[EM] One benefit to nonmonotone methods

[Kristofer Munsterhjelm]

Since it's pretty quiet at the moment, here's another observation. I've 
been testing some methods that pass DMT or DMTC, and I've found out two 
things:

- The "monotonized" contingent vote, where if A is the Plurality winner 
and can give some of his first preferences to some other B to displace 
the other finalist C and get B into the top two instead, A's score 
becomes A>B instead of A>C, is not that much more strategy susceptible 
than the ordinary contingent vote (0.91 vs 0.87 for 97 candidates, 10k 
elections, 32k tries per election).

- But it's much harder to get the true strategy resistance of 
nonmonotone methods, because coalitional strategy is much harder to find 
than two-sided "rank who you're compromising for top, the current winner 
bottom" strategy.

So even if say, a method X and its monotone variant both have strategy 
resistance 0.8, it's often harder to execute strategy against the 
nonmonotone one in practice because you can overshoot.

In a monotone method, if your honest vote is A>B>C>D>E, and the current 
winner is D, and you're compromising for C, then if A>C>B>D>E works, 
then C>A>B>D>E will also work and most likely C>A>B>E>D will also work. 
So you can slam your compromise to the top and your burial target (the 
current winner) to the bottom, and that's a pretty simple strategy.

But in a nonmonotone method, it's possible that A>C>B>D>E will work but 
C>A>B>D>E won't. So even though the honest election is vulnerable to 
strategy with both methods, it's harder to find the correct strategy.

Thus if you absolutely need all the strategy resistance you can get, 
nonmonotone is probably where it's at. I'll still try to find a good 
monotone burial-resistant method, though!

Some stats to show this effect: impartial culture, 97 voters, 5 
candidates, 50k elections, 32k coalitional tries per election:

Smith,Contingent vote:
	Ties:                      0.00612

	Burial without compromise: 0.11049
	Compromise without burial: 0.24123
	Burial and compromise:     0.00278
	Two-sided:                 0.00254
         Other:                     0.51588

	Total susceptibility:      0.87292

Smith,Contingent vote with donation:
	Ties:                      0.00904

	Burial without compromise: 0.13044
	Compromise without burial: 0.23402
	Burial and compromise:     0.01158
	Two-sided:                 0.54180
         Other:                     4e-05

	Total susceptibility:      0.91788

The "Other" category (which contains pushover and pushover-like 
strategy) has been almost entirely emptied, and that strategy has become 
two-sided instead.

(Two-sided is the fraction of elections where neither burial nor 
compromising works, but doing both at the same time works.)

-km