On Fri, 14 Dec 2001, Richard Moore wrote: > 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.
Oops, you're right! > > 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? You're right again! > Maybe you meant > > 45 AB > 30 BC > 16 C > 9 CA > Actually, what I had in mind was 45 AB 30 BC 25 C Any unilateral change made by one voter or one faction will not improve the outcome for that voter or faction. In particular, increasing the C faction's support for A will not help A win without cooperation from the A faction. > > 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 > I think you've described the two way deal dynamics quite well. Forest
