With 3 merit-consideration-dimensions instead of 2, it's: Expected R = (3/4) (R2^4 - R1^4)/(R2^3 - R1^3) ...suggesting a formula: With n dimensions, Expected R = (n/(n+1)) (R2^(n+1) - R1^(n+1) )/(R2^n - R1^n). By that formula, for n = 1: Expected R = (1/2) (R2^2 - R1^2)/(R2 - R1) = (R2 + R1)/2. --------------------------------------------------------------------- Someone might object: "The merit-mean Approval cutoff is justified by the fact that the presence of more candidates in a region implies the presence of more voters in that regiion." Answer: Maybe in an ideal world. In what world do you plan to vote? Do you think that our elections' candidates were chosen by the voters, or that they were chosen on the basis of how well they match the things that the country's people want? If you believe that, then use the merit-mean Approval cutoff. If you use it in a simulation, then your simulation is for a different world, one in which that statement is true, or else you're saying that the voters in your simulation mistakenly believe that it's true. Michael Ossipoff