Evolution and market behavior with endogenous investment rules
Introduction
The Efficient Market Hypothesis (EMH), advanced as a general interpretative and normative framework nearly 40 years ago (Fama, 1970), has grown to become a widely accepted working tool for the economic profession. A reason for the widespread reliance on the EMH is to be found in the market selection argument that supports it: if poorly informed investors are persistently losing wealth in favor of the better informed ones, the formers are driven out of the market by the latters so that, in the long run, prices reflect the best available information about fundamentals.1
Despite its pervasive influence, a general formal proof of the selective capability of financial markets is still lacking. Only fairly recently scholarshiply work has started to investigate this issue. Several behavioral models based on evidence collected from laboratory experiments and real markets (see Barberis and Thaler, 2003, and references therein) contend both the positive and normative aspects of the EMH. Rational and informed behavior does not appear to be a pervasive property of trading, nor does automatically guarantee, even if appropriately implemented, better performances and higher probability to “survive” the speculative struggle. The modeling effort of these studies has been, however, limited to specific examples and no general statements are offered.
A general framework was firstly proposed in Blume and Easley (1992). They investigate wealth-driven selection, and the asymptotic behavior of equilibrium prices, in an i.i.d. economy where traders can use Arrow securities to transfer wealth inter-temporally. Traders are characterized by investment rules, a function of their preferences and beliefs, that specify the fraction of wealth to be invested in each asset. A first result is that the best rule for long term survival is the maximization of expected log utility. This log-rule can be seen as a generalization of the Kelly (1956) rule to equilibrium models. When the trader with the most accurate beliefs employs the log-rule, she gains all the wealth in the long run so that asset prices will ultimately reflect her beliefs. However, when the trader with the most accurate beliefs does not use the log-rule, Blume and Easley are able to construct examples in which she vanishes, suggesting that the market selection argument supporting the EMH deserves further investigation. Two groups of contributions were developed from their analysis.
A first group of works has investigated the outcome of market selection under a more general asset structure when traders use investment rules not necessarily coming from utility maximization (see Evstigneev et al., 2009, for a recent survey). Rules are allowed to depend on histories of exogenous variables, in particular past dividends, but not on endogenous market variables, such as asset equilibrium prices. A robust finding is that knowing the correct probability measure behind the dividend process and investing proportionally to asset expected dividends, a rule named the Generalized Kelly, is the unique dominating rule. The result holds for both short- and long-lived assets with general dividend processes, as shown in Amir et al. (2005) and Evstigneev et al. (2008) respectively. These contributions are silent on the outcome of market selection when a full knowledge of the underlying dividend process is lacking and/or when rules depend on endogenous market variables.
A second group of works has instead focused on selection in a standard general equilibrium framework when traders have rational price expectations but have different beliefs about the dividend process. Given trader-specific preferences and discount factors, investment rules maximize (subjective) discounted expected utility and are functions of beliefs and equilibrium prices. The main objective is to establish if, at least in this idealized market set-up, wealth reallocation selects for traders with the most accurate beliefs, irrespective of preferences. Sandroni (2000) and Blume and Easley (2006) find that when markets are dynamically complete2 the answer is positive. The selection argument behind the EMH is correct: in the long-run only the trader with beliefs “closest” to the truth is selected for, provided that all traders discount future utility at the same rate.
There is however a general question that both groups of contributions leave un-answered: it is not known how wealth-driven selection works when traders use investment rules that depend on endogenous market variables but, at the same time, they do not have rational price expectations. The aim of the present paper is to answer this question. In particular, we build upon the model in Blume and Easley (1992) and provide sufficient conditions for asymptotic survival and dominance of investment rules that depend on current and past equilibrium prices. Our final objective is to move closer to a formal general check of the selective capability of financial markets, even among traders that are not perfectly rational, and understand when there is support for the market selection argument behind the EMH and when not.
The dependence of investment rules on past prices implies a feedback from past market performances to present investment decisions. This effect has already been investigated in several heterogeneous agent models. The main finding is that market instability and asset mis-pricing are in general possible (see Hommes, 2006, LeBaron, 2006). In these works, selection operates according to fitness measures different from relative wealth (Levy et al., 2000, Farmer, 2002, Chiarella and He, 2001). In particular, Brock and Hommes (1998) show that market instability and mis-pricing occur when fitness is determined by short run profitability, while Hommes et al. (2012) study the role of memory in the fitness measure. Moreover, results are often derived for specific investment behaviors and in a partial equilibrium framework. Some gaps have been filled by our previous works, see e.g. Anufriev and Bottazzi (2010), Anufriev et al. (2006), and Anufriev and Dindo (2010), which study wealth-driven selection on the general class of price dependent investment rules in a deterministic and growing economy.
The present paper extends the analysis to stochastic economies. We study market selection by analyzing the random dynamical system that describes prices and wealth dynamics. In Section 3 we introduce our market model. In Section 4 we identify the wealth distributions and prices that are invariant under the wealth-reallocation process and investigate their stochastic stability. We provide general sufficient conditions stating whether any given rule is locally dominating, or locally dominated by, other rules. Our findings apply to investment rules that are general (smooth enough) functions of current and past prices so that a wide spectrum of behaviors can be modeled, including those derived from the maximization of a Constant Relative Risk Aversion (CRRA) expected utility.
In Section 5 we introduce the S-rule, a generalization of the Kelly rule to equilibrium models with a general structure of short-lived assets. When the S-rule trades in the market, it acts as the “local” champion: no matter the set of competing investment rules, it establishes a stable long-run market equilibrium where risky assets are priced proportional to their expected relative dividends. At the same time, it destabilizes all the other long-run market equilibria. However, when the S-rule is not used by any agent, the dependence of investment rules on current and past prices is responsible for multiplicity of stable and unstable equilibria, leading to path dependency and persistent heterogeneity.
A first motivating example is in Section 2 while, in Section 6, different cases of selection failures are reviewed. The discussion clarifies that multiple equilibria and market instability, and thus the failure to reward a unique dominant rule, are related to two mechanisms. Firstly, since average wealth growth rates depend on equilibrium prices, the relative average performance of rules can be different in different market states. For example, it may well happen that, given two rules, the first has the highest average wealth growth rate at the prices determined by the second while the second has the highest average wealth growth rate at the prices determined by the first, so that none can prevail. Alternatively either rule can prevail but only close to those market states where it possesses all the wealth and it determines asset prices. Secondly, instability might be related to price feedback. Even though a given rule has the highest average wealth growth rate for all possible market states, prices feedback may be strong enough to destabilize the market dynamics.
Section snippets
An instructive example
In this section we shall consider the simplest example where market selection on price dependent rules can be fully appreciated.
Time is discrete and uncertainty is modeled as a two states i.i.d. Bernoulli process where is the state of the world at time t. For any t the probability that is π. In each period two Arrow securities can be traded. Security pays 1 if and 0 otherwise. Both securities are in unitary exogenous supply. Assets are traded sequentially in
The model
Given the set of states of the world , we define the set of sequences with elements . For all we denote and ωt all the elements of with the same history till time t. Let be the cylinder σ-algebra and ρ a probability measure on so that is a well-defined probability space. Each is defined as the σ-algebra generated by partial histories ωt. Name θ the shift operator on , so that , and θt the t-times composition of θ, so
Local stability of market selection equilibria
In this section we derive results about the existence and local stability of Market Selection Equilibria for the market dynamics described by (3.8). In presenting our findings it is convenient to treat the case of single survivor equilibria first and consider the multiple survivors case later.
Never vanishing rules
Knowing about the determinants of local stability, the next issue we address is whether there exist rules that never vanish. In Section 5.1 we characterize such a rule, which we name S-rule or . In Section 5.2 we compare it with the Kelly rule and its generalizations. In Section 5.3 we show that in the specific class of investment rules that depend on some given statistics of past prices it is possible to find rules that successfully adapt to .
Throughout this section we work with a pairwise
Examples of selection failure
As discussed in Section 2.3, when rules are constant and assets are Arrow securities, market selection generically rewards a unique rule, and this rule sets prices in the long-run. In this section we show that with price dependent rules this is not the case even if assets are Arrow securities. In Section 6.1 we show that, despite the existence of never vanishing rules, the survival relation on the set of endogenous rules is not transitive and thus rules cannot be ordered according to their
Conclusion
We investigate wealth-driven selection in a repeated market for short-lived assets when traders do not have rational price expectations and use investment rules that depend on current and past equilibrium prices. We derive local stability conditions of long-run Market Selection Equilibria where a rule, or a group of rules, dominates and determines asset prices. A strength point of our results is that they can be applied to any repeated market for short-lived assets as long as their dividend
Acknowledgments
This work benefited from the interaction with Mikhail Anufriev, Larry Blume, David Easley, Cars Hommes, Jan Werner, two anonymous referees, and participants to a number of workshops and seminars. We thank all of them for their feedbacks.
Pietro Dindo is supported by a Marie Curie International Outgoing Fellowship, proposal number 300637, within the 7th European Community Framework Programme. He gratefully acknowledges the hospitality of the Department of Economics at Cornell University. All
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2019, Journal of Mathematical EconomicsCitation Excerpt :Rather, we choose to investigate the evolution of market shares and our interest lies in their terminal values. The topic of market selection is increasingly popular, especially in evolutionary finance, and we refer to Yan (2008), Palczewski and Schenk-Hoppé (2010) and Kogan et al. (2016) for models in continuous time and to Amir et al. (2005), Hens and Schenk-Hoppé (2005), Amir et al. (2013) and Bottazzi and Dindo (2014) for models in discrete time. Earlier seminal models of market selection, e.g., Blume and Easley (1992) and Sandroni (2000), evaluate market selection based on agents’ consumption choices and on the accuracy of their predictions.
Survival in speculative markets
2019, Journal of Economic TheoryCitation Excerpt :If, given a partition, each group has the largest survival index when it has all the endowment, then either group can dominate. Both Theorems 2 and 3 rely on applications of the martingale convergence theorem as used in Bottazzi and Dindo (2014) and Bottazzi and Dindo (2015). In economies with more than two agents, they provide only weak sufficient conditions as, even when one group has null relative consumption, the sign of the survival index difference is likely to depend on the exact consumption distribution within both groups.