Chapters
Anufriev, M. & Hommes, C. 2008, 'Evolutionary switching between forecasting heuristics: An explanation of an asset-pricing experiment', pp. 41-53.
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In this paper we propose an explanation of the findings of a recent laboratory market forecasting experiment. In the experiment the participants were asked to predict prices for 50 periods on the basis of past realizations. Three different aggregate outcomes were observed in an identical environment: slow monotonic price convergence, persistent price oscillations, and oscillatory dampened price fluctuations. Individual predictions exhibited a high degree of coordination, although the individual forecasts were not commonly known. To explain these findings we propose an evolutionary model of reinforcement learning over a set of simple forecasting heuristics. The key element of our model is the switching between heuristics on the basis of their past performance. Simulations show that such evolutionary learning can reproduce the qualitative patterns observed in the experiment. 2008 Springer-Verlag Berlin Heidelberg.
Anufriev, M. & Panchenko, V. 2006, 'Heterogeneous beliefs under different market architectures', pp. 3-15.
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The paper analyzes the dynamics in a model with heterogeneous agents trading in simple markets under different trading protocols. Starting with the analytically tractable model of [4], we build a simulation platform with the aim to investigate the impact of the trading rules on the agents' ecology and aggregate time series properties. The key behavioral feature of the model is the presence of a finite set of simple beliefs which agents choose each time step according to a fitness measure. The price is determined endogenously and our focus is on the role of the structural assumption about the market architecture. Analyzing dynamics under such different trading protocols as the Walrasian auction, the batch auction and the 'order-book' mechanism, we find that the resulting time series are similar to those originating from the noisy version of the model [4]. We distinguish the randomness caused by a finite number of agents and the randomness induced by an order-based mechanisms and analyze their impact on the model dynamics. 2006 Springer-Verlag Berlin Heidelberg.
Anufriev, M. & Dindo, P. 2006, 'Equilibrium return and agents' survival in a multiperiod asset market: Analytic support of a simulation model', pp. 269-282.
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We provide explanations for the results of the Levy, Levy and Solomon model, a recent simulation model of financial markets. These explanations are based upon mathematical analysis of a dynamic model of a market with an arbitrary number of heterogeneous investors allocating their wealth between two assets. The investors' choices are endogenously modeled in a general way and, in particular, consistent with the maximization of an expected utility. We characterize the equilibria of the model and their stability and discuss implications for the market return and agents' survival. These implications are in agreement with the results of previous simulations. Thus, our analytic approach allows to explore the robustness of the previous analysis and to expand its spectrum. 2006 Springer-Verlag Berlin Heidelberg.
Anufriev, M. & Bottazzi, G. 2006, 'Noisy trading in the large market limit' in Mathieu, P., Beaufils, B. & Brandouy, O. (eds), Artificial economics: Agent-based methods in finance, game theory and their applications, Springer, Germany, pp. 137-145.
This paper analyzes to what extent and how the trading activity of a group of heterogeneous agents can be described, in the aggregate, as the result of the investment decision of a single "representative" agent. We consider a two-asset pure exchange economy populated by CRRA traders whose individual demands are functions of the past market history. If individual choices are expressed as noisy versions of a common behavior, and the number of agents is large, one can consider the Large Market Limit of the economy and reduce the model to a low-dimensional stochastic system. We investigate the goodness of this approximation under different market conditions and different agents ecologies. The results of the analysis can be used in the study of the general case with an arbitrary number of heterogeneous agents.
Journal articles
Anufriev, M., Assenza, T., Hommes, C. & Massaro, D. 2013, 'Interest rate rules and macroeconomic stability under heterogeneous expectations', Macroeconomic Dynamics, vol. 17, no. 8, pp. 1574-1604.
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The recent macroeconomic literature stresses the importance of managing heterogeneous expectations in the formulation of monetary policy. We use a simple frictionless dynamic stochastic general equilibrium (DSGE) model to investigate inflation dynamics under alternative interest rate rules when agents have heterogeneous expectations, and update their beliefs based on past performance, as in Brock and Hommes [Econometrica 65(5), 1059-1095 (1997)]. The stabilizing effect of different monetary policies depends on the ecology of forecasting rules (i.e., the composition of the set of predictors), on agents' sensitivity to differences in forecasting performance, and on how aggressively the monetary authority sets the nominal interest rate in response to inflation. In particular, if the monetary authority responds only weakly to inflation, a cumulative process with rising inflation is likely. On the other hand, a Taylor interest rate rule that sets the interest rate more than point for point in response to inflation stabilizes inflation dynamics, but does not always lead the system to converge to the rational expectations equilibrium, as multiple equilibria may persist. Copyright Cambridge University Press 2012.
Anufriev, M., Hommes, C.H. & Philipse, R.H.S. 2013, 'Evolutionary selection of expectations in positive and negative feedback markets', Journal of Evolutionary Economics, vol. 23, no. 3, pp. 663-688.
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An economic environment is a feedback system, where the dynamics of aggregate variables depend on individual expectations and vice versa. The type of feedback mechanism is crucial for the aggregate outcome. Experiments with human subjects (Heemeijer et al., J Econ Dyn Control 33:1052-1072, 2009) have shown that price converges to the fundamental level in a negative feedback environment but fails to do so under positive feedback. We present an explanation of these experimental results by means of a model of evolutionary switching between heuristics. Active heuristics are chosen endogenously, on the basis of their past performance. Under negative feedback an adaptive heuristic dominates explaining fast price convergence, whereas under positive feedback a trend-following heuristic dominates resulting in persistent price deviations and oscillations. 2011 The Author(s).
Anufriev, M., Arifovic, J., Ledyard, J. & Panchenko, V. 2013, 'Efficiency of continuous double auctions under individual evolutionary learning with full or limited information', Journal of Evolutionary Economics, vol. 23, no. 3, pp. 539-573.
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In this paper we explore how specific aspects of market transparency and agents' behavior affect the efficiency of the market outcome. In particular, we are interested whether learning behavior with and without information about actions of other participants improves market efficiency. We consider a simple market for a homogeneous good populated by buyers and sellers. The valuations of the buyers and the costs of the sellers are given exogenously. Agents are involved in consecutive trading sessions, which are organized as a continuous double auction with order book. Using Individual Evolutionary Learning agents submit price bids and offers, trying to learn the most profitable strategy by looking at their realized and counterfactual or "foregone" payoffs. We find that learning outcomes heavily depend on information treatments. Under full information about actions of others, agents' orders tend to be similar, while under limited information agents tend to submit their valuations/costs. This behavioral outcome results in higher price volatility for the latter treatment. We also find that learning improves allocative efficiency when compared to outcomes with Zero-Intelligent traders. 2011 The Author(s).
Anufriev, M. & Tuinstra, J. 2013, 'The Impact of Short-Selling Constraints on Financial Market Stability in a Heterogeneous Agents Model'.
Recent turmoil on global fi nancial markets has led to a discussion on which policy measures should or could be taken to stabilize financial markets. One such a measure that resurfaced is the imposition of short-selling
constraints. It is conjectured that these short-selling constraints reduce speculative trading and thereby have the potential to stabilize volatile financial markets. The purpose of the current paper is to investigate this conjecture
in a standard asset pricing model with heterogeneous beliefs. We model short-selling constraints by imposing trading costs for selling an asset short. We find that the local stability properties of the fundamental rational
expectations equilibrium do not change when trading costs for short-selling are introduced. However, when the asset is overvalued, costs on short-selling increase mispricing and price volatility.
Anufriev, M., Kopnyi, D. & Tuinstra, J. 2013, 'Learning cycles in Bertrand competition with differentiated commodities and competing learning rules', Journal of Economic Dynamics and Control, vol. 37, no. 12, pp. 2562-2581.
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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. 2013 Elsevier B.V.
Anufriev, M. & Hommes, C. 2012, 'Evolution of market heuristics', Knowledge Engineering Review, vol. 27, no. 2, pp. 255-271.
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Abstract The time evolution of aggregate economic variables, such as stock prices, is affected by market expectations of individual investors. Neoclassical economic theory assumes that individuals form expectations rationally, thus forcing prices to track economic fundamentals and leading to an efficient allocation of resources. However, laboratory experiments with human subjects have shown that individuals do not behave fully rationally but instead follow simple heuristics. In laboratory markets, prices may show persistent deviations from fundamentals similar to the large swings observed in real stock prices. Here we show that evolutionary selection among simple forecasting heuristics can explain coordination of individual behavior, leading to three different aggregate outcomes observed in recent laboratory market-forecasting experiments: slow monotonic price convergence, oscillatory dampened price fluctuations, and persistent price oscillations. In our model, forecasting strategies are selected every period from a small population of plausible heuristics, such as adaptive expectations and trend-following rules. Individuals adapt their strategies over time, based on the relative forecasting performance of the heuristics. As a result, the evolutionary switching mechanism exhibits path dependence and matches individual forecasting behavior as well as aggregate market outcomes in the experiments. Our results are in line with recent work on agent-based models of interaction and contribute to a behavioral explanation of universal features of financial markets. Copyright Cambridge University Press 2012.
Anufriev, M. & Bottazzi, G. 2012, 'Asset pricing with heterogeneous investment horizons', Studies in NonLinear Dynamics and Econometrics, vol. 16, no. 4, pp. 1-36.
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We consider an analytically tractable asset pricing model describing the trading activity in a stylized market with two assets. Traders are boundedly rational expected utility maximizers with different beliefs about future prices and different investment horizons. In particular, we analyze the effects of the latter source of heterogeneity on the dynamics of price. We find that in the case with homogeneous agents, longer investment horizons lead to more stable dynamics. This is not true, however, in the case of a mixed population of traders, when the increase of heterogeneity in the investment horizons can introduce instability in the system. Furthermore, the role of heterogeneity turns out to be different for different trading behaviors and its effect on the aggregate dynamics depends on the whole ecology of agents' beliefs.
Anufriev, M. & Hommes, C. 2012, 'Evolutionary selection of individual expectations and aggregate outcomes in asset pricing experiments', American Economic Journal: Microeconomics, vol. 4, no. 4, pp. 35-64.
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In recent "learning to forecast" experiments (Hommes et al. 2005), three different patterns in aggregate price behavior have been observed: slow monotonic convergence, permanent oscillations, and dampened fluctuations. We show that a simple model of individual learning can explain these different aggregate outcomes within the same experimental setting. The key idea is evolutionary selection among heterogeneous expectation rules, driven by their relative performance. The 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 forecasting and aggregate price behavior.
Anufriev, M., Bottazzi, G., Marsili, M. & Pin, P. 2012, 'Excess covariance and dynamic instability in a multi-asset model', Journal of Economic Dynamics and Control, vol. 36, no. 8, pp. 1142-1161.
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The presence of excess covariance in financial price returns is an accepted empirical fact: the price dynamics of financial assets tend to be more correlated than their fundamentals would justify. We advance an explanation of this fact based on an intertemporal equilibrium multi-assets model of financial markets with an explicit and endogenous price dynamics. The market is driven by an exogenous stochastic process of dividend yields paid by the assets that we identify as market fundamentals. The model is rather flexible and allows for the coexistence of different trading strategies. The evolution of assets price and traders' wealth is described by a high-dimensional stochastic dynamical system. We identify the equilibria of the model consistent with a baseline assumption of procedural rationality. We show that these equilibria are characterized by excess covariance in prices with respect to the dividend process. Moreover, we show that in equilibrium there is a positive expected marginal profit in choosing more risky portfolios. As a consequence, the evolutionary pressure generates a trend towards more remunerative strategies, which, in turn, increase the variance of prices and the dynamic instability of the system. 2012 Elsevier B.V..
Anufriev, M. & Bottazzi, G. 2010, 'Market equilibria under procedural rationality', Journal of Mathematical Economics, vol. 46, no. SI6, pp. 1140-1172.
We analyze the endogenous price formation mechanism of a pure exchange economy with two assets, riskless and risky. The economy is populated by an arbitrarily large number of traders whose investment choices are described by means of generic smooth functions of past realizations. These choices can be consistent with (but not limited to) the solutions of expected utility maximization problems. Under the assumption that individual demand for the risky asset is expressed as a fraction of individual wealth, we derive a complete characterization of equilibria. It is shown that irrespectively of the number of agents and of their behavior, all possible equilibria belong to a one-dimensional "Equilibrium Market Curve". This geometric tool helps to illustrate the possibility of different phenomena, as multiple equilibria, and can be used for comparative static analysis. We discuss the relative performances of different strategies and the selection principle governing market dynamics on the basis of the stability analysis of equilibria.
Anufriev, M. & Dindo, P. 2010, 'Wealth-driven selection in a financial market with heterogeneous agents', Journal of Economic Behavior and Organization, vol. 73, no. 3, pp. 327-358.
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We study the co-evolution of asset prices and individual wealth in a financial market with an arbitrary number of heterogeneous boundedly rational investors. Using wealth dynamics as a selection device we are able to characterize the long run market outcomes, i.e., asset returns and wealth distributions, for a general class of competing investment behaviors. Our investigation illustrates that market interaction and wealth dynamics pose certain limits on the outcome of agents' interactions even within the "wilderness of bounded rationality". As an application we consider the case of heterogeneous mean-variance optimizers and provide insights into the results of the simulation model introduced by Levy, Levy and Solomon (1994). 2009 Elsevier B.V. All rights reserved.
Anufriev, M. & Branch, W.A. 2009, 'Introduction to special issue on complexity in economics and finance', Journal of Economic Dynamics and Control, vol. 33, no. 5, pp. 1019-1022.
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Anufriev, M. & Panchenko, V. 2009, 'Asset prices, traders' behavior and market design', Journal of Economic Dynamics and Control, vol. 33, no. 5, pp. 1073-1090.
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The dynamics of a financial market with heterogeneous agents are analyzed under different market architectures. We start with a tractable behavioral model under Walrasian market clearing and simulate it under different trading protocols. The key behavioral feature of the model is the switching by agents between simple forecasting rules on the basis of a fitness measure. By analyzing the dynamics under order-driven protocols we show that the behavioral and structural assumptions of the model are closely intertwined. The high responsiveness of agents to a fitness measure causes excess volatility, but the frictions of the order-driven markets may stabilize the dynamics. We also analyze and compare allocative efficiency and time series properties under different protocols. 2009 Elsevier B.V. All rights reserved.
Anufriev, M. 2008, 'Wealth-driven competition in a speculative financial market: Examples with maximizing agents', Quantitative Finance, vol. 8, no. 4, pp. 363-380.
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This paper demonstrates how both the quantitative and qualitative results of a general, analytically tractable asset-pricing model in which heterogeneous agents behave consistently with a constant relative risk-aversion assumption can be applied to the special case of optimizing behaviour. The analysis of the asymptotic properties of the market is performed using a geometric approach that allows the visualization of all possible equilibria by means of a simple one-dimensional Equilibrium Market Curve. The case of linear (particularly, mean-variance) investment functions is thoroughly analysed. This analysis highlights the features that are specific to linear investment functions. As a consequence, some previous contributions of the agent-based literature are generalized. 2008 Taylor & Francis.
Anufriev, M., Bottazzi, G. & Pancotto, F. 2006, 'Equilibria, stability and asymptotic dominance in a speculative market with heterogeneous traders', Journal of Economic Dynamics and Control, vol. 30, no. 9-10, pp. 1787-1835.
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We consider a pure exchange economy where one risky and one riskless security are traded in discrete time. Individual demands are expressed as fractions of individual wealth and depend on traders' forecasts about future price movement. Introducing the 'Equilibrium Market Line' as the locus of all possible equilibrium returns, we show that, irrespectively of the number of traders and of their investment behavior, the economy possesses isolated equilibria where a single agent dominates the market and continuous manifolds of equilibria where many agents hold finite wealth shares. Moreover, we prove that no global dominance order relation among strategies can be defined. 2006 Elsevier B.V. All rights reserved.
Other
Anufriev, M. & Tuinstra, J. 2013, 'The impact of short-selling constraints on financial market stability in a heterogeneous agents model'.
Recent turmoil on global financial markets has led to a discussion on which policy measures should or could be taken to stabilize financial markets. One such a measure that resurfaced is the imposition of short-selling constraints. It is conjectured that these short-selling constraints reduce speculative trading and thereby have the potential to stabilize volatile financial markets. The purpose of the current paper is to investigate this conjecture in a standard asset pricing model with heterogeneous beliefs. We model short-selling constraints by imposing trading costs for selling an asset short. We find that the local stability properties of the fundamental rational expectations equilibrium do not change when trading costs for short-selling are introduced. However, when the asset is overvalued, costs for short-selling increase mispricing and price volatility. 2013 Elsevier B.V.
Anufriev, M., Tuinstra, J. & Bao, T. 2013, 'Fund Choice Behavior and Estimation of Switching Models: An Experiment'.
We run a laboratory experiment that contributes to the finance literature on "return chasing behavior" studying how investors switch between mutual funds driven by past performance of the funds. The subjects in this experiment make discrete choices between several (2, 3 or 4) experimental funds in multiple periods. The time series of funds' profits are exogenously generated prior to the experiment and subjects are paid for that period according to the profit of the fund they choose. The experimental results show that the investment decision can to a large extent be explained by a discrete choice model ("switching model") with a few lags and a predisposition effect. The intensity of choice parameter \beta in the discrete choice model depends on the structure of the profit time series of the funds, and there is no evidence that it is influenced by experience.
Anufriev, M., Kopnyiz, D. & Tuinstra, J. 2013, 'Learning Cycles in Bertrand Competition with Differentiated Commodities and Competing Learning Rules'.
This paper stresses the importance of heterogeneity in learning rules. We introduce an evolutionary competition between different learning rules and demonstrate that, though these rules can coexist, their convergence
properties are strongly affected by heterogeneity. We consider a Bertrand oligopoly with differentiated goods. Firms do not have full information about the demand structure and they want to maximize their perceived one-period profit
by applying one of two different learning rules: OLS learning and gradient learning. We analytically show that the stability of gradient learning depends on the distribution of learning rules over firms. In particular, as the number
of gradient learners increases, gradient learning may become unstable. We study evolutionary competition between the learning rules by means of computer simulations and illustrate that this change in stability for gradient learning
may lead to cyclical switching between the rules. Stable gradient learning typically gives higher average profit than OLS learning, making firms switch to gradient learning. This however, destabilizes gradient learning which, because
of decreasing profits, makes firms switch back to OLS learning. This cycle may repeat itself indefinitely.
Anufriev, M. & Hommes, C.H. 2009, 'Evolutionary Selection of Individual Expectations and Aggregate Outcomes'.
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 monotonic convergence, permanent oscillations and dampened
fluctuations. We construct a simple model of individual learning, based on
performance based evolutionary selectionor reinforcement learning among
heterogeneous expectations rules, explaining these different 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 forecasting behavior as well as aggregate price behavior.
Anufriev, M. & Branch, W.A. 2009, 'Introduction to the Journal of Economic Dynamics and Control special issue on Complexity in Economics and Finance'.
Anufriev, M. & Bottazzi, G., 'Behavioral Consistent Market Equilibria under Procedural Rationality'.
In this paper we analyze a dynamic, asset pricing model where an arbitrary number of heterogeneous, procedurally rational investors divide their wealth between two assets. Both fundamental dividend process and behavior of traders are modeled in a very general way. In particular, agents' choices are described by means of the generic smooth functions defined on a commonly available information set. The choices are consistent with (but not limited to) the solutions of the expected utility maximization problems.
As a natural rest point of the corresponding dynamics we propose the notion of the Behavioral Consistent Equilibria (BCE) where the aggregate dynamics are consistent with the agents' investment choices. We show that provided that the dividend process is given, all possible equilibria of the system can be characterized by means of one-dimensional Equilibrium Market Line (EML). This geometric tool allows to separate the effects of dividend process and agents' behaviors on the aggregate dynamics. Namely, the precise shape of this line depends on the character of the dividend process, but the realized equilibrium, i.e.~a point on the line, is determined by the ecology of agents' behaviors. We argue that the EML can be useful in investigation of the questions of existence, multiplicity and stability of the BCE and provide corresponding examples. The EML also allows to make the comparative static exercises in a framework with heterogeneous agents and discuss the relative performances of different strategies.
The notion of BCE can be considered as a generalization of the Rational Expectations Equilibrium on the framework with heterogeneous traders. It can be, therefore, useful also in other fields of economics where heterogeneity of actors plays an important role for the aggregate outcome