Hay Voting
Marcello Herreshoff and I talked a bit about voting theory during the Summer of AI, and we came up with our own voting method. In compliance with Stigler’s Law, I’ve decided to call it Hay Voting (after Nick Hay). Hay Voting is, I fear, somewhat impractical, but it has some interesting properties.
Hay Voting is a stochastic voting method; it does not always elect the same candidate given the same votes. Hay Voting is based on the idea of expected utility. It assumes that each voter possesses a utility function which assigns a numeric value to each candidate, and that the voter will vote in such a way as to maximize their expected value for this number. Our intention was to design a voting method such that, when voters follow their optimal strategy, we can deduce their utility functions. You might think of this as a voting method which encourages voters to be “honest.”
A Voting Method that Rewards Honesty
Before I get to Hay Voting, I’m going to talk a bit more about honesty and stochastic methods. Suppose we only wish to know the identity of each voter’s favorite candidate. Here we will give each voter a single vote, which they can cast in favor of any candidate. If we simply count the number of votes for each candidate, and elect the candidate with the most votes (this is called Plurality Voting, and is used by many countries), then we will fail to achieve our goal of discovering which candidate each voter most favors.
That’s because plurality voting introduces voting strategies. Let’s suppose that there is an election between three candidates: A, B, and C. A is really your favorite, but you know that A has no real chance of winning, and the race between B and C is very close. Of those, you prefer B, so voting for B would be a good strategy. So that’s what you do.
Suppose instead that I selected one vote at random. I put all the votes in a hat, pull one out, and elect whichever candidate is lists. If I select a vote other than yours, then your vote had no effect, and if I select your vote, then I elect that candidate. So your optimal strategy is to vote for your favorite candidate – A. So this voting method encourages you to be honest. Of course, all I found out about you was who your favorite candidate was. I don’t know anything else about your preferences.
A Ranked Voting Method that Rewards Honesty
Most voting methods that I know of (e.g. Condorcet Voting, the Borda Count, Instant Runoff Voting (which is awful)) ask each voter to rank each candidate from most to least favorite. All of the methods listed above encourage some strategy; that is, there exist situations in which a voter can achieve a better result by voting something other than vis true preferences. Such situations are supposed to be rare in Condorcet voting methods, but they exist.
It is easy to construct a ranked voting method which encourages voters to be honest – if we assume that the voters are ignorant of the value of a certain variable. Let’s call it X, and let’s say that there are N candidates. We need a probability distribution for X such that p(X=1) > p(X=2) > … > p(X=N).
Now we select a ballot from our hat, and then elect the candidate listed in place X on that ballot. Since that’s most likely to be the first candidate listed, and less likely to be each successive candidate after that, your best bet is to rank the candidates by preference.
The N-Substance Problem
Okay, now we’re almost ready to talk about Hay Voting, but there’s one more preliminary. It’s what I call the “N-substance problem.” Let’s say I’m a merchant, and I have a collection of N different substances which I can sell to you. You have a preference function over these substances, which you are going to try to maximize. I’m making a special assumption about this preference function: it’s a linear combination of the quantities of each substance that you have.
Let’s say Vk is the volume of substance k that you have, for k=1…n. Then your preference function U can be written as:
We can call uk your utility density for substance k.
Let’s say you have 1.00 credit to spend on these substances, but this credit can be divided into arbitrarily small quantities. The N-substance problem is this: how can I devise a pricing scheme such that, based on what you buy, I can determine (up to a positive Affine Transformation) what your utility function is (that is, the values u1…un)?
Not just any pricing scheme will work. For example, consider the obvious scheme in which you can buy one liter of any substance for 1 credit. If the substances are green tea, lemonade, and maple syrup, with utility densities 2, 3, and 20 respectively, your optimal strategy is to spend your entire credit on maple syrup, without buying any green tea or lemonade. But then I can’t find out the relative values you assign to green tea and lemonade, so this pricing scheme fails.
So now let’s specify our pricing scheme. We want a function f which determines the volume of substance you get for a given number of credits. And for any u1, u2, we want U = u1f(x1) + u2f(x2) to be maximized whennever f(x1) / f(x2) = u1 / u2. Furthermore, we will arbitrarily set f(1) = 1.
So let’s say we are spending a total of C credits on x1 and x2. Then we can let x2 = C−x1. Now if we differentiate U with respect to x1 (conveniently assuming f is differentiable), we get:
If U is at a maximum, then this derivative will be 0, so:
whennever:
Combining these two equations, we get:
We set x1 = 1. Recall that we arbitrarily set f(1) = 1, so:
Letting x = C−1 and letting k = f ‘(1):
So f(x) = √x.
Hay Voting
Now we have a pricing scheme which encourages you to purchase quantities of each substance which are proportional to your utility density for that substance – assuming it makes sense to talk about “utility density” (that is, assuming your preferences are linear). When does that assumption hold?
If you’re trying to maximize the expected value of some function, then your preferences are linear over the probabilities assigned to each outcome. So if I sell you probabilities using this pricing scheme, I can discover your utility function!
Under Hay Voting, you receive a ballot which has a √(N-1) units of “voting mass” (called votes) allocated to each candidate, where N is the number of candidates. You get 1.00 voting credit, which you may spend to reallocate some of this voting mass. If you spend x credits to trasnfer mass from candidate A to candidate B, then A loses √x votes and B gains √x votes. The √(N-1) value was selected so that you have just enough voting credit to completely eliminate one candidate from your ballot, or double the voting mass assigned to one candidate.
To elect a candidate, we gather all the ballots together and place them into a hat. One ballot is drawn at random, and then one point of voting mass is selected from that ballot, so that the probability of any candidate being selected is proportional to the amount of voting mass allocated to that candidate.
Now it makes sense to talk about the utility density of a transfer of voting mass from one candidate to another. This density is simply the difference between the utilities you assign to the two candidates. So if you buy these transfers in such a way as to maximize your expected utility, I can see relative sizes of the intervals between the utilities you assign to any two candidates – and that tells me everything there is to know about your utility function over candidates.
Deterministic Hay Voting…?
In Deterministic Hay Voting, the election procedure is as above, but the candidate with the most voting mass is always elected. It may still be possible (at least with idealized agents) to determine the utility functions of the voters by looking at their votes.
Why do we vote? In real life one may vote for any number of reasons: to make a statement, to do one’s patriotic duty, etc. In designing Hay Voting I made the unrealistic assumption that one votes only in order to affect the outcome of the election. I will continue to use that assumption to talk about deterministic voting.
Thus, one’s utility function over votes is dependent upon two things: one’s utility function over candidates, and one’s probability distribution for the effect of one’s vote on the result of the election. The latter can be written in terms of a probability distribution for the actions of the other voters.
As long as it makes sense to talk about the utility density of votes (as long as the size of your own vote is small relative to your uncertainty about how others will vote), we can discover your utility function over votes by using the square-root pricing rule. Then if we knew your probability distribution for the actions of the other voters, we would be able to find the set of all possible utility functions over candidates that would suggest the voting strategy you used.
To discover your probability distribution for the actions of the other voters, we can play a prediction game using a strictly proper scoring rule (you can read about strictly proper scoring rules here). Your reward in this game is based on the probability you assigned to the correct outcome; a strictly proper scoring rule is one such that your optimal strategy is to report your actual beliefs. In this case we would probably prefer a squared-error scoring rule because it is bounded (unlike the logarithm). The reward could be additional votes in the next election.
Deterministic Hay Voting is still a fairly sketchy idea. I’m not sure exactly what conditions the voters’ probability distributions must satisfy in order for it to work.
Again, I feel pretty uncertain about how clear my explanations were here, so please give me feedback and ask questions about anything unclear.
Thanks for reading!
I haven’t fully grasped and thought through your voting scheme, but the nondeterministic voting scheme at least seems vulnerable to cloning. ie if there are a bunch of very similar candidates (clones), the voters von’t be able to reallocate enough voting mass away from them to prevent there from being a very high probablity of one of the clones being elected.
In practice, a scheme as complicated like this is unlikely to be adopted anyway though.
Comment by simon — September 15, 2006 @ 7:26 am
It seems to my tired brain that you should let voting mass itself take the place of the N substances, instead of voting mass transfers. This would at least prevent clones from winning if all the voters hate the clones. However the voting method would still be vulnerable to cloning in the sense that clones would have a higher chance of winning than non cloned-candidates with the same support, if the clones’ supporters split their voting mass between the clones.
Comment by simon — September 15, 2006 @ 7:56 am
but the nondeterministic voting scheme at least seems vulnerable to cloning.
Yup.
It seems to my tired brain that you should let voting mass itself take the place of the N substances
But then the voting mass can no longer be thought of as representing probabilities, but odds, so it’s no longer an N-substance problem. If I pick a random ballot and then pick a random point from it, your ideal strategy would then be to only allocate mass to your favorite candidate. If I pick a random ballot weighted by the total voting mass in each ballot, your optimal strategy now depends on how much voting mass you think the rest of the population is using. In either case I don’t get to see your utility function.
Comment by Peter de Blanc — September 15, 2006 @ 10:24 am
I had in mind the second case. The optimal voting strategy, for a given utility function, would depend not only on how much voting mass others had used but also on which candidates they had distributed it to. I hadn’t realized that before reading your response.
Anyway, with the voting mass transfer method as you literally described it, the amount of credits you take to do a set of transfers with the same final result can depend on what transfers it’s composed of. Perhaps you should rephrase it to something like:
You can assign any voting mass distribution you want, provided that:
The total voting mass must sum to 1
The sum of the squares of (the voting mass assigned to each candidate minus 1/N) must be 1/(N^2-N).
I’m sure something like this is what you had in mind, but the way you said it, it sounds like transfering some voting mass from each of candidates C and D to each of candidates A and B can get you a bigger total transfer than transfering some from C to A and some from D to B (for example).
Comment by simon — September 15, 2006 @ 6:07 pm
I had in mind the second case. The optimal voting strategy, for a given utility function, would depend not only on how much voting mass others had used but also on which candidates they had distributed it to. I hadn’t realized that before reading your response.
Actually, it should only depend on how much voting mass they used. Only one ballot gets selected. You would prefer it to be yours, but whatever you do, you can’t change the relative voting mass of two other ballots.
You can assign any voting mass distribution you want, provided that:
The total voting mass must sum to 1
The sum of the squares of (the voting mass assigned to each candidate minus 1/N) must be 1/(N^2-N).
Hmm, interesting suggestion. It’s not immediately obvious to me what this would do. Where did you get those particular values from?
I’m sure something like this is what you had in mind, but the way you said it, it sounds like transfering some voting mass from each of candidates C and D to each of candidates A and B can get you a bigger total transfer than transfering some from C to A and some from D to B (for example).
That is so. If, for example, you prefer candidates in order A>B>C>D, then you should spend some of your credit on each of the transfers D->C, D->B, D->A, C->B, C->A, B->A. The amount of mass that you transfer in each case should be proportional to your difference in utility between the two candidates.
Comment by Peter de Blanc — September 15, 2006 @ 10:04 pm
Imagine that you’re voting in the system I suggested in comment #2. There are three candidates, and there is only one other voter. You know that the other voter will vote either entirely for candidate A or entirely for candidate C. Either way the total voting mass the other voter uses is the same of course. Your own utility function is 1 if candidate A is elected, 1-epsilon if candidate B is elected and 0 if candidate C is elected. If you expect them to vote for A, you should vote entirely for A also. If you expect them to vote for C, you should split your vote almost equally between A and B.
The system I suggested in comment #4 was equivalent to what I had thought you had meant in your original description of Hays voting. I had misunderstood (largely because I found your system unintuitive, and reinterpreted it as the nearest intuitive-to-me idea), so thanks for clarifying. The values I chose enable you to assign zero or double voting mass to one candidate, as long as you split the remaining voting mass equally between the others.
Comment by simon — September 16, 2006 @ 3:13 am
Let me rephrase the first paragraph of my last comment in a more general and intuitive way: given the voting system of comment #2, the more you expect the other voters to vote in a way you dislike, the more important it is to maximize the voting mass you use.
Comment by simon — September 16, 2006 @ 3:52 am
the more you expect the other voters to vote in a way you dislike, the more important it is to maximize the voting mass you use.
Ah, yes.
Comment by Peter de Blanc — September 16, 2006 @ 6:17 pm
Interesting. I know of only one other voting method that is designed to consider utility, this would be Range Voting. See http://RangeVoting.org. I did not find it easy to understand the explanation. I *think* I understood it, but I’ll agree that it is unlikely to see actual use in public elections because of its complexity. But that doesn’t make it useless. For one thing, if it is an ideal voting method, it would be useful to compare against other systems. With Range as a background, though, it seems inordinately complex, plus it assumes ranking as a basis, and it seems to me that it does not, in fact, discover the voter’s utility function, any more than Range would (where voters directly express their utility function). Why would Hays Voting be, in any way, superior to Range Voting? Participants in the Range Voting mailing list, rangevoting@yahoogroups.com, include mathematicians and others who would be more motivated to follow the math than I. To get me to wade through it, I’d need to see more specifics about how it works.
Does Hays Voting satisfy the Condorcet Criterion or the Majority Criterion? (I’ve written extensively how the Condorcet Criterion and Majority Criterion can fail to indicate the optimal winner of an election, and I would guess, off-hand, that Hays does not satisfy them. But the crucial questions is, under what conditions does it not satisfy them? Can we clearly show that the winner is better than the Condorcet winner?)
Range Voting does not satisfy the Condorcet Criterion, but only when there is a candidate with broader approval, and the majority cannot fail to elect its first preference unless the majority also elects to give weight to the election of another candidate.
I’d think your participation in the Range Voting community would be most welcome. It is quite established in promoting Range Voting, or its elementary version, Approval Voting, in public elections.
Comment by Abd ul-Rahman Lomax — January 20, 2007 @ 5:35 pm
Hay Voting uses numerical ratings, not ordinal rankings. The ranked method at the top of the article was thrown in to give another example of a voting system which rewards honest voters, but Hay Voting is not a ranked voting method.
Hay Voting satisfies neither the Condorcet Criterion nor the Majority Criterion. We can’t actually guarantee anything about the elected candidate, because Hay Voting is a stochastic method.
We really do get to find out the voters’ utility functions, so Hay Voting is ideal in that sense, but it does not always elect the candidate who maximizes aggregate utility; if you did that, then the voters would want to vote strategically.
Comment by Peter de Blanc — January 20, 2007 @ 6:53 pm
Dear Peter de Blanc & Marcello Herreshoff.
I believe your scheme is busted. For why, please see the following web page:
http://rangevoting.org/HayVoting.html
Please contact me by email so we can talk…
warren DOT wds AT gmail DOT com
Comment by Warren D Smith — February 4, 2007 @ 1:22 am
Warren D Smith:
The logarithmic function and the square root function satisfy different criteria. If you use a logarithm, then your optimal strategy is for the amount of credit you spend to be proportional to your utility density for the substance. If you use a square root, then your optimal strategy is for the amount of substance that you purchase to be proportional to your utility density. Both allow your preferences to be extracted. We chose the square root instead of the logarithm because it does not have a vertical asymptote.
Note that you mention scaling so that your least favorite candidate gets 0 units of mass. But if you spend zero credits on a candidate, they would get -infinity units of voting mass under your scheme. How do you scale -infinity to zero?
I think your further criticisms of the LogRand method fail when applied to Hay Voting. No matter how you vote, you don’t change the probability of your ballot being the one selected. If your ballot is not selected, then those other ballots don’t matter, and if one of the other ballots is selected, then it doesn’t matter how you voted. So in no case does your optimal vote depend on how the other voters voted.
Admittedly, it is the case that whether you decide to bother voting at all would depend on how others vote. But given that you do vote, you should always vote the same way regardless of what you believe the other voters are doing.
Comment by Peter de Blanc — February 5, 2007 @ 2:33 am
>Note that you mention scaling so that your least favorite candidate gets 0 units of mass.
But if you spend zero credits on a candidate, they would get -infinity
units of voting mass under your scheme. How do you scale -infinity to zero?
-that is true, that was a screwup by me.
>The logarithmic function and the square root function satisfy different criteria.
If you use a logarithm, then your optimal strategy is for the amount of credit you
spend to be proportional to your utility density for the substance. If you use a square
root, then your optimal strategy is for the amount of substance that you purchase to be
proportional to your utility density. Both allow your preferences to be extracted.
We chose the square root instead of the logarithm because it does not have a vertical asymptote.
–aha.
If you maximize sum(k)of uk*Vk
subject to
sum(k)of Vk = V = fixed
(where the sum-of-volumes=const constraint makes more sense [for your voting application]
than my sum-of-dollars=const constraint) then the solution is to buy only
one substance. Failure.
If you maximize sum(k)of uk*Vk
subject to Vk=sqrt(xk)
sum(k)of xk = X = fixed
then the solution is xk=const*uk^2 and Vk=const*uk.
Correct. Success.
If you maximize sum(k)of uk*Vk
subject to Vk=xk^P
where P>0 is any constant (above was P=1/2, but now allow any)
sum(k)of xk = X = fixed
then the solution is uk=const*xk^(1-P) and Vk=const*uk^(P/(1-P))=xk^P.
Also success; this also allows utilities to be deduced. But the larger P is,
the more juice the voter gets for her money… suggesting you really want to use P=0.999.
(Since values P>=1 do not seem to make sense, think they lead to a U-min not a max.)
>I think your further criticisms of the LogRand method fail when applied to Hay Voting.
No matter how you vote, you the one selected. If your ballot is not selected, then
your vote doesn’t matter. If your ballot slected,
doesn’t depend on how the other voters voted.
–this one worries me. Your argument here ought to be equally correct applied to either your
square-root scheme or the log scheme. I don’t see why that should affect its correctness?
OK, anyhow, maybe this is some classic probability paradox, and
I certainly appear to be an idiot here based on your bust of my bust… but
I’m still confused. Namely your argument sounds valid, but my counter argument also
sounds valid. Huh? Aha, I think I see it, my “counter argument” really asserts
your argument is valid just linearly transformed by factors which had
thrown me off but actually do not matter.
–OK!! Sorry for the criticism and my stupidity!!
I guess I now have to agree with Forest Simmons who
said your Hay Voting scheme was a great new contribution to voting theory.
And note the fact you can do that arbitrary power P to generalize your scheme.
(My log scheme arises in the limit P–>0+, your sqrt-scheme when P=1/2.)
So then you have to ask: what is the best choice of P?
And it seems like the best choice is P only slightly below 1, such as 0.999.
That way you are both honest, and voting almost all your mass for your favorite.
Warren D Smith
http://rangevoting.org
Comment by Warren D Smith — February 5, 2007 @ 3:57 am
[continued]
>That way you are both honest, and voting almost all your mass for your favorite.
–AND, furthermore, the money-constraint now closely resembles the volume=const
constraint which you want for fairness. So I’m pretty sure P=0.999 is where you really want to
be.
wds
Comment by Warren D Smith — February 5, 2007 @ 4:05 am
But the stochasticity of the scheme seems to make it unacceptable in practice.
If Bush is 55 and Gore 45, I want Bush to just win, dammit, not get 55% election
probabilty. So I wish I understood your half baked bit at the end muttering
about determinizing it?
And in the P–>1 limit, this scheme really just morphs into the Gibbard random ballot
scheme (each voter names favorte as vote; random ballot is used to pick winner;
strategic voters will be honest).
That suggests t me your scheme is a generalization of Gibbard random ballot, and
a generalization which sucks (i.e makes it worse) and G.R.B already was bad.
So on third thought, now I’m not liking it anymore!
Jeez.
Comment by Warren D Smith — February 5, 2007 @ 4:22 am
Actually despite my claims P=0.999 is the best member of the P-family,
there is one boast Hay voting’s P=0.5 can make:
It is the only member of the family (and the only voting method in the world,
assuming we don’t count stuff like Clarke-Tideman-Tullock as a “voting method”;
see http://www.rangevoting.org/CTT.html) wich causes the best winner
(maximum utility sum) to be the candidate
most likely to be elected, with strategic voters, always.
wds
Comment by Warren D Smith — February 5, 2007 @ 5:00 am
Hi, Peter. I liked your article on Hay voting. Neat idea, sidesteps Arrow with the random process.
I had a bit of an idea that might help with the cloning problem and the problem where it has a fair probability of electing a candidate that maybe no-one likes much. I call it Multistage Hay voting – I guess Steigler’s law affected me too. }:)
I put a bit of a writeup at http://panix.com/~tehom/essays/hay-extended.txt if you want to look at it.
Tehom (Tom Breton)
Comment by Tehom (Tom Breton) — January 20, 2008 @ 6:34 pm
Nice work, Tehom. You have a good understanding of Hay voting.
My own idea of how to make something like Hay voting produce good election outcomes was to pick some other method you like (say, approval voting), and have each voter allocate mass to every possible way of filling out a ballot for an approval vote. Then pick a random point of mass for each voter to determine what their approval ballot looks like, and enter all of these ballots into an approval vote. (this reveals the voters’ utility function over ballots; if you also quiz them on their predictions about the election, you can use this to figure out their utility function over candidates).
But I like your idea more.
Comment by Peter de Blanc — January 29, 2008 @ 11:16 pm
Update: I made a better writeup with graphic illustrations and put it at
http://www.panix.com/~tehom/essays/hay-extended.html
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