# [EM] Approval Equilibrium

Forest W Simmons fsimmons at pcc.edu
Wed Jun 13 15:36:45 PDT 2007

```Suppose that candidate Y has the greatest pairwise opposition against
candidate X.  Let the letter n represent the number of ballots on which
Y is rated strictly above X, i.e. X's maximum pairwise opposition.

If X is an approval equilibrium winner, then the equilibrium approval
of Y will be at least n.  It may be larger, because there may be some
ballots on which X and Y are rated equal top.

Setting aside the ballots on which X and Y are rated equal, what is the
"cost" of getting an approval of n+1 for X on the remaining ballots.

For those remaining ballots on which X is rated top the cost is zero.
For those remaining ballots on which X is rated k levels down from the
top, the cost is c(k), some increasing function of k.

Choose those (n+1) of the (remaining) ballots for which the total cost
is a minimum.

This total cost T(X) is the cost of making X into an approval
equilibrium.

The candidate X for which T(X) is minimal is the approval equilibrium
winner.

What c(k) function should we use?

Forest

```