[EM] Loggerheads (was New Condorcet Lottery)

[Forest Simmons]

I'm changing the name to "Loggerheads" because (1) our two player game is
based on players with polar opposite preferences, and (2) it's not quite
Condorcet compliant because of its way of handling pairwise ties. However,
if we just go back to the customary zero payoff for pairwise ties, the
method becomes Condorcet compliant.

For now let's leave the pairwise tie payoff open; it may help us
distinguish weak CW's from strong ones .... when the tie handling makes a
difference, it might be a sign of marginal weakness or instability in the
Condorcet winner.

Just to pursue this point a little further, the non-zero payoff for the row
player when both players choose the same candidate X, is the difference
F(X)-R(X), which is positive only when X has more first place than last
place votes. So if X has fewer first than last place votes, then that
diagonal payoff entry will be negative, preventing  X from being the sure
winner, even if X is the Condorcet candidate ... but not impairing the
Condorcet efficiency too much, unless the tie payoff entries are very
negative, which would be very unusual.

So let's keep open the possibility of non-zero pairwise tie payoffs, but
make zero the default payoff for simplicity.

Analogously de-cloned Copeland loses its absolute Condorcet efficiency when
we allow pairwise ties to count other than zero. So let's go back to the
original version there, as default, too:

The (default) De-Cloned Copeland Score of candidate X is ...

The Sum (over all candidates Y pairwise defeated by X) of F(Y)
Minus
The Sum (over all Z that pairwise defeat X) of R(Z)

Now continuing on with "Loggerheads" .... since the two players have polar
opposite preferences, it seems that their optimal strategies must be
maximally resistant to manipulation... your optimal defensive strategy
against your most antagonistic enemy should hold up against lesser foes, as
well!

At least that is my basic heuristic for this method.

The first Condorcet Lottery method that we learned about, nearly two
decades ago, disappointly turned out to be non-monotonic, as did the more
advanced Rivest method that incorporated pairwise defeat scores into the
payoff matrix.

It seems that the problem was the same basic problem we faced when trying
to preserve monotonicity while de-cloning Kemeny-Young, Borda, and Copeland.

Our recent (last week) breakthrough in that context is the impetus for this
Loggerhead method.

One way of looking at the breakthrough is this: making a clear distinction
between passive lack of approval and active disapproval allows us to
de-couple mono-raising of one candidate from lowering (mono or otherwise)
of another candidate.

In our original unsuccessful versions we did not distinguish the role of F
from the role of R.  There we just used "lack of F" as a proxy for R.

Fixing that crucial defect not only made monotonicity possible, but also,
as a pleasant surprise,  made possible the strong reverse symmetry enjoyed
by all of these new methods.

Some people resist lotteries as legitimate election methods, but if, as we
have been assured by our RCV friends the 440 real life elections they
analyzed all enjoyed Condorcet Winners, irrespective of employing a
non-Condorcet compliant method ... almost all of these lotteries will be
zero entropy lotteries ... the possibility of chance serving only as a
deterrent to insincere rankings.

And suppose that a sincere rock, paper, scissors cycle should exist.... it
is comforting to know that the support of the winning lottery is always a
subset of the Dutta Set, a kind of special subset of the better known
Banks, Landau, and Smith sets.

It has often been suggested that in the absence of a sincere CW, the best
thing might be to choose randomly from the Smith Set.  Well, that's
precisely what this Loggerheads method does ... and with probabilities
calculated to make sincere voting optimal.

We'll continue when I get some more free time.

In the mean time, somebody in contact with James Green-Armytage could help
by passing this message along to him ... I seem to remember him expressing
interest in the Rivest Lottery recently. It would be nice to get him, and
others with a game theoretic bent, thinking along these lines.

Forest

El lun., 17 de ene. de 2022 12:17 a. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> I would like to propose this Rivest-like two-player, zero-sum game related
> to the de-cloned versions of Kemeny-Young, Borda, and Copeland that I
> recently posted.
>
> For each candidate k, let F(k) be the random ballot Favorite probability
> of candidate k, and let R(k) be the random ballot favorite of candidate k
> on the Reversed ballots.
>
> Let P be the payoff matrix for the row player defined as follows:
>
> P(i, j) is F(j) if candidate i pairwise defeats j.
> P(i, j) is -R(i) if candidate i is pairwise defeated by j.
> P(i,  j) is F(j)-R(i) if candidates i and j are pairwise tied, including
> the case of i=j.
>
> Remember the game is zero sum, so the column player's payoff is the
> opposite of the row player's payoff.
>
> In general optimal strategies for the players are stochastic mixtures of
> the respective pure deterministic strategies, i.e. they are Lotteries.
>
> Let L and L* be the respective optimal lotteries for the respective row
> and column players.
>
> L(k) and L*(k) are the probabilities with which the respective players
> should bet on row or column k.
>
> For the un-reversed ballots, the method winner is chosen by L.
>
> For the reversed ballots the winner is chosen by L*.
>
> That's the method ... more commentary next time....
>
> Forest
>
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