> From: Kristofer Munsterhjelm > To: [email protected] > Cc: [email protected] > Subject: Re: [EM] Least Expected Umbrage, a new lottery method > Message-ID: <[email protected]> > Content-Type: text/plain; charset=ISO-8859-1; format=flowed > > On 12/19/2011 01:50 AM, [email protected] wrote: > > Let M be the matrix whose row i column j element M(i,j) is the > number> of ballots on which i is ranked strictly above j plus > half the number > > of ballots on which neither i nor j is ranked. > > > > In particular, for each k the diagonal element M(k , k) is > half the > > number of ballots on which candidate k is unranked. > > > > Now think of M as the payoff matrix for the row player in a > zero sum > > game. > > > > Elect the candidate that would be chosen by the optimal > strategy of > > the row player. > > > > [End of Method Definition] > > Is that method strategy-proof? If not, is there any way to > determine > whether a given stochastic method is strategy-proof or not? I > know that > some are (such as Random Pair, for instance).
Random favorite is also strategy proof. I do not know of any general technique for detecting strategy proofness. > > From: Warren Smith > To: election-methods > Subject: [EM] Least Expected Umbrage > Message-ID: > > Content-Type: text/plain; charset=ISO-8859-1 > > can you compare this method with > Rivest-Shen voting? > > http://people.csail.mit.edu/rivest/gt/latest_full.pdf > Rivest-Shen voting has the same relation to MinMax(margins) that LEU voting has to MMPO with Symmetric Completion Bottom. Like MinMax(margins) Rivest Shen satisfies the Condorcet Criterion but fails the FBC. LEU satisfies the FBC but fails the Condorcet Criterion. Rivest Shen fails mono-raise-winner. I hope that LEU is fully monotone. The Rivest Shen probabilities form the optimal mixed strategy for a two person zero sum game based on the margins matrix. In that game the row and column players have the same strategy. The LEU probabilities form the optimal mixed strategy for the column player in the two person zero sum game based on the matrix M whose (i, j) entry M(i,j) is the number of ballots on which candidate i is rated strictly greater than candidate j plus half the number of ballots on which both i and j are rated at minRange. Basically M(i,j) is the opposition to candidate i from the candidate j supporters. Since for different values of i the opposition is different, we must give greater weight to the more credible values of i. The row player's probabilities supply these weights. The column player's probabilities give the optimal response to this threat. This is easiest to see when there is a saddle point so that for some i and j, the value M(i,j) is the min element in row i and the max element in column j. In this case the optimal row strategy is P(i)=100%. the optimal column strategy is P(j)=100%. This is the same result given by the deterministic method MMPO with Symmetric Completion Bottom. Note that i is the biggest offensive threat to j, and j has the best defense against i. So we are pitting the candidate with the best offense against the candidate with the best defense. Electing the candidate with the best defense over-rides the preferences of the fewest voters. When there is no saddle point there is no single best offensive or defensive candidate. That's where the mixed strategies come in. You might say that the MMPO winner is the best defensive candidate, but that is only true if the candidate that scores the most against him/her is an actual credible threat, i.e. a candidate that is strong against other candidates as well. The mixed strategies automatically take this consideration into account. There's probably a simpler way to explain this, but I hope my explanation is somewhat helpful. Forest ---- Election-Methods mailing list - see http://electorama.com/em for list info
