تعادل بازی های پویا با بازیگران زیاد: موجودیت، تقریب و ساختار بازار

In this paper we study stochastic dynamic games with many players; these are a fundamental model for a wide range of economic applications. The standard solution concept for such games is Markov perfect equilibrium (MPE), but it is well known that MPE computation becomes intractable as the number of players increases. We instead consider the notion of stationary equilibrium (SE), where players optimize assuming the empirical distribution of othersʼ states remains constant at its long run average. We make two main contributions. First, we provide a rigorous justification for using SE. In particular, we provide a parsimonious collection of exogenous conditions over model primitives that guarantee existence of SE, and ensure that an appropriate approximation property to MPE holds, in a general model with possibly unbounded state spaces. Second, we draw a significant connection between the validity of SE, and market structure: under the same conditions that imply SE exist and approximates MPE well, the market becomes fragmented in the limit of many firms. To illustrate this connection, we study in detail a series of dynamic oligopoly examples. These examples show that our conditions enforce a form of “decreasing returns to larger states;” this yields fragmented industries in the SE limit. By contrast, violation of these conditions suggests “increasing returns to larger states” and potential market concentration. In that sense, our work uses a fully dynamic framework to also contribute to a longstanding issue in industrial organization: understanding the determinants of market structure in different industries.

A common framework to study dynamic economic systems of interacting agents is a stochastic game, as pioneered by Shapley [40]. In a stochastic game agentsʼ actions directly affect underlying state variables that influence their payoff. The state variables evolve according to a Markov process in discrete time, and players maximize their infinite horizon expected discounted payoff. Stochastic games provide a valuable general framework for a range of economic settings, including dynamic oligopolies—i.e., models of competition among firms over time. In particular, since the introduction of the dynamic oligopoly model of Ericson and Pakes [17], they have been extensively used to study industry dynamics with heterogeneous firms in different applied settings (see [14] for a survey of this literature). The standard solution concept for stochastic games is Markov perfect equilibrium (MPE) [21], where a playerʼs equilibrium strategy depends on the current state of all players. MPE presents two significant obstacles as an analytical tool, particularly as the number of players grows large. First is computability: the state space expands in dimension with the number of players, and thus the “curse of dimensionality” kicks in, making computation of MPE infeasible in many problems of practical interest. Second is plausibility: as the number of players grows large, it becomes increasingly difficult to believe that individual players track the exact behavior of the other agents. To overcome these difficulties, previous research has considered an asymptotic regime in which the number of agents is infinite [30] and [27]. In this case, individuals take a simpler view of the world: they postulate that fluctuations in the empirical distribution of other playersʼ states have “averaged out” due to a law of large numbers, and thus they optimize holding the state distribution of other players fixed. Based on this insight, this approach considers an equilibrium concept where agents optimize only with respect to the long run average of the distribution of other playersʼ states; Hopenhayn [27] refers to this concept as stationary equilibrium (SE), and we adopt his terminology. SE are much simpler to compute and analyze than MPE, making this a useful approach across a wide range of applications. In particular, SE of infinite models have also been extensively used to study industry dynamics (see, for example, [33], [35], [31] and [26]). In this paper, we address two significant questions. First, under what conditions is it justifiable to use SE as a modeling tool? We provide theoretical foundations for the use of SE. In particular, our main results provide a parsimonious collection of exogenous conditions over model primitives that guarantee existence of SE, and ensure that an appropriate approximation property holds. These results provide a rigorous justification for using SE of infinite models to study stochastic games with a large but finite number of players. The second question we address relates to a longstanding topic of research in industrial organization: when do industries fragment, and when do they concentrate? In a fragmented industry all firms have small market shares, with no single firm or group of firms becoming dominant. By contrast, in a concentrated industry, a few participants that hold a notable market share can exert significant market power. In dynamic oligopoly models in particular, this is a challenging question to answer due to the inherent complexity of MPE. Our second main contribution is to draw a significant connection between the validity of SE, and market structure: under the same conditions that imply SE exist and an appropriate approximation property holds, under all SE the market becomes fragmented in the limit of many firms. In particular, we interpret our conditions over model primitives as enforcement of a form of “decreasing returns to larger states” for an individual firm, that yields fragmented industries in the limit. By contrast, as we discuss, violation of these conditions suggests “increasing returns to larger states” and potential market concentration. Our main results are described in detail below. 1. Theoretical foundations for SE: Existence of SE. We provide natural conditions over model primitives that guarantee existence of SE over unbounded state spaces. This is distinct from prior work on SE, which typically studies models with compact state spaces. Crucially, considering unbounded state spaces allows us to obtain sharp distinctions between increasing and decreasing returns to higher states, and the resulting concentration or fragmentation of an industry. In addition, even though SE of a given model may exist over any compact state space, it may fail to exist over an unbounded state space. The reason is that agents may have incentives to grow unboundedly large and in this case the steady-state distribution is not well defined. Hence, a key aspect of our conditions is that they ensure the stability of the stochastic process that describes each agentʼs state evolution, and that the resulting steady-state distribution is well defined. In this way, we guarantee the compactness of an appropriately defined “best-response” correspondence. Our conditions also ensure the continuity and convexity of this correspondence, allowing us to use a topological fixed-point approach to prove existence. 2. Theoretical foundations for SE: Approximating MPE. We show that the same conditions over model primitives that ensure the existence of SE, imply that SE of infinite models approximate well MPE of models with a finite number of players, as the number of agents increases. An important condition that is required for this approximation result to hold is that the distribution of playersʼ states in the SE under consideration must possess a light-tail, as originally observed by Weintraub et al. [45] for a sequence of finite games, and in [46] for a limiting infinite model like the one studied in this paper. In a light-tailed equilibrium, no single agent is “dominant;” without such a condition it is not possible for agentsʼ to rationally ignore the state fluctuations of their dominant competitors. Crucially, the light-tail assumption as used by Weintraub et al. [45] and [46] is an endogenous condition on the equilibrium outcome. A central contribution of this work is to develop exogenous conditions over model primitives that ensure the existence of light-tailed SE. In fact, the conditions that guarantee compactness in the existence result ensure that all SE are light-tailed. Thus approximation need not be verified separately; verification of our conditions simultaneously guarantees existence of SE as well as a good approximation to MPE as the number of agents increases. 3. Market structure in dynamic industries. Our results provide important insights into market structure in dynamic industries. The literature on dynamic oligopoly models has largely study individual industries in which market outcomes are very sensitive to certain model features and parameters [14]. In contrast, our results provide conditions for which we can predict important features of the equilibrium market structure for a broad range of parameters and specifications. In particular, our conditions over model primitives imply that all SE are light-tailed, and therefore, in all SE the industry yields a fragmented market structure and no dominant firms emerge. Moreover, all these SE are valid approximations to MPE. While these conditions cannot pin-down the equilibrium exactly, they guarantee that in all of them the market structure is fragmented. In that sense, our work contributes to the “bounds approach” in the industrial organization literature pioneered by Sutton [42], which aims to identify broad structural properties in industries that would yield a fragmented or a concentrated market structure. A novelty of our analysis compared to previous work is that it is done in a fully dynamic framework. To illustrate the connection between our theoretical results and market structure in dynamic industries, we study in detail a collection of three examples in industrial organization. For each of these examples, we demonstrate that our conditions on model primitives that guarantee existence of light-tailed SE can be interpreted as enforcing “decreasing returns to higher states.” Conversely, our analysis of the examples suggests that when these conditions are violated, the resulting models exhibit “increasing returns to higher states,” and SE are not expected to provide accurate approximations or may not even exist. We note that, as emphasized above, unbounded state spaces are necessary to highlight the difference between increasing and decreasing returns to higher states. The first example we discuss is a quality-ladder dynamic oligopoly model where firms can invest to improve a firm-specific state; e.g., a firm might invest in advertising to improve brand awareness, or invest in R&D to improve product quality [39]. Firmsʼ single period profits are determined through a monopolistic competition model. Through a limiting construction where the number of firms and market size both scale to infinity, we use our conditions to show that light-tailed SE exist and approximate MPE asymptotically if the single period profit function exhibits diminishing marginal returns to higher quality. Next, we discuss a model with positive spillovers between firms [24]. Here our conditions impose a form of decreasing returns in the spillover effect that, together with the decreasing returns to investment condition introduced in the previous model, ensure SE exist and provide good approximations to MPE. When the spillover effect is controlled in this way, the market is more likely to fragment. Finally, we discuss a dynamic oligopoly that incorporates “learning-by-doing,” so that firms become more efficient as they gain experience in the marketplace [20]. In this case, we find that firmsʼ learning processes must exhibit decreasing returns to scale to ensure existence of light-tailed SE. These conditions are consistent with prior observations in the literature that suggest industries with prominent learning-by-doing effects will tend to concentrate; our results compactly quantify such intuition. Indeed, in all these examples, our results validate intuition by providing quantifiable insight into market structure. Industries with increasing returns are typically concentrated and dominated by few firms, so SE would not be good approximations. By contrast, our conditions on model primitives delineate a broad range of industries with decreasing returns that become fragmented under SE in the limit, and for which SE provide accurate approximations. The remainder of the paper is organized as follows. Section 2 describes related literature. Section 3 introduces our stochastic game model, and there we define both MPE and SE. We then preview our results and discuss the motivating examples above in detail in Section 4. In Section 5, we develop exogenous conditions over model primitives that ensure existence of light-tailed SE. In Section 6, we show that under our conditions any light-tailed SE approximates MPE asymptotically. Section 7 revisits the examples in light of the theoretical results provided in the two previous sections. We conclude and discuss future research directions in Section 8. Appendix A, Appendix B, Appendix C, Appendix D and Appendix E contain all mathematical proofs as well as important complementary material.

This paper considered stationary equilibrium in dynamic games with many players. Our main results provide a parsimonious set of assumptions on the model primitives which ensure that an SE exists in a large variety of games. We also showed that the same set of assumptions ensure that SE yield fragmented market structures, and is a good approximation to MPE in large finite games. Through a set of examples, we illustrate that our conditions on model primitives can be naturally interpreted as enforcing “decreasing returns to higher states.” We conclude by noting a couple of extensions that can be developed for the models described here. 1. Connections between SE and oblivious equilibrium in finite models. In some contexts, particularly in empirical settings, it may be more appropriate to work over a model with a finite number of agents. In these cases, as discussed in Appendix B, it is possible to define an “oblivious equilibrium” for finite models [45]. We conjecture that under some additional technical conditions over the model primitives we can prove that a sequence of OE satisfies the AME property. 2. Nonstationary equilibrium. Our focus was on SE because it is of practical interest and has received significant attention in the literature. We conjecture, however, that our results can be extended to nonstationary versions of an equilibrium concept based on averaging effects that could be used to approximate transitional short-run dynamics as oppose to long-run behavior.