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Two economic applications

We are ready to show that the relevance quotient, via the probabilistic evolution described by (29) and the consequent equilibrium distribution (30), can be used in economics. We take two situations into account. The first involves a town with shops and shoppers; the second a stock market trading in a single asset. Other applications of the relevance quotient have been proposed by Masanao Aoki (1996; 2002) in order to model aggregate behavior in economics. What we are doing enables a better understanding of the meaning of the relevance quotient and of invariance.

Customers and shops: Ewens's limit

Consider a system such a town with N inhabitants and d shops. All the inhabitant buy in some shop in the town, so the number of customers is N too. This system is closed: no new inhabitants come into the town and as a consequence the total number of customers is fixed. The strategy of a customer is to buy from one of the d shops. Customers may change strategy, that is, they may decide to cease buying from one shop and to start buying from another one. The state of the system, N = (Nv...,Nj,...,Nd),%f=1Nj=N, specifies the number of customers of the different shops. Hence, P(N;|N) is the probability the number of customers of shop i decreases by 1 while P(Nj|N) is the probability that, the number of customers of shop i having decreased by 1, the number of customers of shop j increases by 1.

As we have seen, the destruction probability is ruled by (27). In this respect it is worth stressing that the probability of decreasing by 1 the number of customers of a shop, being equal to the fraction of customers of the shop, depends only upon the number of customers of the shop, irrespective of the particular shop taken into account. In turn, this holds if and only if all customers have the same probability of being chosen for a destruction irrespective of the shop of which they are customers. In other words, all customers are on a par with respect to destructions or (which is the same) the destruction of a customer in a shop always happens at random regardless of the shop in which the destruction takes place.

If exchangeability holds, then the probability that an inhabitant becomes a new customer of a shop does not depend upon the identity of the customers of the various shops but only on the numbers of its customers. The condition of invariance reduces this dependence on the number of customers of the shop and on the total number of inhabitants of the town. The destruction probability is the probability with which a customer ceases to be customer of a shop. Conversely, the creation probability is the probability with which an inhabitant becomes one of the customers of a shop. Considering this interpretation, one may express doubts about the conditions ruling both probabilities (27) and (15). In fact, one can pose questions like these: Do customers always leave shops at random? Does coming to a shop always depend upon the number of its customers? Does the creation probability not depend upon other factors—for example, which people are customers of the shop? With this interpretation, exchangeability and invariance become empirically testable.

In what we have done the distinctive feature of the system, which is that the number of customers and that of shops are given at the beginning, in particular it is supposed that the number of shops is finite, as is the number of customers. However, it may happen that a shop loses all its costumers or, as we shall say, it becomes void. We wish to analyze what happens in the case in which void shops cannot be reopened, and are possibly replaced by new shops. More exactly, we shall consider the case in which the probability that an empty shop will be reopened is equal to zero, and the probability that a new shop opens is positive. This is a way of saying that all shops that have lost all their customer are replaced by some new shops, the actual number of shops being a random variable.

To implement this scenario, besides the conditions we have already considered we suppose that: the number of shops is d, large but finite; all initial probabilities are equal, that is, p==1/d; X is positive and finite. Now we consider the limit d^», that is, Ewens's limit. In a formal way this amounts to supposing that the probabilistic behaviors of customers are ruled by exchangeability and invariance, that all initial probabilities

pj ^ 0, hence for all j and the real number A, which in this context we denote by в for historical reasons, is finite. We call these assumptions Ewens's limit because they were first considered by Ewens (1972).

If Ewens's limit holds, the number of customers being finite, at a fixed time no more than N shops can be occupied; moreover, the probability that a specific void shop receives a customer is zero, as pj ^ 0. As we have said, the probability that a shop which has lost all his customers will be put in action again is equal to zero. Another consequence of Ewens's limit is that it becomes useless to look at the system through the occupation vectors because these almost all have occupation numbers equal to zero. In order to study the evolution of the system we must consider another way of describing it. Given an occupation vector N, we denote by ZT the number of its occupation numbers equal to r. Thus ZT is the number of clusters of size (occupation number) r. A cluster size vector Z=(Z0,Zj,^,Z„), 2N=0ZT=d, %N=0Z=N, is an (N+1)-dimensional vector specifying the number of shops with exactly r = 0,1,...,N customers. In fact, Z0, is the number of void shops, Z1 is that of the shops with a sole customer, Zr that of shops with r customers and ZN is the number of shops with N customers. Clearly ZN may be either 0 or 1. Moreover, if ZN = 1 in the town there is only one active shop.

To study the effect of Ewens's limit, we consider first a large but finite number of shops of which only the first h are active. This means that the first h shops have at least one shopper while the other d—h are void. Obviously h < n and h

When Ewens's limit holds, (31) becomes

This transition probability shows that, as the initial weights of all shops tends to zero, it is almost certain that sooner or later shops become void; the probability that a well-specified void shop will become active is zero, hence after a shop has become void there is no chance that it will reopen; the probability that some void shop receives a costumer is finite; and with probability 1 all existing active shop close, and the closure is for ever.

To account for the dynamical evolution of the system we consider a sequence of cluster size vectors Z(0), Z(1),...,Z(t),... . Without entering in the details, we note that as before this is a homogeneous Markov chain, focusing on transition probabilities. As in the Ewens's limit Z0 ^ d , we consider only Z = (Z1,...,Zn), with the sole constraint

» ^ N к ^ N

y rZr = N, while y Zr = k is a random variable. Suppose that Z = (Z1,..., ZN) is the cluster size vector of the town with respect to shops and customers. We look for the partition vector after the destruction of a customer in a shop whose occupation number is i and the creation of a customer in a shop whose occupation number is /—1. After the first step, the occupation number of the shop concerned becomes i— 1. This cut has a double effect on the cluster size vector of the town: Zi becomes Z— 1 while Z—1 becomes Zi—1 +1. On the other hand, as a consequence of the second step, Z— 1 becomes Z— 1 —1 while becomes Z/+1. We denote by z/ = (Z0,Z1,...,Z,_1 +1, Z,- -1,...,Z/_1 -1, Zj +1,...,ZN) the partition vector resulting from the transition. The related transition probability can be immediately reached by means of (32), noting that the first step is the Zt disjunction of events whose probabilities are equal to N, and that the second is the disjunction of Z/—1 events whose probabilities are equal to g+r-_1 . Thus we have

The transition probability (33) is that of a homogeneous Markov chain, irreducible and aperiodic. The equilibrium distribution is the Ewens Sampling Formula

According to the derivation we have just shown, (34) can be seen as the fraction of time that the town spends in the state described by Z. Aoki has interpreted the parameter u as a measure of correlatedness of individual agents. Referring to two agents and using [34] he says, "(t)he closer the value of в to zero, the larger is the probability that two randomly chosen agents are of the same type. The larger that value of в, the more likely that two randomly chosen agents are not of the same type" (Aoki, 2002, p. 165). The definition we have stipulated for the parameter в reveals the root of the correlation, which, in turn, depends upon to the value of the relevance quotient.

 
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