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Interacting agents in finance represent a new behavioural, agent-based approach in which financial markets are viewed as complex adaptive systems consisting of many boundedly rational, heterogeneous agents interacting through simple investment strategies, constantly learning from each other as new information becomes available and adapting their behaviour accordingly over time. Simple interactions at the individual, micro level cause sophisticated structure and emergent phenomena at the aggregate, macro level. Recent surveys of this approach are Hommes (2006) and LeBaron (2006).

The traditional approach in finance is based on a representative, rational agent who makes optimal investment decisions and has rational expectations about future developments. Friedman (1953) made an early, strong argument in favour of rationality, arguing that ‘irrational’ agents would lose money whereas rational agents would earn higher profits. This is essentially an evolutionary argument saying that irrational agents will be driven out of the market by rational agents. In a perfectly rational world, information is transmitted instantaneously, asset prices reflect economic fundamentals and asset allocations are efficient. In the traditional view, agents interact only through the price system.

In contrast, Keynes earlier stressed that prices of speculative assets are not solely driven by market fundamentals, but that ‘market psychology’ also plays an important role. Another early critique on perfect rationality is due to Simon (1957), who emphasized that agents are limited in their computing abilities and face information gathering costs. Therefore individual behaviour is more accurately described by simple, suboptimal ‘rules of thumb’. Along similar lines, Tversky and Kahneman (1974) in psychology argued that individual decision behaviour under uncertainty can be better described by simple heuristics and biases. Since the 1990s the traditional view of financial markets has been challenged through developments in bounded rationality (for example, Sargent 1993), behavioural finance (for example, Barberis and Thaler 2003) and computational, agent-based modelling (for example, Tesfatsion and Judd 2006).

Fundamentalists Versus Chartists

Most interacting agents models in finance include two important classes of investors: fundamentalists and chartists. Fundamentalists base their investment decisions upon market fundamentals, such as interest rates, growth of the economy, company’s earnings, and so on. Fundamentalists expect the asset price to move towards its fundamental value and buy (sell) assets that are undervalued (overvalued). In contrast, chartists or technical analysts look for simple patterns, for example, trends in past prices, and base their investment decisions upon extrapolation of these patterns. For a long time, technical analysis has been viewed as ‘irrational’ and, according to the Friedman argument, chartists would be driven out of the market by rational investors. Frankel and Froot (1986) were among the first to emphasize the role of fundamentalists and chartists in real financial markets. Evidence from survey data on exchange rate expectations (for example, Frankel and Froot 1987; Allen and Taylor 1990) shows that at short time horizons (say, up to three months) financial forecasters tend to use destabilizing, trend-following forecasting rules, whereas at longer horizons (say 3–12 months or longer) they tend to use stabilizing, mean- reverting, fundamental forecasts. Frankel and Froot (1986) argue that the interaction of chartists and fundamentalists amplified the strong rise and subsequent fall of the dollar exchange rate in the mid-1980s.

Another simple interacting agent system with chartists and fundamentalists driven by herding behaviour is due to Kirman (1991, 1993). This model was motivated by the puzzling behaviour of ants observed by entomologists. A colony of ants facing two identical food sources distributes asymmetrically, say 80–20 per cent, over the two sources. Moreover, at some point in time the distribution suddenly reverses to 20–80 per cent. Kirman (1993) proposed a simple stochastic model explaining ants’ behaviour and applied it to a financial market setting (Kirman 1991). Agents can choose between two investment strategies – a fundamentalist or a chartist strategy– to invest in a risky asset. Two agents meet at random and with some interaction-conversion probability one agent will adopt the view of the other. There is also a small self-conversion probability that the agent will change her view no matter what the other agent believes. It turns out that, when the interaction- conversion probability is relatively high compared with the self-conversion probability, the distribution of agents is bimodal. The behaviour of the agents is very persistent and the market tends to be dominated by one group for a long time, but then the majority of agents suddenly switches to the other view, and so on.

But what about the Friedman argument? Will not ‘irrational’ technical trading rules be driven out of the market by rational investment strategies? DeLong et al. (1990) presented one of the first models showing that this need not be the case. Their model contains two types of traders, noise traders, with erroneous stochastic beliefs, and rational traders who are perfectly rational and take into account the presence of noise traders. Noise traders create extra risk and risk-averse rational traders are not willing to fully arbitrage away the mispricing. Noise traders bear more risk and can earn higher realized returns than rational traders, and therefore noise traders can survive in the long run. Lux (1995) presents a herding model with fundamentalists and chartists, whose behaviour is driven by imitation and past realized returns, leading to temporary bubbles and sudden crashes. Furthermore, Brock et al. (1992) showed empirically, using 90 years of daily Dow Jones index data, that technical trading rules can generate significant above-normal returns.

Markets as Complex Adaptive Systems

Since the end of the 1980s, multidisciplinary research as done at the Santa Fe Institute (SFI) (for example, Anderson et al. 1988) has stimulated a lot of work on interacting agents in economics and finance. Models of interacting particle systems in physics served as examples of how local interaction at the micro level may explain structure, for example a phase transition, at the macro level. This has motivated economists to study the economy as an evolving complex system.

Arthur et al. (1997) consider the so-called SFI artificial stock market consisting of an ocean of different types of agents choosing among many simple investment strategies. Agents’ investment decisions are affected by their expectations or beliefs about future asset prices. Beliefs affect realized prices, which in turn determine new beliefs, and so on. Prices and beliefs about prices thus co-evolve over time, and agents continuously adapt their behaviour as new observations become available, replacing less successful strategies by more successful ones. Are simple forecasting strategies irrational and will rational traders outperform technical traders in such an artificial market? In general, no. The reason is that a speculative asset market is an expectations feedback system. Imagine a situation where an asset price is overvalued and the majority of traders remains optimistic expecting the rising trend to continue. Aggregate demand will increase and as a result the asset price will rise even further. Optimistic expectations thus become self-fulfilling and chartists will earn higher realized returns than fundamental traders who sold or shortened the asset because they expected a decline in its price. As long as optimistic traders dominate the market and reinforce the price rise, fundamentalists will lose money. Even when the fundamentalists may be right in the long run, there are ‘limits to arbitrage’, for example due to short selling constraints, preventing them from holding their positions long enough against a prevailing optimistic view, as stressed by Shleifer and Vishny (1997).

Emergent Phenomena and Stylized Facts

The interacting agents approach has been strongly motivated by a number of important stylized facts observed in many financial time series (for example, Brock, 1997): (a) unpredictable asset prices and returns; (b) large, persistent trading volume; (c) excess volatility and persistent deviations from fundamental value, and (d) clustered volatility and long memory. According to (a) asset prices are difficult to predict. New information is absorbed quickly in asset prices and there is ‘no easy free lunch’, that is, arbitrage opportunities are difficult to find and exploit. The traditional rational, representative agent framework can explain (a), but has difficulty in explaining the other stylized facts (b)–(d). In particular, in a world with only rational, risk-averse investors with asymmetric information there can be no trade, because no trader can benefit from superior information since other rational traders will anticipate that this agent must have superior information and therefore will not agree to trade (for example, Fudenberg and Tirole 1991). These no-trade theorems are in sharp contrast to the huge daily trading volume observed in real financial markets, which suggests that there must be other types of heterogeneity such as differences in opinion about future movements. Stylized fact (c) means that fluctuations in asset prices are much larger than fluctuations in underlying market fundamentals. This point has been emphasized by, for example, by Shiller (1981). When markets are excessively volatile, prices can deviate from their fundamental values for a long time. Stylized fact (d) means that price fluctuations are characterized by irregular switching between quiet, low volatility phases, with small price fluctuations and turbulent phases of high volatility and large swings in asset prices. Interacting agent models have been able to explain these stylized facts simultaneously (for example, LeBaron et al. 1999; Lux and Marchesi 1999).

Evolutionary Selection of Strategies

Blume (1993) and Brock (1993) present a general probabilistic framework for strategy selection motivated by results from interacting particle systems in physics (see also Föllmer 1974). The probability of agents using strategy h changes over time according to a random utility fitness measure of the general form

$$ {U}_{ht}={\pi}_{ht}+{S}_{ht}+{\varepsilon}_{ht}. $$
(1)

Here πht represents private utility, for example given by (a weighted average of) realized profit, realized utility or forecasting performance. Sht represents social utility measuring herding behaviour or social interactions (see Brock and Durlauf 2001a, b). For example, agents may behave as conformists, that is, they are more likely to follow strategies that are more popular among the population (global interaction) or among their neighbours (local interaction). Agents observe the performance of each strategy with some idiosyncratic errors, represented by εht.

A frequently used model for the probabilities or fractions of the different strategy types is the discrete choice or multinomial logit model

$$ {n}_{ht}={e}^{\beta {U}_{h,t-1}}/{Z}_{t-1}, $$
(2)

where \( {Z}_{t-1}={\sum}_j{e}^{\beta {U}_{j,t-1}} \) is a normalization factor so that the fractions add up to one. When the errors εht in (1) are independently and identically distributed according to a double exponential distribution, the probability of choosing strategy h is exactly given by (2). The crucial feature of (2) is that, the higher the fitness of trading strategy h, the more agents will select strategy h, and therefore it is essentially an evolutionary selection mechanism. Agents are boundedly rational and tend to follow strategies that have performed well in the (recent) past. The parameter β is called the intensity of choice and is inversely related to the variance of the noise εht. It measures how sensitive agents are to selecting the optimal strategy. The extreme case β = 0 corresponds to noise with infinite variance, so that differences in fitness cannot be observed and all fractions will be equal to 1/H, where H is the number of strategies. The other extreme β = +∞ corresponds to the case without noise, so that the deterministic part of the fitness is observed perfectly, and in each period all agents choose the optimal forecast. An increase in the intensity of choice β represents an increase in the degree of rationality concerning strategy selection.

Brock and Hommes (1997, 1998) propose a simple, analytically tractable heterogeneous agent model to show how non-rational strategies can survive evolutionary selection. Brock and Hommes (1997) consider a market with an endogenous evolutionary selection of expectations rules described by the multinomial logit model (2), with fitness given by past realized profits. Agents choose between a set of different forecasting rules and tend to switch to forecasting strategies that have performed well in the recent past. When agents face information gathering costs, because sophisticated rational strategies are more costly to obtain, simple rule of thumb strategies can survive in this market. In Brock and Hommes (1998) this evolutionary selection of strategies is applied to a standard asset pricing model similar to but much simpler than the SFI artificial stock market. Agents choose between fundamentalists’ and chartists’ investment strategies. When the sensitivity to differences in past performance of the strategies is high (that is, the parameter β is high), evolutionary selection of strategies destabilizes the system and leads to complicated, possibly chaotic asset price fluctuations around the benchmark rational expectations fundamental price. The fluctuations are characterized by an irregular switching between a quiet phase with asset prices close to the fundamentals and a more turbulent phase with asset prices following (temporary) trends or bubbles. In contrast with Friedman’s argument, chartists can survive in this evolutionary competition and may on average earn (short-run) profits equal to or even higher than (short-run) profits of fundamentalists.

A common finding in these models is that more rationality, that is, a larger intensity of choice, leads to instability. The intuition is that random choice leads to stability, because agents will be evenly distributed over the strategy space without systematic biases. In contrast, correlated choice may cause instability when, for example, many traders switch to a profitable trend-following strategy. Another common finding is that, when the social interaction effect is strong, multiple equilibria exist and it depends sensitively on the initial state to which of the many equilibria the market system will settle down (for example, Brock and Durlauf, 2001a, b).

Summary and Future Perspectives

Although the approach in finance is relatively new, interacting agent models have been able to explain important stylized facts simultaneously. An interacting agents system acts as a noise filter, transforming and amplifying purely random news about economic fundamentals into an aggregate market outcome exhibiting excess volatility, temporary bubbles and sudden crashes, large and persistent trading volume, clustered volatility and long memory. It should be emphasized that at the aggregate level these asset price fluctuations are highly irregular and unpredictable, there exists no easy free lunch, and arbitrage will be very difficult and risky in such a market.

Much more theoretical work is needed in this area, for example, to find the ‘simplest tractable model’ explaining all important stylized facts. Speculative bubbles have been observed in laboratory experiments of Smith et al. (1988) and more recently in Hommes et al. (2005), showing that coordination on trend-following rules can destabilize a laboratory experimental asset market. Another important topic for future research is estimation of interacting agent models on financial data. Boswijk et al. (2007) is one of the first attempts to estimation of an evolutionary model with fundamentalists versus trend-following chartists using yearly S&P 500 data, suggesting that trend-following behaviour amplified the strong rise in stock prices at the end of the 1990s. More laboratory experiments and estimation of interacting agents models are needed to test the robustness and empirical relevance of the interacting agents approach.

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