[EM] Dollar example

[Steve Eppley]

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 
buying useless toys produced abroad by a hostile nation.  :-)

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)