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Unary moves

As time goes by agents change strategies, and this gives rise to the evolution we want study. The assumptions we have made about destruction and creation probabilities are tantamount to supposing that the probabilistic evolution of the system is regular, exchangeable, and invariant. This evolution is described by a sequence of occupation vectors N(0),N(1),...,N(t),.... As we shall see, this is a homogeneous Markov chain. Moreover, the elements of its stochastic matrix do not depend on time explicitly. A unary move occurs when an agent from i moves on to j. As a consequence of such a move, the state of the system undergoes a transition from to N to Nj. The probability of this transition, that is, P(Nj|N), can be arrived at by considering two steps: a destruction immediately followed by a creation, as analyzed above. The probabilities of these steps are given by (27) and (28). Hence, the transition probability we are looking for is

with the correction stressed by (28) for repeated indices.

In (29) either l > 0, or 1pj is integer and lU < N. Hence, starting from a given state of the system, all states of the system can be reached by repeated applications of (29). Further, all these states are persistent, as no absorbing states exist. Hence, the set of all states is ergodic and there is a (unique) invariant probability, say p(N), on this ergodic set. Because (29) accounts for the case i=j, too, the chain is not periodic and the invariant probability is the equilibrium distribution. By the Chapman- Kolmogorov equation, and by using the detailed balance between any couple of occupation vectors, it can be proved (see Costantini and Garibaldi, 2000) that the equilibrium distribution is the (generalized) Polya (23), that is,

in which the parameters are the same as in (15). We recall that the mean value and variance of this probability distribution are respectively

E(N) = Npi and Var(Nt) = Npt(1-pt)A±N .

A ± 1

To conclude the section, we recall that moves of an order greater than 1 are more complicated but can be handled in the same way as unary moves. The sole difference amounts to which approach rate to equilibrium becomes greater (see Costantini and Garibaldi, 2004). In the extreme case in which all agents are involved in an N-ary move, the equilibrium is reached immediately. In the case in which the equilibrium distribution is reached in this way, it amounts to the juxtaposition of 2N steps of which the first N are destructions, ruled by an hypergeometric probability, and the second N are creations, ruled by an exchangeable and invariant probability. What we have shown in the previous section devoted to statistical mechanics is a case of an N-ary creation. In all cases we may consider the equilibrium distribution is always (23). This illustrates certain relations existing between statistical mechanics and economics. We are persuaded that this analysis deserves more attention.

 
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