Oligopoly games under asymmetric costs and an application to energy production

Oligopolies in which firms have different costs of production have been relatively under-studied. In contrast to models with symmetric costs, some firms may be inactive in equilibrium. (With symmetric costs, the results trivialize to all firms active or all firms inactive.) We concentrate on the linear demand structure with constant marginal but asymmetric costs. In static one-period models, we compare the number of active firms, i.e. the number of firms producing a positive quantity in equilibrium, across four different models of oligopoly: Cournot and Bertrand with homogeneous or differentiated goods. When firms have different costs, we show that, for fixed good type, Cournot always results in more active firms than Bertrand. Moreover, with a fixed market type, differentiated goods result in more active firms than homogeneous goods. In dynamic models, asymmetric costs induce different entry times into the market. We illustrate with a model of energy production in which multiple producers from costly but inexhaustible alternative sources such as solar or wind compete in a Cournot market against an oil producer with exhaustible supply.     

Advertising in a Differential Oligopoly Game

We illustrate a differential oligopoly game where firms compete à la Cournot in homogeneous goods in the market phase and invest in advertising activities aimed at increasing the consumers reservation price. Such investments produce external effects, characterizing the advertising activity as a public good. We derive the open-loop and closed-loop Nash equilibria, and show that the properties of the equilibria depend on the curvature of the market demand function. The comparative assessment of these equilibria shows that the firms advertising efforts are larger in the open-loop equilibrium than in the closed-loop equilibrium. We also show that a cartel involving all the firms, setting both output levels and advertising efforts, may produce a steady state where the social welfare level is higher than the social welfare levels associated with both open-loop and closed-loop noncooperative settings.     

Dynamic strategic interaction between an innovating and a non-innovating incumbent

This paper analyzes the effects of product innovation on the firms’ investment behavior in a dynamic duopoly framework. A differential game setting is considered where initially two firms are active on a homogeneous product market. One of the firms has an option to introduce a new product that is horizontally and vertically differentiated from the established product. The resulting differential game has three states corresponding to three capital stocks: one for each firm to produce the established product, and one for the innovating firm to produce the new product. We numerically derive Markov perfect equilibria. One of the most remarkable results is that in most cases the non-innovating firm benefits when the other firm carries out the innovation option. The intuition is that, to increase demand for the innovative product, the innovative firm reduces capacity on the established market, which increases the price of the established product and thus the payoff of the non-innovating firm.     

Strategic upfront marketing channel integration as an entry barrier

This paper investigates an organizational design problem concerning whether duopolistic firms competing in a product market should vertically integrate or separate their marketing channels in a dynamic noncooperative game setting. Previous operational research models have shown that the separation of the marketing channel with the adoption of a two-part tariff contract is the dominant strategy compared with integration for each firm if the two firms face retail price competition, and thereby constitutes the subgame perfect Nash equilibrium (SPNE). Contrary to this previous insight, this paper demonstrates that if exogenous parameters that characterize fixed costs, product substitutability, and a demand function fall into a specific region, marketing channel integration dominates the separation strategy when one of the two firms is the incumbent firm while the other is a potential entrant. In other words, the well-known result in the price-setting game can be reversed when we take entry threats into consideration. Specifically, we show that upfront vertical integration of the marketing channel enables the incumbent to deter the entry of the potential competitor and to monopolize the market in the SPNE. This result has operational implications for a firm confronting the threat of potential rivals entering the market, in that the firm can use this apparently inferior strategy as a commitment device, which creates a virtual entry barrier.     

Competitive capacity expansion under demand uncertainty

We analyze the capacity expansion behavior of firms in a duopoly faced with an uncertain new market. The market demand may be high or low with a given probability mass function. Firms obtain private information about the market size and build their capacity before the market demand is known. Once the demand is revealed, firms enter a capacity constrained price competition phase which determines their revenues. Two scenarios are considered: first, when firms choose their capacities simultaneously in the investment phase, and second, when they do so sequentially. For each case, we determine the unique symmetric Nash equilibrium. Excess capacity can occur in equilibrium in the industry. It is seen that preempting the competitor in the capacity expansion phase offers first mover benefits. We argue that the sequential moves game is more prone to equilibrium excess capacity compared to the simultaneous case. We show that preemption is a good strategy if the investing environment is either highly optimistic or highly pessimistic. If the industry outlook is only moderately optimistic, a capacity planner is still better off preempting his competitor, however, the industry may encounter overcapacity as a consequence.     

On the equivalence of quantity competition and supply function competition with sunk costs

This paper considers competition in supply functions in a homogeneous goods market in the absence of cost or demand uncertainty. In order to commit to a supply schedule, firms are required to build sufficient capacity to produce any quantity that may be prescribed by that schedule. When the cost of extra capacity (given the level of sales) is strictly positive, any Nash equilibrium outcome of supply function competition is also a Nash equilibrium outcome of the corresponding Cournot game, and vice-versa. Conversely, when the cost-savings from reducing output (given the capacity level) are sufficiently small, any outcome of iterated elimination of weakly dominated strategies in the supply function game is also an outcome of the same process in Cournot, and vice-versa.     

Equilibrium two-part cost structures

We consider an oligopolistic product market in which two competing firms instead of paying a competitive input price choose a two-part tariff. Costs for the input are divided up into upfront fixed costs independent of the output level and reductions in marginal costs. We explore under which competitive settings will such a two-part cost structure correspond to equilibrium behavior in a two stage game. We find that firms in a static model do have an incentive to choose a two-part cost structure when competition in the product market is not too strong and oligopoly rents can be shifted form the rival to the own firm. In a dynamic market when firms use Markov strategies competition is so intense that there are no rents to be shifted and firms do not benefit from two-part cost structures.     

Duopoly information sharing with differentiated products

Consider a model where firms own the same technology in linear Cournot duopolies with differentiated products and the slope of the demand curve facing the firm is unknown, containing an own-price effect and a cross-effect. We discuss as follows: whether there is an incentive to share information when firms are symmetrically informed about the random demand. In a two-stage game, for independent goods and complements, it is a Nash equilibrium for firms to put their private information in a common pool.     

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.     

Feedback Nash Equilibrium for Dynamic Brand and Channel Advertising in Dual Channel Supply Chain

This paper investigates dynamic brand and channel advertising and their effects on market expansion and market share in a dual channel supply chain by using differential game theory. Nash equilibrium of the differential game will be obtained by the feedback approach. A sensitivity analysis was performed to assess the impact of key parameters on optimal strategies and to characterize behaviors of both the manufacturer and retailer over time. Comparative statistics show that a higher compatibility of a product with online marketing leads to a higher advertising effort for the online channel by the manufacturer, an enhanced steady state for demand of the brand as well as greater sales in the steady state through the online channel.     

Non-cooperative joint replenishment under asymmetric information

We consider jointly replenishing n ex-ante identical firms that operate under an EOQ like setting using a non-cooperative game under asymmetric information. In this game, each firm, upon being privately informed about its demand rate (or inventory cost rate), submits a private contribution to an intermediary that specifies how much it is willing to pay for its replenishment per unit of time and the intermediary determines the maximum feasible frequency for the joint orders that would finance the fixed replenishment cost. We show that a Bayesian Nash equilibrium exists and characterize the equilibrium in this game. We also show that the contributions are monotone increasing in each firm’s type. We finally conduct a numerical study to compare the equilibrium to solutions obtained under independent and cooperative ordering, and under full information. The results show that while information asymmetry eliminates free-riding in the contributions game, the resulting aggregate contributions are not as high as under full information, leading to higher aggregate costs.     

寡头市场的一类动态博弈模型

首先讨论了寡头垄断市场中n批厂商分批(每批至少有两个以上的厂商)先后进入某行业各批厂商依次且每批同时选择其产量的动态博弈模型的子博弈精练解及其相关结论,探讨了此结论与有关问题的比较分析,并给出此问题的几种特殊情况,说明了此模型的广泛性和实用性.     

On the feedback solutions of differential oligopoly games with hyperbolic demand curve and capacity accumulation

To safeguard analytical tractability and the concavity of objective functions, the vast majority of models belonging to oligopoly theory relies on the restrictive assumption of linear demand functions. Here we lay out the analytical solution of a differential Cournot game with hyperbolic inverse demand, where firms accumulate capacity over time à la Ramsey. The subgame perfect equilibrium is characterized via the Hamilton–Jacobi–Bellman equations solved in closed form both on infinite and on finite horizon setups. To illustrate the applicability of our model and its implications, we analyze the feasibility of horizontal mergers in both static and dynamic settings, and find appropriate conditions for their profitability under both circumstances. Static profitability of a merger implies dynamic profitability of the same merger. It appears that such a demand structure makes mergers more likely to occur than they would on the basis of the standard linear inverse demand.     

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.      

Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain

Cooperative advertising is an incentive offered by a manufacturer to influence retailers’ promotional decisions. We study a dynamic durable goods duopoly with a manufacturer and two independent and competing retailers. The manufacturer, as a Stackelberg leader, announces his wholesale prices and his shares of retailers’ advertising costs, and the retailers in response play a Nash differential game in choosing their optimal retail prices and advertising efforts over time. We obtain the feedback equilibrium policies for the manufacturer and the retailers in explicit form for a linear demand formulation. We investigate issues, like channel coordination and antidiscriminatory legislation, and also study a case, when the manufacturer sells through only one retailer and the second retailer sells a competing brand.     

Pure equilibria in a simple dynamic model of strategic market game

We present a discrete model of two-person constant-sum dynamic strategic market game. We show that for every value of discount factor the game with discounted rewards possesses a pure stationary strategy equilibrium. Optimal strategies have some useful properties, such as Lipschitz property and symmetry. We also show value of the game to be nondecreasing both in state and discount factor. Further, for some values of discount factor, exact form of optimal strategies is found. For β less than , there is an equilibrium such that players make large bids. For β close to 1, there is an equilibrium with small bids. Similar result is obtained for the long run average reward game.     

价格歧视下四种市场的均衡产量和价格

在线性需求函数条件下,对1个垄断厂商市场情形用微分法,对,1个厂商市场情形用完全信息静态博弈的方法,对实施二度价格歧视时均衡产量和价格的确定进行了研究,给出了市场均衡歧视产量,均衡歧视价格和均衡歧视总收益的统一的计算公式,并分析了其性质.     

A goodwill model with predatory advertising

We investigate the dynamic advertising policies of two competing firms in a duopolistic industry, assuming a predatory phenomenon between their advertising campaigns. The resulting model is a differential game which is not linear-quadratic. We show that there exists a Markovian Nash equilibrium, and that it leads to time constant advertising strategies. According to this model, predatory advertising produces a negative externality: the interference between the advertising campaigns decreases the total demand of the market.     

A dynamic free-entry oligopoly with sluggish entry and exit adjustments

We study a dynamic free-entry oligopoly with sluggish entry and exit of firms under general demand and cost functions. We show that the number of firms in a steady-state open-loop solution for a dynamic free-entry oligopoly is smaller than that at static equilibrium and that the number of firms in a steady-state memoryless closed-loop solution is larger than that in an open-loop solution.     

A differential oligopoly game with differentiated goods and sticky prices

We investigate a dynamic oligopoly game where goods are differentiated and prices are sticky. We study the open-loop and the closed-loop memoryless Nash equilibrium, and show that the latter equilibrium entails a larger level of steady state production as compared to the former; both equilibria entail a larger level of production in steady state than the static game. We also study the effects of price stickiness and product differentiation upon the steady state equilibrium allocation and profits. The per-firm equilibrium output is increasing in both product differentiation and price stickiness, while profits are increasing in both product differentiation and the speed of price adjustment. The steady state social welfare monotonically increases in the speed of price adjustment, and the overproduction entailed by dynamic competition has beneficial effect from a social standpoint.