Appendix 5. Election of the Governing Board.

At each time step, once the updated pollutant level is known, it is possible to calculate the economic return, had any given policy been chosen in the previous time step. Weights for the propagation of agents to the next time step are based on relative policy performance in the past time step (note that this comparison could have been carried out over an arbitrary number of past time steps). The group of agents with the best performing policy receive weight w (0.5 < w 1), and the other group receives weight 1-w. Let Wt be a column vector of these weights and pt-1 be a column vector giving the proportion of agents of each type in the past time step. Then the proportion of agents in the next time step is

pt+1 = [pt-1 Wt] φ / [Σ([pt-1 Wt] φ )]


where φ = [φ 1-φ] is a vector that introduces autocorrelation into the time series of N and Σ(.) denotes the sum of matrix elements. Stochastic propagation of agents is introduced by drawing the agents for the next time step, Nt+1, randomly according to the probabilities pt+1.

The weights Wt control the bias in propagation due to success of one policy relative to the other. If wt is near 1.0, then the group with the more successful policy will be represented heavily in the next time step. If wt is near 0.5, then the two groups will be represented more equally in the next time step. Thus, w sets the magnitude of suppression of the losing group by the winning group.

The weights φ control the autocorrelation of the propagation process. If φ is near 1.0, then the past N is weighted heavily. Thus, the next N will resemble the past N, even if policy performance is reversed. Conversely, if φ is near zero, then agent propagation depends mainly on policy performance in the most recent time step. Simply put, φ determines the tendency of a power balance to persist despite changes in the relative performance of policies.