[Election-Methods] a strategy-free range voting variant?

[Jobst Heitzig]

Dear Warren,

you wrote:
> But I do not fully understand it yet and I think you need to
> develop+clarify+optimize it further...  plus I'd like you to unconfuse me!

I'll try...
> Of course, this is far from being a new idea so far, and it is not yet
> the whole idea since it has an obvious problem: although it obviously
> manages to elect the "better" option (the one with the larger total
> monetary value), it encourages both the seller and the buyer to
> misrepresent their ratings so that the gap between R2(B)-R2(A) and
> R1(A)-R1(B) becomes as small as possible and hence their respective
> profit as large as possible. In other words, this method is not at all
> strategy-free.
> 
> --QUESTION:
> if they make the gap small, then the buyer pays little to the seller.
> Yes, that is better for the buyer.
> But doesn't the seller have the opposite incentive?
> It is not clear to me the "incentive" you say exists here, really does exist.
> If it doesn't, then you do not need to fix this "problem"
> because there is no problem. It'd help to clarify this point.

Isn't that the usual situation when bargaining? Given that the buyer 
would be willing to pay more than the seller would minimally accept as a 
price, the seller tries to maximize the price as long as he thinks the 
buyer is willing to pay it, and the buyer tries to minimize her offer as 
long as she thinks the seller is willing to accept it. So, both work to 
minimize the gap between the demanded and the offered payment.

> 5. Finally, the voting accounts are adjusted like this:
> a) Each deciding voter's account is increased by an amount equal to the
> total rating difference between the winner and the benchmark lottery
> among the *other* deciding voters, minus some fixed fee F, say
> 10*N^(1/2). (Note that the resulting adjustment may be positive or
> negative.)
> 
> QUESTION:
> I'm confused about this whole benchmarking thing.
> 
> You said the "benchmark" voters were being benchmarked, but now you
> say the "deciding"
> voters are being benchmarked.  ???

That might be a language problem for my part. What I mean is this: In my 
thinking, democracy demands equal decision power for every voter. Random 
Ballot accomplishes this in a way, but is not efficient. But the Random 
Ballot lottery can still serve as a "benchmark" for other, more 
efficient choices. In my suggested method, the "benchmark" voters are 
needed only to estimate what the Random Ballot lottery amoung all voters 
would be. The individual ratings for the actual winner of the election, 
who is only determined by the "deciding" voters, is then compared to the 
individual ratings for this "benchmark" (i.e. of the estimated Random 
Ballot lottery) in order to the individual transfers of "voting money". 
The higher a deciding voter rated the benchmark and the lower she rated 
the winner, the more "voting money" is transferred to her account (or, 
rarely, the less is transferred *from* her account).

In mathematical terms: Let p(X) be the probability of X being the 
highest rated option when we draw one of the "benchmark" voter's ballots 
  uniformly at random. (So the p's define our "benchmark lottery")
Let r(i,X) be the rating deciding voter i specified for X. Put
   r0(i) := sum { p(X)*r(X,i) : X }
(over all options X), i.e., the expected rating deciding voter i 
specified for the lottery outcome. Then put
   t(X) := sum { r(X,i) : i }
(over all deciding voters i) and
   t0 := sum { p(X)*t(X) : X }.
Assume W is the range voting winner of the deciding ballots, i.e.,
   t(W) > t(X) for all X other than W
Now the voting account of deciding voter i is changed by this amount:
   sum { r(W,j)-r0(j) : j different from i }
(over all deciding voters j different from i),
which is equal to
   (t(W)-t0) - (r(X,i)-r0(i))
The higher you rated the winner (i.e., the higher your r(X,i)) and the 
lower you rated the average favourite of the benchmark voters (i.e., the 
lower your r0(i)), the less "voting money" you get.

> 
> What does "total rating difference between the winner and the
> benchmark lottery among the *other* deciding voters" MEAN precisely???
>   This is not clear english...  the winner's rating is a number but
> "the benchmark lottery" is not a number.  You need two numbers.

It means
   sum { r(W,j)-r0(j) : j different from i }
(see above).

> The compensating voter's accounts are decreased by the same total
> amount as the deciding voter's accounts are increased, but in equal
> parts. (This may also be positive or negative)
> 
> --this seems to hurt poor voters.  I.e. if there are "rich" voters who
> vote +-100
> and poor voters who vote +-1 then the poor voters will need to pay the same fee
> in 5b as the rich voters.  They may therefore have incentive to avoid
> being in the electorate at all, in which case the electorate will
> become biased (rich-dominated).

Yes, that might be a problem. So, being in the electorate (meaning 
amoung the whole number of N voters) should not be something one can 
choose. In other words, we put N to be the number of all eligible 
voters, no matter whether they choose to abstain or vote, and for those 
who don't show up to vote we add a ballot of zeroes.

Still, we have to make sure the transferred amounts don't become too 
large, which is an issue I admitted already in my last mail.

One alternative choice for the transfer formula could be this: Assume 
deciding voter i's voting account before the decision is b(i). Then 
after the election it will be
   b(i) + sum { r(W,j)-r0(j) : j different from i }
if
   sum { r(W,j)-r(X,j) : j different from i } < b(i)
   for at least one X different from W,
otherwise b(i) will remain the same.
In this way, voter i's account will only be changed iff voter i had a 
chance of influencing the result in the first place, given the other 
deciding voter's ratings. This resemles the Clarke tax condition in some 
way.

> and the
> benchmark lottery will tend to equal the ordinary random ballot lottery
> with all voters.
 > --I do not understand what this means exactly.

I meant the distribution of favourite options amoung the "benchmark" 
voters alone can be used as an estimator of the distribution of 
favourite options in the whole electorate. The first is what I called 
the "benchmark lottery", while the latter is the ordinary Random Ballot 
lottery.

>> In particular, the method is quite efficient.
> --what does "efficient" mean?

You're right, I was quite loose here. By "efficient" I mean that given 
some measure of social utility f, the expected value f(winner of method) 
is high.

> If the fee F is set so that the expected (in a reasonable model)
> adjustments of the individual accounts are all zero
> 
> --well, I'm not sure what that means, but it seems like F can be
> readjusted every election
> so over a long sequence of elections accounts balance.
> Then F would not be "fixed" but it could be fixed in each election
> individually.  Does that matter?

Good idea. It should not matter strategically since it does not affect 
the expected change in an individual's account but only the variance of 
that change.

> Finally, if I'm not mistaken, the method is still strategy-free for
> these reasons: i) A benchmark voter's "favourite" mark does neither
> influence the winner nor the voter's own account, so there is no
> incentive to misstate the favourite. ii) A deciding voter's ratings do
> influence the voter's account only by influencing the winner; the same
> arguments as above show that it is optimal for a deciding voter to
> sincerely state her true ratings. iii) A compensating voter's ratings to
> not affect anything. Since you don't know in which group you end up,
> your optimal vote is the true ratings!
> 
> --I think your basic ideas summarized here seem
> VERY promising, but as above I am
> confused about the details --
> and even when I get unconfused
> there may be modifications
> that may be desirable.
> I would like to post this all as a web page on the CRV web site
> (credited to you of course) when things clarify, if you permit.

Yes, of course, feel free to reformulate all of this in proper English :-)

Yours, Jobst