P-Norm Voting Systems
p=1/2 Voting System
Suppose that each voter is given a ballot with N Candidates, and the voter has 10 points to assign to the candidates and up to 100 votes. The number of votes that are assigned to a candidate from the voter is the square of the number of points assigned to the candidate. For the purposes of this subsection, I am going to introduce the following notation for point allocation and vote awarding. ‘p=(2, 3, 0, 5)’ means 2 points for candidate A, 3 points for candidate B, 0 points for candidate C, and 5 points for candidate D. This ballot yields ‘v=(4, 9, 0, 25)’, or 4 votes for candidate A, 9 votes for candidate B, 0 votes for candidate C, and 25 votes for candidate D. Here are some properties of the system:
1) Positive: There is a penalty for indecisiveness. The indecisive voter will affect the election less, as measured by total number of points given to the candidate pool.
2) Positive: There is a penalty for cut throat behavior. Suppose that there are three viable candidates. If a voter wants candidate A, but is worried about candidate B, the voter candidate can either allocate p=(10, 0, 0) on (A, B, C) or p=(9, 0, 1). In this case the voter has reduced the votes for his candidate by 19 votes, increased the votes on candidate C by only 1 vote. This means that assuming B was going to have more votes than A, the voter increased the number of votes needed by B to win the election by only 1 vote. Thus the vote must ponder “is it worth risking 19 votes on my candidate to only knock down the other candidate by 1 vote?” The downside is that it is hard to stop candidates from being elected. If Hitler were to run for President, he would have a block of undivided full votes. While hopefully he wouldn’t win a presidential election, if implemented in an election of the House of Representatives, there’s a good chance the Nazi party would be making a showing. But I would be willing to pay that price. You have 434 other reps in there to keep anything crazy from happening. You would only have to worry if they were the swing vote on an bill or committee ruling. Logistically, they’re not going to be any bills on concentration camps coming up. The constitution will prevent anything too crazy. The most they would do is little influences of fascism and other non-Jew hating things that they care about that I didn’t feel like researching. But they would have to comprise to affect anything. Which means whatever part of their agenda does get pushed is probably that kernel of logic that is in (most) every idea. And, this would be a very effective means at corrupting (from their point of view) the Nazi party. You make the zealot leader of zealots a politician, and you have a politician, and his political staff, leading zealots. Quadratic voting systems in a body like The House of Representatives can in fact be a method for integrating segments of society previous isolated. It’s a lot easier to hate from the outside.
3) Mixed: similar candidates will take away votes from each other. Pre-filtering of candidates by political parties could help fix this.
4) Positive: The vote stealing can be good because it if there are multiple candidates that have different permutations of stances of a few issues, but there is one candidate that has an unpopular stance on an issue of extreme importance to a few voters (like gay marriage or legalization of marijuana); that candidate will receive the full votes from the small subset of the voters that care about those issues. This aspect is a big bonus when multiple candidates are being elected simultaneously, like for Congress (if The House or Senate were to adopt a national simultaneous voting system). What would happen if The House of Representatives had a national simultaneous election? Suppose that there was a candidate whose stance was “I am going to do everything I can for the state of Michigan that I can.”? That candidate might get full vote value from a good percentage of the voters in Michigan, thus getting in to office. The House could theoretically end up with a similar dynamic, many candidates that all ran on support for a regional base. Probably not, but this demonstrates that the same game theoretic reasoning for the existence of The House of Representatives and its geographical interests is encompassed by a quadratic voting system. The purpose of geographic representation is to give groups of people, with like concerns, that nobody else cares about, a voice in government. Geographic representation has been around as long as republics. It is simply an acknowledgement of the need for support for issues that have strong impacts for a few, and little impact for the rest. Quadratic voting systems do this; single-vote plurality rules does not. faSingle-vote plurality rules always degenerates in to a two-party system where swing issues are the only issues that matter. While Ralph Nader may still not have a chance in a quadratic voting system, the effect of the votes he steals will be felt more and his threat will be greater. And by “felt more”, I mean that mathematically. If a voter gives x points to a candidate, they receive v=x2 votes. If those points are then re-allocated, the rate of vote drop for the candidate will be dv/dx=2x.
5) It’s
The crucial variables are the decrement and increment factors. If a voter has all N points in one candidate and 0 points for another, then transfers a marginal amount of points (x or ‘dx’ if I followed Prescott’s notation) from the favored candidate to the unflavored candidate, the decrement rate on the favored candidate is 2N*x, and the increment rate is 1*x.
Now suppose you had a two-norm voting system. In this system, the sum of the squares of the votes awarded would be constant for each voter. An example would be if the voters had 100 points to distributed, and the votes would be the square-root of the number of points. Thus, a voter could vote v=(10, 0, 0 ,0), v=(5, 5, 5, 5), or (7.07, 0, 5 , 5). Here, the increment and decrement rates are (1/2)(x/N) and x. This means the voter can increase his affect on the election by giving a marginal vote to a candidate that is second best in his mind. Or, another way of looking at it is that it would be good strategy for a staunch democrat to allocate 1 vote to some republican in a presidential election instead of giving everything to a democrat. If candidate A is the democrat, candidate B is the leading republican, and candidate C is the underdog republican that democrats like better, then an incremental vote for candidate C increases candidates C’s votes relative to B more than the decrease in votes in candidate A. Another way of looking at it is that it incents a diversified portfolio of candidates for the voters. This will intern incent more diversity amongst candidates of a given party. Which means less interchangeability and less vote stealing from candidates. If the cube-root was used instead of the square-root, then there would be greater incentive for voters to distribute their votes amongst multiple candidates. Which means a greater incentive for candidates to individualize themselves from their parties. In fact, in the limit as p goes to infinity, the p-root voting system becomes the approval voting system. In the approval voting system, the party platforms become meaningless and candidates basically choose the optimal stance to run on. Therefore the p-value is a number that can be tweaked to any desired level of partisanship. The greater the value of p, the greater the expected diversity of candidates, but the lesser the ability for “strong feelings” to affect things. That is the trade off.
What’s an optimal value for p for a presidential election? I like 1.4. Let’s make it sqrt(2) to be cute. Hopefully the above examples and description of properties has convinced you of the value for p<1 – norm voting for multi-person bodies and p>1-norm voting for single person elections. If not, here’s what Alexander Prescott says:
“For an election of a single candidate, a p-norm voting system with p>1 implies the problem is convex. Points are conserved. Therefore, dv/dp can be written as dv/dcandidate pair, or dv/dcandidate-vector. If there were two issues primary issues of a continuous nature (like tax % and interest rate), you could use the point allocation of a voter to make a contour map with the vertical axis being votes and each relative max being a candidate. This then implies how a candidates votes from that voter would change if they moved in any given direction on any issue (the direction for the two issues is obvious, the location of the other candidates in the 2-D plane is a projection of their position in all of issue space). A single voters contour map is convex, therefore the sum of all voters’ contour maps is convex. Therefore there exists the ability to develop an optimal candidate profile. This means that the voters are sending a coherent signal to the market about what they want. The market will reply, and the elected candidate will be at an equilibrium.”
P-Norm voting systems: complicated yes, appropriate yes, perfect no. But if we are acknowledging that the system is flawed, and that the simplicity of the system fueling flawed results, and are willing to make a leap to a more enlightened system, shouldn’t we go all the way? The flaws in our system are becoming more apparent by the year because of the increasing complexity of society. This complexity will only increase in the future. Therefore, if we take only half a step forward, we are just delaying the inevitable taking of the other half-step. Let’s move forward as much as we can now. Change requires two factors: breaking away from the old, and adopting the new. The better a system is, the harder it is to break away from. Thus, in the future if we need to break away from a much better system than we have today, it will be even harder; even if just as necessary. We should take advantage of the ease of departure from our current system that we have now and implement a system that will remain sufficiently effective for the foreseeable future.
LCM – v.1.0
No comments:
Post a Comment