[EM] Preliminary Chain Climbing investigation

[Kristofer Munsterhjelm]

On 8/15/23 03:20, Forest Simmons wrote:
> As soon as you abandon the monotone agenda formation requirement, chain 
> Climbing (from the nominally unpromising end of the agenda towards the 
> nominally favorable end) loses its validity ... all bets are off.
> 
> In the three candidate Smith set case the most promising candidate that 
> defeats the least promising Smith member is elected.
> 
> The least promising Smith member is usually the sincere CW. The Smith 
> member that beats it is the "bus" under which it was nuried ... so good 
> poetic punishment for the niriers.
> 
> Like most methods good at punishing the buriers, this method needs a 
> sincere runoff to recover the sincere CW.
> 
> The appropriate three candidate runoff is the chain climbing winner W 
> [most likely the bus] versus a runoff between the least promising Smith 
> member Z [probably the sincere CW] and the one that does not defeat Z 
> [the probable burier candidate].
> 
> Rational voters informed on preferences will pass on W, and elect Z.
> 
> Any Condorcet method that almost always positively punishes the buriers 
> (by electing the "bus"rather than either the buried CW or burier) will 
> look just as strange. In particular it will fail symmetry reversal when 
> restricted to a three member Smith set.  Hence the importance of the 
> sincere runoff for any such method, not just chain climbing.

That's not quite what I'm saying. The candidate who is elected by a 
burial-resistant method may leave something to be desired if burial is 
done anyway - see my other reply. But what seems to be particular to 
chain climbing - at least on agendas that aren't strongly monotone - is 
that it warps the regions where good methods lie and where the bad 
methods lie so thoroughly that nothing makes sense anymore.

It's not just that strategy-resistant methods produce bad outcomes. But 
rather, methods that were originally strategy resistant stop being so; 
but methods that weren't strategy resistant don't gain resistance either.

I ran the chain climbing composition of just about every method 
available to quadelect (some methods excluded due to bugs or 
neutrality/anonymity violations), using a multi-armed bandit search to 
find the best. The best "plain" strategy-resistant method is usually 
something like Smith,IRV or Smith//IRV where strategy succeeds around 
45-50% of the time under impartial culture (97 voters, 5 candidates), 
disregarding Antiplurality whose results seem to just be an artifact of IC.

With the chain climbing composition, the best methods are 
Antiplurality-CC and Coombs-CC, at 65% or so. And I suspect they only 
get that much because of impartial culture's weak spot to antiplurality, 
although I'm surprised that doesn't get warped away too.

Then I did a three candidate search over 15k linear methods and 100k 
nonlinear ones on three candidates, showing that strategy resistance 
still exists, but the methods make no sense.

So perhaps what we can say is, and what I wanted to say was:
	- Chain climbing doesn't work on methods that aren't strongly
		monotone.
	- For such methods, strategy resistance is neither improved
		nor preserved through composition.
	- There do exist non-strongly-monotone methods that when
		composed with CC are strategy-resistant.
	- However, designing them is near-impossible.

Which I guess is what you implied by "all bets are off". I just don't 
think the need for a runoff is related to the bizarre behavior of chain 
climbing in particular.

-km