Thursday, January 18, 2007

Voting with real numbers

In a comment, Yi suggested that voters use real numbers, say, in [-1,1]. This raises a number of interesting questions. Are our voting schemes now $f : [-1,1]^n \rightarrow \{-1,1\}$? Presumably here a vote of -1 means you really really like candidate -1, a vote of .9 means you really like candidate 1, a vote of .01 means you are basically indifferent with a very slight preference towards candidate 1, etc. (It's a bit like rating the candidates on the Hot Or Not scale.)

In some cases (where $f$ can be expressed as a multilinear polynomial) this is basically the same as usual voting; one can just have the voting machine convert a vote of $y$ to a random vote from $\{-1,1\}$ chosen to have expectation $y$. But in other cases it's not the same.

The KKL theorem can be extended (with an appropriate new definition of "influence") to the case of functions $f : [-1,1]^n \to \{-1,1\}$, but certain corollaries cease to hold. We may see this result in the class; it's by "BKKKL" -- the B is Fields Medalist is Jean Bourgain and the extra K is Yitzhak Katznelson.

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