One-to-many non-cooperative matching games

We study a strategic model of wage negotiations between firms and workers. First, we define the stability of an allocation in an environment where firms can employ more than one worker. Secondly, we develop a one-to-many non-cooperative matching game, which is an extension of Kamecke’s one-to-one non-cooperative matching game. The main result shows the equivalence between the stable allocations and the outcomes of the subgame equilibria in the matching game: for any stable allocation in this game there is a subgame perfect equilibrium which induces the allocation on the equilibrium path, and every subgame perfect equilibrium induces a stable allocation on the equilibrium path. Furthermore, as for the existence of a stable allocation, we argue that a stable allocation, as with a subgame perfect equilibrium, does not always exist, but it exists under some conditions, using Kelso and Crawford’s modelling.     

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.     

Stackelberg equilibria in managerial delegation games

Managerial compensation packages do not only influence managers’ behavior, but also have an impact on competing firms. In a managerial delegation game investigating the latter aspect, it is shown that the inherent prisoner’s dilemma situation can be resolved (without changing the normally studied setup or timing). In the first stage, owners choose an incentive function for their managers, in the second stage they choose the weights assigned to that function besides profits and in the third stage managers play a Cournot game. Solving this continuous optimization problem with the implicit function theorem shows that choosing an incentive from the set of “multiplicative incentives”, i.e. any generalized affine transformation of the product of both firms’ quantities, which includes e.g. relative profit, ensures that the Stackelberg outcome is among the set of equilibrium outcomes. Furthermore, it is the unique outcome if the rival owner opts for one of the well-known incentives like sales, revenue or market share. The general approach used allows demonstrating that with no other linear incentive a Stackelberg outcome results and that incentives like profit-to-cost ratio should be avoided. Selecting a multiplicative incentive is a dominant strategy of the game.     

Probabilistic stable rules and Nash equilibrium in two-sided matching problems

We study many-to-many matching with substitutable and cardinally monotonic preferences. We analyze stochastic dominance (sd) Nash equilibria of the game induced by any probabilistic stable matching rule. We show that a unique match is obtained as the outcome of each sd-Nash equilibrium. Furthermore, individual-rationality with respect to the true preferences is a necessary and sufficient condition for an equilibrium outcome. In the many-to-one framework, the outcome of each equilibrium in which firms behave truthfully is stable for the true preferences. In the many-to-many framework, we identify an equilibrium in which firms behave truthfully and yet the equilibrium outcome is not stable for the true preferences. However, each stable match for the true preferences can be achieved as the outcome of such equilibrium.     

双寡头时间限制型免费试用策略竞争分析

目前,提供限时免费试用已经成为软件厂商降低消费者对产品质量不确定性及促进新产品扩散的常见做法。现有研究多局限于垄断情况,对现实中常出现的寡头竞争情况鲜有探讨。在考虑消费者学习异质性的基础上,基于Hotelling模型研究了双寡头企业免费试用策略博弈均衡及其影响因素。结果表明:(1)如果两家企业的期望用户体验都比较好,都应该提供免费试用;如果期望用户体验都比较差,都不提供。(2)否则,均衡结果进一步取决于消费者对他们之间水平差异的敏感性。如果很不敏感,则两家企业必须采取相反的免费试用提供策略,并且只有当它们的期望用户体验差距很大时,优势企业一定提供而劣势企业不提供;否则,保持相反即可。如果非常敏感,则两家企业只需在免费试用提供策略上保持一致即可。(3)消费者完全学习软件所需时间越长,两家企业“都不提供”和“只需一致”的可能性增大,而“都提供”和“必定相反”的可能性降低。这是对免费试用策略竞争问题进行的首次探索,所得结果对相关理论研究具有补充和促进作用,也能为相关实践提供科学的决策依据。     

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.     

Nonlinear Cournot oligopoly games with isoelastic demand function: The effects of different behavior rules

The analysis of asymptotical convergence for the oligopoly game has always been important to characterize the firms’ long-term behavior. In the nonlinear oligopoly competition possibly involving chaotic fluctuations, non-convergent trajectories are particularly undesirable since the resulting behavior will become unpredictable. In this paper, consistent with a traditional assumption that the firms update their outputs simultaneously, we at first construct an adjustment process and discuss the convergence to the equilibrium for a nonlinear Cournot duopoly game with the isoelastic demand function. We indicate that the tendency to instability does rise with the number of firms and the adjustment speeds. In particular, we alter this assumption from simultaneous decisions to sequential decisions so that the latter firms are able to observe the former ones at every time periods. We finally arrive at a conclusion that the unique equilibrium is convergent as long as the adjustment speeds are less than a fixed threshold, no matter what the number of the firms. Our findings show that the firms with sequential decisions can achieve the equilibrium more easily.     

Perfect equilibria in simultaneous-offers bargaining

A generalization of the Nash demand game is examined. Agents make simultaneous offers in each period as to how a pie is to be divided. Incompatible offers send the game to the next period, while compatible offers end the game with a split-the-difference trade. The set of perfect equilibria of this game includes any individually rational outcome, including inefficient outcomes and even including the outcome of perpetual disagreement. We suggest a stronger equilibrium concept of universal perfection, which requires robustness against every rather than just one sequence of perturbed games. The set of universally perfect equilibria also includes all individually rational outcomes. The results provide useful insights into both simultaneous-offers bargaining and the nature of the perfect equilibrium and similar concepts (such as stability and hyperstability) in infinite games.     

Cooperation by indirect revelation through strategic behavior

The paper deals with a one-shot prisoners' dilemma when the players have an option to go to court but cannot verify their testimonies. To solve the problem a second stage is added to a game. At the first stage the players are involved in the prisoners' dilemma and at the second stage they play another game in which their actions are verifiable. In such a setup the information about the actions chosen at the prisoners' dilemma stage can be revealed through strategic behavior of the players during second stage. A mechanism for such revelation in the extended game is described. It provides an existence of a unique sequential equilibrium, which may be obtained by an iterative elimination of dominated strategies and has a number of desirable properties.     

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.     

On implementation via demand commitment games

A simple version of the Demand Commitment Game is shown to implement the Shapley value as the unique subgame perfect equilibrium outcome for any n-person characteristic function game. This improves upon previous models devoted to this implementation problem in terms of one or more of the following: a) the range of characteristic function games addressed, b) the simplicity of the underlying noncooperative game (it is a finite horizon game where individuals make demands and form coalitions rather than make comprehensive allocation proposals and c) the general acceptability of the noncooperative equilibrium concept. A complete characterization of an equilibrium strategy generating the Shapley value outcomes is provided. Furthermore, for 3 player games, it is shown that the Demand Commitment Game can implement the core for games which need not be convex but have cores with nonempty interiors. Received March 1995/Final version February 1997     

Finding location equilibria for competing firms under delivered pricing

We address the problem of finding location equilibria of a location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. Assuming that firms set the equilibrium prices in the second stage, the game is reduced to a location game for which a global minimizer of the social cost is a location equilibrium if demand is completely inelastic and marginal production cost is constant. The problem of social cost minimization is studied for both a network and a discrete location space. A node optimality property when the location space is a network is shown and an Integer Linear Programming (ILP) formulation is obtained to minimize the social cost. It is also shown that multiple location equilibria can be found if marginal delivered costs are equal for all competitors. Two ILP formulations are given to select one of such equilibria that take into account the aggregated profit and an equity criterion, respectively. An illustrative example with real data is solved and some conclusions are presented.     

Costly information acquisition under horizontal competition

We analyze endogenous acquisition of costly information for two firms that sell homogeneous products. Prior to determining its production quantity, either firm has an opportunity to acquire a costly forecast. There exists a correlation between errors in the acquired forecasts. We model the problem as a two-stage game in which the firms first decide whether to acquire their respective forecasts and then decide their production quantities. We derive the equilibrium outcome on information acquisition and production quantity.     

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.     

Sequential share bargaining

This paper presents a new extension of the Rubinstein-St?hl bargaining model to the case with n players, called sequential share bargaining. The bargaining protocol is natural and has as its main feature that the players’ shares in the surplus are determined sequentially rather than simultaneously. The protocol also assumes orderly voting, a restriction on the order in which players respond to a proposal. The bargaining protocol requires unanimous agreement for proposals to be implemented. Unlike all existing bargaining protocols with unanimous agreement, the resulting game has unique subgame perfect equilibrium utilities for any value of the discount factor. The result builds on the analysis of so-called one-dimensional bargaining problems. We show that also one-dimensional bargaining problems have unique subgame perfect equilibrium utilities for any value of the discount factor.     

Internal correlation in repeated games

This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.     

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.     

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.     

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.