a “fictitious” player of the game. We also explore and strengthen the connection between an equivalent martingale measure for option valuation selected by an equilibrium state of the stochastic differential game and that arising from a regime switching version of the Esscher transform. When the stock price process is governed by a Markovian regime-switching GBM, the pricing measures chosen by the two approaches coincide. When the stock price process is governed by a Markovian regime-switching jump-diffusion model, we identify the condition under which the pricing measures selected by the two approaches are identical.     

Controlled predatory pricing in a multiperiod Stackelberg game: an MPEC approach

We analyze a multiperiod oligopolistic market where each period is a Stackelberg game between a leader firm and multiple follower firms. The leader chooses his production level first, taking into account the reaction of the followers. Then, the follower firms decide their production levels after observing the leader’s decision. The difference between the proposed model and other models discussed in literature is that the leader firm has the power to force the follower firms out of business by preventing them from achieving a target sales level in a given time period. The leader firm has an incentive to lower the market prices possibly lower than the Stackelberg equilibrium in order to push the followers to sell less and eventually go out of business. Intentionally lowering the market prices to force competitors to fail is known as predatory pricing, and is illegal under antitrust laws since it negatively affects consumer welfare. In this work, we show that there exists a predatory pricing strategy where the market price is above the average cost and consumer welfare is preserved. We develop a mixed integer nonlinear problem (MINLP) that models the multiperiod Stackelberg game. The MINLP problem is transformed to a mixed integer linear problem (MILP) by using binary variables and piecewise linearization. A cutting plane algorithm is used to solve the resulting MILP. The results show that firms can engage in predatory pricing even if the average market price is forced to remain higher than the average cost. Furthermore, we show that in order to protect the consumers, antitrust laws can control predatory pricing by setting rules on consumer welfare.     

Strategic Manipulation Approach for Solving Negotiated Transfer Pricing Problem

This paper suggests an approach for solving the transfer pricing problem, where negotiation between divisions is carried out considering the manipulation game theory model for a multidivisional firm. The manipulation equilibrium point is conceptualized under the Machiavellian social theory, represented by three concepts: views, tactics and immorality. In this approach, we are considering a non-cooperative model for the transfer pricing problem: a game model involving manipulating and manipulated players engaged cooperatively in a Nash game, restricted by a Stackelberg game. The cooperation is represented by the Nash bargaining solution. The transfer pricing problem is conceptualized as a strong Stackelberg game involving manipulating and manipulated divisions. This structure established conditions of unequal relative power among divisions, where high-power divisions tend to be abusive and less powerful divisions have a tendency to behave compliantly. For computation purposes, we transform the Stackelberg game model into a Nash game, where every division is able of manipulative behavior to some degree: the Nash game relaxes the interpretation of the manipulation game and the equilibrium selection for the transfer pricing problem. The manipulation dynamics and rationality proposed for the transfer pricing problem correspond to many real-world negotiation situations. We present an example, that illustrates how manipulation can be employed to solve the transfer pricing problem in a multidivisional firm.     

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.     

Search allocation game

This paper deals with a two-person zero-sum game called search allocation game (SAG), where a searcher allocates his searching resources in a search space to detect a target while the target takes a path running across the space to evade the searcher. We consider the discrete SAG and the continuous SAG defined on the discrete search space and the continuous one, respectively. In a general way, we prove an existence theorem of equilibrium points for both the SAGs and elucidate that an equilibrium of the continuous SAG is given by a convergence point of equilibria of the discrete SAG. After then we develop a method to solve a large size of the discrete problem with specific feasibility conditions. As one of numerical examples, we take so-called flaming datum search game, which is adequate to demonstrate the convergence theorem.     

Optimal production,purchasing and pricing: A differential game approach

This paper deals with a problem of determining optimal production and pricing policies of a manufacturing firm which is supplying a retailer, The latter faces a price-dependent demand function towards the final consumers and wishes to determine optimal purchasing and pricing policies. Both firms carry inventories and backlogging is permitted. The problem is modelled as a two-player nonzero-sum differential game with the inventory levels as the state variables. The controls are the rates of production and purchasing as well as the prices. Assuming linear production costs, but convex ordering and holding costs, open-loop Nash controls are characterized by using switching point analysis as well as phase diagrams.     

具随机折现的博弈期权定价问题

对具随机折现的博弈期权定价问题进行了研究,在满足一个可积性条件的情况下,借用过份函数等工具给出了期权价格的表达式和买卖双方的最优停止策略.对于不满足可积性条件的情况,推广了相关文献的结果,并给出了τ＊存在的条件.最后给出了一个例子.     

A zero-sum stochastic game model of duopoly

We consider a discrete time zero-sum stochastic game model of duopoly and give a partial characterization of each firm's optimal pricing strategy. An extension to a continuous time model is also discussed.     

乘客策略行为下考虑需求学习的航空客运动态定价策略研究

研究航空公司在需求学习下的动态定价策略。通过假设乘客到达率不确定以及具有策略等待行为,运用贝叶斯理论和博弈论对航空公司需求学习下的多周期动态定价问题进行建模,探讨了机票最优定价策略的充分条件,并通过分析航空公司收益函数的性质,得到了最优定价随时间和已出售机票数量的变化趋势。最后应用算例分析了需求学习的效果,得出:需求学习能够缓解需求不确定带来的损失,但不能完全消除;乘客策略程度越大,需求学习效果越明显。     

A differential game model of a market of substitutable products

Modern industrial organization often classifies groups of differentiated products that are fairly good substitutes to belong to the same market. This paper develops a differential game model of a market of substitutable products. To avoid the problem of time-inconsistency, we solve a feedback Nash equilibrium solution for the game. A set of state-dependent equilibrium strategies is derived. Extensions of the model to a stochastic formulation and to an infinite time horizon specification are also provided.     

A game theoretic approach to coordinate pricing and vertical co-op advertising in manufacturer-retailer supply chains

Vertical cooperative (co-op) advertising is a marketing strategy in which the retailer runs local advertising and the manufacturer pays for a portion of its entire costs. This paper considers vertical co-op advertising along with pricing decisions in a supply chain; this consists of one manufacturer and one retailer where demand is influenced by both price and advertisement. Four game-theoretic models are established in order to study the effect of supply chain power balance on the optimal decisions of supply chain members. Comparisons and insights are developed. These embrace three non-cooperative games including Nash, Stackelberg-manufacturer and Stackelberg-retailer, and one cooperative game. In the latter case, both the manufacturer and the retailer reach the highest profit level; subsequently, the feasibility of bargaining game is discussed in a bid to determine a scheme to share the extra joint profit.     

Robust non-zero-sum stochastic differential reinsurance game

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) who encounter model uncertainty and seek the optimal reinsurance decision under relative performance concerns. Each AAI manages her own risks by purchasing reinsurance with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her counterparty. The two AAIs’ decisions influence each other through the insurers’ relative performance concerns and the correlation between their surplus processes. We establish a general framework of Nash equilibrium for the associated non-zero-sum game with model uncertainty. For the representative case of exponential utilities, we solve the equilibrium strategies explicitly. Numerical studies are conducted to draw economic interpretations.     

Monitoring cooperative equilibria in a stochastic differential game

This paper deals with a class of equilibria which are based on the use of memory strategies in the context of continuous-time stochastic differential games. In order to get interpretable results, we will focus the study on a stochastic differential game model of the exploitation of one species of fish by two competing fisheries. We explore the possibility of defining a so-called cooperative equilibrium, which will implement a fishing agreement. In order to obtain that equilibrium, one defines a monitoring variable and an associated retaliation scheme. Depending on the value of the monitoring variable, which provides some evidence of a deviation from the agreement, the probability increases that the mode of a game will change from a cooperative to a punitive one. Both the monitoring variable and the parameters of this jump process are design elements of the cooperative equilibrium. A cooperative equilibrium designed in this way is a solution concept for a switching diffusion game. We solve that game using the sufficient conditions for a feedback equilibrium which are given by a set of coupled HJB equations. A numerical analysis, approximating the solution of the HJB equations through an associated Markov game, enables us to show that there exist cooperative equilibria which dominate the classical feedback Nash equilibrium of the original diffusion game model.This research was supported by FNRS-Switzerland, NSERC-Canada, FCAR-Quebec.     

On-line solution of a stochastic pursuit-evasion game

In this paper, the concepts of prior-commitment and delayed-commitment strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are applied to a detailed example of a pursuit-evasion game. Quasilinearization is used to solve the nonlinear two-point boundary-value problem of the prior-commitment game. A closed form solution is obtained for the delayed-commitment strategies. Comparison of the payoff functionals confirms the relationships discussed in Ref. 1.     

On the core: complement-reduced game and max-reduced game

This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to “complement-reduced game” and converse consistency with respect to “max-reduced game”. The other is based on consistency with respect to “max-reduced game” and weak converse consistency with respect to “complement-reduced game”. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non-emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.     

Volume discount pricing strategy in the VMI supply chain with price sensitive demand

This paper develops a volume discount scheme to coordinate Vendor Managed Inventory (VMI) supply chains with multiple heterogeneous retailers, in which the supply chain is modelled as a Stackelberg game with price sensitive demand. The paper proposes a method to construct a volume discount price scheme and shows that, any volume discount can be represented as a piecewise constant function of demand. We provide the game formulations of VMI supply chains and develop algorithms to solve this type of game problems, including finding the optimal volume discount scheme. Through a numerical study comparing the results of applying a volume discount strategy with the profits from a single wholesale price strategy, we show that the volume discount pricing strategies can be used to improve profits for all participants in the VMI supply chain in comparison with single price strategies.     

灾后运输网络中的最短路修复合作博弈

在最短路修复合作博弈中,当灾后运输网络规模较大时,最优成本分摊问题难以直接求解。基于拉格朗日松弛理论,提出了一种最短路修复合作博弈成本分摊算法。该算法将最短路修复合作博弈分解为两个具有特殊结构的子博弈,进而利用两个子博弈的结构特性,可以{高效地}求解出二者的最优成本分摊,将这两个成本分摊相加,可以获得原博弈的一个近乎最优的稳定成本分摊。结果部分既包含运输网络的随机仿真,也包含玉树地震灾区的现实模拟,无论数据来源于仿真还是现实,该算法都能在短时间内为最短路修复合作博弈提供稳定的成本分摊方案。     

A dynamic Cournot–Nash game: a representation of a finitely repeated feedback game

This paper studies market outcome equivalence of two dynamic production-capital investment games under uncertainty. One is played under complete information, while the other, feedback (FB) game, is played under incomplete information about the opponents’ costs and market demand. The FB game structure may be observed in some newly initiated industries, in which a homogeneous good is exchanged via an auction mechanism. In that case, the FB game setting may predict the complete information equilibrium market outcomes.      

维修保障船与修船厂维修远海受损舰船的定价博弈

维修保障船能够实现远海受损舰船的现场维修,当其进入舰船维修市场时将与修船厂形成一定程度的竞争。为解决维修保障船模式与修船厂模式维修远海受损舰船的最优定价问题,通过分析这两种维修模式在时空以及成本方面的差异,建立维修保障船与修船厂的定价博弈双层规划模型,并设计迭代算法实现模型的求解。以维修远海船体受损的舰船为例,求得了两者的最优定价、最大收益以及船东对两种维修模式的需求,验证了模型与算法的有效性,并分析了调遣距离、舰船受损程度、海上维修技术难度等因素对维修保障船与修船厂定价博弈的影响。该方法能够为维修保障船与修船厂的经营者提供最优定价策略,为两种维修模式的定价博弈提供理论支持。