# [EM] another lottery method

Simmons, Forest simmonfo at up.edu
Wed Nov 9 14:30:36 PST 2005

```Ballots are ordinal with approval cutoffs (or some other way of inferring approval).

Let's say that a subset of candidates is "dense" in the set of all candidates iff each candidate outside of the subset is beaten pairwise by some member of the subset.

Similarly, let's say that a subset of candidates is "majority dense" in the set of candidates iff each candidate outside of the subset is majority defeated by some member of the subset.

A subset is "minimal dense" iff it is dense, but no proper subset of it is dense.

Let K be the minimal dense subset whose approval sum is the greatest.

Choose from K by random ballot.

The motivation for this method is obvious:  plain old random ballot has lots of nice properties, but it is too promiscuous with the probability; a candidate with one first place vote has a chance of being elected on that basis alone.

We want to choose randomly from a dense set so that any candidate that complains that he pairwise beat the winner can be told that either he lost by bad luck in the lottery or else was beaten pairwise by some candidate that had bad luck in the lottery.

The reason for minimality is to horde the probability as much as possible consistent with this kind of immunity from loser complaints.

Of course there are many variations on this theme.  For example ...

Use minimal majority dense instead of minimal dense.

Let K be the minimal majority dense subset with minimal cardinality.  Ties are broken by comparing approval of candidates in the two subsets, starting with the highest approval candidates.

Instead of random ballot, give probability 1/2^k to the k_th member of K counting from greatest approval to least, with the least approval member of K getting the leftover probability 1/2^(#K).

Forest

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