ABSTRACT
Let us consider a simplified situation that only two states of opinions, such as expectations of a bull or bear market or liberal or conservative agenda, are subject to public choice. The two opinions are denoted by the symbols of plus and minus. The formation of an individual’s opinion is influenced by the presence of the fellow community members with the same or the opposite opinion. We may use a transition probability to describe the changes of opinions. We denote the transition probabilities by
p n n+− + −( , ) and p n n−+ + −( , )
where n+ and n-are the numbers of individuals holding the corresponding opinions + and –, respectively; p+– denotes the opinion changes from state + to –, and p-+ denotes the opposite changes from – to +. We also denote the probability distribution function by f(n+, n-; t). The master equation can be derived as follows:
df n n t dt
n p n n f n n t( , ; ) ( ) ( , ) ( , ; )+ − + +− + − + −= + + − + −1 1 1 1 1 +
( ) ( , ) ( , ; )n p n n f n n t− −+ + − + −+ − + − +1 1 1 1 1 –
[( ) ( , ) ( ) ( , )] ( , ; )n p n n n p n n f n n t+ +− + − − −+ + − + −+ (9.1)
We may simplify the equation by introducing new variables and parameters:
Total population: n n n= ++ −
Order parameter: q n n
n = −+ −
So we have:
n n q+ = +
1 2
n n q− = −
1 2
w q n p n n n q p q+− + +− + − +−= = +
( ) ( , ) ( )1 2
w q n p n n n q p q−+ − −+ + − −+= = −
( ) ( , ) ( )1 2
where, n is the total number of the community members, q measures the difference ratio and can be regarded as an order parameter, w+–(q) and w-+(q)are the new function describing the opinion change rate which is a function of order parameter q. This equation can describe collective behavior such as formation of public opinion, imitation, fashion, and mass movement following a crowd. The problem
is still unsolved about the transition probabilities p+–(n+, n-) and p-+(n+, n-). The form of transition probability is closely associated with the assumption of communication patterns in human behavior.
