## Tuesday, January 16, 2007

### Voting power

The slides from Lecture 1 are on the course web site now.

I mentioned John F. Banzhaf III in class; he's got quite a web site there, in which he proudly declaims about being the "Osama bin Laden of torts", among other things. I guess he's famous for suing McDonald's (over obesity) and tobacco companies. Apparently he's part of the reason for smoking bans in the UK and other countries. He also shows up in the news these days as a commenter on the Duke Lacrosse happening.

Anyway, he's frequently credited with describing the notion of influences of voters on elections we discussed in class. That came about when he sued the Nassau County Board for having an unfair voting scheme; they passed proposals according to the following boolean function:

\$f(x)\$ = 1 iff \$9x_1 + 9x_2 + 7x_3 + 3x_4 + x_5 + x_6 \geq 16\$.

You can check that for this \$f\$, \$Inf_4(f)\$ = \$Inf_5(f)\$ = \$Inf_6(f)\$ = 0.

More interesting info on measuring the power of voters is here:

Banzhaf Power Index

Voting Power

Aaron said...

\$Inf_4(f)\$ = \$Inf_5(f)\$ = \$Inf_6(f)\$ = 0.

Its worse than this, right? 4, 5, and 6 have no influence even as a coallition, so there really is no reason for them to show up. In order for something to pass, it needs a majority of \$\{x_1,x_2,x_3\}\$.

Ryan said...

True. This one's pretty egregious; on the other hand, Naussa County's voting is not the most significant thing in the world.

For an example with the qualities reversed (subtler problem, more important situation), search for "Luxembourg" here.

yi said...

it seems if every one can write down a number in [-1,1] will make the voting better to reflecting the weighting of different voters.