Representation of relative merits of alternatives
We offer three interpretations for our specifications of the functions η in the transition rates of the last chapter. The first is based on some approximate calculations of the perceived difference of the expected utilities, or advantages of one choice over the other.
A second interpretation is based on discrete choice theory as in Anderson et al. (1993) or McFadden (1972, 1974). These are related to the extreme-value distributions of type I, which are discussed in the Section 6.3 and Section A.6. See also Aoki (1996a, Chap. 3). The third is via constrained maximization of entropy or minimization of the Kullback-Leibler divergence, subject to some constraints such as the size of the model and other macroeconomic signals. All three explain heuristically how distributions called Gibbs distributions arise, and how they are related to error functions.Let us start with the first interpretation. Denote by u1(x) the expected return from choice 1, given that fraction x has selected choice 1. For definiteness, think of the discounted present value of the benefit stream based on the assumption that fraction x remain the same over some planning horizon. Define u2(x) analogously. Let
Next, assume that the difference ∆n(x) = u1(x) - u2(x) is approximately distributed as a normal random variable with mean g(x) and variance φ2. We calculate the probability that it is nonnegative,
where the error function is defined by
and 
with the same X introduced in Chapter 5. This offers one interpretation of β that appears in the transition rates: large variances mean large uncertainty in the expected difference of the alternative choices.
Alternatively put, we may interpret g(x) as the conditional mean of a measure that choice 1 is better than choice 2, conditional on the fraction x having decided on choice 1.
See Aoki (1996a, Chaps. 3, 8) on the partition functions and how β arises as a Lagrange multiplier to incorporate macrosignals as constraints. The parameter β is related to the elasticity of the number of microeconomic configurations with respect to macrosignals. Small values of β mean that the number of microeconomic configurations responds little when macroeconomic signals change. See Aoki (1996a, p. 216).
6.2