[EM] My Recent New Methods Summary with Corrections

[Forest Simmons]

Since all of these new methods depend (by default) on the benchmark lottery
f and its reverse counterpart f*, we begin by defining these probability
densities.

The context is a set beta of (voted) ranked choice ballots.

For each candidate X, let f(X) be the probability that the random ballot
favorite candidate is X.  A thought experiment defines this probability: if
a randomly drawn ballot has more than one contender for favorite,
additional ballots are drawn sequentially to narrow down to a single
favorite. The probability that X is the resulting favorite is the value of
f(X).

Similarly, f*(X) is the random ballot anti-favorite probability for X. if a
ballot randomly drawn from beta has more than one contender for
anti-favorite, additional ballots are drawn sequentially to narrow down to
a single anti-favorite. The probability that X is the resulting
anti-favorite is the value of f*(X).

Note that if the ballot rankings are all reversed f and f* swap places.

The salient (i.e. sufficient for what follows) properties of f and f* in
this context are ...
1. They are both probability density functions on the set of candidates.
2. If X (and only X) is raised on one or more ballots, then ...
 f(X) does not decrease, nor does f*(X) increase
AND
for Y not equal to X, f(Y) does not increase nor does f*(Y) decrease.
3. If X (and only X) decreases on one or more ballots, then ...
f(X) does not increase, nor does f*(X) decrease
AND
for Y not equal to X,  f(Y) does not decrease, nor does f*(Y) increase.
4. Both f a d f* respect clone sets, which means (in the case of f), if
candidate X is replaced by a clone set chi, then
f(X)=Sum(over x in chi) of f(x).
5. The procedure that yields f when applied to the ballot set beta, yields
f* when applied to the reversed ballot set beta*.
[Condition 5 is not essential, except for the strong reverse symmetry
property.]

Now that we are all set up, we can define the four methods ... de-cloned
Kemeny-Young, de-cloned Borda, de-cloned Copeland, and the Loggerheads
Condorcet Lottery.

I'm going to break here to save what we have so far ...







El lun., 17 de ene. de 2022 1:44 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> 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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