[EM] fpA-max fpC better than sum? Generalizations, stats

[Kristofer Munsterhjelm]

A follow-up on the last post; here are some stats for the different 
simple fpA-fpC generalizations ("IFPP-like", fpA-max fpC, and fpA-sum 
fpC). I haven't implemented Friendly Cover yet.

What's surprising is that fpA-max fpC is better than fpA-sum fpC, but 
after thinking a bit about it, I think I get it. The heuristic says that 
whether A wins instead of B shouldn't depend on A>C (to the degree we 
can do so). Suppose A is beaten by C and beats B and D. Then if we have 
fpA-sum fpC, we can always perturb A's score by reversing the A>D 
defeat. But under fpA-max fpC, we can only do so if D has more first 
preferences than C. That's my hunch, at least.

(I used to prefer fpA - sum fpC because it's more naturally cloneproof. 
But IIRC, Kevin Venzke preferred fpA - max fpC because it had fewer 
Plurality failures. Guess I was wrong and Kevin was right, in a sense :-)

Anyway, the stat categories denote the fraction of elections where a 
particular strategy is possible. The strategies are:

"Burial, no compromise" (BnC): Burial is possible but compromising isn't.
"Compromise, no burial" (CnB): Compromise is possible but burial isn't.
"Burial and compromise" (BaC): Either works.
"Two-sided" (2S): Neither of the above, but putting the candidate you 
want to win first and the current winner last (simultaneous burial and 
compromise) works.
"Other coalition strats" (Misc): None of the above, but brute-forcing a 
coalitional strategy works. There are a max of three coalitions, i.e. 
each strategist may employ any of up to three different ballots.
"Manipulable": Some strategy worked

97 voters, 4 candidates, impartial culture. 50k elections per method, 
512 coalitional strategy tests per election.

                    BnC    CnB    BaC    Two-sided  Misc  Manipulable
Smith,IFPP-like    0.234  0.130  0.024  0.507      0     0.895
Smith,fpA-sum fpC  0.354  0.095  0.070  0.374      0     0.893
Smith,fpA-max fpC  0.132  0.148  0.015  0.395      0     0.691
Smith,IFPP         0.091  0.170  0.006  0.015      0.05  0.331
Smith,IRV          0.056  0.174  0      4e-05      0.05  0.280

or collapsing the burial types:

                    Burial   Just comprmise Two-sided  Misc  Manipulable
Smith,IFPP-like    0.2582   0.1296         0.5073     0     0.8950
Smith,fpA-sum fpC  0.4232   0.0954         0.3744     0     0.8930
Smith,fpA-max fpC  0.1477   0.1480         0.3950     0     0.6906
Smith,IFPP         0.0969   0.1700         0.0148     0.05  0.3310
Smith,IRV          0.0560   0.1738         4e-05      0.05  0.2799


Note that the test system may underestimate the nonmonotone methods' 
burial rates because the burial strategy just tries to put the current 
winner at the bottom; these methods may reward lowering the candidates 
some of the way but not all the way to the bottom. But any such 
underestimates should be captured by "Misc".