This query has lead me to some interesting ideas: Approval Ballots. Two to elect: For each pair of candidates {X1, X2}, let p1 and p2 be the respective probabilities that X1 or X2 would be selected by a random approval ballot drawing (restricted to our pair of candidates), and let p0 be the probability that a random ballot would approve neither X1 nor X2. Elect the pair with the greatest value of min(p1, p2) - p0. This actually gives two methods, since there are two natural ways of selecting a candidate by random approval ballots. The first way is to select ballots at random until the approval set for one of them has non-empty intersection with the set from which we are to select a winner. The names of the candidates are drawn randomly from a hat. The first name drawn of a candidate in the intersection set is the name of the winner. The second way starts out as above, but once the first non-empty intersection set is determined, additional ballots are drawn as needed to narrow down the intersection to one candidate, the winner. More later ... From: Forest Simmons <fsimmons@pcc.edu> > To: EM <election-methods@lists.electorama.com>, Andy Jennings > <elections@jenningsstory.com> > > > How about thie following ideas? > > Elect the pair that covers the most voters (i.e. that leaves the fewest > voters with nobody that they approved elected). In case of ties, among > tied pairs elect the one whose weaker member has the most approval. > > Or this variant: If no pair covers more than 70 percent of the voters, > elect the pair that covers the greatest number of voters. Otherwise > consider all pairs that cover at least 70 percent of the voters to be > tied. Then among tied pairs, elect the one whose weaker member has the > greatest approval. > > > From: Andy Jennings <elections@jenningsstory.com> > > To: Election Methods <election-methods@electorama.com> > > Subject: [EM] Approval ballots. Two to elect. Best method? > > > > SPAV? > > 1. Candidate with most approvals wins. > > 2. That candidate's voters have their voting weight halved (or multiplied > > by 1/3). > > 3. Remaining candidate with most points wins. > > > > STV-like? > > 1. Choose quota Q = one-third (or one half) of voters. > > 2. Candidate with most approvals wins. (T = # of approvals) > > 3. That candidate's voters have their voting weight multiplied by > > max(1-(Q/T), 0) > > 4. Remaining candidate with most points wins. > > > > Monroe-like? > > 1. For each pair of candidates, find the voter-assignment which maximizes > > the number of voters assigned to a candidate they approved, such that no > > more than half the voters are assigned to one candidate. > > 2. Elect the pair which satisfies the most voters. > > > > Others? Toby, what are your favorite PR methods at the moment? Can you > > give a short explanation of how Phragmen/Ebert would work with only two > to > > elect? > > > > > > > > Specifically, I'm worried that in practically every approval-ballot PR > > method, if there is a candidate you really like, but are sure that she > can > > get elected without your vote, you gain an advantage by not approving > > them. Is there any method that minimizes that incentive? > > > > ~ Andy > > >

Continuing as promised .. Suppose that there are to be w winners, and the ballots are still approval style: For each w_tuple of candidates, let p1, p2, ... be the respective probabilities of selection of the respective candidate by random ballot (restricted to the w-tuple), and let p0 be the probability that a random ballot would not approve any of the candidates in the w-tuple. Elect the w-tuple with the largest value of min(p1, p2, ...) - p0. To Be Continued ... On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > > > This query has lead me to some interesting ideas: > > Approval Ballots. Two to elect: > > For each pair of candidates {X1, X2}, let p1 and p2 be the respective > probabilities that X1 or X2 would be selected by a random approval ballot > drawing (restricted to our pair of candidates), and let p0 be the > probability that a random ballot would approve neither X1 nor X2. Elect > the pair with the greatest value of min(p1, p2) - p0. > > This actually gives two methods, since there are two natural ways of > selecting a candidate by random approval ballots. > > The first way is to select ballots at random until the approval set for > one of them has non-empty intersection with the set from which we are to > select a winner. The names of the candidates are drawn randomly from a > hat. The first name drawn of a candidate in the intersection set is the > name of the winner. > > The second way starts out as above, but once the first non-empty > intersection set is determined, additional ballots are drawn as needed to > narrow down the intersection to one candidate, the winner. > > More later ... > > From: Forest Simmons <fsimmons@pcc.edu> >> To: EM <election-methods@lists.electorama.com>, Andy Jennings >> <elections@jenningsstory.com> >> >> >> How about thie following ideas? >> >> Elect the pair that covers the most voters (i.e. that leaves the fewest >> voters with nobody that they approved elected). In case of ties, among >> tied pairs elect the one whose weaker member has the most approval. >> >> Or this variant: If no pair covers more than 70 percent of the voters, >> elect the pair that covers the greatest number of voters. Otherwise >> consider all pairs that cover at least 70 percent of the voters to be >> tied. Then among tied pairs, elect the one whose weaker member has the >> greatest approval. >> >> >> From: Andy Jennings <elections@jenningsstory.com> >> > To: Election Methods <election-methods@electorama.com> >> > Subject: [EM] Approval ballots. Two to elect. Best method? >> > >> > SPAV? >> > 1. Candidate with most approvals wins. >> > 2. That candidate's voters have their voting weight halved (or >> multiplied >> > by 1/3). >> > 3. Remaining candidate with most points wins. >> > >> > STV-like? >> > 1. Choose quota Q = one-third (or one half) of voters. >> > 2. Candidate with most approvals wins. (T = # of approvals) >> > 3. That candidate's voters have their voting weight multiplied by >> > max(1-(Q/T), 0) >> > 4. Remaining candidate with most points wins. >> > >> > Monroe-like? >> > 1. For each pair of candidates, find the voter-assignment which >> maximizes >> > the number of voters assigned to a candidate they approved, such that no >> > more than half the voters are assigned to one candidate. >> > 2. Elect the pair which satisfies the most voters. >> > >> > Others? Toby, what are your favorite PR methods at the moment? Can you >> > give a short explanation of how Phragmen/Ebert would work with only two >> to >> > elect? >> > >> > >> > >> > Specifically, I'm worried that in practically every approval-ballot PR >> > method, if there is a candidate you really like, but are sure that she >> can >> > get elected without your vote, you gain an advantage by not approving >> > them. Is there any method that minimizes that incentive? >> > >> > ~ Andy >> > >> >

Continuing as promised... Now let's allow range ballots with greater resolution than mere approval ballots. For each w-tuple of candidates we ... (1) first divide the ballots into two sets, the set S of ballots that have positive support for at least one candidate of our w-tuple, and the complementary set S' of ballots that rate every member of our w-tuple at zero. (2) Then we imagine an experiment of drawing a ballot B at random from the entire set of ballots. Let p0 be the probability that B is a member of S', and let p1, p2, ... pw be the respective probabilities for the choices of the respective members of our w_tuple given that B is a member of S. [Multiplication by (1-p0) would give the respective unconditioned probabilities.] (3) Let a1, a2, ... be the respective candidate score averages for ballots in S. (4) elect the w-tuple with the greatest value of the expression min(a1*p1, a2*p2, ...) - p0. There are at least three good ways of defining the details of the random ballot experiment. (1) The value p3, say, is the probability that candidate 3 would be the highest rated candidate (from our w-tuple) on a randomly drawn ballot from the set S. If several candidates are tied in this respect, divide up the probability (from Ballot B) equally among them.. (2) The positive scores on every ballot B are normalized with respect to the members of our w-tuple. These normalized scores are averaged over S to get the respective "random ballot" probabilities. (3) Use a Toby Pereira transformation to convert each range style ballot into one hundred approval ballots. Then use a random approval ballot lottery on that set. I'll explain this in my next post ... On Thu, Dec 3, 2015 at 1:44 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > Continuing as promised .. > > Suppose that there are to be w winners, and the ballots are still approval > style: > > For each w_tuple of candidates, let p1, p2, ... be the respective > probabilities of selection of the respective candidate by random ballot > (restricted to the w-tuple), and let p0 be the probability that a random > ballot would not approve any of the candidates in the w-tuple. > > Elect the w-tuple with the largest value of min(p1, p2, ...) - p0. > > To Be Continued ... > > > On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > >> >> >> This query has lead me to some interesting ideas: >> >> Approval Ballots. Two to elect: >> >> For each pair of candidates {X1, X2}, let p1 and p2 be the respective >> probabilities that X1 or X2 would be selected by a random approval ballot >> drawing (restricted to our pair of candidates), and let p0 be the >> probability that a random ballot would approve neither X1 nor X2. Elect >> the pair with the greatest value of min(p1, p2) - p0. >> >> This actually gives two methods, since there are two natural ways of >> selecting a candidate by random approval ballots. >> >> The first way is to select ballots at random until the approval set for >> one of them has non-empty intersection with the set from which we are to >> select a winner. The names of the candidates are drawn randomly from a >> hat. The first name drawn of a candidate in the intersection set is the >> name of the winner. >> >> The second way starts out as above, but once the first non-empty >> intersection set is determined, additional ballots are drawn as needed to >> narrow down the intersection to one candidate, the winner. >> >> More later ... >> >> From: Forest Simmons <fsimmons@pcc.edu> >>> To: EM <election-methods@lists.electorama.com>, Andy Jennings >>> <elections@jenningsstory.com> >>> >>> >>> How about thie following ideas? >>> >>> Elect the pair that covers the most voters (i.e. that leaves the fewest >>> voters with nobody that they approved elected). In case of ties, among >>> tied pairs elect the one whose weaker member has the most approval. >>> >>> Or this variant: If no pair covers more than 70 percent of the voters, >>> elect the pair that covers the greatest number of voters. Otherwise >>> consider all pairs that cover at least 70 percent of the voters to be >>> tied. Then among tied pairs, elect the one whose weaker member has the >>> greatest approval. >>> >>> >>> From: Andy Jennings <elections@jenningsstory.com> >>> > To: Election Methods <election-methods@electorama.com> >>> > Subject: [EM] Approval ballots. Two to elect. Best method? >>> > >>> > SPAV? >>> > 1. Candidate with most approvals wins. >>> > 2. That candidate's voters have their voting weight halved (or >>> multiplied >>> > by 1/3). >>> > 3. Remaining candidate with most points wins. >>> > >>> > STV-like? >>> > 1. Choose quota Q = one-third (or one half) of voters. >>> > 2. Candidate with most approvals wins. (T = # of approvals) >>> > 3. That candidate's voters have their voting weight multiplied by >>> > max(1-(Q/T), 0) >>> > 4. Remaining candidate with most points wins. >>> > >>> > Monroe-like? >>> > 1. For each pair of candidates, find the voter-assignment which >>> maximizes >>> > the number of voters assigned to a candidate they approved, such that >>> no >>> > more than half the voters are assigned to one candidate. >>> > 2. Elect the pair which satisfies the most voters. >>> > >>> > Others? Toby, what are your favorite PR methods at the moment? Can >>> you >>> > give a short explanation of how Phragmen/Ebert would work with only >>> two to >>> > elect? >>> > >>> > >>> > >>> > Specifically, I'm worried that in practically every approval-ballot PR >>> > method, if there is a candidate you really like, but are sure that she >>> can >>> > get elected without your vote, you gain an advantage by not approving >>> > them. Is there any method that minimizes that incentive? >>> > >>> > ~ Andy >>> > >>> >> >

I'll try to explain the Toby transformation in the five minutes remaining on my account: Let ballot B be a range style ballot with scores in the range between zero and 100 inclusive. For each k between zero and 99 (including zero and 99) we form an approval ballot B_k that approves all of the candidates rated above k by ballot B. This gives us 100 approval ballots to use in place of ballot B. Do this for every ballot B in the set of ballots, and give them equal weight, say 1/100 if you are worried about the "one man one vote" fanatics. To be continued ... On Thu, Dec 3, 2015 at 2:42 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > Continuing as promised... > > Now let's allow range ballots with greater resolution than mere approval > ballots. > > For each w-tuple of candidates we ... > > (1) first divide the ballots into two sets, the set S of ballots that have > positive support for at least one candidate of our w-tuple, and the > complementary set S' of ballots that rate every member of our w-tuple at > zero. > > (2) Then we imagine an experiment of drawing a ballot B at random from the > entire set of ballots. Let p0 be the probability that B is a member of S', > and let p1, p2, ... pw be the respective probabilities for the choices of > the respective members of our w_tuple given that B is a member of S. > [Multiplication by (1-p0) would give the respective unconditioned > probabilities.] > > (3) Let a1, a2, ... be the respective candidate score averages for > ballots in S. > > (4) elect the w-tuple with the greatest value of the expression > > min(a1*p1, a2*p2, ...) - p0. > > There are at least three good ways of defining the details of the random > ballot experiment. > > (1) The value p3, say, is the probability that candidate 3 would be the > highest rated candidate (from our w-tuple) on a randomly drawn ballot from > the set S. If several candidates are tied in this respect, divide up the > probability (from Ballot B) equally among them.. > > (2) The positive scores on every ballot B are normalized with respect to > the members of our w-tuple. These normalized scores are averaged over S to > get the respective "random ballot" probabilities. > > (3) Use a Toby Pereira transformation to convert each range style ballot > into one hundred approval ballots. Then use a random approval ballot > lottery on that set. I'll explain this in my next post ... > > On Thu, Dec 3, 2015 at 1:44 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > >> Continuing as promised .. >> >> Suppose that there are to be w winners, and the ballots are still >> approval style: >> >> For each w_tuple of candidates, let p1, p2, ... be the respective >> probabilities of selection of the respective candidate by random ballot >> (restricted to the w-tuple), and let p0 be the probability that a random >> ballot would not approve any of the candidates in the w-tuple. >> >> Elect the w-tuple with the largest value of min(p1, p2, ...) - p0. >> >> To Be Continued ... >> >> >> On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons@pcc.edu> wrote: >> >>> >>> >>> This query has lead me to some interesting ideas: >>> >>> Approval Ballots. Two to elect: >>> >>> For each pair of candidates {X1, X2}, let p1 and p2 be the respective >>> probabilities that X1 or X2 would be selected by a random approval ballot >>> drawing (restricted to our pair of candidates), and let p0 be the >>> probability that a random ballot would approve neither X1 nor X2. Elect >>> the pair with the greatest value of min(p1, p2) - p0. >>> >>> This actually gives two methods, since there are two natural ways of >>> selecting a candidate by random approval ballots. >>> >>> The first way is to select ballots at random until the approval set for >>> one of them has non-empty intersection with the set from which we are to >>> select a winner. The names of the candidates are drawn randomly from a >>> hat. The first name drawn of a candidate in the intersection set is the >>> name of the winner. >>> >>> The second way starts out as above, but once the first non-empty >>> intersection set is determined, additional ballots are drawn as needed to >>> narrow down the intersection to one candidate, the winner. >>> >>> More later ... >>> >>> From: Forest Simmons <fsimmons@pcc.edu> >>>> To: EM <election-methods@lists.electorama.com>, Andy Jennings >>>> <elections@jenningsstory.com> >>>> >>>> >>>> How about thie following ideas? >>>> >>>> Elect the pair that covers the most voters (i.e. that leaves the fewest >>>> voters with nobody that they approved elected). In case of ties, among >>>> tied pairs elect the one whose weaker member has the most approval. >>>> >>>> Or this variant: If no pair covers more than 70 percent of the voters, >>>> elect the pair that covers the greatest number of voters. Otherwise >>>> consider all pairs that cover at least 70 percent of the voters to be >>>> tied. Then among tied pairs, elect the one whose weaker member has the >>>> greatest approval. >>>> >>>> >>>> From: Andy Jennings <elections@jenningsstory.com> >>>> > To: Election Methods <election-methods@electorama.com> >>>> > Subject: [EM] Approval ballots. Two to elect. Best method? >>>> > >>>> > SPAV? >>>> > 1. Candidate with most approvals wins. >>>> > 2. That candidate's voters have their voting weight halved (or >>>> multiplied >>>> > by 1/3). >>>> > 3. Remaining candidate with most points wins. >>>> > >>>> > STV-like? >>>> > 1. Choose quota Q = one-third (or one half) of voters. >>>> > 2. Candidate with most approvals wins. (T = # of approvals) >>>> > 3. That candidate's voters have their voting weight multiplied by >>>> > max(1-(Q/T), 0) >>>> > 4. Remaining candidate with most points wins. >>>> > >>>> > Monroe-like? >>>> > 1. For each pair of candidates, find the voter-assignment which >>>> maximizes >>>> > the number of voters assigned to a candidate they approved, such that >>>> no >>>> > more than half the voters are assigned to one candidate. >>>> > 2. Elect the pair which satisfies the most voters. >>>> > >>>> > Others? Toby, what are your favorite PR methods at the moment? Can >>>> you >>>> > give a short explanation of how Phragmen/Ebert would work with only >>>> two to >>>> > elect? >>>> > >>>> > >>>> > >>>> > Specifically, I'm worried that in practically every approval-ballot PR >>>> > method, if there is a candidate you really like, but are sure that >>>> she can >>>> > get elected without your vote, you gain an advantage by not approving >>>> > them. Is there any method that minimizes that incentive? >>>> > >>>> > ~ Andy >>>> > >>>> >>> >> >

Now I just have time to point out that in place of the random ballot lottery we could use any proportional lottery, such as PAV or The Ultimate Lottery. More on next time ... On Thu, Dec 3, 2015 at 2:42 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > Continuing as promised... > > Now let's allow range ballots with greater resolution than mere approval > ballots. > > For each w-tuple of candidates we ... > > (1) first divide the ballots into two sets, the set S of ballots that have > positive support for at least one candidate of our w-tuple, and the > complementary set S' of ballots that rate every member of our w-tuple at > zero. > > (2) Then we imagine an experiment of drawing a ballot B at random from the > entire set of ballots. Let p0 be the probability that B is a member of S', > and let p1, p2, ... pw be the respective probabilities for the choices of > the respective members of our w_tuple given that B is a member of S. > [Multiplication by (1-p0) would give the respective unconditioned > probabilities.] > > (3) Let a1, a2, ... be the respective candidate score averages for > ballots in S. > > (4) elect the w-tuple with the greatest value of the expression > > min(a1*p1, a2*p2, ...) - p0. > > There are at least three good ways of defining the details of the random > ballot experiment. > > (1) The value p3, say, is the probability that candidate 3 would be the > highest rated candidate (from our w-tuple) on a randomly drawn ballot from > the set S. If several candidates are tied in this respect, divide up the > probability (from Ballot B) equally among them.. > > (2) The positive scores on every ballot B are normalized with respect to > the members of our w-tuple. These normalized scores are averaged over S to > get the respective "random ballot" probabilities. > > (3) Use a Toby Pereira transformation to convert each range style ballot > into one hundred approval ballots. Then use a random approval ballot > lottery on that set. I'll explain this in my next post ... > > On Thu, Dec 3, 2015 at 1:44 PM, Forest Simmons <fsimmons@pcc.edu> wrote: > >> Continuing as promised .. >> >> Suppose that there are to be w winners, and the ballots are still >> approval style: >> >> For each w_tuple of candidates, let p1, p2, ... be the respective >> probabilities of selection of the respective candidate by random ballot >> (restricted to the w-tuple), and let p0 be the probability that a random >> ballot would not approve any of the candidates in the w-tuple. >> >> Elect the w-tuple with the largest value of min(p1, p2, ...) - p0. >> >> To Be Continued ... >> >> >> On Thu, Dec 3, 2015 at 1:30 PM, Forest Simmons <fsimmons@pcc.edu> wrote: >> >>> >>> >>> This query has lead me to some interesting ideas: >>> >>> Approval Ballots. Two to elect: >>> >>> For each pair of candidates {X1, X2}, let p1 and p2 be the respective >>> probabilities that X1 or X2 would be selected by a random approval ballot >>> drawing (restricted to our pair of candidates), and let p0 be the >>> probability that a random ballot would approve neither X1 nor X2. Elect >>> the pair with the greatest value of min(p1, p2) - p0. >>> >>> This actually gives two methods, since there are two natural ways of >>> selecting a candidate by random approval ballots. >>> >>> The first way is to select ballots at random until the approval set for >>> one of them has non-empty intersection with the set from which we are to >>> select a winner. The names of the candidates are drawn randomly from a >>> hat. The first name drawn of a candidate in the intersection set is the >>> name of the winner. >>> >>> The second way starts out as above, but once the first non-empty >>> intersection set is determined, additional ballots are drawn as needed to >>> narrow down the intersection to one candidate, the winner. >>> >>> More later ... >>> >>> From: Forest Simmons <fsimmons@pcc.edu> >>>> To: EM <election-methods@lists.electorama.com>, Andy Jennings >>>> <elections@jenningsstory.com> >>>> >>>> >>>> How about thie following ideas? >>>> >>>> Elect the pair that covers the most voters (i.e. that leaves the fewest >>>> voters with nobody that they approved elected). In case of ties, among >>>> tied pairs elect the one whose weaker member has the most approval. >>>> >>>> Or this variant: If no pair covers more than 70 percent of the voters, >>>> elect the pair that covers the greatest number of voters. Otherwise >>>> consider all pairs that cover at least 70 percent of the voters to be >>>> tied. Then among tied pairs, elect the one whose weaker member has the >>>> greatest approval. >>>> >>>> >>>> From: Andy Jennings <elections@jenningsstory.com> >>>> > To: Election Methods <election-methods@electorama.com> >>>> > Subject: [EM] Approval ballots. Two to elect. Best method? >>>> > >>>> > SPAV? >>>> > 1. Candidate with most approvals wins. >>>> > 2. That candidate's voters have their voting weight halved (or >>>> multiplied >>>> > by 1/3). >>>> > 3. Remaining candidate with most points wins. >>>> > >>>> > STV-like? >>>> > 1. Choose quota Q = one-third (or one half) of voters. >>>> > 2. Candidate with most approvals wins. (T = # of approvals) >>>> > 3. That candidate's voters have their voting weight multiplied by >>>> > max(1-(Q/T), 0) >>>> > 4. Remaining candidate with most points wins. >>>> > >>>> > Monroe-like? >>>> > 1. For each pair of candidates, find the voter-assignment which >>>> maximizes >>>> > the number of voters assigned to a candidate they approved, such that >>>> no >>>> > more than half the voters are assigned to one candidate. >>>> > 2. Elect the pair which satisfies the most voters. >>>> > >>>> > Others? Toby, what are your favorite PR methods at the moment? Can >>>> you >>>> > give a short explanation of how Phragmen/Ebert would work with only >>>> two to >>>> > elect? >>>> > >>>> > >>>> > >>>> > Specifically, I'm worried that in practically every approval-ballot PR >>>> > method, if there is a candidate you really like, but are sure that >>>> she can >>>> > get elected without your vote, you gain an advantage by not approving >>>> > them. Is there any method that minimizes that incentive? >>>> > >>>> > ~ Andy >>>> > >>>> >>> >> >