Open binary models
There are six admissible state transitions in an open binary choice model: entry of each of two types of agents, departure of each of two types, a change of strategy by a type one agent into a type two, and the reverse transition of an agent from type two to type one.
These transitions are conveniently expressed by defining two vectors
and
Entry and exit by a type i agent are the events denoted by state transition
for i = 1, 2. The number of agents in the market changes by one in these two transitions. Change of strategy by an agent from strategy i to j is denoted by 
participants changes into a type one agent. Note that unlike entries or exits, the total number of agents in the market remains the same in the type changes.
Now we posit, as in Kingman (1969),4
Equations (4.2) through (4.4) specify that an agent of either type may enter or change his or her type at rates that are influenced by the number of agents of the same type in the market.
In general, stationary probability distributions cannot be obtained analytically. When the equality holds term by term, that is, when we assume that the detailed-balance conditions hold, we can sometimes solve for the equilibrium distributions.
This is the case with K = 2. Although there are six separate equations when we impose the detailed-balance condition, the solution of any one of the six equations turns out to satisfy the remainder of the equations with suitable conditions imposed on the model transition parameters. For example, we start with (4.2):
for i = 1, 2, or with (4.2) or (4.3). It does not matter how we begin.
We first verify for the case with K = 2 that the stationary probabilities in product form satisfy the detailed-balance condition. That is, we posit
This is the same as what Kingman calls simple transition rates. The detailed- balance condition using (4.2) yields the first-order difference equations
is the departure rate when the number of type i agents is ni + 1. This is a first-order difference equation for the probabilities. Iterating this relation, we obtain
where bi is the normalizing constant.
As a special linear case, we may use φ(ni) = dini, di = 0, and ψi(ni) = ai + cini. This specification is used in Kelly (1979, p. 139). The specified process is sometimes called a birth-death-with-immigration process, since there is a term ai that is independent of ni in the entry transition rate.
In economic applications, the constant term in the entry rate is important. It may represent exogenous (policy) effects either to encourage or facilitate entry in the case of positive ai, or to discourage or retard entry in the case of negative ai. Sutton (1997) speaks of niche effects, such as increases in demand that affect only a subset of the sectors of economy - that are not economy-wide. Positive as produce (generalized) Polya distributions, and negative ones produce hypergeometric distributions. We return to these distributions later, and discuss one example.
4.3.1 Examples
4.4