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Oligopoly Game: Price Makers Meet Price Takers
Abstract
The paper studies an oligopoly game, where firms can choose between price-taking and price-making strategies. On a mixed market price takers are always better off than price makers, though the profits of both types decline in the number of price takers. We investigate and confront two possibilities of firms’ decisions about their types: forward-looking equilibrium reasoning and backward-looking individual learning. We find that the Cournot outcome is the only equilibrium prediction and it is learnable if firms are sufficiently sensitive to profit differences. However, with a larger number of firms, a unilateral deviation from Cournot behavior becomes profitable. Under learning this incentive creates a space for permanent oscillations over different markets with a positive but low number of price takers.
... This allows to classify Nash equilibria in evolutionary stable and evolutionary unstable ones and study the (non-)sustainability of cooperation from a Darwinian point of view. An interesting result in this direction shows that the Walrasian equilibrium (where firms are price-takers) is an evolutionary stable equilibrium, while the more profitable Cournot-Nash equilibrium is not, see on this Vega-Redondo (1997), Radi (2017) and Anufriev and Kopányi (2018). ...
In this paper, we reconsider the model in Bischi and Lamantia (J Econ Interact Coord 17:3–27, 2022) and reformulate it in a two-population context. There, the Cournot duopoly market examined is in equilibrium (Cournot-Nash-equilibrium quantities are produced) conditionally to the players’ (heterogeneous) attitudes toward cooperation. To accommodate players’ attitudes, their objective functions partly include the opponent’s profit, resulting in greater (partial) cooperation or hostility toward the opponent than in the standard duopoly setting. An evolutionary selection mechanism determines the survival of cooperative or competitive strategies in the duopoly. The game is symmetric and Bischi and Lamantia (J Econ Interact Coord 17:3–27, 2022) assumes that the two players involved start the game by choosing the same strategic profile. In this way, the full-fledged two-population game simplifies in a one-dimensional map. In this paper, we relax this assumption. On one hand, this approach allows us to investigate entirely the dynamics of the model and the evolutionary stability of the Nash equilibria of the static game that is implicit in the evolutionary setup. In fact, the model with only one population partially represents the system dynamics occurring in an invariant subset of the phase space. As a remarkable result, this extension shows that the steady state of the evolutionary model where all players are cooperative can be an attractor, although only in the weak sense, even when it is not a Nash equilibrium. This occurs when firms have a very high propensity to change strategies to the one that performs better. On the other hand, this approach allows us to accommodate players’ heterogeneity (non-symmetric version of the game), whose analysis confirms the main insights attained in the homogeneous setting.
... Cerboni Baiardi et al. (2015) build an evolutionary exponential replicator oligopoly model, focusing on the coexistence of strong and weak attractors. Similar switching principles are used in Bischi et al. (2015), Anufriev and Kopányi (2018), Lamantia and Radi (2018) and Bischi and Lamantia (2022). ...
We develop a nonlinear duopoly model in which the heuristic expectation formation and learning behavior of two boundedly rational firms may engender complex dynamics. Most importantly, we assume that the firms employ different forecasting models to predict the behavior of their opponent. Moreover, the firms learn by leaning more strongly on forecasting models that yield more precise predictions. An eight-dimensional nonlinear map drives the dynamics of our approach. We analytically derive the conditions under which its unique steady state is locally stable and numerically study its out-of-equilibrium behavior. In doing so, we detect multiple scenarios with coexisting attractors at which the firms’ behavior yields distinctively different market outcomes.
... Radi (2017) demonstrates that the high-rationality best reply rule can be evolutionary dominant and the learning framework leads an oligopoly market to the Nash equilibrium. Similar results are obtained by Anufriev and Kopányi (2018). Vallée and Yıldızoğlu (2009) provide detailed analytical results on genetic learning algorithm and claim that the coordination on the Nash equilibrium is only possible if learning firms discover the decreasing relationship that prevails between observed market price and their quantities. ...
This paper analyzes the learning behavior of firms in a repeated Cournot oligopoly game. Literature shows the degree of information and cognitive capacity of learning firms is a key factor that determines long run outcome of an oligopoly market. In particular, when firms possess the knowledge of market demand and are capable of computing the optimal production quantity given the output of other firms, the resulting market outcome is the Nash equilibrium. On the other hand, imitation that assumes low behavioral sophistication of firms generally favors higher output and converges to the Walrasian equilibrium. In this paper, a reinforcement learning algorithm with low cognitive requirement is adopted to model firms’ learning behavior. Reinforcement learning firms observe past production choices and fine tune them to improve profits. Analytical result shows that the Nash equilibrium is the only fixed point of the reinforcement learning process. Convergence to the Nash equilibrium is observed in computational simulations. When firms are allowed to imitate the most profitable competitor, all states between the Nash equilibrium and the Walrasian equilibrium can be reached. Furthermore, the long run outcome shifts towards the Nash equilibrium as the length of firms’ memory increases.
... This result is partially confirmed by experiments in Apesteguia et al. (2007) and by a nonlinear evolutionary oligopoly in Radi (2017) where the Walrasian equilibrium results to be asymptotically stable, but coordination problems may destabilize this market-equilibrium configuration and the Cournot-Nash equilibrium may gain stability, at least in Milnor's sense (see Milnor 1985). Further relevant contributions on this issue are Vriend (2000) and Anufriev and Kopányi (2018). ...
This paper offers an overview of the literature on the economic and financial applications of theory of nonlinear dynamics, especially bifurcation theory. After a short introductory discussion of the first nonlinear dynamic models in social sciences and the economic relevance of the zoo of bifurcations and complicated dynamics that such models can generate, we present an overview of the literature on nonlinear dynamic models in the areas of underdevelopment, environmental poverty traps, the management of common goods, industrial organization and financial markets. The review of the literature is enriched by reflections and ideas for future research.
Purpose
Over the last years, powerful advances in the area of dynamic games have enriched game theory and made it more applicable to the modeling of real-world competitive strategies. The study of strategic behaviors of firms in an oligopoly market has received little attention, even though real firms have been shown to compete in output and in price in a single industry. The purpose of this study is to propose a game-theoretic approach to studying strategic behaviors of firms in an oligopoly market structure.
Design/methodology/approach
This approach was developed to study market dynamics and pricing strategic behavior of firms that have the possibility of deciding to be one of the two types (price-maker or price-taker) and reconsider the choice overtime on the basis of their current insights and knowledge and their experience. Firms try to improve their performance in the competitive market in a strategic way, by considering their steady-state profits and choosing the best type given the other firms’ types, actions and interactions.
Findings
The results of the present study confirm the previous study that the Cournot market is a stable market, where each firm can be a price-maker and enjoy individual learning as well as social learning. On the contrary, the market with price-takers only is never stable, and, therefore, the Walrasian equilibrium may not be supported in some instances. The Cournot market loses its stability as the number of firms in the market increases due to the fact that it will be more profitable for a firm to switch to price-taking when the number of firms is high enough. In such a situation, when the number of price-takers increases, there are no stable markets and price dynamics are destabilized.
Originality/value
The study and modeling of real-world competitive strategies would enhance the understanding of oligopoly markets. The study of strategic behaviors of firms in an oligopoly market has received little attention, even though real firms have been shown to compete in output and in price in a single industry as price-takers and price-makers.
In this foreword to the Special Issue “Nonlinear dynamics and game-theoretical modeling in economics and finance” we review the contributions in the issue highlighting the economic results and the connections with aspects of dynamic analysis. Indeed, the common theme of the contributions is the focus on system dynamics and the latest analytical techniques. This issue is devoted to celebrating the 70th birthday of Professor Laura Gardini, who inspired a generation of scholars in the study of discrete-time systems and global methods of analysis.
An evolutionary oligopoly game, where firms can select between the best-reply rule and the Walrasian rule, is considered. The industry is characterized by a finite number of ex-ante homogeneous firms that, characterized by naïve expectations, decide next-period output by employing one of the two behavioral rules. The inverse demand function is linear and all firms have the same quadratic and convex cost function (decreasing return to scale). Based upon realized profits, the distribution of behavioral rules is updated according to a replicator dynamics. The model is characterized by two equilibria: the Cournot-Nash equilibrium, where all firms adopt the best-reply rule and produce the Cournot-Nash quantity, and the Walrasian equilibrium, where all firms adopt the Walrasian rule and produce the Walrasian quantity. The analysis reveals that the Walrasian equilibrium is globally stable as long as the rate of change of marginal cost exceeds the sum of residual market price sensitivities to output. If not, the Walrasian equilibrium loses stability and an attractor, representing complicated dynamics with evolutionary stable heterogeneity, arises through a bifurcation. As the propensity of firms to select the more profitable behavioral rule increases, the attractor disappears through a global bifurcation and the Cournot-Nash equilibrium can become a global Milnor attractor. To sum up, the best-reply rule can be evolutionary dominant over the Walrasian rule and this can lead an oligopoly to select the Cournot-Nash equilibrium.
We consider a model of evolutionary competition between adjustment processes in the Cournot oligopoly model and investigate the effect of increasing the number of firms. Our focus is on Nash play versus a general short-memory adaptive adjustment process. We find that, although Nash play has a stabilizing influence, a sufficient increase in the number of firms in the market tends to make the Cournot-Nash equilibrium unstable. This shows that the famous result by Theocharis (Rev Econ Stud 1960), that Cournot oligopoly markets are unstable for more than three firms, is robust, although the instability threshold increases in the presence of Nash firms. We establish that both the existence and the level of this threshold depend on the information costs associated with Nash play. Moreover, the interaction between adjustment processes naturally leads to the emergence of complicated endogenous fluctuations as the number of firms increases, even when demand and costs are linear.
An evolutionary oligopoly game, where firms can select between the best-reply rule and the Walrasian rule, is considered. The industry is characterized by a finite number of ex-ante homogeneous firms that, characterized by naïve expectations, decide next-period output by employing one of the two behavioral rules. The inverse demand function is linear and all firms have the same quadratic and convex cost function (decreasing return to scale). Based upon realized profits, the distribution of behavioral rules is updated according to a replicator dynamics. The model is characterized by two equilibria: the Cournot-Nash equilibrium, where all firms adopt the best-reply rule and produce the Cournot-Nash quantity, and the Walrasian equilibrium, where all firms adopt the Walrasian rule and produce the Walrasian quantity. The analysis reveals that the Walrasian equilibrium is globally stable as long as the rate of change of marginal cost exceeds the sum of residual market price sensitivities to output. If not, the Walrasian equilibrium loses stability and an attractor, representing complicated dynamics with evolutionary stable heterogeneity, arises through a bifurcation. As the propensity of firms to select the more profitable behavioral rule increases, the attractor disappears through a global bifurcation and the Cournot-Nash equilibrium can become a global Milnor attractor. To sum up, the best-reply rule can be evolutionary dominant over the Walrasian rule and this can lead an oligopoly to select the Cournot-Nash equilibrium.
Price-taking has long been mistakenly regarded as an inferior firm behavior in an imperfectly competitive market. This scenario is challenged when a “Naiver’s Paradox” is shown to exist in an oligopolic market where all firms produce the same product with the same technology (cost structure). It is shown that a firm behaving as a naive price-taker with ignorance of its output impact on the market will perform no worse or even better than its rivals in terms of profits achieved, where the latter are assumed to take “Cournot”, “relative profit” or other more advanced strategies. More significantly, when the number of firms in the market is large, a price-taker may achieve higher profit not only in a relative sense, but also in an absolute sense. Such paradoxical outcome is generic, for it results from neither ad hoc assumptions on market structure nor on information sets, but from the conventionally granted “convexity” assumption on cost functions. An analogous phenomenon is observed for oligopsony market.
In this paper we study a model of a quantity-setting duopoly market where firms lack knowledge of the market demand. Using a misspecified demand function firms deter-mine their profit-maximizing choices of their corresponding perceived market game. For illustrative purposes we assume that the (true) demand function is linear and that the reaction functions of the players are quadratic. We then investigate the global dynamics of this game and characterize the number of steady states and their welfare properties. We study the basins of attraction of these steady states and present situations in which global bifurcations of their basins occur when model parameters are varied. The eco-nomic significance of our result is to show that in situations where players choose their actions based on a misspecified model of the environment, additional self-confirming steady states may emerge, despite the fact that the Nash-equilibrium of the game under perfect knowledge is unique. As a consequence the long run outcome of the game and overall welfare is highly dependent upon initial conditions.
In recent 'learning to forecast' experiments with human subjects (Hommes, et al. 2005), three different patterns in aggregate asset price behavior have been observed: slow mono-tonic convergence, permanent oscillations and dampened fluctuations. We construct a simple model of individual learning, based on performance based evolutionary selection or reinforcement learning among heterogeneous expectations rules, explaining these dif-ferent aggregate outcomes. Out-of-sample predictive power of our switching model is higher compared to the rational or other homogeneous expectations benchmarks. Our results show that heterogeneity in expectations is crucial to describe individual forecast-ing behavior as well as aggregate price behavior.
Trade among individuals occurs either because tastes (risk aversion)differ, endowments differ, or beliefs differ. Utilising the concept of`adaptively rational equilibrium' and a recent framework of Brock and Hommes[6, 7] this paper incorporates risk and learning schemes into a simplediscounted present value asset price model with heterogeneous beliefs. Agentshave different risk aversion coefficients and adapt their beliefs (aboutfuture returns) over time by choosing from different predictors orexpectations functions, based upon their past performance as measured byrealized profits. By using both bifurcation theory and numerical analysis, itis found that the dynamics of asset pricing is affected by the relative riskattitudes of different types of investors. It is also found that the externalnoise and learning schemes can significantly affect the dynamics. Comparedwith the findings of Brock and Hommes [7] on the dynamics caused by change ofthe intensity of choice to switch predictors, it is found that many of theirinsights are robust to the generalizations considered: however, the resultingdynamical behavior is considerably enriched and exhibits some significantdifferences.
A paradoxical phenomenon where irrationality or ignorance lead to higher profits is shown to prevail in an oligopolistic market where three firms produce a homogeneous product with identical technologies. Under the conventional assumption on the convexity of cost function, the profit made by a naive price-taker will always be higher than or at least as much as the other two rivals, no matter what kind of strategies the latter may take and whether they form a coalition or not. By sticking to the price-taking strategy, a firm always gains a positive profit improvement, should any one of its rivals try to upgrade from price-taker to the Cournotor. With these incentives, a firm prefers to remain as a naive price-taker.
We propose an oligopoly game where quantity setting firms have incomplete information about the demand function. At each time step they solve a profit maximization problem assuming a linear demand function and ignoring the effects of the competitors’ outputs. Despite such a rough approximation, that we call “Local Monopolistic Approximation” (LMA), the repeated game may converge at a Nash equilibrium of the game played under the assumption of full information. An explicit form of the dynamical system that describes the time evolution of oligopoly games with LMA is given for arbitrary differentiable demand functions, provided that the cost functions are linear or quadratic. In the case of isoelastic demand, we show that the game based on LMA always converges to a Nash equilibrium. This result, compared with “best reply” dynamics, shows that in this particular case less information implies more stability.
This paper contributes empirically to our understanding of informed traders. It analyzes traders' characteristics in a foreign exchange electronic limit order market via anonymous trader identities. We use six indicators of informed trading in a cross-sectional multivariate approach to identify traders with high price impact. More information is conveyed by those traders' trades which--simultaneously--use medium-sized orders (practice stealth trading), have large trading volume, are located in a financial center, trade early in the trading session, at times of wide spreads and when the order book is thin.
Challenging the conventional belief that sophistication in strategy is always better, it was found in W.
Huang (2002a) that a price-taker who adopts the Cobweb strategy yields higher profits than those who adopt
more sophisticated strategies. This study explores the possibility of improving further the relative profit
advantage that the price-taker has over its counterparts through incorporating the growth-rate adjustment
strategy. A linear heterogenous oligopoly model is used to illustrate the merits of such strategy in the case of
disequilibrium. It is shown in theory and supported with numerical simulations that the adoption of growth-rate
adjustment strategy together with price-taking strategy confers on the price-taker the stabilization power
in a dynamically unstable market in addition to better relative performance in terms of major performance
measures.
Conventional economic assumption that more sophistication in decision making
is better than less is challenged with a profitability analysis conducted with
an oligopolistic model consisting of a naive firm and a group of sophisticated
firms. While the naive firm is assumed to adopt a simple Cobweb strategy by
equating its marginal cost of current production to the last period's price,
the sophisticated firms can take either individually or collusively any
conventional sophisticated strategy such as Cournot and Stackelberg
strategies. Contrary to the economic intuition, it is not the sophisticated firms but the
naive firm who triumphs in equilibrium as well as during the dynamical transitionary
periods, no matter what strategies the sophisticated firms may take. Moreover, when the economy turns cyclic or chaotic, a combination of the Cobweb strategy with a cautious adjustment
strategy could also bring relative higher average profits for the naive firm
than its rivals.
We examine the Cournot oligopoly model in the context of social and individual learning. In both models of learning, firms update their decisions about how much to produce via variants of the genetic algorithm updating procedure. Arifovic (1994) found that both models of social and individual learning converged to the Walrasian, competitive equilibrium. Vriend (2000) reports that the model of social learning converges to the Walrasian equilibrium outcome, while the model of individual learning converges to the Cournot–Nash equilibrium. We revisit the issue and conduct simulations varying elements of the updating algorithms, as well as of the underlying economic model. In the analysis of the outcomes of our simulations, we conclude that the convergence to the Cournot–Nash equilibrium is due to two things: the specific way in which production rules’ performance is evaluated coupled with a specific cost function specification. Copyright Springer 2006
Adaptively rational equilibrium is introduced, where agents adapt their beliefs by choosing from a finite set of predictor functions. Agents make a rational predictor choice, based upon a publically available performance measure such as realized past profits. This results in an adaptive belief system, where predictor choice is coupled to the market equilibrium dynamics. As a typical example, the cobweb model with rational versus naive expectations is analyzed. If the market is locally unstable and rational expectations are costly to obtain, a high intensity of choice for predictor selection leads to chaos and strange attractors.
Growing out of a conference on Expectations Formation and Economic Disequilibrium held in New York City in 1981, the papers in this volume provide a complex view of market processes in which individual rationality is no guarantee of convergence to the 'correct' model and the equilibrium coordination of agents' plans. They reject the 'optimality' argument for the rational expectations hypothesis, opening the door to other hypotheses of optimal expectations of agents in the decentralized market economy.
In this paper, we propose an evolutionary model of oligopoly competition where agents can select between different behavioral rules to make decisions on productions. We formalize the model as a general class of evolutionary oligopoly games and then we consider an example with two specific rules, namely Local Monopolistic Approximation and Gradient dynamics. We provide several results on the global dynamic properties of the model, showing that in some cases the attractor of the system may belong to an invariant plane where only one behavioral rule is adopted (monomorphic state). The attractors on the invariant planes can be either strong attractors or weak attractors. However, we also explain why the system can be in a state of Evolutionary Stable Heterogeneity, where it is more profitable for the agents to employ both heuristics in the long term (polymorphic state).
In this paper we analyze a dynamic game of Cournot competition with heterogeneous firms choosing between two different adaptive behavioral rules in deciding output strategies. The underling oligopoly structure is standard: using a constant returns to scale technology, N firms produce homogeneous goods, which are sold in a market characterized by constant price elasticity. In this setup, we assume that a fraction of firms employs a quite rough rule of thumb, the so-called Local Monopolistic Approximation (LMA), whereas the complementary fraction plays Best Reply (BR), a more demanding strategy in terms of information and computation requirements. The model is first considered with exogenously fixed fractions of firms in the two complementary groups. Then it is generalized by considering an endogenous evolutionary switching process between the two behavioral strategies based on profit-driven replicator dynamics. The role of the number of firms, information costs and inertia (or anchoring attitude) in production decisions is analyzed, as well as the influence in the evolutionary process of random noise in the demand function and memory of past profits. Global properties of the oligopoly with evolutionary pressure between behavioral rules are discussed, with particular regard to cases in which the Nash equilibrium is unstable.
Within the seminal cobweb model of Brock and Hommes, firms adapt their price expectations by a profit-based switching behavior between free naïve expectations and costly rational expectations. Brock and Hommes demonstrate that fixed-point dynamics may turn into increasingly complex dynamics as the firms’ intensity of choice increases. We show that policy-makers are able to manage such rational routes to randomness by adjusting profit taxes. In particular, policy-makers should increase (decrease) profit taxes if destabilizing expectations generate higher (lower) profits than stabilizing expectations to alter the composition of applied expectation rules and thereby to promote market stability. Our results are not restricted to cobweb models: a huge body of literature demonstrates that rational routes to randomness may emerge in many different markets.
I. Introduction, 227. — II. Akerman's argument, 229. — III. Reformulation in terms of adaptive expectations, 230. — IV. An
alternative but equivalent formulation, 233. — V. Some empirical results, 236. — VI. More complicated models, 238. — VII.
Conclusions, 240.
In order to explain fairly simply how expectations are formed, we advance the hypothesis that they are essentially the same as the predictions of the relevant economic theory. In particular, the hypothesis asserts that the economy generally does not waste information, and that expectations depend specifically on the structure of the entire system. Methods of analysis, which are appropriate under special conditions, are described in the context of an isolated market with a fixed production lag. The interpretative value of the hypothesis is illustrated by introducing commodity speculation into the system.
This paper stresses the importance of heterogeneity in learning. We consider a Bertrand oligopoly with firms using either least squares learning or gradient learning for determining the price. We demonstrate that convergence properties of the rules are strongly affected by heterogeneity. In particular, gradient learning may become unstable as the number of gradient learners increases. Endogenous choice between the learning rules may induce cyclical switching. Stable gradient learning gives higher average profit than least squares learning, making firms switch to gradient learning. This can destabilize gradient learning which, because of decreasing profits, makes firms switch back to least squares learning.
In ‘experience-weighted attraction’ (EWA) learning, strategies have attractions that reflect initial predispositions, are updated based on payoff experience, and determine choice probabilities according to some rule (e.g., logit). A key feature is a parameter δ that weights the strength of hypothetical reinforcement of strategies that were not chosen according to the payoff they would have yielded, relative to reinforcement of chosen strategies according to received payoffs. The other key features are two discount rates, φ and ρ, which separately discount previous attractions, and an experience weight. EWA includes reinforcement learning and weighted fictitious play (belief learning) as special cases, and hybridizes their key elements. When δ= 0 and ρ= 0, cumulative choice reinforcement results. When δ= 1 and ρ=φ, levels of reinforcement of strategies are exactly the same as expected payoffs given weighted fictitious play beliefs. Using three sets of experimental data, parameter estimates of the model were calibrated on part of the data and used to predict a holdout sample. Estimates of δ are generally around .50, φ around .8 − 1, and ρ varies from 0 to φ. Reinforcement and belief-learning special cases are generally rejected in favor of EWA, though belief models do better in some constant-sum games. EWA is able to combine the best features of previous approaches, allowing attractions to begin and grow flexibly as choice reinforcement does, but reinforcing unchosen strategies substantially as belief-based models implicitly do.
Abstract In a quantity-competition oligopoly, previous studies have shown that a price-taking firm can outperform any rival with identical technology that produces at a different output level at any intertemporal equilibrium. This research seeks to examine this seemingly counter-intuitive fact in a heterogeneous duopoly consisting of an adaptive price-taker and a dynamic optimizer who dynamically optimizes its either absolute or relative profit stream over an infinite planning horizon. With conventional economic assumptions, the Hamiltonian system always converges to a saddle point. If goal of the dynamic optimizer is absolute profit, the price-taker produces more than the dynamic optimizer at the equilibrium, around which there exists a long-run relative profitability portion for the price-taker. Consequently, the price-taker unconsciously makes higher relative profit than the dynamic optimizer in the long-run. In contrast, if the dynamic optimizer pursues the relative profit, both firms produce at the competitive level at the equilibrium while the whole optimal path lies in the relative profitability regime of the dynamic optimizer so that it will enjoy a higher profit than the price-taker before reaching equilibrium. These results provide economical justification for ever-growing evolutionary game theoretical literatures in appreciating price-taking behavior and additionally clarify possible misunderstandings in interpreting their conclusions.
The dynamic behavior of the output in nonlinear oligopolies is examined when the equilibrium is locally unstable. Continuously
distributed time lags are assumed in obtaining information about rivals’ output as well as in obtaining or implementing information
about the firms’ own output. The Hopf bifurcation theorem is used to find conditions under which limit cycle motion is born.
In addition to the classical Cournot model, labor managed and rent seeking oligopolies are also investigated.
This paper studies an explicitly dynamic evolutionary model of Cournot oligopoly in which the behavior of firms is based on imitation of success and experimentation. It is proved that only (evolutionary) stable sets are selected by the unique invariant measure of the induced Markov chain as the probability of experimentation tends to zero. Evolutionary stability refers to a finite population set-up here. We show that stable singletons and symmetric Walrasian equilibria coincide in general. Furthermore, Walrasian equilibria can be approximated by stable proper intervals. These results generalize and extend the approach of Vega-Redondo [Vega-Redondo, F., 1997. The evolution of Walrasian behavior. Econometrica 65, 375–384].
A new approach has been proposed to analyze the strategy-switching dynamics in a discrete model or a non-cooperative game with the following characteristics: (i) the agents are motivated by absolute payoffs rather than relative ones; (ii) the payoffs vary as the composition of the population changes; (iii) the agents are boundedly rational in the sense that they can only predict the outcomes of their own actions. Rich dynamic phenomena are exhibited for such discrete dynamic models or games. The approach is then applied to explore the relative-profitability of price-taking behavior in a quantity-competing heterogeneous oligopoly.
Drawing a conclusion from recent insights in evolutionary game theory, we show that a so-called spite effect implies that there is an essential difference between individual and social learning. We illustrate its consequences for the choice of computational tools in economics and social settings in general by analyzing two variants of a Genetic Algorithm.
This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their ‘beliefs’ in each period in a boundedly rational way, according to a ‘fitness measure’ such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (‘rational animal spirits’). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.
An evolutionary game theoretic model of Cournot competition is investigated. Individuals choose from a finite set of different behavioral rules. Each rule specifies the quantity to be produced in the current period as a function of past quantities. Using more sophisticated rules may require extra information costs. Based upon realized payoffs, the fractions of the population choosing a certain behavioral rule are updated according to the replicator equation with noise. The long-run behavior of the evolutionary system consisting of the population dynamics coupled with the quantity dynamics of the Cournot game may be complicated and endogenous fluctuations may arise. We consider a typical example where firms can choose between two rules: the Nash rule and the best-reply rule. We show that a homoclinic tangency between the stable and unstable manifold of the equilibrium occurs as evolutionary pressure increases, implying bifurcation routes to complicated dynamics and strange attractors.
Dynamic oligopolies are examined without full information on the price function, with each firm using a perceived price function that usually differs from the actual inverse demand function of the market. It is assumed that firms experience time lags in obtaining and implementing information on the price and also on output. Under realistic assumptions we show that without time lags the steady state is always locally asymptotically stable whereas in the presence of time lags situations of local instability may occur. The possibility of limit cycles is examined, and some special cases are considered.
Recent literature shows that learning in oligopoly games might in the long run result in the Cournot or in the Walrasian equilibrium. Which outcome is achieved seems to depend on the underlying learning dynamics. This paper analyzes the forces behind the learning mechanisms determining the long-run outcome. The apparent difference between social and individual learning is caused by different degrees of rationality of the learning agents: Learning the Cournot strategy requires the agents to acquire a large amount of information and behavioral sophistication, while the Walrasian strategy can be shown to be a particular "low-rationality result."
We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.
This paper explores the impact of memory in Cournot oligopolies where firms learn through imitation of success as suggested in Alchian [J. Polit. Econ. 57 (1950) 211] and modeled in Vega-Redondo [Econometrica 65 (1997) 375]. As long as memory includes at least one period, the long-run outcome corresponds to any quantity between the Cournot one and the Walras one. The (evolutionary) stability of the walrasian outcome relies on interfirm comparisons of simultaneously observed profits (i.e., whether a firm earns more than others in a given period), whereas the stability of the Cournot-Nash equilibrium is derived from intertemporal comparisons of profits for each given firm (i.e., whether a deviation pays off for that firm).
Consider an evolutionary context where a given number of quantity-setting oligopolists tend to mimic successful behavior, occasionally experimenting with some small probability. In this context, it is shown that the unique long-run outcome of the process has all firms playing Walrasian, i.e., choosing an output that maximizes profits when taking the market-clearing price as given. (This abstract was borrowed from another version of this item.)
This paper investigates whether agents can learn how to form rational expectations using standard econometric techniques in the case of a linear stochastic supply and demand model with a production lag. This model has a unique rational expectations equilibrium in which the expected price is a linear function of an observable exogenous random variable. Outside of rational expectations equilibrium agents predict the price by using a regression of past prices on the exogenous random variable where the regression is estimated by either ordinary least squares or Bayesian methods. If the agents are Bayesians, they may have diverse prior beliefs on the mean of the estimated parameter, but all have the same precision. This estimation procedure would be appropriate for an outside observer estimating the parameters of the model in rational expectations equilibrium the coefficient of the equation relating the mathematical conditional expectation of the price to the exogenous variable is constant through time. Outside rational expectations equilibrium this coefficient, which changes each time new data change the regression coefficient. The data are generated by a time-varying parameter model where the varying parameter is determined by past data and the estimation procedure. Agents fail to take this feedback into account and so are estimating a misspecific model.
Conventional economic beliefs that 'equilibrium' is better than 'disequilibrium' and 'stability' is better than 'fluctuation' are challenged with a heterogeneous oligopolistic model that consists of a naive firm and a group of sophisticated firms. The naive firm is assumed to adopt a simple Cobweb strategy while the sophisticate firms, who command all market information, form a collusion and best respond the naive firm's current action. When the market equilibrium is unstable, the naive firm is able to turn an explosively diverging market into a bounded but chaotic one by adopting simultaneously a cautious adjustment strategy (that is, limiting the growth rate of output). There exists an upper-bound such that as long as the growth rate does not exceed this bound, the average profits made by all oligopolistic firms are higher than their respective equilibrium profits. Moreover, the average economic surplus can also be higher than the equilibrium surplus. In this sense, chaos is beneficial not only to all oligopolistic firms but also to the economy as a whole.
This paper presents the cobweb model in which competitive firms, in a market for a single good, use a genetic algorithm to update their decision rules about next-period production and sales. The results of simulations show that the genetic algorithm converges to the rational expectations equilibrium for a wider range of parameter values than other algorithms frequently studied within the context of the cobweb model. Price and quantity patterns generated by the genetic algorithm are also compared to the data of experimental cobweb economies. It is shown that the algorithm can capture several features of the experimental behavior of human subjects better than three other learning algorithms that are considered.
In this paper we give a theoretical model of buyers' behaviour on a market for a perishable good where no prices are posted. We show that if buyers learn from their own previous experience there is a sharp division between those who learn to be loyal to certain sellers and those who continue to "shop around". This feature remains in more general models which are simulated and is consistent with empirical data from the Marseille fish market.
This experiment was designed to test various learning theories in the context of a Cournot oligopoly. The authors derive theoretical predictions for the learning theories and test these predictions by varying the information given to subjects. The results show that some subjects imitate successful behavior if they have the necessary information and, if they imitate, markets are more competitive. Other subjects follow a best reply process. On the aggregate level, the authors find that more information about demand and cost conditions yields less competitive behavior, while more information about the quantities and profits of other firms yields more competitive behavior.
The cobweb model: its instability and the onset of chaos
- Chiarella
Convergence in the finite Cournot oligopoly with social and individual learning
- Vallée