Critical points and hazard function
Here, we relate the critical points for the potential to the notion of hazard function. In the current model, we have seen that the stationary probability distribution for the fraction of agents with choice 1 is of the form exp{- β NU (x)}, where U (x) is a potential of the form
4 In a pioneering paper of Kirman (1993), the transition rates he used are η1(x) = ∈ + (1 — δ)x and η2(x) = ∈ + (1 — δ)(1 — x), inthe notation of this section.
He used a/N for ∈ and 2a/N for δ. In other words, in Kirman, βg(x) = ln{x + (a/N)(1 — 2x)} is the key equation. Note that in his model β plays no role and there is only one critical point. His model does not exhibit interplay of uncertainty with the number of equilibria.
Putting this differently, at a zero point of g, the potential is still increasing if the zero is between 0.5 and 1, and the potential is decreasing if the zero is between 0 and 0.5.
To interpret the condition for minimal potential with smaller values of β where the zero of the function g is not the minimizing point, we adapt the notion of hazard function (rate) in the reliability literature.5 Cox and Miller (1965, p. 253) define the hazard function for a random variable Y as
[1] In this literature, the notion of hazard rate is applied to the life of durable goods such as light bulbs. The hazard rate gives the conditional probability that a light bulb fails in the next hour, given that it has lasted 1000 hours, say. In our application here, we look for the conditional probability that an agent switches his or her choice in response to a (perceived) small increase in the return difference, conditional on the current choice at a current fraction x.
Now, let tg(x' )be our Y, and use the probabilities conditional on x *. We start with
as our conditional distribution. Then, the conditional hazard function is given by
Alternatively, the conditional hazard function is approximately given by
In words, given that
where ei∙ stands for
some errors in perception or observation noise, a slight increase in the difference by υ > 0 will switch the preferred choice from choice 0 to 1:
This switching of the choices, or crossing of the boundary between the choices, occurs when x = x*.
In our context, the potential minimization condition can be put as
from which we conclude that
This equation shows that for small values of β the conditional hazard function is approximately equal to β, that is, β may be considered as the conditional hazard rate in the range where β is small. We may state that the potential is minimized at a fraction x at which the conditional hazard function is approximately equal to β.
5.5