[EM] The Nash Equilibrium method for counting rankings. Other "player" definition.

[Rob Speer]

On Mon, Jan 05, 2004 at 04:07:08PM +0000, MIKE OSSIPOFF wrote:
> 
> I believe it was probably Alex who once suggested a method that chooses the 
> candidate who wins at Nash equilibrium.
> 
> Alex, could you repeat that method definition again?

There's probably been more than one method like this. I brought up a
method that involved treating the "margins" of victory or defeat (I put
"margins" in quotes because winning-votes could probably be used too) as
a score table for a hypothetical two-player game. Then you find the
Nash-equilibrium strategy for that game.

A Condorcet winner will be the only one chosen by a Nash-equilibrium
strategy; but if there is no Condorcet winner, you get a probability
distribution among some odd number of candidates. You could elect
the candidate with the highest probability, or use everyone with a
non-zero weight as the "Nash set" for a further method.

The problem: the weights assigned to the candidates are very much
non-monotonic. Even Nash Sequential Dropping ended up being shown by
Markus to be non-monotonic, I believe.

> But I suggest a more natural definition of a "player" in voting: A set of 
> voters who can improve on the outcome for themselves.

So the players are the voters (or groups of voters, but I think your
sets of voters are fuzzy enough that you'd just want to use individual
voters as players, who might happen to use the same strategy.)

Does anybody here know how to find a Nash equilibrium strategy in a game
of more than two players? Even in the two-player case, finding the
equilibrium of five or more possible moves takes some serious
number-crunching by something like Matlab.

How do you assign scores to the outcomes? For example, a player might
determine that a certain strategy gets them a 20-percent chance of
electing their 1st choice candidate, while a different one gets them a
40-percent chance of electing their 2nd choice candidate. How do you
determine which strategy they choose?

-- 
Rob Speer