[EM] Hay voting bust, busted

[Peter de Blanc]

Hi. Thanks for all the new feedback.

On Mon, 2007-02-05 at 08:09 -0500, raphfrk at netscape.net wrote:
>  From: wds at math.temple.edu
> >
> > Sorry, I appear to have been an idiot.  Peter de Blanc
> > answered my complaints at
> > http://www.spaceandgames.com/?p=8
> >
> > and it looks to me like I NOW have to agree with Forest Simmons that
> > this IS a great new contribution to voting theory.
> > Also, I showed there in my comment how to generalize their scheme
> > by adding a parameter P.  It looks liek the best P is P=0.99 or
> > so, not P=0.5 (their value) or P=0+  (my value from before).

Yes, this makes sense. We had been thinking rather narrowly when we came
up with the square root function. Actually it looks like any increasing,
differentiable function f(x) such that f'(x)->infinity as x->0 will
allow your preferences to be extracted.

> I wonder if it might be worth having the voter write their honest
> utilities on the ballot.  The 'backend' would then convert that
> vote into an optimised ballot for inclusion.

Yes, this would be necessary. Making the voter do all the calculations
is not very user-friendly (although I'd still add in an "expert mode,"
though I can't see why anyone would want to use it).

> If it truely is the optimal strategy to be honest, then there would
> be no incentive to lie.  However, it would mean that very low
> probability of winning candidates would not get lost in the noise.
>                                                                                                                                                              
> For example, they could just have people write their truthful
> utilities.
> The backend would then just get the square-root of all their utilities
> and then renormalise so that it sums to unity.  This would give the
> probabilities for each candidate.
>                                                                                                                                                              
> What about something like the following for a non-random method.
>                                                                                                                                                              
> Prior to the election:
>                                                                                                                                                              
> Determine the probability of each candidate having a total vote
> within a% of the highest other candidate.
>                                                                                                                                                              
> If a is low, then this should be proportional to the probability
> of the candidate ending up in a tie.  It seems reasonable, but
> is that true ?  Anyway, call this probability Pc for the cth
> candidate.
>                                                                                                                                                              
> Each voter votes a number of votes for each candidate and the
> candidate with the highest total wins.
>                                                                                                                                                              
> The benefit of giving a vote for a candidate is the probability
> of the candidate being in a tie times the utility of the candidate.

I had been thinking of something along these lines. I had an idea which
makes no assumptions about these probabilities, but which still isn't
quite deterministic (but the global results are very predictable).

The idea is that you would allocate mass according to a Hay Voting-like
rule, and then one point of voting mass will be randomly selected to
represent your entire ballot. Then the ballots are entered into a
plurality vote. The utility density of voting mass for any candidate is
the expected utility of a (full) vote for that candidate.

The expected utility of a vote for candidate C1 is:

sum over all k: P(Ck is winning but a vote for C1 will cause C1 to win)
* [U(C1) - U(Ck)]

Ties can be settled with a coinflip or whatever, and the probability
mentioned above simply has to take that into account. Also note that
these probabilities will never be zero (regardless of how certain you
are of how others will vote), because of the small amount of randomness
in the method itself. This means that there should be a one-to-one
correspondence between utility functions and voting strategies.

Again I like the idea of using a strictly-proper scoring rule prediction
game to assess the voter's beliefs about the election outcome. I like
this because the voter's beliefs are what actually influences how ve
actually votes. The judgment of the prediction market can be made
available to the voter as evidence. (In the front-end, you'd probably
have a 'use market predictions' option and a 'make my own predictions'
option)

If you do it this way, then as the parameter P->1-, and as
population->infinity then strategic voting approaches Plurality voting,
so you know what to expect as far as election results go. But you still
get all the juicy information about voter utility functions.

It's less obvious to me what election outcomes would end up looking like
with P=0.5 or whatever. You could find out by running simulations. Use
the results of each round to update the market predictions for the next
round and then vote again.