&circlef; A stream of people arrives at the bar. Each new arrival pairs up with someone already in the bar and that couple leaves, provided each member of the couple is an acceptable match for the other. If not, the new arrival joins the people already in the bar and waits for an acceptable date.
&circlef; On average each person finds that 1 person in p is an acceptable partner. Such pairings are independent events and p ≥ 1 and is the same for each person.
M: male population of the bar, r: proportion of men in the incoming stream of people.
F: female population of the bar, s: proportion of women in the incoming stream of people.
There are several questions we would like to answer:
&circlef; For given values of r and s do the male and female populations in the bar increase, decrease, or remain constant over time?
&circlef; For what values of p, r, and s does the total population in the bar remain greater than 0 (so the bar does not empty out) but does not grow unbounded (so that crowding keeps newcomers from visiting?
The probability that a man (or woman) arriving at the bar finds a woman (or man) already in the bar to be an acceptable date is . Since the object of desire also has a probability of of finding the newcomer acceptable, the probability that a new couple will form (and then leave the bar) is The complementary event is that a new person arrives at the bar and does not find a mutually acceptable date. The probability of this event is
1-. Since this expression is used a great deal in this model, we will define the symbol:
Suppose a man enters the bar. If there are F women in the bar, the probability that the man does not find an acceptable date in the bar is Thus the male population of the bar will increase by 1 with a probability of r
The male population of the bar will decrease by 1 if a woman arrives and finds an acceptable date. If there are M men in the bar, this event occurs with a probability of 1- Thus the male population of the bar will decrease by 1 with a probability of s(1-).
By a similar argument, the probability that the female population of the bar increases by 1 is sand the probability it decreases by 1 is given by r (1-).
Thus the probability the population of the bar increases by 1 (by adding either 1 man or 1 woman) is given by the function f(M,F) = r+ s.
The expected change in the male population is described by the difference between the probability that a man arrives and does not find a date and the probability that a woman arrives and does find a date. Algebraically this is
r- s(1-) = f(M,F) - s.
The male population will stay constant (in other words be in equilibrium) if f(M,F) = s.
By the same argument, the expected change in the female population is described by the difference between the probability that a woman arrives and does not find a date and the probability that a man arrives and does find a date. Algebraically this is
s- r(1-) = f(M,F) - r.
The female population will stay constant if f(M,F) = r.
Since r + s = 1, then both the male and female populations can simultaneously be in equilibrium if r = s = Therefore we should consider the curve f(M,F) =
If f(M,F) < (i.e. the bar's population is expected to decrease), M and F will decrease. This will cause f(M,F) to increase. On the other hand if f(M,F) > (i.e. the bar's population is expected to increase), M and F will increase. This will cause f(M,F) to decrease. Thus this curve is stable.
Now suppose that r > s.
If f(M,F) < s then F will decrease and M will increase. This will cause an increase in f(M,F). If f(M,F) > r then F will increase, but M will also increase. Thus the region between the stability curves of F and M is stable.