[EM] SEC quickly maximizes total utility in spatial model

[Jobst Heitzig]

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:


> 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.)


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.

Yours, Jobst