[EM] Some Examples of Equilibria

[Forest Simmons]

I want to head off potential confusion by pointing out that the first
example below is cast in terms of approval voting, given pundit
information, to show the rationale behind the counting method for the
interval ballots in the later examples.

The method I have in mind takes as input the interval ballots (without
worrying about pundits or polls) and calculates the "Ideal Equilibrium
Distribution" from those interval ballots, and then uses that distribution
to  determine how many balls to put in the bag for each candidate.

The owner of the randomly drawn ball is the winner of the election.

Also I want to explain why the ideal equilibrium distribution should arise
(perhaps as a limiting distribution) from (at least marginally) stable
equilibria;  otherwise it would be more vulnerable to manipulation.

Forest


On Mon, 20 Oct 2003, Forest Simmons wrote:

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 (2) Fifteen hundred of each candidate's marbles are withdrawn (or all of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 the marbles are withdrawn for those candidates that have less than fifteen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