In each time step, target pollutant inputs are calculated as a weighted average of the input rates preferred by the two groups. Each group's preferred input rate is calculated from the state of the lake and the forecasting model, using the optimization procedure of Carpenter et al. (1999). The preferred input rates differ because of the differences in discount factor used in the calculation. The weights ** V** are calculated as

- 0.5)]}N |
(A.6.1) |

or, equivalently,

) / [exp(2 Ψ N) + exp(Ψ)].N |
(A.6.2) |

The target input rate *a*_{targ} is

)V |
(A.6.3) |

where ** a** is a column vector of the input rates preferred by each group of agents. The parameter Ψ controls the intensity of the shift toward the policy favored by the numerically dominant group of agents. If Ψ = 0, each policy is weighted equally, so the policy is an average of the two preferences, regardless of the number of agents in each group. If Ψ = 1, weights are linearly proportional to the numbers of each group of agents, so the resulting policy is a compromise, shifted toward the preferences of the most abundant group in proportion to relative abundance. The outcome is similar to that of the information market model (Appendix 4), in which the total input is the sum of individual actions. As Ψ grows larger than 2, the preference of the most abundant group receives progressively heavier weight. For Ψ greater than about 100, the outcome is "winner take all:" the policy of the most abundant group is adopted with no accomodation for the preferences of the less abundant group.