Mike wrote: >That hadn't occurred to me--that there's another non-reversed >defensive strategy that always thwarts offensive order-reversal >in the wv methods: equal-top-ranking of the CW. > >Of course the Nader voters might insist that the Gore voters >announce publicly that they're going to defensively truncate, >because that's an easier or more mild strategy. Given that announcement, >the Bush voters wouldn't order-reverse.
For reference, Mike is referring to my example, where the sincere preferences are: 49: Bush>Gore>Nader 12: Gore>Bush>Nader 12: Gore>Nader>Bush 27: Nader>Gore>Bush ...making Gore the Condorcet winner. But the Bush voters are threatening to reverse: 49: Bush>Nader>Gore 12: Gore>Bush>Nader 12: Gore>Nader>Bush 27: Nader>Gore>Bush ...which gives the election to Bush (N>G 76, B>N 61, G>B 51). I suggested that the Nader voters could equal-rank Gore: 49: Bush>Nader>Gore 12: Gore>Bush>Nader 12: Gore>Nader>Bush 27: Nader=Gore>Bush (B>N 61, G>B 51, N>G 49), and now Gore wins in a winning-votes method. Mike suggests that another way for the Gore camp to achieve their desired results is if the Gore>Bush>Nader voters announce their intention to truncate. If the Bush voters still order reverse, you get 49: Bush>Nader>Gore 12: Gore>Bush=Nader 12: Gore>Nader>Bush 27: Nader>Gore>Bush (N>G 76, G>B 51, B>N 49) and now Nader wins. Since this is the least-desired result of the Bush camp, they will fear such an outcome and lose their incentive to order-reverse, and will in stead vote honestly: 49: Bush>Gore>Nader 12: Gore>Bush=Nader 12: Gore>Nader>Bush 27: Nader>Gore>Bush And now Gore is once again the Condorcet winner. So, which one of our two equilibriums is more meaningful? Both are Nash equilibria, even if we define the players to be entire sets of voters. (I believe Mike coined the phrase "many-voter equilibrium" for this.) I'd say that both equilibria are illustrative, for different reasons. Mike's is nice because it only involves one simple truncation at the bottom of the rankings, by one camp, and everyone else votes honestly. Nobody has any need to insincerely rank another candidate equal to their favorite. It is realistic that you could convince these Gore voters to drop Bush from their ballots, since Nader is unlikely to win anyway. The reason I also like the other equilibrium, however, is that it is more free of concern about what the other camp does. This makes it more stable. If the Gore>Bush voters are told to bullet vote Gore, but the Bush voters announce that they plan to order-reverse anyway, then you get into a game of strategic brinksmanship. Both camps would prefer each other to Nader, but if they protect against Nader (by not truncating in the Gore camp, or by not reversing in the Bush camp) they run the risk that the other side will use the more aggressive strategy and win the election. Looking at this as a two-player game sheds some light on the situation. The players are the 49% Bush>Gore>Nader voters, and the 12% Gore>Bush>Nader voters. Each has two strategies. The Bush camp's strategies are Bush>Nader>Gore and Bush>Gore>Nader ("reverse" and "no reverse"), while the Gore camp's strategies are Gore>Bush=Nader and Gore>Bush>Nader ("truncate" and "no truncate"). The outcomes of the game are: Truncate | No Truncate |---------+-------------| | | | | Nader | Bush | Reverse | | | |---------+-------------|------------ | | | | Gore | Gore | No Reverse | | | ------------------------- (I fully realize that the above table looks horrible in most readers. Hopefully you get the idea. If it's too garbled to make out, cut and paste it into a simple text viewer and it will look fine.) There are two Nash Equilibria here: (no-reverse, truncate) and (reverse, no truncate). One gives the election to Gore, and the other to Bush. Hence, the game of brinkmanship to see which camp will be able to bully the other camp into letting them win to stay away from Nader. That's not all the information we can garner from this table, however. Game theorists refer to a strategy as a "dominated strategy" if, for every response the other player gives, one strategy is as good or better than another. In this case, truncation is a dominated strategy for the Gore camp. If the Bush camp reverses, they do better by not truncating, and if the Bush camp does not reverse, then they do just as well by not truncating as by truncating. This suggests that the Bush camp has an advantage over the Gore camp in the game of brinkmanship. I fully recognize that this does not mean the Bush camp will succeed in convincing the Gore camp to give in and rank Bush second. The leaders of the Bush camp seem to have a much more difficult job, trying to convince their supporters to uprank Nader, even though this runs the risk of handing Nader the election. The leaders of the Gore camp just have to convince their supporters to bullet vote, which seems to me a much easier job. Furthermore, it seems that the Bush voters would be even more averse to Nader than the Gore voters, which puts them at a disadvantage in the brinkmanship/bullying contest. This won't necessarily be true in a more general case, though. At any rate, back to my original point. This is why I like the other equilibrium, where the Nader voters announce that they plan to equal-rank Gore and Nader. This effectively nullifies all offensive strategy on the part of the Bush camp. Of course, it requires insincere equal first-ranking, so it has problems of its own. But it has none of the stability issues of the other equilibrium. -Adam ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em