Since the issue of CR and Approval being strategically equivalent has come up here a bit in the last week or so, here's another attempt to resolve the issue:
If all Nash equilibria for CR involve giving every candidate either zero or full points then Approval and CR are strategically equivalent. The methods are also equivalent if a faction of votes with identical preferences (treated as a single player for these purposes) is indifferent between the outcome obtained by giving a candidate an intermediate rating and the outcome obtained by giving that candidate the top or bottom rating. Here's a sketch of the argument, but maybe not a rigorous proof: Say that we have a Nash equilibrium where candidate B wins, and a faction which prefers A to B has given A an intermediate rating. If that faction alone could cause candidate A to win by giving A the top rating then we wouldn't have a Nash equilbrium. Conversely, if that faction cannot cause A to win by giving him the top rating, then the faction is indifferent between giving him an intermediate rating versus giving him either the top or bottom rating. Also, if candidate B wins, but A would win if a faction preferring A to B gave B a zero rating instead of an intermediate rating, then we wouldn't be at a Nash equilibrium. Moreover, if that faction cannot cause A to lose by giving B a zero rating then the faction is indifferent between giving him an intermediate rating versus giving him either the top or bottom rating. So, I now concede the strategic equivalence of Approval and CR when payoffs are considered for a single election only and voters have perfect information. However, I still maintain that payoffs calculated over multiple elections may give incentives to give intermediate ratings, and that intermediate ratings may be optimal in some rare cases with imperfect information. The long-term case: I used the Greens as an example. I argued that if the Greens give the Democrats an intermediate rating, they are a valued constituency that cannot be written off easily. Moreover, they have both a stick to punish transgressions and a carrot to entice further concessions from the Dems. Mike argued that if the Greens want to influence the Dems they should use the biggest stick possible, and what stick is bigger than a zero rating? My reply: First, consider a person from a GOP core constituency who regularly gives money to the GOP. This person would presumably give the Dems a zero rating in CR elections. He is using the biggest stick possible. Apart from a few old-school conservative Southern Democrats, most Dems will write this person off as unreceptive and seek support from more moderate or liberal voters. Now consider Greens who consistently refuse to support the Dems. After a few elections the Dems may write them off as unreachable, just as they disregard conservative constituencies. Greens then have no leverage over the Dems, because they can't make their stick any bigger and the Dems think that pursuing the Green carrot is a waste of time. However, if the Green give partial support they have a stick that can always be made larger and a carrot that interests the Dems. Note: I use the Greens and Dems only to illustrate a strategic question, not to take sides in the debate between Greens and Dems. If a Green believes that the Dems are unreformable then withholding the carrot and using the biggest stick is of course the rational and optimal strategy. The above argument is predicated on the assumption that both parties are capable of finding common ground, an assumption that fails if one of the parties is unreformable. Alex ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
