[EM] Simulation of political identity space in voting
Ka-Ping Ye did some excellent work which inspired me to replicate it. Given a two axis system of candidate and voter space, plot the results of population of voters centered at points on the plane voting based on distance to candidate. The original is here, and was discussed on this list many months ago: http://zesty.ca/voting/sim/ My new results are here: http://bolson.org/voting/sim_one_seat/www/spacegraph.html Mostly I've independently verified the results, but I've added my favorite pet method, Instant Runoff Normalized Ratings (IRNR) into the mix. This method is great because it makes behaviors of the election method readily apparent visually. I used to claim that IRNR would be free of IRV's oddities because IRNR considered the whole ballot and used continuous ratings. Someone here cleverly found a counter case, but graphically it jumps out of the picture that IRNR does have irregularities. On the plus side, they're much smaller than IRV's problems. :-) I wish it were easier to test all the different methods that have been proposed here. I already had a simulation framework for testing social utility which will run lots of tests under different numbers of candidates and voters and varying error rate. The same voting implementation also gets used by this new graphical test. It would be great to get more systems built in and tested. There's a pretty simple C++ interface to code to when implementing a new election method. I've made my source available in the past and will do so again if anyone wants to also work on this. I understand that most of you aren't computer scientists and quick to program up new tests, but I'm excited about this testing right now and if you'll just implement your favorite election method in _some_ language, C, C++, java, javascript, perl, python, heck I'll even accept PHP, LISP or FORTRAN, I'll translate it and fit it into the test harness. Anyway, mostly I wanted to share the pretty graphs I made of simulated elections. An ounce of data is worth a pound of theorizing? Brian Olson http://bolson.org/ election-methods mailing list - see http://electorama.com/em for list info
[EM] [Fwd: Re: Apportionment (biased?) let me add some more confusion to the mix :)]
What is Hamilton's method? Before exploring that, my proposal had two parts: EVERY state SHALL have at least the number of seats they earn with fractions ignored. While I may have got to Hamilton with my words as to disposing of fractions, I suggest that others may be able to do better with this part. I question whether monotonicity, etc., can justify failing to honor the part I demand. While labels can be convenient, we too often have different definitions behind the names. For example, in "Method of Apportionment", www.census.gov says that Hamilton was used 1850-1900, and includes a paragraph describing it. While they mention the problem of low population states elsewhere, they do not mention such in their description of Hamilton. While Israel could have trouble with small parties, we do not have enough small states to need better than what we have - if they got to be a problem, combining a few such states would be a possibility. New York disposes of too small parties by refusing to give any support to those that are too weak. DWK On Sun, 10 Dec 2006 22:30:05 -0500 Joseph Malkevitch wrote: > Dear David, > > In essence what you are describing is Largest Remainder or Hamilton's > Method. If the house size is allowed to change this method does not obey > house monotonicity. It also violates various "population monotonicity" > axioms. The house size is currently fixed so that is not a problem. If > you do not mind that population monotonicity can be violated then you > can promote Largest Remainder. There is also some consideration whether > or not this method may or may not advantage "smaller" vs. "larger" > states. Israel used to us this system and had lots of troubles with many > small parties. Eventually it moved to D'Hondt. In the European version > of the apportionment problem there are typically requirements that > parties get a minimum percentage of the vote to get any seats. > > Regards, > > Joe > > > > > > On Dec 10, 2006, at 9:22 PM, Dave Ketchum wrote: > >> I suggest the following as a proposed LIMIT on the fancy finagling: >> >> Calculate persons per district as (total persons)/(legislature size). >> >> Since this is for Congress, every state earning less than one whole >> seat gets one, with no consideration as to fractions - period. >> >> Each other state gets the whole seats they have earned - period. >> >> I CLAIM that each state has earned the above and should get ALL of >> that - period. >> >> So there are some leftover fractions we can debate, but debate limited >> to these - trying to avoid Alabama and other paradoxes is restricted >> to allocation of these fractions. >> >> I propose, without arguing against whoever may claim to do better: >> Sort the fractions as to size, with largest sizes each getting >> one of the leftover seats. If the end of this requires deciding among >> identical fractions, assign among these randomly. -- [EMAIL PROTECTED]people.clarityconnect.com/webpages3/davek Dave Ketchum 108 Halstead Ave, Owego, NY 13827-1708 607-687-5026 Do to no one what you would not want done to you. If you want peace, work for justice. election-methods mailing list - see http://electorama.com/em for list info
Re: [EM] Apportionment (biased?) let me add some more confusion to the mix :)
On Dec 11, 2006, at 12:38 , [EMAIL PROTECTED] wrote: > > From: [EMAIL PROTECTED] > > > > One more tool that can be useful in some situations is the > > hierarchical structure of the states/parties. To guarantee that > > certain set of states/parties will not be underrepresented they > could > > form a team/alliance. When seats are allocated to that team they > > could lose (in typical allocation methods) only one seat to rounding > > errors instead on many of them losing a seat. Geographic alliances > > would maybe be more natural than e.g. an alliance of small states. > > What about sorting the States based on population and then splitting > them into 2 groups such that the total population in each group is as > equal as possible. > > The fractional seat is then split between the 2 groups based on > (Webster?) > ... or maybe Webster should be used directly? > > This is then applied to each group recursively. Ok. A binary tree like structure makes the division quite balanced (maybe even more than necessary for most practical needs). > If any State ends up with zero seats, it is removed from the process > and given a seat directly. The process is then re-run, until it > completes with all remaining States getting at least 1 seat. Careful with this. There is a risk that the calculation rules steal seats from the states that have slightly more than one seat worth of inhabitants. > This pretty much is forced to be unbiased between small and large > States > size. However, perhaps it would be biased in other ways. > > An additional rule could then be that States are allowed to form > groups > 'manually', and manual groups cannot be split in two by the algorithm > (until the group being processed is the manual group itself). It is possible to support multiple "proportionalities". It is possible to make more than one of them "exact" (=all divisions followed to the accuracy of rounding errors that are smaller than one seat) at the same time or "approximate" (= one division based on one rule, then another rule applied in each group (of the first division), e.g. first the manual groups and then the automatic size based groups within the manual groups). A more typical situation would be to use some more orthogonal measures like party/ideology proportionality and regional proportionality. The ideological and regional proportionality requirements are the most common ones. What others could there be? The state size based one was already discussed. Countries that have clear ethnical or religious division lines could use such additional proportionality rules. Maybe also different age or sex groups could be guaranteed a proportional share of the seats. It is quite straight forward to develop methods that respect such criteria either exactly or approximately in hierarchy. The number of seats should be large and the criteria should be as orthogonal as possible if we want to use several of them (to avoid situations where there for example are no female catholic candidates left in Hawaii when we would need one). Strong requirements on exact divisions also lead to pushing the rounding errors to some less critical areas but in a way that makes them very visible (e.g. (exact) ideological/party proportionality in 50 states with (exactly) one seat in each state would probably lead to electing a green candidate in some state that has only a 5% minority of green votes). > > > > I already mentioned the different voting power. A simple method in > > that direction would be to elect one representative from every state > > and give her voting power in relation to the number of people she > > represents. Or maybe large states would be given n seats with 1/n of > > the voting power of the state etc. Maybe the building where these > > representatives will work has a fixed number of physical seats => > > fill those seats and allocate voting power according to that. > > The logistics of this would make the legislature less efficient. One > possible rule would be that all Representatives must have voting > strengths between 0.9 and 1.1 and a detailed count only happens if > the vote is close (or if there is a motion demanding it). I agree that some reasonable restrictions should be applied. Also my scenarios where the seats were divided in time may lead to too short times in office. If parties are allowed to fill the seats as they wish, one could also consider terms that last longer than one election period and terms that need not end and start at election time. Everything is ok as long as the party has exactly the agreed number of representatives active at any given time. I however think that also a method that gives two votes to a candidate that got two quotas of votes and so on would be quite ok (i.e. the maximum number of votes could be infinite or some fixed limit instead of 1.1). Giving very long terms to candidates that get lots of votes is no
[EM] Webster vs Bias-Free
Let me define a few terms. S(q) is an allocation's function, seats as a function of quotas. dS is the amount by which S is above the 1-seat-per-quota line (of the graph that I posted about the other day). Yes, dS should stand for an infinitessimal change in S, but most computers don't have the letter "delta", and so I'm saying dS for that finite difference. Of course if ds is negative, then S is below the one-seat-per-quota line. The horizontal part of the step function below the 1-seat-per-quota line is the "low section". The high section is similarly defined. A low section and its subsequent high section is a "cycle" of the step function. This morning I looked at Webster's justification for proportionality, it's a sort of blockwise approach that looks at a whole cycle at a time. In each cycle, in Webster, dS sums to zero. So whether a cycle is in the high-population or low-population part of the range, the cycles have no overal net dS--lif they did, then the varying value of q would cause different ends of the range to have different overall S/q. So, in that rough sense, Webster is unbiased. Well, when I looked at that this morning, I said, "You can't say that about Bias-Free (the method that I defined last night). Its dS doesn't sum to zero over each cycle. So Webster's rough unbias argument doesn't work for Bias-Free. So I hurried to the computer to pretty much retract what I'd said about Bias-Free (BF) being the genuinely unbiased quota & rounding method. I spoke too soon. Though Webster's rough argument doesn't work for BF, a finer and more accurate one does. Here's why I said (correcly) last night that Webster has a liittle bias: In a partilcular cycle, consider a point on the low-section, and the corresponding point on the subsequent high-section. They both have the same dS. But they don't both have the same q. The piont that's on the high section of course has a larger q. So The point on the low section's negative dS/q has a greater absolute value than the positive dS/q of the corresponding point on the high section. That's true in every cycle. There's a deficit of dS/q, a net negative sum. And that's more pronounced at the low-population end of the range, because, there, q is less, and so the dS/q is more negative. So there's less S/q at the low end of the range. As I said, the difference is very slight, and Webster's unbias is very slight. But it's there. That's why I devised Bias-Free. Sum dS/q over a cycle, set it equal to zero, and solve for R, the rounding point. In that way, you make dS/q sum to zero over the cycle. That's Bias-Free. No nonzero sum of dS/q in any cycle. No S/q deficit in any cycle. That's a more accurate and detailed unbias than Webster has. I showed last night that Webster is very close to BF. Hill is significantly different from BF. Hill's rounding points are considerably lower. Or look at the graphs that I described a few days ago. From the above argument, or from the graphs, Hill gives significantly more seats per quota to the smallest states. That is not "equal proportions". Yes, Hill advocates justify it from how it rounds off. But the meaningful consisderation is how the states' S/q compare. In Hill, the proportions are _not_ equal. The name Equal Proportions acknowledges equal S/q as the important thing, but Hill doesn't deliver. Webster of course is very close to BF. Webster, BF. and Hamilton are all adequately unbiased. BF & Hamilton are completely unbiased. But Hamilton is capricious. As you know, states get very annoyed when they lose a seat for no good reason. For that reason we want BF or Webster. BF would be better, but Webster is simpler, more naturally obvious, and has precedent. Mike Ossipoff _ Share your latest news with your friends with the Windows Live Spaces friends module. http://clk.atdmt.com/MSN/go/msnnkwsp007001msn/direct/01/?href=http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mk election-methods mailing list - see http://electorama.com/em for list info
Re: [EM] Apportionment bias
At 11:39 PM 12/10/2006, Warren Smith wrote: > claim by this same (standard) definition, all other apportionment >methods so far discussed, generically exhibit bias. Note that Asset Voting with precinct-based vote transfers produces virtual districts and practically exact proportional representation, some of which would be effectively multimember districts, with no bias at all. (Or, more accurately, the "bias" is that the gerrymandering is done by the voters through their proxies, the candidates they vote for.) (It is PR because any faction of voters who care to act as a faction can create seats belonging to the faction. But it also represents independent, non-affiliated voters. Essentially, the "factions" are those supporting a particular candidate or set of candidates. This is far more sophisticated than anything else I've seen proposed. It makes Range Voting, per se, not necessary for representative elections; Range still makes sense for single-winner officer elections.) election-methods mailing list - see http://electorama.com/em for list info
[EM] Webster bias?
When I posted last night, I'd looked at Webster and noticed that each cycle of the step function has overall seats per quota that's a little less than one seat per quota. I saw that as bias, and found a different rounding formula by which the sum, over a step-function cycle, of the function's deviatian from 1 seat per quota, would be zero. But now, looking at it again, it occurs to me that there's nothing wrong with Webster having net deviation from 1 seat per quota in each cycle, as long as it's the same in each cycle. Now it seems that all that is needed is that the sum of the seats(quotas) function's summed displacement from the 1-seat-per-quota line be zero. And Webster achieves that. So, last night, I was making it more complicated than it is. I shouldn't have so quick to conclude that Balinski & Young were mistaken about Webster being unbiased. Anyway, I retract my statement that SL/Webster has bias. The other roundoff formula that I posted last night would be the unbiased one if we wanted the sum of s(q)/q's summed displacement from 1 to be zero. But if we instead want the sum of s(q)'s displacement from the 1-seat-per-quota line to be zero, we have an easier problem, a simpler formula, and it's Webster. By the way, my demonstration that LR/Hamilton is unbiased used the same assumption that I've used with other methods. But it's a reasonable assumption. For instance, if a state's quotas are between two and three, I've assumed that it could equally well be anywhere between two and three. So I also retract what I said about LR's unbias being less distribution-dependant than that of Webster. Mike Ossipoff _ Stay up-to-date with your friends through the Windows Live Spaces friends list. http://clk.atdmt.com/MSN/go/msnnkwsp007001msn/direct/01/?href=http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mk election-methods mailing list - see http://electorama.com/em for list info
[EM] Webster bias?
When I posted last night, I'd looked at Webster and noticed that each cycle of the step function has overall seats per quota that's a little less than one seat per quota. I saw that as bias, and found a different rounding formula by which the sum, over a step-function cycle, of the function's deviatian from 1 seat per quota, would be zero. But now, looking at it again, it occurs to me that there's nothing wrong with Webster having net deviation from 1 seat per quota in each cycle, as long as it's the same in each cycle. Now it seems that all that is needed is that the sum of the seats(quotas) function's summed displacement from the 1-seat-per-quota line be zero. And Webster achieves that. So, last night, I was making it more complicated than it is. I shouldn't have so quick to conclude that Balinski & Young were mistaken about Webster being unbiased. Anyway, I retract my statement that SL/Webster has bias. The other roundoff formula that I posted last night would be the unbiased one if we wanted the sum of s(q)/q's summed displacement from 1 to be zero. But if we instead want the sum of s(q)'s displacement from the 1-seat-per-quota line to be zero, we have an easier problem, a simpler formula, and it's Webster. By the way, my demonstration that LR/Hamilton is unbiased used the same assumption that I've used with other methods. But it's a reasonable assumption. For instance, if a state's quotas are between two and three, I've assumed that it could equally well be anywhere between two and three. So I also retract what I said about LR's unbias being less distribution-dependant than that of Webster. Mike Ossipoff _ Get the latest Windows Live Messenger 8.1 Beta version. Join now. http://ideas.live.com election-methods mailing list - see http://electorama.com/em for list info
Re: [EM] Apportionment (biased?) let me add some more confusion to the mix :)
> From: [EMAIL PROTECTED] > > One more tool that can be useful in some situations is the > hierarchical structure of the states/parties. To guarantee that > certain set of states/parties will not be underrepresented they could > form a team/alliance. When seats are allocated to that team they > could lose (in typical allocation methods) only one seat to rounding > errors instead on many of them losing a seat. Geographic alliances > would maybe be more natural than e.g. an alliance of small states. What about sorting the States based on population and then splitting them into 2 groups such that the total population in each group is as equal as possible. The fractional seat is then split between the 2 groups based on (Webster?) ... or maybe Webster should be used directly? This is then applied to each group recursively. If any State ends up with zero seats, it is removed from the process and given a seat directly. The process is then re-run, until it completes with all remaining States getting at least 1 seat. This pretty much is forced to be unbiased between small and large States size. However, perhaps it would be biased in other ways. An additional rule could then be that States are allowed to form groups 'manually', and manual groups cannot be split in two by the algorithm (until the group being processed is the manual group itself). > > I already mentioned the different voting power. A simple method in > that direction would be to elect one representative from every state > and give her voting power in relation to the number of people she > represents. Or maybe large states would be given n seats with 1/n of > the voting power of the state etc. Maybe the building where these > representatives will work has a fixed number of physical seats => > fill those seats and allocate voting power according to that. The logistics of this would make the legislature less efficient. One possible rule would be that all Representatives must have voting strengths between 0.9 and 1.1 and a detailed count only happens if the vote is close (or if there is a motion demanding it). Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection. election-methods mailing list - see http://electorama.com/em for list info
[EM] Webster has a little bias. The Unbiased Rounding method.
When I said that my definition of bias, a systematic disparity in seats per quota, opens up a can of worms when it's applied, what I meant was that it shows bias for Webster. Very, very little bias. Sometimes it's best to open a can of worms. Is there a quota and roundoff method that's free of bias? For quota and roundoff methods, such as Webster, Hill, etc., freedom from bias is only possible with some particular probability density disrtribution for the states' populations or their numbers of population quotas. So let's say that that distribution is uniform. The quota and roundof method that is unbiased is the one that has, as its roundoff point (between the integers a & b): (b**b/a**a)(1/e) The first of its successive roundoff points (to the nearest hundredth) are: 1.47, 2.48, 3.49, 4.49, 5.49, 6.49, 7.49, 8.5, 9.5 These roundoff points are much closer to those of Webster than to those of Hill, suggesting that Webster is the least biased of the 5 standard quota and roundoff methods. Maybe the above-described method has already been described, but if not, or if it hasn't been named, I'll call it the Unbiased Method, the Bias-Free method, or (more descriptively) Unbiased Roundoff. Largest-Remander/Hamilton is the only distribution-independent unbiased method. Mike Ossipoff _ WIN up to $10,000 in cash or prizes enter the Microsoft Office Live Sweepstakes http://clk.atdmt.com/MRT/go/aub0050001581mrt/direct/01/ election-methods mailing list - see http://electorama.com/em for list info
