[EM] Least Expected Umbrage, a new lottery method

[fsimmons at pcc.edu]

Let M be the matrix whose row i column j element M(i,j) is the number of ballots on which i is ranked 
strictly above j plus half the number of ballots on which neither i nor j is ranked.

In particular, for each k the diagonal element M(k , k) is half the number of ballots on which candidate k 
is unranked.

Now think of M as the payoff matrix for the row player in a zero sum game.

Elect the candidate that would be chosen by the optimal strategy of the row player.

[End of Method Definition]

Remarks:

If there is a saddle point (i, j) such that the element M(i,j) in that position is the lowest in its row and the 
highest in its column, then the game is deterministic, and the winner is candidate j.  

In this case candidate j is the same as the MMPO winner under the Symmetric Completion Bottom rule, 
i.e. the Least Resentment Voting (LRV) winner.

However in general the optimal strategies are mixed, which means that the players' moves are 
determined by probability distributions or "lotteries."  In this case, the column player's optimal lottery is 
used to pick the winner.

By definition this method chooses the winner in a way that minimizes its expected opposition, so on 
average it accomplishes more completely the heuristic justification of LRV than LRV itself does.

In other words use of this method will (on average) distress the opposition less than any other method.

So let's call it the Least Expected Umbrage method or LEU.

We need to check that it passes all of the tests and satisfies all of the properties that we think are most 
important.

I know that most people are prejudiced against chance, so determinism is high on their list of 
importance. But I hope that future generations will be more enlightened on this score, and embrace the 
judicious use of chance.

When they look back and see that we anticipated some of their ideas, they might forgive us for some of 
our other oversights.

Forest