Capacity optimization and competition with cyclical and lead-time-dependent demands

Capacity acquisition is often capital- and time-consuming for a business, and capacity investment is often partially or fully irreversible and difficult to change in the short term. Moreover, capacity determines the action space for service/production scheduling and lead-time quotation decisions. The quoted lead-time affects the customer’s perceived service quality. Thus, capacity acquisition level and lead-time quotation affect a firm’s revenue/profit directly or indirectly. In this paper, we investigate a joint optimization problem of capacity acquisition, delivery lead-time quotation and service-production scheduling with cyclical and lead-time-dependent demands. We first explore the structural properties of the optimal schedule given any capacity and lead-time. Then, the piecewise concave relationship between the delay penalty cost and the capacity acquisition level is found. Thereby, an efficient and effective polynomial time algorithm is provided to determine the optimal capacity acquisition level, delivery lead-time quotation and service/production schedule simultaneously. Furthermore, a capacity competition game among multiple firms is addressed. The numerical studies show that capacity equilibrium often exists and converges to a unique solution.     

Persuasive advertising under Bertrand competition: A differential game

We investigate a linear state differential game of advertising, under Cournot and Bertrand competition. A unique saddlepoint equilibrium exists if the marginal cost of advertising is sufficiently low. Bertrand competition entails more intense advertising than Cournot competition, since increasing market size is more important to firms when competition is tough.     

Pricing and strategy selection in a closed-loop supply chain under demand and return rate uncertainty

Closed-loop supply chain (CLSC) decision-making involves many uncertainties, which makes the decision-making process more complex and diversified. This study considered a two-stage CLSC consisting of an original manufacturer and a third-party recycler. Without any government policy support, considering the effects of market demand, product return rate, and consumer perceived value, a CLSC decision model based on market demand with a [0,1] distribution was established. The model analyzes three situations—a manufacturer monopoly, the Cournot duopoly game, and the Stackelberg competition game—and solves them. The optimal values of decision variables such as optimal pricing, market demand, and all parties’ profits in the CLSC are obtained, and a strict mathematical proof is given. Through the model-solving process, the effects of product return rate and consumer perceived value on decision variables are analyzed; then, the profit allocation between the original manufacturer and the third-party recycler under different cooperation modes is analyzed. In addition, the four combinations of competition and cooperation are analyzed based on game theory. The Nash equilibrium solution and Pareto optimal solution of the four modes are analyzed by drawing a bimatrix Nash equilibrium table. The results indicate that the cooperation–cooperation mode is difficult to produce automatically, and government policy guidance and support are often needed to achieve Pareto optimality. Finally, a numerical example is given to validate the proposed model. In this way, the proposed model provides reliable theoretical support for the decision-making of both sides in a CLSC.

     

Network capacity management under competition

We consider capacity management games between airlines who transport passengers over a joint airline network. Passengers are likely to purchase alternative tickets of the same class from competing airlines if they do not get tickets from their preferred airlines. We propose a Nash and a generalized Nash game model to address the competitive network revenue management problem. These two models are based on well-known deterministic linear programming and probabilistic nonlinear programming approximations for the non-competitive network capacity management problem. We prove the existence of a Nash equilibrium for both games and investigate the uniqueness of a Nash equilibrium for the Nash game. We provide some further uniqueness and comparative statics analysis when the network is reduced to a single-leg flight structure with two products. The comparative statics analysis reveals some useful insights on how Nash equilibrium booking limits change monotonically in the prices of products. Our numerical results indicate that airlines can generate higher and more stable revenues from a booking scheme that is based on the combination of the partitioned booking-limit policy and the generalized Nash game model. The results also show that this booking scheme is robust irrespective of which booking scheme the competitor takes.     

Nonlinear duopoly games with positive cost externalities due to spillover effects

A Cournot duopoly game is proposed where the interdependence between the quantity-setting firms is not only related to the selling price, determined by the total production through a given demand function, but also on cost-reduction effects related to the presence of the competitor. Such cost reductions are introduced to model the effects of know-how spillovers, caused by the ability of a firm to take advantage, for free, of the results of competitors' Research and Development (R&D) results, due to the difficulties to protect intellectual properties or to avoid the movements of skilled workers among competing firms. These effects may be particularly important in the modeling of high-tech markets, where costs are mainly related to R&D and workers' training. The results of this paper concern the existence and uniqueness of the Cournot–Nash equilibrium, located at the intersection of non-monotonic reaction curves, and its stability under two different kinds of bounded rationality adjustment mechanisms. The effects of spillovers on the existence of the Nash equilibrium are discussed, as well as their influence on the kind of attractors arising when the Nash equilibrium is unstable. Methods for the global analysis of two-dimensional discrete dynamical systems are used to study the structure of the basins of attraction.     

Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition

We consider two game-theoretic models of the generation capacity expansion problem in liberalized electricity markets. The first is an open loop equilibrium model, where generation companies simultaneously choose capacities and quantities to maximize their individual profit. The second is a closed loop model, in which companies first choose capacities maximizing their profit anticipating the market equilibrium outcomes in the second stage. The latter problem is an equilibrium problem with equilibrium constraints. In both models, the intensity of competition among producers in the energy market is frequently represented using conjectural variations. Considering one load period, we show that for any choice of conjectural variations ranging from perfect competition to Cournot, the closed loop equilibrium coincides with the Cournot open loop equilibrium, thereby obtaining a ‘Kreps and Scheinkman’-like result and extending it to arbitrary strategic behavior. When expanding the model framework to multiple load periods, the closed loop equilibria for different conjectural variations can diverge from each other and from open loop equilibria. We also present and analyze alternative conjectured price response models with switching conjectures. Surprisingly, the rank ordering of the closed loop equilibria in terms of consumer surplus and market efficiency (as measured by total social welfare) is ambiguous. Thus, regulatory approaches that force marginal cost-based bidding in spot markets may diminish market efficiency and consumer welfare by dampening incentives for investment. We also show that the closed loop capacity yielded by a conjectured price response second stage competition can be less or equal to the closed loop Cournot capacity, and that the former capacity cannot exceed the latter when there are symmetric agents and two load periods.     

Inventory competition and allocation in a multi-channel distribution system

Consider a supply chain involving one manufacturer and one independent retailer. The manufacturer distributes her product to the end consumer through the independent retailer as well as through her direct channel. Each of the two channels faces a stochastic demand. If one channel is out of stock, a fraction of the unsatisfied customers visit the other channel, which induces inventory competition between the channels. Under the scenario described above, will the manufacturer ever undercut the retailer’s order when the capacity is infinite? What are the equilibria of the game? How does a capacity constraint affect the equilibrium outcome? What is the optimal inventory allocation strategy for the manufacturer? Using a game theoretic model we seek answers to the above questions. Both the capacitated and the infinite capacity games are considered. We establish the necessary condition for a manufacturer to undercut a retailer’s order and show that a manufacturer may deny the retailer of inventory even when the capacity is ample. We show that there can be an equilibrium in the capacitated game where a manufacturer might not use the entire capacity and still deny a retailer inventory. We also show that a mild capacity constraint may make both parties better off and thereby increase the total supply chain profit. We develop a simple yet practical contract called the reverse revenue sharing contract and show that along with a fixed franchise fee this contract can coordinates our decentralized supply chain.     

Duopoly competition in networks

This paper presents a new approach to modeling competition between firms in network-based industries, i.e. industries where the firms' technology decisions correspond to choices of networks. Industries having this structure include transportation, telecommunications, and some service industries. Competition is studied between two firms who make both network design decisions and price decisions for services. This situation is modeled as a game, an equilibrium solution corresponding to a Nash equilibrium is defined, and properties of the solution are characterized. Necessary and sufficient conditions are shown for equilibrium solutions and existence of equilibrium solutions is demonstrated. Among the results is that each firm will maximize its own profit by minimizing total industry cost of providing services. An example demonstrating results is presented.     

竞争环境下基于退换货的网购供应链动态均衡研究

研究竞争环境下基于退换货的网购供应链动态均衡模型.此供应链包含多个生产商、电商、快递商及需求市场.将快递商的运输速度作为竞争的一个重要因素进行研究.通过正弦函数说明,网购供应链的市场需求也呈季节性变化.利用纳什均衡及变分不等式得到各层决策者的竞争均衡解.通过分析换货比重得出电商应减少消费者的退货率,以提高整条供应链的利润和竞争能力.并利用数值算例说明模型的正确性与合理性.     

Supply chain network sustainability under competition and frequencies of activities from production to distribution

In this paper, we develop a competitive supply chain network model with multiple firms, each of which produces a differentiated product by brand and weights the emissions that it generates through its supply chain network activities in an individual way. The supply chain network activities of production, transport and distribution, and storage have associated with them distinct capacities and the firms seek to determine their optimal product flows and frequencies of operation so that their utilities are maximized where the utilities consist of profits and weighted emissions. Multiple production, storage, and transport mode options are allowed. The governing equilibrium concept is that of Cournot–Nash equilibrium. We provide both path and link flow variational inequality formulations of the equilibrium conditions and then propose an algorithm, which, at each iteration, yields closed form expressions for the underlying variables. Numerical examples illustrate the generality of the model and the information provided to managerial decision-makers and policy-makers. This paper adds to the growing literature on sustainable supply chains through the development of a computable general competitive supply chain network game theory model, which brings a greater realism to the evaluation of profit and emission trade-offs through the incorporation of frequencies.     

Complex dynamic properties of Cournot duopoly games with convex and log-concave demand function

A Cournot duopoly game is proposed where the interdependence between firms depends on convex and log-concave demand function. In this paper, a model of two rational firms that are in competition and produce homogeneous commodities is introduced. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation and chaos are investigated. Furthermore, a multi-team Cournot game is introduced. Through simulation the dynamical characteristics of the equilibrium points of this game are illustrated.     

A Shared-Constraint Approach to Multi-Leader Multi-Follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over this shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization problem are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.     

Competition among providers in loss networks

Communication networks are becoming ubiquitous and more and more competitive among revenue-maximizing providers, operating on potentially different technologies. In this paper, we propose to analyze thanks to game theory the competition of providers playing with access prices and fighting for customers. Considering a slotted-time model, the part of demand exceeding capacity is lost and has to be resent. We consider an access price for submitted packets, thus inducing a congestion pricing through losses. Customers therefore choose the provider with the cheapest average price per correctly transmitted unit of traffic. The model is a two-level game, the lower level for the distribution of customers among providers, and the upper level for the competition on prices among providers, taking into account what the subsequent repartition at the lower level will be. We prove that the upper level has a unique Nash equilibrium, for which the user repartition among different available providers is also unique, and, remarkably, efficient in the sense of social welfare (with a so-called price of anarchy equal to one). Moreover, even when adding a higher level game on capacity disclosure with a possibility of lying for providers, providers are better off being truthful, and the unique Nash equilibrium is thus unchanged.     

Competition analysis of a triopoly game with bounded rationality

A dynamic Cournot game characterized by three boundedly rational players is modeled by three nonlinear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and a complex chaotic behavior occurs. Numerical simulation results show that complex dynamics, such as, bifurcations and chaos are displayed when the value of speed of adjustment is high. The global complexity analysis can help players to take some measures and avoid the collapse of the output dynamic competition game.     

Nonzero-sum stochastic games with unbounded costs: Discounted and average cost cases

We treat non-cooperative stochastic games with countable state space and with finitely many players each having finitely many moves available in a given state. As a function of the current state and move vector, each player incurs a nonnegative cost. Assumptions are given for the expected discounted cost game to have a Nash equilibrium randomized stationary strategy. These conditions hold for bounded costs, thereby generalizing Parthasarathy (1973) and Federgruen (1978). Assumptions are given for the long-run average expected cost game to have a Nash equilibrium randomized stationary strategy, under which each player has constant average cost. A flow control example illustrates the results. This paper complements the treatment of the zero-sum case in Sennott (1993a).     

Strategic bidding in continuous electricity auctions: an application to the Spanish electricity market

In this paper we introduce an asymmetric model of continuous electricity auctions with limited production capacity and bounded supply functions. The strategic bidding is studied with this model by means of an electricity market game. We prove that for every electricity market game with continuous cost functions a mixed-strategy Nash equilibrium always exists. In particular, we focus on the behavior of producers in the Spanish electricity market. We consider a very simple form for the Spanish electricity market: an oligopoly consisting just of independent hydro-electric power production units in a single wet period. We show that a pure-strategy Nash equilibrium for the Spanish electricity market game always exists.     

不完全信息多维Cournot模型及其分析

根据现实中的博弈现象,本文首先提出了多维博弈概念,然后描述了其特征、策略型式,并将Nash均衡扩展为多维Nash均衡．最后,讨论了两个企业关于具有一定替代性的两种产品的不完全信息静态多维Cournot多维博弈模型,并进一步分析得到两个企业对两种产品单独博弈的均衡策略劣于对两种产品多维博弈的均衡策略．     

Side-payment contracts in two-person nonzero-sum supply chain games: Review,discussion and applications

This paper investigates supply chain coordination with side-payment contracts. We first summarize specific side-payment contracts and present our review on the literature that developed general side-payment schemes to coordinate supply chains. Following our review, we discuss two criteria that a proper side-payment contract must satisfy, and accordingly introduce a decision-dependent transfer payment function and a constant transfer term. We present the condition that the transfer function must satisfy, and use Nash arbitration scheme and Shapley value to compute the constant transfer term and derive its closed-form solution. Next, we provide a five-step procedure for the development of side-payment contract, and apply it to four supply chain games: Cournot and Bertrand games, a two-retailer supply chain game with substitutable products and a one-supplier, one-retailer supply chain. More specifically, for the Cournot game, we construct a linear transfer function and a constant side-payment to coordinate two producers. For the Bertrand game, we build a nonlinear transfer function which is equivalent to a revenue-sharing contract, and show that the constant term is zero and two firms in the game equally share the system-wide profit. For a supply chain with substitutable products, we present a side-payment contract to coordinate two retailers. For a two-echelon supply chain, we develop a proper side-payment scheme that can coordinate the supply chain and also help reduce the impact of forward buying on supply chain performance.     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

A <Emphasis Type="Italic">d</Emphasis>-person Differential Game with State Space Constraints

We consider a network of d companies (insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d -person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous; there are d interlinked control problems with state constraints; each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.