Dynamic Bertrand Oligopoly

We study continuous time Bertrand oligopolies in which a small number of firms producing similar goods compete with one another by setting prices. We first analyze a static version of this game in order to better understand the strategies played in the dynamic setting. Within the static game, we characterize the Nash equilibrium when there are N players with heterogeneous costs. In the dynamic game with uncertain market demand, firms of different sizes have different lifetime capacities which deplete over time according to the market demand for their good. We setup the nonzero-sum stochastic differential game and its associated system of HJB partial differential equations in the case of linear demand functions. We characterize certain qualitative features of the game using an asymptotic approximation in the limit of small competition. The equilibrium of the game is further studied using numerical solutions. We find that consumers benefit the most when a market is structured with many firms of the same relative size producing highly substitutable goods. However, a large degree of substitutability does not always lead to large drops in price, for example when two firms have a large difference in their size.     

Strategic delegation under cost asymmetry revised

We revisit the Cournot duopoly game with strategic delegation and asymmetric costs of Delbono et al. (2016). In particular, the authors claim that a Prisoner Dilemma always arises. However, we show that, by setting properly the admissible parameter set, if the firms are sufficiently different, the efficient firm is better off when both firms delegate production. Therefore, in contrast with the traditional view, we claim that a Prisoner Dilemma is not an inevitable outcome in a strategic delegation game.     

A finite step algorithm via a bimatrix game to a single controller non-zero sum stochastic game

Given a non-zero sum discounted stochastic game with finitely many states and actions one can form a bimatrix game whose pure strategies are the pure stationary strategies of the players and whose penalty payoffs consist of the total discounted costs over all states at any pure stationary pair. It is shown that any Nash equilibrium point of this bimatrix game can be used to find a Nash equilibrium point of the stochastic game whenever the law of motion is controlled by one player. The theorem is extended to undiscounted stochastic games with irreducible transitions when the law of motion is controlled by one player. Examples are worked out to illustrate the algorithm proposed.The work of this author was supported in part by the NSF grants DMS-9024408 and DMS 8802260.     

Ex-post price stability with convex costs

Homogeneous goods markets with convex costs, do not generally possess Bertrand-Nash equilibria in pure strategies. In order to identify ex-post stable prices in such markets, the set of outcomes feasible in Bertrand competition are analysed as a non-transferable utility coalitional game. The market-clearing price is shown to always implement a strict-core outcome. Moreover, where at least two sellers compete, the strict-core converges to only admit market-clearing outcomes. The analysis has implications for a number of prominent models of oligopoly competition. When firms engage in capacity pre-commitment, the set of ex-post stable prices converges to the corresponding Cournot prices. This result holds for arbitrary capacity choices and a general class of rationing rules. Conversely, double-marginalisation is never ex-post stable.     

A private contributions game for joint replenishment

We study a non-cooperative game for joint replenishment by n firms that operate under an EOQ-like setting. Each firm decides whether to replenish independently or to participate in joint replenishment, and how much to contribute to joint ordering costs in case of participation. Joint replenishment cycle time is set by an intermediary as the lowest cycle time that can be financed with the private contributions of participating firms. We characterize the behavior and outcomes under undominated Nash equilibria.     

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.     

Repeated congestion games with bounded rationality

We consider a repeated congestion game with imperfect monitoring. At each stage, each player chooses to use some facilities and pays a cost that increases with the congestion. Two versions of the model are examined: a public monitoring setting where agents observe the cost of each available facility, and a private monitoring one where players observe only the cost of the facilities they use. A partial folk theorem holds: a Pareto-optimal outcome may result from selfish behavior and be sustained by a belief-free equilibrium of the repeated game. We prove this result assuming that players use strategies of bounded complexity and we estimate the strategic complexity needed to achieve efficiency. It is shown that, under some conditions on the number of players and the structure of the game, this complexity is very small even under private monitoring. The case of network routing games is examined in detail.     

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.     

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.     

IPO市场条件下风险投资退出时机的抉择

本文利用博弈论和信息经济学的理论知识。通过建立一个两阶段基本模型，给出了风险企业的退出条件。由于风险投资家要重复地参与资本市场。又将两阶段模型推广到无限期重复博弈，通过建立声誉机制可解决不同类型风险企业的退出时机问题。对低质量的企业继续增加投资的目的不是为了建立声誉，而是为了增加收益。对年轻的风险投资家而言，要建立自己的声誉可通过使企业价格偏低来实现。     

不完全信息动态二维价格博弈模型及其分析

单位生产成本为不完全信息条件下,本文首先讨论了两个企业关于具有一定替代性的两种产品价格的动态二维博弈模型,并求得其精练Bayes均衡.然后分析了当两种产品不存在替代关系时,企业对这两种产品价格进行动态博弈的精练Bayes均衡相当于对这两种产品单独进行博弈的精练Bayes均衡的简单组合.     

Game Options Analysis of the Information Role of Call Policies in Convertible Bonds

Abstract

In debt financing, existence of information asymmetry on the firm quality between the firm management and bond investors may lead to significant adverse selection costs. We develop the two-stage sequential dynamic two-person game option models to analyse the market signalling role of the callable feature in convertible bonds. We show that firms with positive private information on earning potential may signal their type to investors via the callable feature in a convertible bond. We present the variational inequalities formulation with respect to various equilibrium strategies in the two-person game option models via characterization of the optimal stopping rules adopted by the bond issuer and bondholders. The bondholders’ belief system on the firm quality may be revealed with the passage of time when the issuer follows his optimal strategy of declaring call or bankruptcy. Under separating equilibrium, the quality status of the firm is revealed so the information asymmetry game becomes a new game under complete information. To analyse pooling equilibrium, the corresponding incentive compatibility constraint is derived. We manage to deduce the sufficient conditions for the existence of signalling equilibrium of our game option model under information asymmetry. We analyse how the callable feature may lower the adverse selection costs in convertible bond financing. We show how a low-quality firm may benefit from information asymmetry and vice versa, underpricing of the value of debt issued by a high-quality firm.     

考虑相互依赖性的信息系统安全投资及协调机制

考虑信息系统安全相互依赖情形下最优化信息系统连续时间安全投资水平是一个值得研究的问题。首先讨论了非合作博弈下信息系统安全投资的最优策略选择,在此基础上讨论了安全投资效率参数、黑客学习能力、传染风险对信息系统脆弱性及信息系统安全投资率的影响。其次,在推导出两企业在合作博弈情形下最优策略选择的基础上,对比两种情形下的博弈均衡结果,得出合作博弈下的投资水平高于非合作博弈下的投资水平。原因是两个企业的相互依赖关系隐含着企业投资的负外部性,从而导致企业投资不足。最后,构建一种双边支付激励机制消除企业投资不足问题,从而使企业达到合作博弈下的最优投资水平,提高两个企业的收益。     

Technology selection and capacity investment under uncertainty

We analyze the problem of technology selection and capacity investment for electricity generation in a competitive environment under uncertainty. Adopting a Nash-Cournot competition model, we consider the marginal cost as the uncertain parameter, although the results can be easily generalized to other sources of uncertainty such as a load curve. In the model, firms make three different decisions: (i) the portfolio of technologies, (ii) each technology’s capacity and (iii) the technology’s production level for every scenario. The decisions related to the portfolio and capacity are ex-ante and the production level is ex-post to the realization of uncertainty. We discuss open and closed-loop models, with the aim to understand the relationship between different technologies’ cost structures and the portfolio of generation technologies adopted by firms in equilibrium. For a competitive setting, to the best of our knowledge, this paper is the first not only to explicitly discuss the relation between costs and generation portfolio but also to allow firms to choose a portfolio of technologies. We show that portfolio diversification arises even with risk-neutral firms and technologies with different cost expectations. We also investigate conditions on the probability and cost under which different equilibria of the game arise.     

Finding all pure strategy Nash equilibria in a planar location game

In this paper, we deal with a planar location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. If firms set the equilibrium prices in the second stage, the game is reduced to a location game for which pure strategy Nash equilibria are studied assuming that the marginal delivered cost is proportional to the distance between the customer and the facility from which it is served. We present characterizations of local and global Nash equilibria. Then an algorithm is shown in order to find all possible Nash equilibrium pairs of locations. The minimization of the social cost leads to a Nash equilibrium. An example shows that there may exist multiple Nash equilibria which are not minimizers of the social cost.     

Dynamic realization games in newsvendor inventory centralization

Consider a set N of n (> 1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et al. 2000, Games Econ Behav 31:26–49; Muller et al. 2002, Games Econ Behav 38:118–126). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror 2005, IIE Trans Scheduling Logistics 37:93–107). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a, Int J Game Theor 31:341–351). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer’s rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players’ demands at different stages.     

Dynamic pricing with uncertain production cost: An alternating-move approach

This article studies a two-firm dynamic pricing model with random production costs. The firms produce the same perishable products over an infinite time horizon when production (or operation) costs are random. In each period, each firm determines its price and production levels based on its current production cost and its opponent’s previous price level. We use an alternating-move game to model this problem and show that there exists a unique subgame perfect Nash equilibrium in production and pricing decisions. We provide a closed-form solution for the firm’s pricing policy. Finally, we study the game in the case of incomplete information, when both or one of the firms do not have access to the current prices charged by their opponents.     

Security investment and information sharing in the market of complementary firms: impact of complementarity degree and industry size

We study a differential game of information security investment and information sharing in a market consisting of n complementary firms. Two game approaches, the non-cooperative game and the totally cooperative game, are employed to investigate the steady state strategy of each firm. Under certain conditions, a unique steady state can be obtained for both games. We find that the steady state security investment and information sharing level are not always less in the non-cooperative game than that in the totally cooperative game. In addition, some theoretical analyses are made on the impacts of the complementarity degree and industry size on firms’ steady state strategies for both games. Finally, some numerical experiments are conducted to give some insights related to the instantaneous profit in the steady state. It can be found that a firm will obtain more instantaneous profit in the steady state of the totally cooperative game than that of the non-cooperative game, which emphasizes the importance of coordinating strategies. The effects of the complementarity degree and industry size on the instantaneous profits in the steady state are also obtained through the numerical experiment results.     

Spatial competition and collaboration networks

We model the formation of collaboration networks among firms that are located in a circular city as a two-stage game. In the first stage, the firms form collaboration links, and in the second stage, they engage in price competition. If two firms form a link, their production costs in the second stage are reduced. The second stage is a generalization of Salop??s (Bell J Econ 10(1):141?C156, 1979) circular city model. We provide a complete characterization of equilibrium prices of the model. We show that a firm prefers forming a link with a more distant firm if the cost-reducing effects are the same. We discuss the stability and social efficiency of the collaboration networks. When link costs are small, there is no conflict between efficiency and stability of networks. When link costs are significant, there is a conflict between efficiency and stability of networks. We also examine the average distances between linked firms.     

Equilibrium Pricing and Advertising Strategies in a Marketing Channel

This paper is concerned with conflict and coordination in a two-member channel of distribution. We propose a differential game model that includes carryover effects of advertising, expressed by a retailer-specific stock of advertising goodwill. Pricing and advertising strategies for both firms are identified under channel conflict as well as coordination. Dynamic advertising policies are designed as stationary Markov perfect strategies. In a symmetric case, these strategies can be determined in closed form, taking into consideration explicitly nonnegativity constraints on advertising rates. We establish a global result for the relationship between the advertising strategies of the two firms under conflict and coordination.