Two kinds of state vectors

Here, we begin by introducing two types of state vectors, called empirical distributions (frequencies) and partition vectors. Since the latter type is not used in the economic literature, we discuss it here and indicate why it is needed.

With K choices or types, where K is larger than 2, detailed information on the choices by individual agents is provided by the vector s, where s_i now takes on one of K possible values, and choice patterns may be represented by the vector of demographic fractions, or by (n₁, n₂,..., n_κ), where n.j is the total number of agents making the j th choice.

As touched on in Section 1.2, there is, however, another possibility when types or choices do not have any inherent labels. The key is whether agents or categories are distinguishable or not. Do labels carry intrinsic information, or do they serve as mere labeling? The situation is exactly the same as that of the occupancy problem in which distinguishable or indistinguishable balls are to be placed in distinguishable or indistinguishable boxes. With identical-looking balls in boxes with no labels or distinguishing marks, how do we count the number of possible patterns of ball placements?

Suppose that K boxes, into which agents with the same choices are allocated, are indistinguishable. Let a_i be the number of boxes with i agents in them. With n agents distributed into K boxes, we have ∑"=₁ jaj = n and ∑"=₁ aj = K. The first equation counts the number of agents, and the second the number of boxes. In Section 1.2 we introduced the partition vector. The partition vector a with these a’s as components is a state vector for some purposes. It is called the partition vector by Zabell (1992) in the statistics literature, and is called the allelic vector by Kingman (1980) in the population-genetics literature. Sachkov (1996, p. 82) refers to it as the secondary specification of states. We give several examples in later chapters.

In other examples our interest in modeling lies not so much in the manner n agents are partitioned or clustered among different groups or types, which is captured by the frequencies, as in some structural properties of the patterns of partitions, for example, in the patterns of frequency variations. These are what are called frequencies of frequencies in the literature of ecology or population genetics.

We use probabilities that are invariant with respect to permutations of agents, because we regard agents as interchangeable. Empirical distributions are in­variant under permutations of agents. Random partitions are equiprobable when their partition vectors are the same. These matters will be taken up in this chapter, as well as some technical points on the existence of stationary or equi­librium distributions for the master equations.

2.2