[EM] The Nash Equilibrium method for counting rankings. Other "player" definition.

[MIKE OSSIPOFF]

I believe it was probably Alex who once suggested a method that chooses the 
candidate who wins at Nash equilibrium.

Alex, could you repeat that method definition again?

It sounds like a promising method. I was suggesting that, for Nash 
equilbrium for voting, a "player"  be a set of same-voting voters.

But when that definition of a player is used, the Nash equilibrium method 
isn't truncation resistant.

40: A
25: BA
35: C

A has 65 votes, B has 25 votes, and C has 35 votes.

A wins. No one set of same-voting voters can improve on that outcome for 
themselves.
.
If the B voters withdraw their vote for A, then A still wins, with 40 votes 
to C's 35.

If the C voters vote for B, then A still wins, beating B 65 to 60.

So that outcome is a Nash equilibrium, by the "player" definition that I 
quoted above. A wins by truncation at Nash equilibrium.

But I suggest a more natural definition of a "player" in voting: A set of 
voters who can improve on the outcome for themselves.

Someone here once suggested a "player" definition different from the one I 
suggested, and I didn't like it for some reason. I don't know if that other 
definition is the one that I'm suggesting now.

So a Nash equilibrium then is an outcome that no set of voters can improve 
on for themselves.

With that definition, and that example, that set is the B voters and the C 
voters. The B voters withdraw their vote for A, and the C voters vote for B. 
So now that A victory isn't a Nash equilibrium.

Using the Nash equilibrium definition in the paragraph before last, let me 
define this method:

Nash Equilibrium Selection (NES):

Balloting: Rank balloting, allowing truncation and equal rankings.

Elect the candidate who can win at Nash equilibrium in an Approval election 
between the candidates, based on the voters preferences expressed in their 
ballot rankings.

Outcome A keeps outcome B from being a Nash equilibrium if some set of 
voters can improve on B for themselves to get A.

If there's no Nash equilibrium, because every outcome is kept from being a 
Nash equilibrium by some othe outcome, then the tie is the set of candidates 
who win in outcomes that keep other outcomes from being Nash equilibrium.

If there's more than one Nash equilibrium, then the tie is the set of 
candidates who win in those outcomes.

Ties are solved by having all the rankings give an Approval vote to each of 
their ranked candidates.

That tie solution seems to get rid of some strategy incentive.

Someone suggested Approval Completed Condorcet. I don't remember exactly how 
it went. Alex, if that was you, would you re-post it?

If one solves circular ties by having each ranking give an Approval vote to 
each of its ranked candidates, that isn't truncation resistant.

The advantage of NES is that it only uses that tie solution when there isn't 
a truncation cycle.

With 3 candidates, NES seems to have strategy properties similar to wv 
Condorcet. I don't know if that remains so with more candidates.

All the new method proposals are more complicated to evaluate than 
Condorcet, and so it's difficult to say how they do in terms of strategy. 
But NES looks good at first glance, with 3 candidates.

So does Forest's and Rob LG's Ballot-By-Ballot (BBB?), and Lorie Cranor's 
DSV, Declared Strategy Voting (when DSV uses Approval with the 
best-frontrunner strategy).

But I only took a quick look at them. The new methods tend to be more 
complicated and therefore more difficult to evaluate.

Those two are even more difficult to evaluate than NES, but with one 
particular 3-candidate example that I checked, they both are truncation 
resistant and, with both, offensive order-reversal can be thwarted by 
defensive truncation.

Interestingly, with BBB, the defensive truncation elected the CW, unlike 
with other methods that I know of. That could be considered a good thing, 
because the defenders are improving the immediate outcome for themselves. 
But it also means that, to the extent that that happens, defensive 
truncation isn't a deterrent to offensive order-reversal. That could mean 
that the defensive strategy could be needed more often. But, even without 
defensive truncation, offensive order-reversal can backfire, and so it 
always has some deterrent if the would be order-reversers aren't sure of the 
faction numbers.

With BBB, the offensive order-reversal failed in my example. Apparently BBB 
has more difficult faction-numbers requirements for successful offensive 
order-reversal.

But all nonprobabilistic 1-balloting methods are vulnerable to offensive 
order-reversal. The method has no way of determining if a circular tie is 
strategic or natural. If it's strategic, the method has no way of 
determining who the CW is and who is order-reversing, or which majorities 
are sincere.

So how can it be expected to elect the CW when there's order-reversal? How 
can it be expected to not reward the refersal if it can't determine who's 
reversing?

So, even with fancier or more deluxe nonprobabilistic 1-balloting methods,  
methods, we've still got offensive order-reversal and, when that strategy is 
used, the need for defensive truncation.

Condorcet wv apparently is very difficult to improve on. Difficult but maybe 
possible, in some minor way. But not in terms of the gross strategy problems 
that 1-balloting methods all have.

Maybe there are other simpler ways of detecting truncation circular ties, 
maybe in wv, so that only the other circular ties are solved by the 
apparently strategy-reducing Approval tiebreaker.

I suggest that someone test the new methods with computer simulation in 
which truncation takes place, and in which offensive order-reversal takes 
place. And in which offensive order-reversal is countered by defensive 
truncation. In that way they can be tested with more candidates.

It's worth doing if there's a chance of one or more of those methods 
bringing a little improvement.

So the anti-order-reversal enhancements that I suggested in a previous 
posting some time ago would be useful in any nonprobabilistic 1-balloting 
method.

I suggested a few, but I'd like to add a few more:

Spectrum-order-estimate:

The method can't recognize order-reversal because it knows nothing about 
anyone's preference ordering and nothing about the political spectrum 
ordering. But the voter can often identify order-reversal if s/he knows 
something about the political spectrum ordering. So the voter could have the 
option of marking his spectrum order estimate on the ballot, authorizing the 
method to drop from his ballot any candidate whose voters apparently 
order-reverse and cause a strategic circular tie.

When any of these anti-order-refersal options are chosen, they'd be carried 
out only if there's a circular tie whose each member has a majority defeat.

A previous suggestion was that the voter have the option to indicate a line 
in his ranking, so that, if there's a circular tie all of whose defeats are 
majority defeats, and with members above and below the line, that voter 
wants to drop the below-line candidates from his ranking.

I also tentatively suggested an option for the voter to use a method of 
detecting some order-reversals based on the pattern of unanimity and 
nonunanimity of the lower choices of voters who share a 1st choice. It makes 
some sense with 3 candidates, but I don't know if it work with more 
candidates.

As before, it would automatically drop the apparent reversers' candidate 
from the ranking of the voter who chose that option.

But, as I said, I don't believe that order-reversal will be a problem with 
wv, for the reasons that I stated in that earlier posting. So these 
anti-order-reversal enhancements wouldn't reallly be needed, though they 
could be proposed at some time after wv enactment, if people became 
concerned about strategy.

So far, all the new methods, NSE, BBB, & DSV and others aren't sufficiently 
tested to be proposed. Some or all of them seem to complicated for a first 
proposal. But if there are any other new method proposals that are expected 
to bring improvement, could someone re-post them or tell the year and day on 
which they can be found in the archives? And these methods should be more 
thoroughly tested in computer simulations with more candidates, with 
truncation and offensive order-reversal, and offensive order-reversal 
countered by defensive truncation.

Mike Ossipoff

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