In recent years, there has been a growing interest in studying the behavior and effects of nonrational investors, who misperceive the distribution of asset values, in financial markets. In these studies, there is an implicit assumption that nonrational investors are relevant and even critical for the study of financial markets.

There are, however, two opposing views on the relevance of this issue in the literature. On the one hand, Black (1986) argues that if all investors are rational and perceive their information correctly, then there will be very little trading in individual assets since it is in the interest of informed traders not to trade with each other (see also Milgrom and Stokey (1982)). Furthermore, if there is little trading and liquidity in individual assets, then it will be difficult to price index funds and derivative assets.

Therefore, the entire financial market cannot function properly without liquidity in individual assets. Black then posits that noise traders, who misperceive their noise as information, provide the necessary liquidity to the market. As a result, informed traders now have incentive to trade and their information is thus incorporated into prices. In essence, the whole structure of financial markets depends on the very presence of these noise traders.

On the other hand, Friedman (1953) argues that nonrational investors are irrelevant because they will be driven out of the market by rational investors eventually in the process of natural selection. In this paper, we draw a line between the two opposing views by examining the viability of nonrational investors and, in particular, the survival of overconfidence and investor sentiment. If nonrational investors could not survive in the long run, then their impact on asset prices and markets would be at best transient.

On the other hand, if nonrational investors could survive, it would then lend support to the relevance of the psychology of investors in studying financial markets. Thus, the key question we confront is whether nonrational investors can survive in the long run. In order to examine the survival issue under a natural selection process, we consider the approach of evolutionary game theory since it is designed to analyze the survival of interactive agents in the evolutionary sense (Maynard Smith (1982) and Friedman (1991)).

In our model, the choice of rational or nonrational types is formulated as a pure strategy in the evolutionary game. The evolution of the population of investor type is essentially driven by the relative fitness of the two strategies in terms of their current payoffs. In this context, we examine the longrun steady state of the population distribution between rational and nonrational investors in the market. While the general evolutionary framework can apply to a variety of cases, we focus on two common scenarios. In the first scenario, we consider pairwise contests where every round of interaction involves two randomly matched individual investors who play a bilateral game in normal form. This scenario captures the strategic element in markets with imperfect competition.

This setup is appropriate for analyzing the kind of markets where there exist a few big players with signifi- cant market power. In the second scenario, we examine playing-the-field contests where a large number of investors interact jointly in the market and yet none of them have market power. This scenario is relevant for studying the competitive market where all investors are price takers. In this paper, we adopt the first scenario to analyze the survival of overconfident traders with market power as described in Kyle and Wang (1997) and the second scenario for the survival of noise traders without market power as described in De Long, Shleifer, Summers, andWaldmann (henceforth DSSW) (1990).

Both Kyle and Wang (1997) and DSSW (1990) deal with static models in which the population share is fixed, and hence the models are inadequate to address the long-run survival issue. In this paper, we extend these static models into evolutionary game models and examine the resulting population dynamic of nonrational traders according to their relative fitness in the game. The population dynamic that emerges from the two models yields remarkably similar results regarding the survival of nonrational traders as a group. First, nonrational traders with negative sentiment will never survive in the long run. This applies to underconfidence in Kyle andWang (1997) and to bearishness in DSSW (1990). Second, nonrational traders with extremely positive sentiment may not survive either. This applies to excessive overconfidence in Kyle and Wang (1997) and to excessive bullishness in DSSW (1990).

Third, nonrational traders with moderately positive sentiment tend to dominate the market, particularly when the variance of the risky asset’s value (which we call the ”fundamental” risk) is large. Note that aggressive trading tends to create a large price impact. Moderate sentiment and large fundamental risk both serve to reduce the adverse price impact. As a result, individual nonrational traders may bankrupt sooner than individual rational traders because of the price risk (Samuelson (1971, 1977)).

But, nonrational traders as a group with a higher expected return can still accumulate wealth at a higher speed than rational traders and hence increase in population. It is in this spirit that we demonstrate the survival of nonrational traders as a group, rather than as individuals. It is important to note that the economic rationale for the survival of overconfi- dence in Kyle andWang (1997) is different from that for the survival of the bullish sentiment in DSSW (1990). In the latter case, the bullish sentiment causes noise traders to hold more of the risky asset than their rational opponents, thus gaining a higher expected return.

In the former case, however, there is no such risk premium, given that all traders are assumed to be risk neutral. Instead, overconfidence leads investors to buy more of the asset when the traders receive good signals and to sell more of the asset with bad signals. As a result, the demand differential between the overconfident traders and their rational opponents tends to be positively correlated with the asset’s value, thus yielding a higher expected return to the overconfident traders. Some recent attempts addressing the survival issue include DSSW (1991), Blume and Easley (1992), Palomino (1996), Wang (1998), and Hirshleifer and Luo (2001). DSSW (1991) study the wealth accumulation process for traders, but they assume the risky assets’ supplies to be infinitely elastic and the returns to be exogenously given. As a result, while investors’ beliefs affect their demands for risky assets, they do not affect the prices of the risky assets.

This is in sharp contrast to our model where both investors’ demands and the equilibrium price are influenced by their beliefs. Blume and Easley (1992) find that nonrational traders can survive better than rational traders if nonrational traders’ utility is closer to log-utility than their rational opponents’. The survival of nonrational traders in this model is due to systematic differences in utility functions. On the one hand, our results capture the same effect of the utility-based argument in the sense that overconfidence and bullish sentiment make nonrational traders trade more aggressively like log-utility traders. On the other hand, the aggressiveness in our model comes from irrational beliefs, rather than different utility functions. Essentially, our paper shows that given the same utility functions, nonrational investors can still survive if their irrational beliefs make them trade more aggressively in the right way.

Moreover, in the case of overconfidence (Kyle andWang (1997)), both rational and nonrational traders are risk-neutral and hence both trade more aggressively than log utility traders do, but still we show that moderate overconfident traders can survive better than their rational opponents. Palomino (1996) finds that spiteful noise traders may earn a higher expected utility than their rational opponents do and eventually dominate the market. In contrast, our model does not assume spiteful behavior for nonrational investors.

Wang (1998) extends Kyle (1985) by incorporating overconfidence into the dynamic model of insider trading. He shows that the overconfident insider trades more aggressively than he or she would if he or she were rational. In anticipating such aggressive informed trading, market makers reduce liquidity. This, in turn, generates greater profits for the overconfident insider at the expense the liquidity traders. This result implies that overconfidence can help a monopolistic insider amass even greater wealth and power, thus strengthening his or her dominance in the market. Hirshleifer and Luo (2001) consider a population dynamic based on imitation of the recent profit in a competitive market.

They find that risk-averse overconfident traders take on more risk and hence earn higher profits than rational traders do. Both Wang (1998) and our paper show that overconfident traders can still make higher profits without such a risk premium and eventually come to dominate the market. Most important, while our survival analysis is much in the spirit of the previous literature, this paper takes a further step forward by explicitly modeling the wealth accumulation process that emerges from the market competition between the group of rational investors and the group of nonrational investors in a large economy. As a result, the population dynamic examined in this paper does not depend on individual adaptation as has often been assumed in previous literature.

The endogenously determined group wealth accumulation process thereby distinguishes the current paper from much of the previous literature with exogenous imitation processes. The plan of this paper is as follows. Section I develops a general population dynamic between rational and nonrational traders in a large economy. The population dynamic conforms to the replicator dynamic in a standard evolutionary game. Section II examines the survival of overconfidence in a pairwise contest based on the trading mechanism in Kyle andWang (1997). Section III examines the survival of investor sentiment in a playing-the-field contest based on the trading mechanism in DSSW (1990). Section IV discusses the robustness and implications of our models and analysis. Section V concludes. All proofs are in the Appendix.

Prof. F. Albert Wang

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