[EM] R. B. MacSmith

[Forest Simmons]

> Date: Mon, 28 Feb 2005 23:37:05 +0100
> From: Jobst Heitzig <heitzig-j at web.de>
> Subject: [EM] R.B.MacSmith

<snip>

>
> Anti-strategic properties:
>   I did not yet test many anti-strategy criteria, but the main
> anti-strategic feature is that, due to the above-mentioned
> randomization, in every majority which thinks about producing a fake CW
> by voting strategically there is someone who takes the risk of actually
> getting worse off by doing so.
>

Unfortunately there is still some strategic incentive for inducing a beat
cycle where there was none. [See examples below.]

Accordingly I propose the following variation on Jobst's idea:

1. List all of the candidates in order of decreasing approval.

2. Go down this approval list crossing out candidates that are covered by
(remaining) candidates higher up on the list.

3. Strike off the ballots all of the candidates that were crossed off the
approval list (but do not change approvals even if all remaining
candidates on a ballot are in the same approval category).

4. (optional) Iterate steps 2 and 3 until no remaining candidate is
strongly covered among the remaining candidates.

5. Choose the winner by random ballot from among the remaining candidates
(ignoring the stricken candidates).

As the following examples will show, non-Smith members may have positive
probabilities.  As the examples also show, this is essential for
discouraging insincere order reversal:

Example 1:

Sincere preferences are

45 A>>C>B
30 B>C>>A
25 C>A>>B .

Any Smith method will give the win to C with certainty.  Then there is an
incentive for the first faction to reverse the order of C and B, since any
cycle resolution method is going to give some (if not all) of the
probability to A.

Let's see how my (non Smith) variation plays out:

With sincere ballots, candidate B is the only strongly covered candidate
(even after the optional iteration since A's approval remains greater than
C's).

The respective probabilities for A and C in this sincere case are 45
percent and 55 percent.

Now suppose that the first faction (insincerely) considers voting
either  A>>B>C  or  A>B>>C .

This introduces a beat cycle ABCA, so no candidate covers (let alone
strongly covers) another, so no candidates are eliminated.

The respective winning probabilities would be 45, 30, and 25 percent,
which would leave A's chances the same but share C's chances with B.

Therefore the contemplated order reversal is unrewarding for those
contemplating it.


Example 2:

Sincere preferences are

49 C>>A=B
24 B>>A>C
27 A>B>>C

Approval order is BCA, while (pairwise) A beats both B and C, and C is
beaten by both A and B.  So C is strongly covered by B.  Once C is
eliminated we have

24 B>>A
27 A>B

so neither of the two remaining candidates strongly covers the other.

The respective winnning probabilities are 24/51 and 27/51 .

Now suppose that the second faction contemplates truncating A so that
their ballots become  B>>A=C .

Then a beat cycle ABCA comes into play, so there canot be any kind of
covering of one candidate over another.

The respective winning probabilities are 27, 24 and 49 percent for A, B,
and C, which would punish the faction contemplating insincere truncation.

[To see this, notice that A and B still have the same odds relative to
each other, but now their worst alternative C has gained a positive chance
of winning.]


[End of examples]


Properties:

1. Some member of the Smith set will have positive probability since no 
member of the Smith set can be strongly covered by any non Smith 
candidate.

2. Pareto is satisfied, since (as Jobst pointed out) if A is ranked above
B on every ballot, and B is not strongly covered, then A is not strongly
covered, so when random ballot time comes, if B is still on the ballot, A
will still be there also, so B cannot be picked.

Pareto comes out slightly better if we define strong covering by saying
that A strongly covers B iff A covers B and A has at least as much
approval as B.  Then we can say that our method is Independent of Pareto
Dominated Alternatives, since Pareto domination would then imply strong
covering, and all strongly covered candidates are stricken from the 
ballots in the course of the procedure.

3. Jobst's clone proof argument still works.

4. Monotonicity.  Candidate A moving up in the ranks relative to the other
candidates cannot decrease its approval (and increasing wouldn't hurt) nor
can it decrease the number of pairwise beats, nor can it decrease A's
chances of being chosen at the random ballot stage. Furthermore it cannot 
increase the number or candidates that are not strongly covered. Putting 
all of this together, we see that A's winning probability cannot be 
decreased by moving A up relative to the other candidates.

5. Independence of zero probability alternatives. If every candidate is in 
top position on at least one ballot (by voting for self, say), then this 
independence from zero probability alternatives condition holds, since 
after all but the strongly covered candidates have been stricken, then 
every remaining candidate will have a positive chance of being chosen by 
random ballot.

More examples of potential order reversal incentives are needed for
testing this method.

Thanks for your inspiring work Jobst!


Forest