[EM] Dyadic Rated Pairs

[Forest Simmons]

It is well known that Ranked Pairs doesn't have absolute immunity from
strategic incentives to rank one's favorite below some compromise
candidate. The leading advocates of the method do not claim that it
satisfies the Favorite Betrayal Criterion.

The reason for not satisfying the FBC is simple: the strength of pairs is
measured by the margin of win.  If you vote your favorite over your
compromise, you are increasing the margin of win of your favorite over
compromise (should your compromise perchance lose to your favorite).

Of course, all would be well if your favorite had enough support to go on
and win the whole kaboodle.  But that is not the only possibility. Your
favorite may lose to some third candidate, and badly enough to lose the
race.

Well then, you say, your compromise would have lost also.

Not necessarily.

To make a long story short, what we need is a way of measuring strength of
pairs that doesn't depend on the size of margins or winning votes.

One of the most natural ways of doing this is through dyadic ballots.

The ballot  A > B >> C > D  naturally imbues some pairs with a stronger
preference relation than others, without reference to margins or winning
votes.

It seems natural to me to reckon the weak relations (A > B and C > D) at
half the rate of the stronger relations (the four that straddle the double
inequality).

Suppose that we average the strength of each of the six pairs over all
ballots in the four candidate election. This will give us six numbers
between zero and one, assuming that the strongest relation is normalized
to have unit strength.

To find our winner we lock in place the win of the strongest pair, no
matter how small the margin of winning or how few winning votes. (The
strength of each pair is determined only by the average of that pair's
relation strength ratings over all of the ballots.)

Then we successively lock into place other pairs in descending order of
strength, skipping a pair only when that pair would contradict the partial
order implicit in the previously locked in relations.

If no contradiction is ever encountered, then the winner is a "beats all"
winner, i.e. a Relative Condorcet Winner with respect to the ballot type.

If full dyadic ballots are used (allowing A >>> B > C >> D , for example,
so that no ballot can put two candidates in the same slot), then the
Relative Condorcet Winner is a true CW.  The Condorcet Criterion is fully
satisfied. 

The method generalizes in an obvious way to any number of candidates.


Forest