Unifying Random Ballot and Score Voting

In honor of today, some silly results (believed to be correct but probably 
useless) about elections with cardinal ballots: it's possible to unify 
random ballot with score voting via a single parameter.

First, the naive way: interpolate between the random ballot lottery and 
score voting's (degenerate) lottery. The parameter is the weight to give one 
of them.

There's also a more interesting way: pick a nonnegative parameter alpha. 
Draw a weight vector for the n voters via a symmetric n-dimensional 
Dirichlet(alpha/n) distribution. Select the candidate with the highest 
weighted score.

The limit as alpha goes to zero is random ballot. The limit as alpha goes to 
infinity is score voting. This comes from the behavior of the Dirichlet 
distribution, which gets "spikey" for very small parameters and converges to 
an equal weighting for very large parameters.

This process gives a realization from an underlying lottery, which can be 
thought of as the expectation over the entire Dirichlet distribution

The 1/n scaling is to preserve the lottery if you duplicate the voters. [1]

Fun property: all of these methods, for both the naive approach and the 
interesting one, satisfy participation. To see this for the latter, pretend 
that a trusted oracle draws and commits to the weight vector in advance of 
the election, but keeps it secret until after the votes are cast. For any 
given weight vector, it's always in a voter's interest to participate.

Also, the second family is not just a repackaging of the first. Consider:

   A  B  C  D  E
  10  0  0  7  6
   0 10  0  7  6
   0  0 10  0  6

Random ballot gives 1/3 probability to each of A, B, and C. Score voting 
gives 100% to E. However, for intermediate values of alpha, there's a 
nonzero probability that D is chosen, since the first two voters could get 
almost all of the total weight.



[1] You can see this from the method of generating a realization from an 
n-dimensional Dirichlet distribution via n realizations from a gamma 
distribution. The sum of the gammas has a shape parameter equal to the sum 
of the individual gammas' shapes, since their scales are all equal. The 1/n 
scaling factor means that if each voter in election A corresponds to m 
identical voters in election B, then each voter in election A's weight is 
distributed the same as the sum of m voters' weights in election B.
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