Multiplicity—An aspect of random combinatorial features

Here, we return to the factor η₁∕η₂ introduced in (5.1) that multiplies the bino­mial coefficients and interpret it. Suppose we are interested in the fraction of agents with choice 1.

with x_i = 1 or 0. How many different ways can a given value of f be realized? To put this differently, how many microeconomic patterns of choices are com­patible with the average value f ? We suppose agents are symmetrical average value in their choices, i.e., agents are exchangeable in the technical sense to be described later. What this assumption implies is that any agent may choose choice 1 or 0 with the same probabilities, and hence μ = λ = 1/2,

and

Substituting these into (5.1), we derive the stationary distribution as

We recognize here the binomial coefficient for choosing n out of N, denoted by Cn,n, i.e.,

with p = q = 1/2. More generally, if an agent chooses 1 with probability p and 0 with probability 1 - p := q, then the above formula holds without p and q being equal to 1 /2.

Given the fraction f, then, there are C_n,_n∕ ways of getting that fraction. Put differently, there are this many ways of realizing the fraction f. Thus, we are led naturally to assess the magnitude of the binary coefficient for large N. We use Stirling’s formula

where the second expression on the right ignores the factor √2π N when we write N! as exp{N(lnN!/N)}.

Applying this approximation to the binomial coefficient, we derive

which becomes

where

with P standing for the discrete distribution (p, 1 - p), and Q for (q, 1 - q), bothfor x = 1and x = 0.Thisexpressionisaspecialcaseoftherelativeentropy D(P; Q) of one distribution P with respect to another Q. We can show that

the relative entropy, also called the Kullback-Leibler divergence or information measure, is nonnegative and equals zero if and only if the two distributions are equal. See, for example, Dupuis and Ellis (1997, p. 32) or Kullback and Leibler (1951).