# [EM] Another kind of approval equilibrium

Forest Simmons fsimmons at pcc.edu
Tue Dec 2 15:28:02 PST 2003

```I have been considering various kinds of approval equilibria which have
the following common features:

(1) The candidates C1, C2, ... are considered to have respective
probabilities, P1, P2, ... of winning.

(2) Voter rankings or ratings are converted to approval ballots based on
these probabilities.

(3) The resulting approval ballots somehow confirm the prior probabilities
P1, P2, ... .

This kind of equilibrium is stable if it can be obtained by a process of
iteration of these three steps, where the "confirmation" in step three
improves despite round-off and other perturbations.

The hardest part is figuring out how to get winning probabilities out of a
set of approval ballots, since most of the time one candidate will have
greater approval than the others, and that would seem to make the
posterior winning probability for that candidate 100%, which turns out to
be problematic for various reasons that I won't bore you with in this
message.

For a while I was considering making winning probabilities proportional to
the tenth power of the candidates' approvals, so that a candidate with any
approval at all would have some positive chance of winning.

After rejecting that, I considered having winning probabilities
proportional to the number of approval votes above some quota, which could
even be tied to the highest approval: any candidate that gets within 10%
of the highest approval candidate gets a positive chance of winning, for
example.

These methods require a final drawing in which the respective candidates'
probabilities of winning are P1, P2, etc.

Until recently I didn't see anyway of avoiding this drawing, since just
picking the candidate with the highest probability would not be a true
equilibrium solution except in the case of P = 100 percent.

Why not just trust the polls?

Well polls don't really tell us winning probabilities; they just give us
approval counts from samples.

Is there any way to determine winning probabilities from mere approval
counts?

Only if there are no significant correlations among candidate preferences.

What if the polls accurately report the correlations as well, "30% of
those who approved A also approved B, etc?"

That would be better, especially if the polled voters are reporting their
approvals after hearing accurate approval results (including correlations)
from previous polls.

In other words, this is the kind of thing that improves with iteration.

So why not just collect the voters' CR ballots, and simulate the
iteration of polls until an equilibrium is reached (if there is one)?

That is roughly my current idea, but (as they say), "The devil is in the
details."

How does this avoid the drawing at the end?

Instead of "enforcing" the probabilities with a drawing, we just interpret
the non-zero probabilities as saying that in statistically similar
populations of voters, these other candidates have significant chances of
winning.

To Be Continued ...

Forest

```