# [EM] another lottery method

Paul Kislanko kislanko at airmail.net
Wed Nov 9 15:01:27 PST 2005

```I'm not sure what you are saying here.

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From: election-methods-bounces at electorama.com
[mailto:election-methods-bounces at electorama.com] On Behalf Of Simmons,
Forest
Sent: Wednesday, November 09, 2005 4:31 PM
To: election-methods at electorama.com
Subject: [EM] another lottery method

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.

Do we need two different versions of the definition of "dense"? It seems to
me that "majority dense" is just a rewording of the definition of Smith Set.

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

Why introduce "majority dense" and not use that?

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

How do you define "approval sum" in this context? Throw out some ballots and
calculate it from the remainder?

Choose from K by random ballot.

I can guarantee you that 99+ percent of us will never vote for a method that
includes the phrase "random ballot." Those of us who understand EM's might
well recognize that any such are trying to get around Arrow's proof by
choosing to accept a dictactor.

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.

The "nice properties" are???? You say "random" and I think "dictator".

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

And who cares about what the candidate whines about. I think this is
ignoring the will of the VOTERS, and the smart ones will see that it never

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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