Transition rates

We use both empirical distributions and partition vectors to describe the state of the market. This section employs empirical distributions. We first give the transition rates in terms of the state vector.

Suppose that there are potentially a large number K of types of agents who participate in a market. For the moment, suppose that the value of K is known. Then the vector n = (n₁, n₂,..., n_κ) describes how n agents are distributed over K types, n = n₁ + n₂ +----- + n_κ.In this section we use this

vector as state vector, then switch to the partition-vector description in the next section.

Suppose that we use the same set of transition rates as these in Sections 7.4 and 10.2,

forwhere e_k is the vector with unit element in the kth position and zero

elsewhere,

and λ j_k = λ_kj for all j, k pairs. The first transition rate specifies the entry rate to the market by agents of type k, the second the exit or departure rate from the market by type j agents, and the last the transition intensity of changing types by agents from type j to type k, that is, switching of trading rules by agents. In the specification of the entry transition rate, the term c_kn_k stands for attractiveness of a large group, such as a network externality that makes it easier for others to join the cluster or group. The other term, c_kh_k, represents new entry to the market, which is independent of cluster size. It represents the attractiveness of the strategy to outsiders, irrespective of the number of agents who are currently using it.

The jump Markov process thus specified has the steady state or stationary distribution

where

where gj = Cj /dj. These expressions are derived straightforwardly by applying the detailed-balance conditions to the transition rates.

For simplicity, suppose that gj = g for all j. Then, as we show in Chapter 10, the joint probability distribution is expressible as

11.2