"Technical Analysis in Financial Markets"

F_j,t^h=F_j,t^h+ε_j,t^h, (14)

where F_j,t^h is the deterministic part of the fitness measure and ε_j,t^h represents personal observational noise. If ε_j,t^h ≠ 0, then this model means that agent j cannot observe the true fitness F_{j, t}^h of belief h perfectly, but only with some observational noise. Assuming that the noise ε_j,t^h is iid drawn across beliefs h=1,...,H and across agents j=1,...,N from a double exponential distribution, then the probability that agent j chooses belief h is equal to

q_j,t^h=

exp(β_j F_{j, t-1}^h)

∑

k=1

exp(β_j F_{j, t-1}^k)

. (15)

Here β_j is called the intensity of choice, measuring how sensitive agent j is to selecting the optimal belief. The intensity of choice β_j is inversely related to the variance of the noise terms ε_j,t^h. If agent j can perfectly observe the fitness of each belief in each period, then V(ε_j,t^h) ↓ 0 and β_j → ∞ and the agent chooses the best belief with probability 1. If agent j cannot observe differences in fitness, then V(ε_j,t^h)→ ∞ and β_j ↓ 0 and the agent chooses each belief with equal probability 1/H.

The excess profit of an agent following strategy h in period t is equal to (P_t+D_t-R P_t-1) z_t-1^h. Therefore the fitness measure of strategy type h as observed by agent j is defined as

F_{j, t}^h=(P_t+D_t-R P_t-1)z_t-1^h-C_j^h+η_j F_{j, t-1}^h.

Here 0 ≤ η_j ≤ 1 is the personal memory parameter and C_j^h is the average per period cost of obtaining forecasting strategy h for agent j. If η_j=1, the memory of the agent is infinite and F_j,t^h is equal to the cumulative excess profits of belief h until time t. In this case F_{j, t}^h measures the total excess profit of the belief from the beginning of the process. If η_j=0, the agent has no memory and F_{j, t}^h is equal to the excess profit on time t-1. If 0 < η_j < 1, then F_j,t^h is a weighted average of past excess profits with exponentially declining weights. The higher the costs C_j^h, the more costly it is for the agent to obtain and invest according to belief h, and the more unlikely it will be that the agent chooses belief h.

BH assume that β_j=β, η_j=η and C_j^h=C^h for all agents, so that F_j,t^h=F_t^h and q_j,t^h=q_t^h are equal for all agents. This means that all agents have the same intensity of choice, have the same memory and face the same costs for trading. Under this assumption, in the limit, as the number of agents goes to infinity, the fraction of agents who choose to invest according to belief h converges in probability to q_t^h. Thus in the equilibrium price equation (6.13) n_t^h can be replaced by q_t^h. Furthermore, it is assumed that all agents have

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