Dynamic Models of Oligopoly

D. Fudenberg, J. Tirole

• Publisher: Routledge
• Year: 2013
• Print ISBN: 9780415269179
• eBook ISBN: 9781136456121
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

Fudenberg and Tirole use the game-theoretic issues of information, commitment and timing to provide a realistic approach to oligopoly.

Dynamic models of oligopoly†

DREW FUDENBERG and JEAN TIROLE

1. INTRODUCTION

THIS MONOGRAPH surveys some recent work on dynamic oligopoly. While we have tried to cover a selection of topics that is a fairly representative sample of current research, these notes are not intended as an exhaustive survey of the field. As will become clear, we have to a large extent selected topics closely related to our own work.

As the title suggests, we focus mainly on the formalization of strategic relationships. We should acknowledge the literature describing the behavior of a competitive firm as a dynamic control problem (e.g., Gaskins [35], Jacquemin [47], Lesourne [58]). This literature considered the dynamic problem of a firm in a market environment that reacts in a specified exogenous way to the firm’s actions. While this approach was useful in developing some preliminary intuition, it did not endogenize the firm’s rivals’ reactions. This endogenization is precisely the object of the game-theoretic models discussed in this monograph. Our methodological approach is complementary to the more thematic ones which can be found in the industrial organization section of this encyclopedia.

We have greatly benefited from the recent Kreps and Spence [54] survey, “Modelling the Role of History in Industrial Organization and Competition.” Our work should be viewed as a complement to theirs: we cover a somewhat different set of models, and develop them in more detail. We have also geared our presentation to a more technically-minded audience.

Following Kreps and Spence, we can identify two reasons for employing dynamic models of oligopoly in preference to static ones. First, nonstationary industries, whether growing or declining, require explicitly dynamic models. Second, “of equal or perhaps greater importance is that the behavior and performance of a mature industry depend crucially on the history of that industry.” We believe that this history-dependence is best modelled in explicitly dynamic models.

Let us begin with the second issue, that of the influence of history in mature industries. Even when the environment is stationary, dynamic considerations are still quite important.

The oldest and best-known approach to modelling history-dependence in oligopolies is supergame analysis. The basic idea, which goes back to Chamberlin [16], is that oligopolists realize their interdependence and thus do not choose outputs (or prices) under the assumption that their decision will have no effect on the actions of their competitors. Instead, they realize that a price cut may provoke retaliation, and thus have an incentive to behave more “cooperatively.” Thus firms can attain a collusive payoff without the need for an explicit (contractual) agreement, because that payoff can be obtained in a noncooperative equilibrium. This is the game-theoretic explanation of “tacit collusion.” As the supergame story is well known (see Friedman [27] for a detailed analysis, or Kreps and Spence, pp. 16–18, for a sketch), we will not work through it here. We will, however, discuss the recent work of Green and Porter [41] which extends the model to allow for random shocks in demand. In their model, a game-theoretic version of Stigler’s [97] paper, firms cannot observe rivals’ output but only the market price. This gives firms an extra incentive to cheat, because the resulting lower price may be blamed on the demand shock. We also discuss the paper by Brock and Scheinkman [12], on supergames with capacity constraints.

The problem with the supergame approach is that it is all too successful in producing tacit collusion. Supergames have lots of equilibria. While one might expect that the history of the industry would be the main determinant of which outcome would in fact occur, history has no direct role in supergame analysis, in the sense that given the strategies (which are common knowledge) each firm can compute its payoff to any sequence of choices. In supergames, the dependence of the equilibrium strategies on history is a sort of “bootstrap” phenomenon—each firm’s strategy depends on the past only because the other firms’ strategies do. Moreover, many different types of history-dependence are equilibria. The problem may be that supergame models are too abstract to predict oligopoly behavior. As Schelling [82] suggested, in moving from the detail of real games to the formalism of strategy spaces and payoffs, one may leave behind information that is crucial to predicting which equilibrium will occur. Schelling’s theory of “focal points” emphasizes the importance of the names of the strategies (“twelve noon” is a focal point, 2:23 isn’t) and cultural expectations (whether “left” or “right” is the focal point may depend on which side of the street one drives on) in determining behavior. Another, related, drawback of the supergame approach is that most games have a finite horizon. With a fixed finite horizon, backwards induction rules out the bootstrapping which supports most of the supergame equilibria. While supergame analysis easily extends to oligopolies in which there’s a constant and small probability that each period is the last (so that the game does end with probability one), the set of equilibria is much closer to that with a fixed finite horizon if the final period is almost certain to be within a small band.

An alternative strategy for modelling tacit collusion and dynamic coordination is to incorporate directly into the model some reason for history to matter. The hope here is that including more of the “nitty-gritty” of oligopolistic competition may allow for more specific predictions. We will discuss two such alternative approaches—the work of Maskin and Tirole on short-run commitments, and the work of Kreps, Milgrom, Roberts and Wilson on games with incomplete information. Maskin and Tirole [63], [64] study oligopolies in which output (or price) is fixed in the very short run. Such markets are not totally static, because history matters, but the history-dependence is of the stationary variety which might be expected in a mature industry. Thus despite the commitments, the model fits under our second heading of the influence of history in mature industries.

In the incomplete information literature, the past matters, not because of a physical link between past and current variables, but rather because the past conveys information about some unknown characteristics of the other firms. This literature makes two methodological points. First, if the past is thought to matter because it signals future intentions, then the oligopoly model used can and should include uncertainty and inference. Second, incomplete information can explain how collusion can be an equilibrium outcome in repeated games with a finite horizon, by breaking the chain of backwards induction which, for example, yields “always cheat” as the only finite-horizon equilibrium of the repeated prisoner’s dilemma.

We will call this form of incomplete-information model, which uses incomplete information to break a chain of backwards induction, one of “reputation effects.” The distinguishing feature of a reputation-effects model is that while the inference procedure is clearly intended to be descriptive, the incomplete information itself need not be so interpreted. In the “reputation effects” models such as Kreps et al. [52], there is typically a very small amount of incomplete information, i.e., each player’s unknown characteristic almost certainly has a particular, (“normal”) value. This contrast with incomplete-information models such as Milgrom and Roberts [67] in which the incomplete information is itself descriptive, and nonnegligible probability is assigned to the various possible values of each player’s characteristic. The distinction is one of emphasis. The “reputation-effects” models stress the difference that an epsilon of incomplete information can make. These papers may thus be viewed as testing the robustness of the associated complete-information model. Because the incompleteness of the information is small in reputation-effect models, it only matters if the game is repeated a number of times.¹

The reputation-effects approach to repeated games, at least in its existing incarnations, has had the virtue of making fairly specific predictions as to equilibrium outcomes. Thus this approach “permits one … to ‘explain’ the equilibrium that arises as a consequence of the formally specified initial beliefs” (K-S). However, the “equilibrium that arises is extremely sensitive to the initial assessments” (beliefs). In fact, Fudenberg and Maskin [29] have recently shown that if one is willing to accept all types of incomplete information (in arbitrarily small amounts) then by varying the initial beliefs one can obtain any individually rational outcome as an equilibrium, so that in a formal sense we are back to the supergame case as far as predictions go. Of course, not all initial beliefs are equally plausible, so that the reputation-effects models can still be useful for choosing between equilibria. By recasting the problem as one of choosing between different beliefs, these models allow more information to be used in making predictions.

Next we come to the other broad group of dynamic models of oligopoly, those concerned with growing, immature industries, or more generally those concerned with explicitly non-stationary environments. The typical paper of this group identifies one particular “tangible” variable, such as the levels of the firms’ capital stocks, defines (flow) payoffs for each level of that variable, and proceeds to analyze competition over that one variable in isolation. For example, if the variable in question were capacity, one specifies how the firms’ profits in the product market at any point in time depend on the levels of the capital stocks. This approach thus assumes that the manner in which the capital stocks were acquired has no effect on the current situation. The entire “mature industry” problem we’ve just discussed, namely the determination of output in a stationary environment, is thus side-stepped, and subsumed into the specific “profit function.”

We call this “black-boxing” of the product market and all elements of the history except the levels of the capacity (or other “tangible” variable) the “state-space” assumption. The idea is that in an exceedingly complex environment, firms simplify their decision problems by only conditioning their behavior on the variables which directly influence payoffs. This “state-space” assumption greatly simplifies the analysis. It permits the use of continuous-time models, by avoiding the “curse of dimensionality”: Were the strategies to depend on the entire history of play their domains would be infinite-dimensional.

As the work on competition over tangible variables is simpler than the work on mature industries and perhaps less familiar, we will examine it first and in greater detail. Section 2a reviews the continuous-time capacity-choice problem of Spence [93] and Fudenberg and Tirole [30] to introduce the idea of a state-space game and to explain the difference between precommitment (“open-loop”) and perfect equilibria. Section 2 then examines two other models of the competition in tangible variables: the accumulation of “experience” and investment in advertising.

Section 3 treats three examples of competition in lumpy investments: R&D, technology adoption, and location in a growing spatial market. These examples illustrate the idea of preemption, which is an extreme form of the first-mover advantage introduced in Section 2.

Section 4 studies work by Eaton and Lipsey [24] and Maskin and Tirole [63] on entry deterrence and fixed costs in natural monopolies.

Section 5 discusses the “classic” problem of the determination of output in a mature oligopoly. We will discuss the Green and Porter paper on trigger strategies in the presence of fluctuating demand, Brock and Scheinkman on supergames with capacity constraints, and Maskin and Tirole on short-run commitments.

Section 6 discusses incomplete-information models of entry and exit. Here again we use the state-space assumption but in such models the state of the competition does not correspond to the level of a tangible variable but rather to the information each firm possesses about its rivals. In this section we discuss work on predation, limit pricing, and cutthroat competition.

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†This survey is the outgrowth of lectures we gave at the Institute for Mathematical Studies in the Social Sciences (IMSSS), Stanford University, in the summer of 1983. Our research was supported by National Science Foundation grants SES 81–04769 and SES 82–01373 at the IMSSS, and by SES 82–07925 at the University of California, Berkeley. The authors would like to thank John Geanakoplos, David Kreps, Eric Maskin, John Roberts, Jose Scheinkman and Robert Wilson for helpful conversations, and Mordecai Kurz for encouraging us to undertake this project.

1. In a recent survey, Wilson [100] uses “Reputation Effects” more broadly to include all signalling in games of incomplete information. We believe the distinction between “epsilon” and “very” incomplete information is useful, and have been able to think of no better name than “reputation effects” for the “epsilon-incomplete” case, although reputation is certainly an issue in, for example, the limit-pricing papers.

2. COMPETITION IN TANGIBLE VARIABLES

2a. Introduction to state-space games

This section illustrates the use of state-space games by sketching the work of Spence [93] and Fudenberg and Tirole [30] on the use of irreversible investment as a strategic weapon. Earlier papers by Spence [92] and Dixit [20, 21] should be acknowledged, but as these assumed once-and-for-all choices of capital levels they are less suited to our expository purposes (for comprehensive treatments of barriers to entry, see also Encaoua et al. [26], Geroski and Jacquemin [37], and Von Weizsäcker [99]). Also, as we will see, the assumption that the incumbent may not invest after entry has occurred is very restrictive.

Thus, consider a duopoly, with firms indexed by i = 1,2. Flow profits at any time (i.e., profits gross of investment expenditures) are determined by functions πⁱ[K₁(t), K₂(t)], where K_i(t) is firm i’s capital stock at time t. Assume that

where the subscripts i, j denote partial differentiation with respect to K_i and K_j respectively.

Πⁱ(K_i, K_j) is the reduced form for short-run competition, which is presumably price competition. Let us explain why the assumption

, which is the basis for downward-sloping reaction curves, is fairly plausible. Capital may take two forms in our model. Either it increases capacity or it decreases costs (formally, increasing capacity is a way to decrease costs). Let us first interpret capital as capacity. Kreps and Scheinkman [53] have shown that, for a particular rationing scheme, the...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Introduction to the Series
• 1. Introduction
• 2. Competition in Tangible Variables
• 3. Preemption
• 4. Short-run Commitment, Fixed Costs, and Natural Monopolies
• 5. Price Wars and Tacit Collusion
• 6. Investing in Disinformation
• References
• Index