Forest Simmons wrote: >>19 A >>26 AB >>30 BC >>13 C >>12 CA >>(57 A, 56 B, 55 C) >> >>As it happens, this is a stable solution, in the sense that the A voters >>can't do anything to improve their outcome (their favorite is already >>a winner), the B voters can't do anything to improve their outcome >>(they've already done everything they can to support their compromise), >>and the C voters can't do anything to improve their outcome (they >>already did improve their outcome). >> > > It seems to me that the C voters could improve their outcome by voting > straight C, reducing A's approval by 12, and giving the win to C.
That just takes us back to the first scenario, where B won. One reason the A voters might have chosen their 19/26 split, by the way, is their intention of keeping C from winning. 26 votes for B from the A faction is just enough to lock C out, assuming the B voters and the C voters all vote for C (and all the B voters vote for B, of course). > Here's the (marginally) stable configuration that I had in mind for a B > win: > > 45 AB > 30 BC > 25 CA > > If the first faction drops B, then C wins, which is worse for that the > first faction. Wait, if the first faction drops B, then 45 A 30 BC 25 CA A wins, right? Maybe you meant 45 AB 30 BC 16 C 9 CA So B wins, and the A voters can drop (at most) 19 B votes without risking a C win. But this is still not stable, since the C faction could always increase their A votes. I also don't see a stable solution in which C wins. The A voters can always prevent C from winning by voting a sufficient number of B votes. In fact, this power gives them a strong influence over the C faction, so they can probably persuade the C voters to vote CA. > In this example any two groups can work together to defeat the other > group. This requires one of the two cooperating groups to support their > compromise candidate. In actual politics this effect is probably strong > enough to overcome the unilateral marginal stability of the solution I > gave above. The winner will be determined by the two groups that can make > the best deal with each other. Based on my last observation, I would say the two groups most likely to cooperate are the A and C factions. If the A and B voters agree to cooperate to elect candidate B: 45 AB 30 B 25 CA then it would certainly be tempting for the A voters to back out of the deal at the ballot box. If the B and C voters agree to cooperate to elect candidate C: 45 AB 30 BC 25 C then the B voters have a lot to gain by backing out of the deal. Of course there are other considerations (outside of electoral mathematics). First, the deal-breakers may hurt their ability to negotiate compromises in future elections, since they won't be trusted. Another possibility is that one party could use something besides votes as a bargaining chip. For example, if B's party has control of the legislative branch, B's party could negotiate a deal with A's party wherein B's party promises passage of a legislative package favored by A's party only if A's party helps B win. -- Richard
