Quantity-setting Game Research Articles

Quantity-setting games with a dominant firm

We consider a possible game-theoretic foundation of Forchheimer’s model of dominant-firm price leadership based on quantity-setting games with one large firm and many small firms. If the large firm is the exogenously given first mover, we obtain Forchheimer’s model. We also investigate whether the large firm can emerge as a first mover of a timing game.

Static and Dynamic Quantity-Setting Games: An In-Class Experiment

This paper illustrates the results of a case study on teaching economics issues employing an experiment where students were made to play quantity-setting games employing the Stackelberg and Cournot theory of oligopoly. A strictly theoretical approach to the study of the oligopolistic market structure is replaced by a discovery-learning method. The goal of the in-class experiment is both to illustrate to students the economics theory through learning by doing approach and to allow the instructors to discover how students act when they have to develop their own strategies, placing them in a role similar to that of firms aiming to maximize the profits. The main finding shows how students converged toward the Nash-equilibrium quantity. Several firms or groups of students, who were producing high output level at the beginning of the game later on reduced their output since they realized that their profit could increase by just producing less. At the end of the experiment, students have emphasized that they have really learned what it is like to interact in a market structure where firms can influence the market variables but not absolutely control them.

PRIVATISATION AND TIMING IN A MIXED OLIGOPOLY WITH BOTH FOREIGN AND DOMESTIC FIRMS*

This paper models a mixed oligopoly with both a domestic and a foreign private firm and examines the resulting timing in the quantity setting game. We demonstrate that with a single simultaneous pre‐game delay stage, the resulting endogenous timing has either the public firm leading or the two private firms leading. An alternative characterisation of the pre‐game stage results in the single timing in which the two private firms lead and the public firm follows. For all timings that emerge endogenously, we show that privatisation will always lower domestic welfare but its influence on global welfare is ambiguous.

Mixed Oligopoly under Demand Uncertainty

In this paper we introduce product demand uncertainty in a mixed oligopoly model and reexamine the nature of sub-game perfect Nash equilibrium (SPNE) when firms decide in the first stage whether to lead or follow in the subsequent quantity-setting game. In the non-stochastic setting, Pal (1998) demonstrated that when a public firm competes with a domestic private firm, multiple equilibria exist but the efficient equilibrium outcome is for the public firm to follow. Matsumura (2003a) proved that when the public firm's rival is a foreign private firm, leadership of the public firm is both efficient as well as SPN equilibrium. Our stochastic model shows that when the leader must commit to output before the resolution of uncertainty, multiple SPNE is possible. Whether the equilibrium outcome is public or private leadership hinges upon the degree of privatization and market volatility. More importantly, Pareto-inefficient simultaneous production is a likely SPNE. Our results are driven by the fact that the resolution of uncertainty enhances the profits of the follower firm in a manner that is well known in real option theory.

Nash equilibria in electricity markets with discrete prices

In this paper we analyse the equilibrium structure for a particular type of electricity market. We consider a market with two generators offering electricity into a pool. Generators are centrally dispatched, with cheapest offers used first. The pool price is determined as the highest-priced offer dispatched, and both generators are paid this price for all the electricity they provide. First generators set their price points (at which bids will later be made) and these are announced. Then each generator chooses the quantities to offer at each price. This reflects the behaviour of the Australian electricity market in which prices are set for 24-hours at a time, but different quantities can be offered within each half-hour period. The demand for electricity is uncertain when offers are made (and is drawn from a probability distribution known to both players). We begin by analysing an example of this two stage game for a simple case where only one price can be chosen. The main results of the paper concern the structure of a Nash equilibrium for the quantity-setting sub-game in which each player aims to maximise their expected profit when prices have already been announced. The distribution of demand plays an important role in the existence of a Nash equilibrium. In the quantity setting game there may be Nash equilibria which are not stable. We show that, under certain circumstances, if the equilibrium offers are sufficiently close to the generators’ marginal costs, then the equilibrium will be stable.

Cournot Vs Stackelberg Equilibria with a Public Enterprise and International Competition

This paper examines the validity of alternative assumptions about public enterprise strategies in the presence of both domestic and international competition. It extends the quantity-setting game to a preplay stage and endogenizes the firms' order of moves to show that i) Cournot competition is not the subgame perfect Nash equilibrium of the extended game, ii) the only SPN equilibria are sequential and iii) for some values of structural parameters, public Stackelberg leadership is the unique SPN equilibrium solution of the game. This has a significant consequence for debates over privatisation because with a public Stackelberg leadership game, there is no distortionary effect associated with the operation of a public enterprise in the domestic market.

Using Market Structure to Regulate a Vertically Integrated Monopolist

A natural monopolist whose cost is private information produces a good which is combined with another good that can be produced by the monopolist or by other firms. The agency that regulates the monopolist can impose any of several different market structures in the industry: integrated monopoly, vertical separation with free entry downstream, or liberalization downstream (both integrated and independent production). When several firms produce downstream, a Cournot quantity-setting game with free entry determines the market price. We derive the optimal contracts to offer the monopolist under all three market structures and examine the influence of downstream cost differences on access prices. We then study the optimal regulatory policy where the regulator can condition the downstream market structure on the monopolist's cost report to the regulator. The optimal regulatory policy awards a monopoly to a low-cost upstream firm, but requires free entry downstream if the monopolist reports high upstream costs. Thus, the choice of market structure is an additional tool to limit rent extraction by the monopolist. Simulation analysis reveals the possibility of significant welfare gains from this additional regulatory tool.

Capacity constraints and voluntary output cutback in dynamic Cournot competition

Abstract This paper considers Cournot competition in which duopolists play a quantity-setting game with sticky prices and capacity constraints. A feedback (subgame-perfect) equilibrium is derived and analyzed. With sufficient differentiability, the paper shows the existence and the uniqueness of symmetric feedback equilibrium. We show that there arises a range of constraints defined by two threshold values in which the firms' outputs are below the constraint level in the steady state. Yet the outputs depend nontrivially on the constraint. For outside observers, the constraint does not seem to be binding. But the constraint is binding in that a marginal change in the level of capacity constraint affects firms' steady-state outputs. The result implies that under this constraint the firms cut back their steady-state outputs voluntarily more than stipulated by the constraint. We also show that the procedure to construct a constrained solution from an unconstrained solution by truncation may not always be appropriate. This paper's analysis utilizes the concept of connectable points associated with the solutions to the auxiliary equation of the Hamilton-Jacobi-Bellman equation. Since the concept of connectable points and their characterization does not directly rely upon the linear-quadratic structure of this game, the method can be modified and applied to other differential games with restrictions on the control space(s).

Fulfilled Expectations of Entry

I develop a two-period model of entry deterrence with endogenous market demands such that the consumer's expectation about the future market structure is fulfilled in equilibrium. I find that both expectations (duopoly and monopoly) are fulfilled in the incumbent's price-setting game, while there is a unique equilibrium in the quantity-setting game. I also find that quantity is more entry-deterring than price in the sense that the maximum value of the entry cost that allows entry in the price-setting game is higher than or equal to that of the quantity-setting game.

Incentives to Form Coalitions with Bertrand Competition

In this article we investigate the incentive to merge when firms that produce differentiated products engage in price competition. We demonstrate that mergers of any size are beneficial and are so increasingly: large mergers yield higher profits than smaller ones. This is in contrast to the result that mergers tend to be disadvantageous in quantity-setting games. This qualitative difference follows from the fact that reaction functions are typically upward sloping in price games but downward sloping in quantity games. Thus, the reaction of outsiders reinforces the initial price increase that results from the merger.