[Election-Methods] Participation Failure Probability - results & open questions

[Warren Smith]

PARTICIPATION FAILURE PROBABILITY
---------------------------------

Call an election situation a "participation failure scenario"
if there exists a vote Q, such that adding some number T>0 of
honest Q-voters, will cause the election result to worsen in their view.

(This is a "no-show paradox" - these extra voters are better off staying home.)

The "random election model" is V voters, V-->infinity,
all independently casting random votes (all votes equally likely).

IRV-3: I did some analysis and concluded the probability that a random
3-candidate IRV election is a participation failure scenario, is 16.2%. 

COND-4:  I can prove the probability P than a random
4-candidate Condorcet election, is a participation failure scenario,
is bounded below by a positive constant independent of which-flavor
of Condorcet you use.

For two particular Condorcet methods, I estimated P by monte-carlo
and it is safe to say  0.5% < P < 5%  and my best guess is 2.5%.
(My program does not compute P exactly, it only finds high-confidence
bounds on it.  If I were less lazy I could tighten the bounds...)

COOL OPEN QUESTIONS
-------------------

I suspect:

COND-INFINITY:
Random C-candidate Condorcet elections are
participation failure scenarios with probability-->1
when C is made large.

IRV-INFINITY:
Random C-candidate IRV elections are
participation failure scenarios with probability-->1
when C is made large.

I have not proven either.  I have got something close to a
proof for three particular Condorcet methods
(Copeland, Simpson-Kramer MinMax, and basic Condorcet)
although even in these cases my proof could be attacked as not
really being a proof (argument is pretty convincing, but not
fully a proof).

For IRV, P is easily seen to be a non-decreasing function of C
so it must approach a limit.

IMPORTANCE:
The (conjectured) fact that this pathology is 100% common if
the number of candidates is made large, seems important...

Warren D. Smith
http://rangevoting.org