John Nash's idea for trying to salvage multiplayer game theory, was the so-called "Nash equilibrium." A situation is an Nash eq. if each player cannot improve her expected utility (payoff at end of game) by altering her strategy (with all other player strategies assumed to stay fixed). Nash's theorem is a Nash eq. always exists. They gave him the Nobel Prize in Economics for that.
Problem: The Nash equilibrium is a nearly worthless idea when applied to voting & elections (viewed as an N-player game where there are N voters). For example, consider a 2-way election Gandhi vs Hitler in which everybody votes for the (unanimously agreed to be) worst choice: Hitler. Well, that is a "Nash equilibrium" because no single voter can change the election result! Indeed, essentially every possible vote pattern in every possible large election, is a Nash equilibrium. So Nash says almost nothing about voting. It is worthless. But now here is a very simple and highly effective fix, apparently suggested here for the first time (and thus proving the stupidity of all voting theorists including me). Have each voter cast, not "one vote" but rather each voter casts "a standard gaussian random variable" number of votes of each possible type. The voter does not get to control her vote, she only gets to control the mean of the Gaussians. So for example, in the Gandhi-Hitler example, she can use the mean +1 or -1 (and fixed variance) and that is all. In a rank-order 3-way ballot (6=3! choices), we could make her employ mean=1 for the vote she likes, and mean=0 for the 5 votes she does not. This "Smith fix of Nash" seems to work for any election method based on a finite number of kinds of "vote totals." In my Gandhi-Hitler scenario, 100% Hitler votes now is NOT a Nash eq, and the only Nash eq is 100% Gandhi votes. This idea is really a pretty excellent fix. It gets rid of all the huge number of "stupid" Nash equilibria and seems to leave only the "sensible" ones. In the DH3 scenario http://rangevoting.org/DH3.html I am not sure what the Nash equilibrium (or equilibria?) are, but I am sure that honest voting is not it, because each individual voter finds burial to be an "improvement." Presumably the Nash strategy in that scenario will be a probability-mixture of honest and strategic votes. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) and math.temple.edu/~wds/homepage/works.html ---- Election-Methods mailing list - see http://electorama.com/em for list info
