[EM] [RangeVoting] A Consensus Seeking Method Based on Range Ballots

[fsimmons at pcc.edu]

----- Original Message -----
From: Jobst Heitzig 
Date: Sunday, February 15, 2009 3:17 am
Subject: Re: [RangeVoting] A Consensus Seeking Method Based on Range Ballots
To: Raph Frank 
Cc: RangeVoting at yahoogroups.com, election-methods at lists.electorama.com, fsimmons at pcc.edu

> Hi friends,
> 
> Raph wrote:
> > On Sat, Feb 14, 2009 at 7:29 PM, wrote:
> >> Note well that in the case of a tie among lotteries the tie 
> is broken by a
> >> voter, not the voter's ballot, and the tie breaking voter 
> only decides among
> >> the
> >> tied lotteries, not directly picking the winner.
> > 
> > It might be worth collecting 2 ballots from each voter then. 
> A voter
> > could give a strategic ballot and an honest ballot.
> 
> To avoid this, the tie-breaking rule could also be this: Amoung those
> lotteries in the tie (i.e., those maximizing the product of all ballot
> expectations), use the one with the smallest entropy or smallest 
> sum of
> ballot variances or something like that.

I like the smallest entropy idea.

> 
> > This would mean you don't have to find the voter later to ask 
> them to
> > pick the winning lottery.
> > 
> >> Under this ULVM, rational voters in the above scenario will vote
> >>
> >> 60: [100, 0, 60]
> >> 40: [0, 100. 40],
> >>
> >> which turns out to be the Nash equilibrium with the highest voter
> >> expectations.
> > 
> > Is this true in the general (3 candidate) case, i.e. is the Nash
> > equilibrium always for the highest utility candidate to win?
> 
> What do you mean by "highest utility"?

If you mean highest average rating, then the answer is no, because this method has the anti-tyranny 
property that gives proportional probability to candidates receiving "bullet votes" even when they have a 
low average rating.

It turns out, however, that if you replace "product" with "sum" in the description of ULVM, then the lottery 
always elects the candidate with the greatest average rating. 

> 
> > Also, sometimes with Nash equilibria, it can be possible to 
> obtain a
> > better result by committing to a stance. The result of the other
> > 'players' adjusting to your solid position can yield higher results
> > for you then that Nash equilibrium.
> 
> But that would require the other voters to be quite ignorant 
> about your
> true utilities, otherwise they would not adjust to your solid 
> position.This is because then they would know that if they did 
> adjust to your
> "solid" position, they would produce a situation which is not a Nash
> equilibrium and in which you would have an incentive to deviate from
> your claimed "solid" position.
> 
> > I wonder if the whole process could be implemented as a DSV method.
> > You give your honest utilities and then method casts your vote 
> for you
> > based on those utilities. It then adjusts your vote based on how
> > others vote. The rule would be to find the best Nash equilibrium.
> 
> "Best Nash equilibrium" in what sense?

I think he means best by the standard of ULVM, i.e. the highest product of ballot expectations.

We once talked about this kind of thing (putting DSV on the front of a method that takes Range Ballots, 
but where optimal strategy usually requires adjustments) under the heading of Small Voting Machine, 
suggested by a Alex Small.  See for example

http://lists.electorama.com/pipermail/election-methods-electorama.com/2002-May/008262.html

This brings up a related question.  Is it always possible to find a Nash equilibrium ballot set that 
replicates the ULVM winning lottery for a given set of ballots?  In other words, suppose that ULVM yields 
lottery L for the ballot set S, but this L is not a Nash equilibrium for S.  Is there a way to alter S to S' 
without order reversals so that L is a Nash equilibrium for S'?