[EM] Some Examples of Equilibria

[Forest Simmons]

Suppose that there are ten thousand voters, and three candidates, A, B,
and C.

Suppose that the normalized sincere utilities are as indicated in the
following table:

4500 u(A)=1, u(C)=0, and u(B) uniformly distributed between 0 and 1.
4000 u(B)=1, u(A)=0, and u(C) uniformly distributed between 0 and 1.
1500 u(C)=1, u(B)=0. and u(A) uniformly distributed between 0 and 1.

Suppose that the voters are told that they are to submit approval ballots,
and that the winner will be chosen by the following procedure:

(1) The approval counts a, b, and c determine how many marbles marked A,
B, and C, respectively, will be placed in a bag.

(2) Fifteen hundred of each candidate's marbles are withdrawn (or all of
the marbles are withdrawn for those candidates that have less than fifteen
percent approval).

(3) From the remaining marbles, one is drawn out at random to determine
the winner.


Furthermore, suppose that a trusted political pundit tells the voters that
she is quite confident that A, B, and C will get approximately 40%, 40%,
and 20% approval respectively.


Suppose that all voters trust this pundit and decide to approve above
their expected utility.

Then one third of the 4500 ABC voters (those who rate B above 66%) find
that B is above their expected utilities, one half of the 4000 BCA voters
find that C is above their expected utilities, and two thirds of the CAB
voters (those who rate A above 33 percent) find that A is above their
expected utilities.

The approval counts are 4500+1000 for A, 4000+1500 for B, and 1500+2000
for C.

When these amounts are reduced by fifteen hundred, then A, B, and C are
left with 4000, 4000, and 2000, marbles, respectively, in the bag.

The pundit was right! The respective probabilities for wins in the random
marble drawing are 40%, 40%, and 20% .

Was this just self fulfilling prophecy?

In a way, yes, but the pundit had an oracle that knew the unique "fixed
point" (equilibrium) distribution of probabilities associated with this
set of utilities and this "quota" value of fifteen hundred.

Furthermore, the pundit knew that the equilibrium was a stable
equilibrium, so that if the voters believed in slightly different
probabilities, the marble distribution would be closer to the
equilibrium than their original beliefs.

Repeated balloting using previous distributions as best estimates of final
distribution would bring about convergence to the equilibrium
distribution.


Now if the quota of fifteen hundred is upped to 3300, then the equilibrium
distribution is approximately 51.7%, 35.6%, and 12.7% for the respective
candidates.  However, this equilibrium is not a stable equilibrium, and so
it would not be a likely result of repeated balloting.

But if we back off the quota slightly to 3250, then we get a stable
distribution of 51.066%, 36.002%, and 12.932%, respectively.

The upper limit of stable distributions (itself marginally stable) would
be somewhere between these two.  That's the distribution that I would like
to be called the "Ideal Equilibrium Distribution" for this set of
utilities, if somebody else hasn't already discovered it and given it a
more deserving name.

This set of utilities is equivalent to the following set of Interval CR
ballots:

9 CR(A)=1 > CR(B) > 0=CR(C)
8 CR(B)=1 > CR(C) > 0=CR(A)
3 CR(C)=1 > CR(A) > 0=CR(B)

Now for another example, let's change the numbers of the respective
interval ballots to 10, 9, and 1.

Here is a result for this new set of numbers:

If the quota q is between 7 and 8.5, then there will be a stable
equilibrium where C has zero probability, and the A:B odds of winning are

                11-q to 9-q.

The upper limiting distribution as q approaches 8.5 is what I would
call the "Ideal Equilibrium Distribution:"

         (5/6, 1/6, 0)

Most methods except Borda would give the win to A outright.  This method
says that B should have one chance in six of winning.

Forest