[EM] Detailed stats for the ordinal methods

[Kristofer Munsterhjelm]

On 2024-05-09 23:56, Closed Limelike Curves wrote:
> Hi Kristofer! Thanks for this :)
> 
> I do want to ask though, do you think the rate of manipulable elections 
> is a good measure of the "general strategy resistance" of an electoral 
> method? The resistant set certainly seems to reduce that rate, but for 
> all I know that 7.5% is all turkey-elections.

The narrow sense of turkey elections - exploiting nonmonotonicity to do 
pushover - must, for ordinal methods, belong to the "Other" strategy 
category. Not every Other strategy need to be pushover, but pushover 
must be an Other strategy. Let's consider Resistant,Borda again:

>>          Ties: 0.001 (5)
>>          Of the non-ties:
>> 
>>          Burial, no compromise:        123     0.0246246
>>          Compromise, no burial:        72      0.0144144
>>          Burial and compromise:        147     0.0294294
>>          Two-sided:                    48      0.00960961
>>          Other coalition strats:       1       0.0002002
>>          ================================================
>>          Manipulable elections:        391     0.0782783

There's only one out of 4995 elections with "other" strats. So pushover 
manipulability is very low. (I've designed resistant set methods that 
seem to have no pushover at all, even though they fail monotonicity. I 
haven't been able to prove why certain resistant set constructions make 
pushover impossible, though.)

I can think of two ways to formalize the broader category of 
turkey-raising as mentioned on Electowiki. They would be:
	- Supporters of candidate A encourage a candidate C to enter, so that 
C>B>A voters express their honest opinion; but that makes A win instead.
	- By making C appear to have more support than he actually does, 
supporters of candidate A trick strategic B>C>A voters to compromise for 
C. As a result, the winner changes from B to A.

Neither effect is captured in my simulations: the first would be a form 
of strategic nomination, and the second is a strategic play under 
imperfect information. As the number of candidates doesn't change, and 
the simulation involves a fully honest election followed by 
full-information strategy, it doesn't capture either.

Strategic nomination can indeed be a problem and should be investigated 
more closely. James-Green Armytage showed that IRV has greater exit 
incentive than the Condorcet-IRV methods do, for instance. I haven't 
written code to do this, and thus my stats don't provide any information 
about strategic nomination.

Taking the second effect into account would be very difficult, as we're 
then moving into a repeated game of imperfect information. But, as a 
heuristic, if the voters know that the method has low (ordinary) 
manipulability, then it would probably be harder to get them to engage 
in a self-destructive strategy as well, since it would be harder to get 
them to engage in strategy in general.

And a final caveat: if you combine a strategy resistant method with 
primaries or other parts, the composition could have turkey strategies. 
I imagine that the likelihood of this happening depends on the strategic 
nomination incentives for the method: so you could see it with IRV but 
it would be less likely with Smith-IRV.

-km