Potentials
With these transition rates, the expression for the equilibrium distribution becomes
where CNk is the binomial coefficient N!/k!( N — k)!.
We see that the binomial coefficient we encountered in the earlier model is now modified by the product of the ratios of η's.We express πk as an exponential function
where Z is a normalizing constant
and is assumed to be finite, and where β is a nonnegative parameter we have introduced. It plays a key role in our analysis from now on. Informally, it represents the effect of uncertainty or incompleteness of information in making the choices. We return to this in the next subsection. The function U introduced above is called potential, and is related to the ∣∣'s by
with q = λ∕(λ + μ).
This function U in the exponent of πk is called a potential, since it is independent of the path from state 0 to state k (or k/N, to be more precise), and depends only on the initial and the current state. This is known as the Kolmogorov criterion; see Kelly (1979, p. 23).
To proceed further we need to specify the η,s more explicitly. Suppose that they are given by
and
with
Ior
A nonnegative function g( f) expresses the relative merits of
alternative choices in a sense that we will explain shortly.
The parameter β incorporates into the transition rates the (intrinsic or extrinsic) uncertainty or incompleteness of information that surrounds agents' decision-making processes.
As specified above, larger values of β with positive values of g( f) cause η1( f) to be larger than η2( f), and make transitions associated with rn more likely to occur than those associated with ln. One the other hand, near-zero values of β make η1 and η2 nearly equal to a half. This makes the choice between the alternatives nearly a fair coin toss by independent agents.Next, we show heuristically that g( f) may be interpreted as the difference of the perceived benefits of the two choices, conditional on the value of the fraction f, when we approximate the uncertain consequences of choices as normally distributed. Alternative interpretations are also possible, and they are discussed later.
Returning to the definition of the potential, we approximate the binomial coefficient by the entropy expression to derive
5.4