[EM] SEC quickly maximizes total utility in spatial model

[peter barath]

Jobst Heitzig wrote:

> Dear folks,
>
> earlier this year Forest and I submitted an article to Social Choice and
> Welfare (http://www.fair-chair.de/some_chance_for_consensus.pdf)
> describing a very simple democratic method to achieve consensus:


I looked at it, and have to admit that my math knowledge is
not enough to follow it fully in reasonable time.

>> Simple Efficient Consensus (SEC):
>> =================================
>>
>> 1. Each voter casts two plurality-style ballots:
>>   A "consensus ballot" which she puts into the "consensus urn",
>>   and a "favourite ballot" put into the "favourites urn".
>>
>> 2. If all ballots in the "consensus urn" have the same option ticked,
>>   that option wins.
>>
>> 3. Otherwise, a ballot drawn at random from the "favourites urn"
>>   decides.
>
>
> This method (called the "basic method" in our paper) solves the problem
> of how to...
>
>> make sure option C is elected in the following situation:
>>
>>   a%  having true utilities  A(100) > C(alpha) > B(0),
>>   b%  having true utilities  B(100) > C(beta)  > A(0).
>>
>> with  a+b=100  and  a*alpha + b*beta > max(a,b)*100.
>> (The latter condition means C has the largest total utility.)


Still, I have the very strong feeling that that claim is not
part of your above mentioned paper and also it is not true.

Counter-example:  a = 40  b = 60  alpha = 10  beta = 99

the condition is true:

max(a,b)*100 = 60*100 = 6000

a*alpha + b*beta = 40*10 + 60*99 = 400 + 5940 = 6340

So C does have the largest total utility. Can be sure option C
is elected? As far as I remember, the paper doesn't say anything
about the decision-making mechanisms in such situations. It always
assumes that enough participants prefer this or that above the lottery.
But here in your post you didn't say "above the lottery", you said
"has the largest total".

And I think in such situation many "A" voters including myself
would prefer the lottery with  40%  chance to the  100 value option
over the sure  10 value. So C wouldn't be elected.

> Since then I looked somewhat into spatial models of preferences and
> found that also in traditional spatial models, our method has the nice
> property of leading to a very quick maximization of total utility (the
> most popular utilitarian measure of social welfare):
>
> Assume the following very common spatial model of preferences: Each
> voter and each option has a certain position in an n-dimensional issue
> space, and the utility a voter assigns to an option is the negative
> squared distance between their respective positions. Also assume that
> voters can nominate additional options for any "in-between" position (to
> be mathematically precise, any position in the convex hull of the
> positions of the original options).
>
> Traditional theory shows that, given a set of voters and options with
> their positions, total utility is maximized by the option closest to the
> mean voter position, but many traditional voting methods fail or
> struggle to make sure this option is picked.
>
> With our method SEC, however, total utility will be maximized very
> quickly: If the "optimal" option X located at the mean voter position is
> already nominated, every voter will have an incentive to tick X on her
> "consensus ballot" since she will prefer X to the otherwise realized
> fall-back lottery that picks the favourite of a randomly drawn voter. If
> X is not already nominated, every voter will have an incentive to
> nominate X for the same reason. This makes sure X is elected and thus
> total utility is maximized.


Still I can't comprehend the full mathemathic background, but look
at this example:

An economic community with a common wealth decides about their future:

Option "Dismiss": dismiss the community by sharing equally the
wealth, and everyone does what she wants with it.

Option "Salary": work as a cooperative, still common wealth, but
members get different payment by their work.

Option "Equality": work as a classic kibbutz, equal living conditions,
no money.

The utility for the 40 "Dismissists": Dismiss(100) Salary(10) Equality(0)

For the 20 "Salarists": Dismiss(10) Salary(100) Equality(30)

For the 40 "Equalists": Dismiss(0) Salary(80) Equality(100)

For me it looks here the Salarists are the median voters, and
also the "Salary" option has the largest total. And again, it
looks that a typical Dismissist will go for the 40% lottery
instead of accepting the low-value compromise.

All these don't make the proposals necessarily look bad in my
eyes. It looks promising wherever high-value compromises
exist, and it looks logical they often do.

Peter Barath

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