Re: [EM] Proportional Representation from Ratings Ballots
(Ahem), actually it's: Stage 1, (Election stage) select k1 so that k1*[ w(a)*r(a) + w(b)*r(b) + . ] = 1 Each candidate gets k1*w(x)*r(x) Stage 2, (Elimination stage) select k2 so that (k2)^2 * ([w(c)*r(c)]^2 + [w(d)*r(d)]^2 + ... ) = 1 - k1*[w(a)*r(a) + w(b)*r(b)] Each candidate gets k2*[w(x)*r(x)] Eliminate the remaining candidate who scores the lowest. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
On Thu, Nov 19, 2009 at 12:49 PM, Raph Frank wrote: > If no candidate is elected, a different rule is used, each ballot is > scaled so that > > [w(a)*r(a)]^2 + [w(b)*r(b)]^2 + ... = 1 > > The running candidate the the lowest score is then eliminated. Stage 1, (Election stage) select k1 so that k1*[ w(a)*r(a) + w(b)*r(b) + . ] = 1 Each candidate gets k1*w(x)*r(x) Stage 2, (Elimination stage) select k2 so that k2*(w(c)*r(c)^2 + w(d)*r(d)^2 + ... ) = 1 - k1*[w(a)*r(a) + w(b)*r(b)] Each candidate gets k2*w(x)*r(x)^2 Eliminate the remaining candidate who scores the lowest. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
Stealing the Meek's method proof :). This is the existance proof: A weighting vector is defined as "feasible" if all elected candidates have a score >= the quota (Q) A candidate receives from each ballot (B) a score (S) equal to S(B) = r(c)*w(c)/(sum_over_x(r(x)*w(x)) r(x) is the rating for the candidate on that ballot w(x) is the global weighting of that candidate The candidate's total T(c) is the sum of the vote received from each ballot Theorom: Replacing an elected candidate's weight by w(c)*k will not convert a feasible vector into an non-feasible one, if k>=Q/T(c) Proof: The candidate will receive from each ballot S_new(B) = r(c)*w_new(c)/(sum_over_x(r(x)*w_new(x)) Only 1 term has changed in the sum Since w_old(x) >= w_new(c) and r(c) >= 0, then the sum cannot increase, i.e. sum_over_x(r(x)*w_new(x)) <= sum_over_x(r(x)*w_old(x)) Thus, S_new(B) >= r(c)*w_new(c)/(sum_over_x(r(x)*w_old(x)) replacing w_new(c) with w_old(c)*k as required, gives S_new(B) >= r(c)*w_old(c)*k/(sum_over_x(r(x)*w_old(x)) Re-arranging: S_new(B) >= k*[r(c)*w_old(c)/(sum_over_x(r(x)*w_old(x)) S_new(B) >= k*S_old(B) Thus, the candidate will receive form each ballot a score that is at least k times as large as before the change. Thus, T_new(c) >= T_old(c)*k Assuming, k>=Q/T_old(c) T_new(c) >= T_old(c)*[Q/T_old(c)] T_new(c) >= Q Thus, the candidate in question who had his weighting decreased will still have at least a quota. All other candidates will at worst have their vote totals remain static, and will likely increase. (The same proof, except their weight doesn't actually decrease). This means that if the vector was feasible initially, then it will still be feasible after updating the weight of candidate c. This means that the process can be applied over and over. In each step, the weighting of one of the elected candidates will be decreased, but never increase. This means that the total vote held by the elected candidates will decrease. This also gives an algorithm. You can apply that rule to each elected candidate in turn. In fact, you can update them all in 1 go (set a weights w(c) = Q/T(c) for elected candidates). If you update them in order, then all the other candidate's who haven't being reduced yet will have totals that will have increased slightly (or stayed the same). Thus, k = Q/T(c) for all the other candidates will drop (or stay the same). If you use the k from before the update, it will still be greater or equal to the k that would have been used if they were done in order, which is all that is required. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
On Thu, Nov 19, 2009 at 4:51 AM, Brian Olson wrote: > Oh, that is a problem. It gets the right answer if I use L1 norm instead of > L2. I think L2 norm is going to work better for single-seat IRNR but L1 norm > better for multi-seat. L2 inflates the amount of vote that winds up getting > applied to multiple choices. The L1 norm mean that each voter always gets to cast exactly 1 vote (ratings add to 1). Thus the total number of votes cast is always constant. This means that a quota can be easily determined. You could use a different rule for eliminating than you use for electing (and I think that is a good idea anyway). For example, for electing, each ballot is scaled so that w(a)*r(a) + w(b)*r(b) + . = 1 All eliminated candidates have a w(x) = 0 and all non-elected candidates have a weighting of 1. Elected candidates have weighting so that they have exactly a quota of the votes. If any candidate meets the Droop quota, that candidate is declared elected and the next round is started. If no candidate is elected, a different rule is used, each ballot is scaled so that [w(a)*r(a)]^2 + [w(b)*r(b)]^2 + ... = 1 The running candidate the the lowest score is then eliminated. (The weights are based on the L1 calculation) This process has the nice feature that a group of voters equal to a Droop quota will decide their candidate using the L2 single seat (L2) version of the process. (This assumes that they rate all non-party candidates at zero and all voters outside the group rate their candidates at zero). Also, there is also a question if the weights assigned in step 1 will always yield a unique set of weights. Hopefully there is a Meek's method like proof. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
On Nov 5, 2009, at 12:37 PM, Raph Frank wrote: Party A (65 voters): A1: 0.7 A2: 0.7 B: 0 Party B (35 voters) A1: 0 A2: 0 B: 1 Round 1 A1: 45.5 A2: 45.5 B: 35 All 3 have exceeded the quota. A1 or A2 will be elected and that will cause the 2nd unelected member of that party to increase his vote since they exceeded the quota. Thus after round 2, both A1 parties will be elected. Oh, that is a problem. It gets the right answer if I use L1 norm instead of L2. I think L2 norm is going to work better for single-seat IRNR but L1 norm better for multi-seat. L2 inflates the amount of vote that winds up getting applied to multiple choices. Warren Smith pointed out some pathological cases like: what if everyone voted (1,0,0,0,0) or (0,1,1,1,1), but I just can't get worked up over how if everyone voted weirdly they might get a weird answer. I'm working up a set of election space diagrams for STV, IRNRP and an artificial method that maximizes happiness where every voter gets a maximum of 1 unit of happiness (no point in giving any one voter too many of their favorite choices). What I want out of a PR election method: 1. Evaluate the whole ballot at once (not just first place vote like IRV/STV). This tends to find better solutions. 2. No wasted vote. Choices that don't win (disqualified in intermediate rounds) don't detract from a voter's ability to elect the best choice they can. Voting for a very popular choice leaves you some vote left for less preferred choices. (STV with Meek redistribution does this OK) 3. Poportionality. Colloquially: An N% constituency should get about N % of the seats. 4. Utility. Maximize the sum happiness of the population. 5. Fairness. No one or no group should be unnecessarily shut out. 6. Strategy resistance. For any voter or bloc of voters, honest voting maximizes expected utility. Brian Olson http://bolson.org/ Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
On Thu, Nov 5, 2009 at 6:38 PM, Kristofer Munsterhjelm wrote: > How does IRV work in this situation? I'd guess that IRNR would work the same > way. Unless I'm mistaken, the ballots would be considered exhausted and thus > "removed from play", as it were. If everyone bullet voted, then the N candidates with the most votes would win the seats. Any candidate which exceeded the quota would win immediately, but there would be no transfers. Thus the candidates would be eliminated one by one from the lowest candidate until the remaining candidates are deemed elected without hitting the quota. In a single seat case, the winner would be the plurality winner. Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] Proportional Representation from Ratings Ballots
Warren Smith wrote: On 11/5/09, Warren Smith wrote: --Olson's method fails. Suppose voter #1 votes (1,0,0,0,...0) "plurality style". If canddt#1 is eliminated, the ballot then becomes unnormalizable. I assume Olson deals with that by throwing it in the garbage. More seriously: If canddt#1 instead is declared a winner with over-quota, then Olson reweights canddt#1 and renormalizes ballot #1 so its sum-of=squares is still 1. However, it is impossible to do both if ALL ballots happen to be plurality-style, the reweighting and renormalization conflict and the conflict is irreconciliable. How does IRV work in this situation? I'd guess that IRNR would work the same way. Unless I'm mistaken, the ballots would be considered exhausted and thus "removed from play", as it were. Election-Methods mailing list - see http://electorama.com/em for list info
