# [EM] Dollar example

Steve Eppley SEppley at alumni.caltech.edu
Wed Feb 23 12:37:29 PST 2000

```Bart Ingles wrote:
> This was not intended to address any particular method.  Instead, this
> deals with standards for who actually _should_ win.  There was a debate
> recently about whether the CW was always the better choice for society,
> or whether some consideration should be given to social utility.

I think most economists would look at Bart's example and say
that all three proposals provide equal social utility (\$75,000).

> A community is trying to decide how best to distribute a windfall of
> \$75,500.  There is a dispute over how much each member deserves.  Group
> A wants to give each of its own members \$1000 for every \$500 that each
> non-member receives.  Group C wants the opposite.  Group B wants to give
> each member of another group \$550, and distribute the rest among its own
> members.
>
> Group A and C are each close to half of the population, with group B
> only a small minority.  Exact populations are unknown.

So it's unknown which is the condorcet winner.  But at least one
of them is, since the preferences can't cycle.

> ~50   A(\$1000)    B(\$550)    C(\$500)
>  ~1   B(\$20500)            A=C(\$500)
> ~50   C(\$1000)    B(\$550)    A(\$500)
>
> Converted to vN-M utilities, we have:
>
> ~50  A(1.0)    B(0.1)    C(0)
>  ~1  B(1.0)            A=C(0)
> ~50  C(1.0)    B(0.1)    A(0)
>
> Since A and C are in a dead heat, it is clearly in those voters'
> interest to truncate and accept the AC lottery rather than settle for
> the \$50.  Note that this strategy doesn't depend on cooperation between
> the A and C groups -- if either group votes sincerely B wins, so there
> is no penalty for using the strategy unilaterally.
>
> It is hard to see how a win for B would benefit society as a whole,
> since it mainly concentrates wealth into the hands of a small minority.

It's possible that the minority would invest productively
whereas the majority would fritter away their found money by

Something Bart's recent examples haven't taken into account is
that, given a good voting method, additional good candidates
have an incentive to compete.  What happens in this example
if someone proposes D = "Distribute the windfall evenly" or
D'= "Distribute the windfall evenly among groups A&C and none
to group B"?  (Why wouldn't someone in group B propose D, if
s/he is concerned that choice B has little chance of winning?)

Then choice B could not be the condorcet winner, and D or D'
would be the condorcet winner if the A group and the C group
are minorities.

~50   A(\$1000)    D'~D(\$750)    B(\$550)       C(\$500)
~1   B(\$20500)   D(\$750)     A=C(\$500)       D'(\$0)
~50   C(\$1000)    D'~D(\$750)    B(\$550)       A(\$500)

I'm not sure how to convert to "vN-M" utilities, so I'll guess
that these are the vN-M utilities:

~50   A(1.0)    D'~D(0.5)       B(0.1)         C(0)
~1   B(1.0)       D(0.0366)  A=C(0.0244)     D'(0)
~50   C(1.0)    D'~D(0.5)       B(0.1)         A(0)

Since for groups A&C both choice D and choice D' are equivalent
to the AC lottery, the A&C voters lose nothing by approving or
ranking D and D'.  Since many people are risk averse, they will
(perhaps irrationally) prefer D with certainty more than the AC
lottery.  The B group may rank D also, if they estimate that
choice B's chance of winning is small.

I agree with Mike's comment that the scenarios Bart is concerned
about would be rare, given a good voting method.  Bart's
examples leave a vacuum which could easily be filled by a
good candidate, which would win and improve social utility.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

```