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Convergence of Bayesian learning to general equilibrium in mis-specified models
Abstract
A central unanswered question in economic theory is that of price formation in disequilibrium. This paper lays the groundwork for a model that has been suggested as an answer to this question in, particularly, Arrow [Toward a theory of price adjustment, in: M. Abramovitz, et al. (Ed.), The Allocation of Economic Resources, Stanford University Press, Stanford, 1959], Fisher [Disequilibrium Foundations of Equilibrium Economics, Cambridge University Press, Cambridge, 1983] and Hahn [Information dynamics and equilibrium, in: F. Hahn (Ed.), The Economics of Missing Markets, Information, and Games, Clarendon Press, Oxford, 1989]. We consider sellers that monopolistically compete in prices but have incomplete information about the structure of the market they face. They each entertain a simple demand conjecture in which sales are perceived to depend on the own price only, and set prices to maximize expected profits. Prior beliefs on the parameters of conjectured demand are updated into posterior beliefs upon each observation of sales at proposed prices, using Bayes’ rule. The rational learning process, thus, constructed drives the price dynamics of the model. Its properties are analysed. Moreover, a sufficient condition is provided, relating objectively possible events and subjective beliefs, under which the price process is globally stable on a conjectural equilibrium for almost all objectively possible developments of history.
... Turning to the Bayesian learning side of our story, we remark that there is an extensive literature on Bayesian learning in finance and economics in which agents update their beliefs as they observe data. Work includes Hautsch & Hess (2004), Kandel & Pearson (1995), Schinkel et al. (2002), Kalai & Lehrer (1993) each of whom uses this Bayesian learning in quite different setups. For example , Schinkel et al. (2002) apply Bayesian learning to n competitive firms who set prices but do not know the demand function. ...
... Work includes Hautsch & Hess (2004), Kandel & Pearson (1995), Schinkel et al. (2002), Kalai & Lehrer (1993) each of whom uses this Bayesian learning in quite different setups. For example , Schinkel et al. (2002) apply Bayesian learning to n competitive firms who set prices but do not know the demand function. They observe demand at each step and use this to update their posterior belief for the state of the world, which then impacts their perceived demand function. ...
... Turning to the Bayesian learning side of our story, we remark that there is an extensive literature on Bayesian learning in finance and economics in which agents update their beliefs as they observe data. Work includes Hautsch & Hess (2004), Kandel & Pearson (1995), Schinkel et al. (2002), Kalai & Lehrer (1993 each of whom uses this Bayesian learning in quite different setups. For example, Schinkel et al. (2002) apply Bayesian learning to n competitive firms who set prices but do not know the demand function. ...
This paper will examine a model with many agents, each of whom has a different belief about the dynamics of a risky asset. The agents are Bayesian and so learn about the asset over time. All agents are assumed to have a finite (but random) lifetime. When an agent dies, he passes his wealth (but not his knowledge) onto his heir. As a result, the agents never become sure of the dynamics of the risky asset. We derive expressions for the stock price and riskless rate. We then use numerical examples to exhibit their behaviour. Comment: 6 figures
... A large stream of economics literature is concerned with the long-term behavior of price adjustment processes in oligopolistic settings, cf. Cyert and DeGroot (1970), Kirman (1975), Aghion et al. (1993), Mirman et al. (1993b), Fishman and Gandal (1994), Harrington (1995), Bergemann and Valimaki (1996), Gallego (1998), Alepuz and Urbano (1999), Rassenti et al. (2000), Belleflamme and Bloch (2001), Schinkel et al. (2002), Keller and Rady (2003), Dimitrova and Schlee (2003), Tuinstra (2004). One usually assumes that firms are using a certain specific learning scheme, and then studies whether the selling prices converge to a Nash equilibrium. ...
... Another form of misspecification is incorrectly assuming that there are no competitors present. Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004Bischi et al. ( , 2007, Isler and Imhof (2008), Cooper et al. (2014), Anufriev et al. (2013) study the effect of this error on the resulting equilibria in various (linear) models, elaborating on earlier work by Kirman (Kirman, 1975, Brousseau and Kirman, 1992. In an airline revenue-management setting, Cooper et al. (2006) show that incorrectly assuming high-fare and low-fare class passengers behave independently can be detrimental for the firm's revenue. ...
The topic of dynamic pricing and learning has received a considerable amount of attention in recent years, from different scientific communities. We survey these literature streams: we provide a brief introduction to the historical origins of quantitative research on pricing and demand estimation, point to different subfields in the area of dynamic pricing, and provide an in-depth overview of the available literature on dynamic pricing and learning. Our focus is on the operations research and management science literature, but we also discuss relevant contributions from marketing, economics, econometrics, and computer science. We discuss relations with methodologically related research areas, and identify directions for future research.
... Turning to the Bayesian learning side of our story, we remark that there is an extensive literature on Bayesian learning in finance and economics in which agents update their beliefs as they observe data. Work includes Hautsch & Hess (2004), Kandel & Pearson (1995), Schinkel et al. (2002), Kalai & Lehrer (1993) each of whom uses this Bayesian learning in quite different setups. For example, Schinkel et al. (2002) apply Bayesian learning to n competitive firms who set prices but do not know the demand function. ...
... Work includes Hautsch & Hess (2004), Kandel & Pearson (1995), Schinkel et al. (2002), Kalai & Lehrer (1993) each of whom uses this Bayesian learning in quite different setups. For example, Schinkel et al. (2002) apply Bayesian learning to n competitive firms who set prices but do not know the demand function. They observe demand at each step and use this to update their posterior belief for the state of the world, which then impacts their perceived demand function. ...
... As an illustrative example, we assume that demand is linear and players' reaction functions are quadratic (where the specific functional form of the reaction functions is obtained by suitably specifying the cost functions). We then go on to show that in this example for demand misspecificationàmisspecification`misspecificationà la Léonard and Nishimura it is possible that new steady states b In a recent paper Schinkel et al. (2002) consider an oligopolistic price setting model where firms do not know the market demand but have demand conjectures instead. Beliefs are updated in a Bayesian way. ...
... This is an obvious and important extension of the model developed here, but one that takes us beyond the aims and the scope of this paper, so it is left for future research. Attempts to incorporate learning into models with misspecified demand include Schinkel et al. (2002), Szidarovszky and Krawczyk (2003) and Wenzelburger (2003). Finally we make the point that although we have provided a specific example with a linear demand function and quadratic reaction function, we may conjecture that the phenomena observed here would occur more generally in models with downward sloping demand functions and unimodal reaction functions as those are the ingredients that give rise to the multiple steady states and the noninvertible map. ...
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.
... A main take-away from this strand of literature is the importance of having the 'right' amount of price experimentation. The importance of incorporating competition into these learning-and-earning models, and the potential detrimental effect of ignoring competition, has been demonstrated by Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004Bischi et al. ( , 2007, Isler and Imhof (2008), Cooper et al. (2014), and Anufriev et al. (2013), building forth on earlier work by Kirman (Kirman, 1975, 1983, Brousseau and Kirman, 1992. ...
This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29-30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.
... Detrimental outcomes of ignoring competition in pricing strategies are shown by Anufriev et al. (2013), Bischi et al. (2004, Isler and Imhof (2008), Schinkel et al. (2002), andTuinstra (2004). The negative effects are even more harmful in fierce competitive settings such as situations with a high number of competitors or price sensitive customers (van de Geer et al. 2019). ...
Past reviews of studies concerning competitive pricing strategies lack a unifying approach to interdisciplinarily structure research across economics, marketing management, and operations. This academic void is especially unfortunate for online markets as they show much higher competitive dynamics compared to their offline counterparts. We review 132 articles on competitive posted goods pricing on either e-tail markets or markets in general. Our main contributions are (1) to develop an interdisciplinary framework structuring scholarly work on competitive pricing models and (2) to analyze in how far research on offline markets applies to online retail markets.
... The importance of incorporating competition into these learning-and-earning models, and the potential detrimental effect of ignoring competition, has been demonstrated by Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004), Bischi et al. (2007), Isler and Imhof (2008), Cooper et al. (2014) and Anufriev et al. (2013), building forth on earlier work by Kirman (1975Kirman ( , 1983Kirman ( , 1995 and Brousseau and Kirman (1992). ...
This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29-30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.
... Kirman (1995) provides an overview of these papers, and also surveys related work. Recent papers that consider the dynamics of duopoly or oligopoly price competitions and that focus on some variation of modeling error include Schinkel et al. (2002), Tuinstra (2004), and Chiarella and Szidarovszky (2005). ...
We consider repeated pricing games in which two competing sellers use mathematical models to choose the prices of their products. Over the sequence of games, each seller attempts to estimate the values of the parameters of a demand model that expresses demand as a function only of its own price using data comprised only of its own past prices and demand realizations. Thus, as is often the case in practice, the sellers' models do not explicitly account for other sellers. We study the behavior of the sellers' prices and parameter estimates under various assumptions regarding the sellers' knowledge and estimation procedures. We identify situations in which (a) the sellers' prices converge to the Nash equilibrium associated with knowledge of the correct demand model, (b) the sellers' prices converge to the cooperative solution, and (c) the sellers' prices converge to other values that are neither the Nash equilibrium nor the cooperative solution and that depend on the initial prices.
In this paper we consider firms that learn about market conditions by estimating the demand function using past market data. We show that learning may lead to suboptimal outcomes even when the estimated demand function perfectly fits the observations used in the regression and firms thus perceive to have learned the demand function correctly.
We consider the Salop model with three firms and two types of consumers that differ in their sensitivity to product differences. Firms do not know the demand structure and they apply least squares learning to learn the demand function. In each period, firms estimate a linear perceived demand function and they play the perceived best response to the previous-period price of the other firms.
This learning rule can lead to three different outcomes: a self-sustaining equilibrium, the Nash equilibrium or an asymmetric learning-equilibrium. In this last equilibrium one firm underestimates the demand for low prices and it attracts consumers with high sensitivity only. This type of equilibrium has not been found in the literature on least squares learning before. Both the Nash equilibrium and the asymmetric learning-equilibrium are locally stable therefore the model has coexisting stable equilibria.
The book focuses on the dynamics of nonlinear oligopoly models. It discusses the classical Cournot model with a large variety of demand and cost functions that illustrate the many different types of possible best response functions and it shows the existence of unique and multiple equilibria. Particular emphasis is placed on the influence of nonnegativity and capacity constraints. Dynamics are introduced under various assumptions for the adjustment process. An introduction to the analysis of global dynamics is given through some specific examples. The book also considers concave and general oligopolies and gives conditions for the local asymptotic stability of their equilibria, and it investigates global dynamics in some special cases. Other oligopolies examined include market share attraction games, labor-managed oligopolies, partially cooperating firms and models with intertemporal demand attraction. Local/global stability analyses are carried out for these models and the impact of constraints is discussed. The book contains a number of technical appendices that summarize techniques of global dynamics not easily accessible elsewhere. © Springer-Verlag Berlin Heidelberg 2010. All rights are reserved.
We review the recent literature on the dynamics of the tâtonnement process, especially from the perspective of the theory of nonlinear dynamics. We show that complicated dynamical phenomena emerge naturally in the tâtonnement process. We also discuss some conceptual drawbacks of the tâtonnement process and directions for future research.
In the wide attention for the so-called ' new economy' , two, not necessarily compatible, issues meet the eye. Firstly, most discussions apply macro-economic concepts, yet secondly their general gist is that the new economy demands new tools for analysis. In this paper, the existing micro-economic tool-kit is searched for theories with which a grip on economics associated with digitalized information is possible. To that end we introduce the distinction between commodity-information, information-commodities and information-infrastructure. It allows for an application of micro- economic insights in various market structures and their welfare consequences to the new economy. Commodity-information is likely to facilitate coordination issues, and thereby has the potential to increase welfare. To fully exploit this potential, however, concerns related to the reliability of information and the confidence in buyers and sellers have to be dealt with. Information- commodities, on the other hand, carry characteristics that may create natural monopolies. The same is true for aspects of the information-infrastructure. Market developments in both categories are to be kept in check by government, so that the great world-wide-welfare potential of the new economy can materialize.
We consider a price adjustment process in a model of monopolistic compe-tition. Firms have incomplete information about the demand structure. When they set a price they observe the amount they can sell at that price and they observe the slope of the true demand curve at that price. With this information they estimate a linear demand curve. Given this estimate of the demand curve they set a new optimal price. We investigate the dynamical properties of this learning process. We Þnd that, if the cross-price effects and the curvature of the demand curve are small, prices converge to the Bertrand-Nash equilibrium. The global dynamics of this adjustment process are analyzed by numerical sim-ulations. By means of computational techniques and by applying results from homoclinic bifurcation theory we provide evidence for the existence of strange attractors.
This paper introduces an agent-based micro-simulation model of housing market processes. The model describes aggregate housing market developments, such as price and turnover, as the outcome of households’ decisions to search for a new dwelling, accept an offered dwelling or sell their dwelling. An important feature of the model is that households’ decisions are based on perceptions of housing market probabilities. Households update these perceptions based on observed bargaining outcomes in the market. The model was tested in a simulation experiment and appeared to respond plausibly to different market settings in terms of prices and households’ perception of the market.
I study the problem of a monopolist maximizing a sum of discounted profits facing a linear demand curve whose slope and intercept are unknown. I show that if the monopolist has a mis-specified model, i.e., if the true slope and intercept lie outside of the support of the monopolist's prior beliefs, then actions and beliefs may cycle on every sample path. This behavior is shown to be robust to perturbations in the prior, true parameter, and actions. Such behavior is not possible if the agent's model is correctly specified; instead actions and beliefs necessarily converge.
Subjective utility maximizers, in an infinitely repeated game, will learn to predict opponents' future strategies and will converge to play according to a Nash equilibrium of the repeated game. Players' initial uncertainty is placed directly on opponents' strategies and the above result is obtained under the assumption that the individual beliefs are compatible with the chosen strategies. An immediate corollary is that, when playing a Harsanyi-Nash equilibrium of a repeated game of incomplete information about opponents' payoff matrices, players will eventually play a Nash equilibrium of the real game, as if they had complete information. Copyright 1993 by The Econometric Society.
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.
The most common mode of analysis in economic theory is to assume equilibrium. Yet, without a proper theory of how economies behave in disequilibrium, there is no foundation for such a practice. The necessary step in proposing a foundation is the formulation of a theory of stability, and in this 1984 book, Professor Fisher is primarily concerned with this subject, although disequilibrium behavior itself is analyzed. The author first undertakes a review of the existing literature on the stability of general equilibrium. He then proposes a more satisfactory general model in which agents realize their state of disequilibrium and act on arbitrage opportunities. The interrelated topics of the role of money, the nature of quantity constraints, and the optimal behaviour of arbitraging agents are extensively treated.
This chapter focuses on learning by firms about the demand conditions. Firms are, in general, imperfectly aware of their environment. The firm's picture of the world may be erroneous in two ways. In the first place, its description of the structure of the system may be correct but it may have a false estimate of the parameters. Secondly, it could have an incorrect model of the true structure, and might persist in this view, attributing its prediction errors to its incorrect estimation of the parameters. A very simple duopoly problem in which firms are in error in the sense that they specify an incomplete model and add a random error term is considered. It is shown from examples that in very special cases, the firms by learning will arrive at the equilibrium position that they would have reached had they been fully informed. In other cases, however their learning process leads to equilibria which are different from the true equilibrium. Their behavior is rather perverse in that, far from informing themselves about the true mechanism at work, they construct an incorrect picture of the world but one in which their behavior would be optimal.
Applying the concepts of Nash, Bayesian, and correlated equilibria to the analysis of strategic interaction requires that players possess objective knowledge of the game and opponents' strategies. Such knowledge is often not available. The proposed notions of subjective games and of subjective Nash and correlated equilibria replace essential but unavailable objective knowledge by subjective assessments. When playing a subjective game repeatedly, subjective optimizers converge to a subjective equilibrium. We apply this approach to some well known examples including single- and multi-person, multi-arm bandit games and repeated Cournot oligopoly games. Journal of Economic Literature Classification Numbers: C73 and C83.
The most common mode of analysis in economic theory is to assume equilibrium. Yet, without a proper theory of how economies behave in disequilibrium, there is no foundation for such a practice. The necessary step in proposing a foundation is the formulation of a theory of stability, and Professor Fisher is primarily concerned with this subject, although disequilibrium behavior itself is analyzed. The author first undertakes a review of the existing literature on the stability of general equilibrium. He then proposes a more satisfactory general model in which agents realize their state of disequilibrium and act on arbitrage opportunities. The interrelated topics of the role of money, the nature of quantity constraints, and the optimal behaviour of arbitraging agents are extensively treated.
We study a dynamic market process in which traders condition their beliefs about payoff-relevant parameters on past endogenously generated market data and current exogenous data. We say that a market process is informative if the beliefs of traders who receive only endogenously generated market data converge almost surely to the true parameter value. Our main result is that under standard regularity hypotheses, the generic market process is informative. We define an equilibrium as the limit points of the market process.
Optimal control of a linear process with unknown parameters is considered when the horizon is infinite and rewards are discounted. Active learning strategies are considered, i.e., agents consider the information value of possible actions, as well as current reward. Distributional assumptions are minimal in that no restriction to conjugate families is made. Convergence of beliefs and actions is established. Copyright 1989 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
We study the dynamical system of expectations generated by a simple general equilibrium model of an exchange economy in which each agent considers a finite collection of models, each of which specifies a relationship between payoff-relevant information and equilibrium prices. One of the models under consideration is a correct description of the rational expectations equilibrium. We find that under a Bayesian type of learning process the rational expectations equilibrium is locally stable, but that nonrational equilibria may also be locally stable.
This lecture, delivered in Edinburgh in 1987, proposes that a useful concept of equilibri um requires a theory of the behavior of economic agents out of equili brium. Particular attention is paid to the learning of agents. Bayesi an learning is assumed and several examples are studied. They reveal that, in general, equilibrium will be history dependent. It is sugges ted that recent recognition of the pervasiveness of multiple equilibr ia of the Walrasian kind makes such history dependence more acceptabl e since the learning process on which it is based will single out one among many equilibria if there is convergence. Copyright 1987 by Scottish Economic Society.
The problem of controlling a stochastic process, with unknown parameters over an infinite horizon, with discounting is considered. Agents express beliefs about unknown parameters in terms of distributions. Under general conditions, the sequence of beliefs converges to a limit distribution. The limit distribution may or may not be concentrated at the true parameter value. In some cases, complete learning is optimal; in others, the optimal strategy does not imply complete learning. The paper concludes with examination of some special cases and a discussion of a procedure for generating examples in which incomplete learning is optimal. Copyright 1988 by The Econometric Society.
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.
This paper argues that some of the pathologies identified by the social learning literature are not robust. Incorrect herds need indivisibilities and signals of bounded precision to arise. In smooth models convergence to the correct action and full revelation of information obtains. However, in the presence of noise convergence is slow. Two robust properties of learning from others are identified. The first, a self-correcting property, responsible for the convergence (self-enhancing facet) at a slow rate (self-defeating facet). The second, the existence of an information externality responsible for herding and underinvestment in public information and relevant from the welfare point of view. The results imply that convergence to full-information equilibria in rational expectations market models may be slow. Nevertheless, this does not apply to models in which learning is mostly from the environment. Furthermore, appropriate market mechanisms may speed up convergence even when learning is from others.
Learning in oligopoly: theory, simulation, and experimental evidence
- A P Kirman
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Mistaken beliefs and resultant equilibria Individual Forecasting and Collective Outcomes
- A P Kirman
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Rational learning and rational expectations Arrow and the Ascent of Modern Economic Theory
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