David wrote: > Thanks for the information. So am I right in thinking that strategy A > gets to the Condorcet winner by a process of iteration. In response to a > series of Approval polls the voters alter their choices and end up voting > in such a way that they elect the Condorcet winner. Or is it more > complex than this in theory (I know it's more complex in reality)?
That's correct. Following Lorrie Cranor's Declared Strategy Voting (http://lorrie.cranor.org/dsv.html), I distinguish between ballot-by-ballot mode and batch mode. You've been using batch mode (all voters react to the last results at the same time), which in this example indeed leads to cycling between winners A and B. Strategy A doesn't always lead to an equilibrium in batch mode even when a Condorcet winner exists (see http://groups.yahoo.com/group/election-methods-list/message/9713), but it's extremely likely to when voters have fully-ranked preferences. When many voters don't, as in your example, equilibria are less common and don't always elect the Condorcet winner. My simulations sometimes generated tied preferences but not often enough to produce this kind of situation. Ballot-by-ballot mode (voters take turns reacting to the latest results) would eventually find an equilibrium, but it won't necessarily elect the Condorcet winner either: A 380 approve A A>B 28 approve A A>C 9 approve AC B 80 approve B B>A 2 approve B B>C 133 approve B C 4 approve C C>A 13 approve CA C>B 351 approve CB This equilibrium elects B (A 430, B 566, C 377). Strategy A is still optimal here in the sense that none of the nine blocs can change its vote and improve the result from its perspective. In fact, as far as I can see, there's no coalition of blocs that can band together and change the result to the coalition's advantage, which makes it a strong Nash equilibrium. Steven Brams has proved that every Approval strong Nash equilibrium elects a Condorcet winner . . . when all voters have fully-ranked preferences. When they don't, obviously strange things can happen. Consider a simpler example: 45:Reagan>Anderson=Carter 20:Anderson>Carter>Reagan 35:Carter>Anderson>Reagan Now there are two equilibria, one that elects Anderson and one that elects Carter. When the three blocs are considered players, this election reduces to the game of chicken. The Reagan voters are effectively sitting the election out; this strangeness goes away when the Reagan voters discover a preference between Carter and Anderson and become kingmakers. As far as I can tell, strategy A does as well for a voter as any other Approval strategy that considers only current "poll" results, own preferences and last own vote. If anyone has a better one, or even an interesting new one, please let me know. ===== Rob LeGrand, psephologist [EMAIL PROTECTED] Citizens for Approval Voting http://www.approvalvoting.org/ __________________________________ Do you Yahoo!? Free Pop-Up Blocker - Get it now http://companion.yahoo.com/ ---- Election-methods mailing list - see http://electorama.com/em for list info