On Tue, Feb 10, 2009 at 10:53 PM, Jobst Heitzig <[email protected]> wrote: > For example, consider our example in which the true utilities are > 55: A(100) > C(70) >> B(0) > 45: B(100) > C(70) >> A(0). > With the suggested method, the resulting approval score of C will not be 100 > (which would be desirable and would be the result under D2MAC, e.g.) but > would rather be only 56. This is because if it would be larger than 56, any > voter who approves of C will have an incentive to not approve of C as this > would transfer some probability from C to her favourite.
I don't think that is correct. Assuming that only 1 of B's faction compromises, the votes would be 55: A*, C 1: B*, C 44: B* C wins round 1 and C's pile becomes 55: A* + C 1: B*+C The 50% rule kicks in as the A* ballots make up more than 50%. 54 of the A*+C ballots are removed from the pile to balance it. This is converted to 1: A*+C 1: B*+C The remaining ballots are 54: A*+C 44: B* These just form 2 piles as there is no common candidates between the 2 sets of ballots. The result is A: 54% B: 44% C: 2% The utility for faction B is: 0.54*0 + 0.44*100 + 0.02*70 = 45.4 Assume that 2 of them compromise, the votes become 55: A*, C 2: B*, C 43: B* C's pile becomes 55: A* + C 2 B*+C This is converted to 2: A*+C 2: B*+C Remaining 53: A*+C 43: B* These just form 2 piles, so the result is A: 53% B: 43% C: 4% The utility for faction B is: 0.53*0 + 0.43*100 + 0.04*70 = 45.8 This is an improvement for faction B. Thus there is an incentive to compromise for the B faction. In general, assume that x of them compromise, the votes become 55: A*, C x: B*, C 45-x: B* C wins with 55+x approvals. C's pile becomes 55: A* + C x B*+C This is converted to x: A*+C x: B*+C Remaining 55-x: A*+C 45-x: B* These just form 2 piles, so the result is A: 55-x% B: 45-x% C: 2*x% The utility for faction B is: (55-x)*0 + (45-x)*100 + (2*x)*70 = 4500 - 100*x + 140*x = 4500 +40*x Thus, it is in the interests of the B faction to set x as high as possible. The effect is that all voters in both factions will compromise. Ofc, that doesn't result in 100% probability though. C: 90% A: 10% Also, the last 10 voters from A could decide not to also approve C without any effect. ---- Election-Methods mailing list - see http://electorama.com/em for list info
