Destruction and creation probabilities
While a unary move taken as a whole does not modify the size of the system, a destruction taken from itself decreases by 1 the size of the system while a creation increases it by 1. We first take destructions into account. N being the state of the system, if the n-th agent is destroyed in i, the description of the decreased system could be the following:
X1 = Ь...Дп_1 = jn-V Xn+1 = jn+y-,XN = jN. We denote by Ni =
Nt — 1,...,Nd) the corresponding occupation vector, that is, the state of a system in which any agent whatsoever is destroyed in i. Moreover, we denote by P(Ni | N) the destruction probability, that is, the probability that the state (occupation vector) of the system changes from N to N;. We assume that the destruction probability is exchangeable and invariant, with parameters X(d)p(f) = —Ni (d is for destruction). As a consequence we have
This probability can be dealt with as a case of direct inference. In fact, (27) is the probability of drawing from a population whose statistical distribution is N a sample of size l, that is, an agent, and the drawn agent is in i. But it is clear that to reach the value of a destruction probability one has to pose in (15) a negative value for X, that is X(d > = —N, and pif)=Ni/N. Thus we suppose that destruction is "universal", that is, it does not depend on the type of element involved.
N; being the state after the destruction, a creation in j restores the size of the system, and Nij is the state (occupation vector) of the resulting system. The probability of such a creation is P(Nj|Ni). We suppose that creation probabilities are exchangeable and invariant, with parameters 1(c)p(c) = Ipj + N, where Nj is the actual occupation number, pj is the probability of accommodation when the cell is empty and c is for construction. Now X is free to depend on the type of correlation among elements. It follows that for these probabilities the main theorem holds. Therefore, the probability we are interested in is (15). Considering a creation probability, we do not compel the creation to occur in a strategy other than that in which destruction has occurred. More clearly, destruction and creation may involve the same strategy. As we have said, if this is the case, the state of the system does not change. Hence, in general the creation probability we shall consider is
where the sign of X induces the sign of accommodation correlations. In both formulae the denominator N— 1 accounts for the size of the system after a destruction in i, and N, —1 for j = i follows from the fact that, after the destruction in i, the occupation number of this cell is Nt —1.