As an application of the cluster size distribution, this chapter models the be­havior of price differences in a share market in which a large number of shares of a holding company are traded.[XVIII]

Agents in the market employ various strategies or trading rules. When we put into the same group or cluster all agents with the same strategy, trading rule, excess demand function, or the like, there typically will be many clusters in the market.

Clusters evolve over time as agents enter or exit the market, and also as they switch their decision rules or behavioral patterns in response to chang­ing economic environments, such as changing market sentiments. In modeling markets, it is important to realize that it is impossible to say in advance how many clusters are going to be present in the market at any given time. We can only sample agents, that is, take a snapshot or freeze the time, and sample some numbers of agents and count the number of different strategies being used at that particular time. There can be, in principle, an infintely many potential strate­gies. For example, random combinations of two basic algorithms in different proportions, say, produce different strategies, because they will have different expected performance and variances or risk characteristics. New decision or trading rules will be invented in the future, and so on. This problem of not knowing the types of agents present in a market is exactly the same as the so- called sampling-of-species problem faced by statisticians in species sampling. See Zabell (1992).

To analyze the behavior of such markets, we consider order statistics of shares of types, that is, we derive distributions of the normalized sizes of the clusters in nonincreasing order, and concentrate on the largest several clusters of agents, if the size distributions are such that most probabilities are concentrated on the first few order statistics. Examining a few such large clusters will give us the approximate behavior of markets as a whole, as we show later in this chapter.

In Chapter 10, the distribution of the partition vector, which describes ran­dom partition patterns of agents over the set of trading rules, is shown to con­verge to the Ewens sampling formula as the number of available rules becomes very large. We examine conditions under which a large number of participants in the market form two groups on opposite sides of the market. We derive sta­tionary distributions of clusters of agents, and look for conditions under which two dominant subgroups nearly occupy 100 percent of the shares of the mar­ket. We assume a certain entry and exit rates for a jump Markov process, and use the order statistics of the sizes of clusters, and their distributions, derived in Chapter 10 to calculate the expected shares and other macroeconomic variables.

These distributions are used, then, to infer behavior of the price differences by assigning specific trading rules for the dominant subgroups. By switching assignment of the rules to the two dominant groups, we also explain switches of the behavior of volatilities of the price differences.

We then concentrate on situations where market participants are positively correlated, which corresponds to small values of the parameter θ in the Ewens distribution. In these cases about 95 percent of the market participants belong to two largest subgroups of agents with two trading rules. Contributions of the remaining 5 percent or so of participants are ignored in examining the market behavior as a whole.

In this way, we can examine market excess demand, and price dynamics. At the end of this chapter a possibility for the existence of a power law is raised.

Our approach thus provides a stochastic generalization of a deterministic model of a share market such as Day and Huang's (1990),^[XIX] by providing a microscopic probabilistic process for agents, changing their strategies.

11.1