[EM] WDSC (and the rest)

[Martin Harper]

The problem I have about Mike's criteria (and some others) is that they 
talk about reversal of preferences, and failure to vote preferences, and 
recently I've begun to doubt the usefulness of these ideas in general.

Specifically, if I define IRV(for eg) as "place marks on this piece of 
paper", then I can claim that the criteria doesn't apply to me, so 
nah-nah. "I'm not asking for the expression of preferences" I say, "You 
can read preferences into the ballots if you like, but that's just your 
*interpretation".

I think this is a reasonable complaint. Although it's clear in general 
how to extract pairwise preferences from the marks on the paper, it's 
not always the case. There are also problems with methods like Approval 
which don't allow the expression of all preferences - eg, Approval fails 
SDSC almost by default - before the votes are even counted, Approval 
fails to pass - that feels unfair, even if (as it turns out) it isn't.

One workaround for this is to only define criteria on ranked ballots, 
and specifically define what a "sincere" vote is on such a ballot, along 
with what a "preference" is. That's a bit of a kludge, and it doesn't 
help us compare, eg, Approval or CR or methods which take Dyadic votes. 
Here's a possible fix...

Define: Sometimes-good vote.
A "sometimes-good" Vote is a vote which, in at least one set of 
circumstances, will improve the expected value of the election to the 
voter, over the expected value if he/she had not votes.

Define: Sometimes-bad vote, always-good, always-bad, never-good, 
never-bad :- in the obvious way.

Define: safe/risky/dumb vote. A safe vote is a sometimes-good and 
never-bad vote. A risky vote is a sometimes-good, sometimes-bad vote. A 
dum vote is a never-good vote.

Define: Naive vote. The naive vote for a method is defined by the 
method. It should be similar to the 0-info strategy, but (if needed) 
simplified. For example, a Naive Condorcet vote would be a full ranking, 
a Naive Approval vote would be approving those of above mean utility, 
and a Naive Plurality vote would be approving your favourite candidate.

: If a method claims to have no naive vote, then laugh at it for being 
impossible to vote without a degree in higher Maths. I think this is a 
reasonable demand.

Define: Sincere Smith Set - as per usual.

--
Now, the criteria...

Martin's Monotonicity Criterion (example):
A Naive vote should always be a Safe vote.

: Other criteria can be similarly worded, such as Condorcet, Majority, 
ec. On the other hand, I can't see a way to reword Favorite Betrayal - 
it explicitly refers to voting one candidate above another, and I can't 
see how to get round that.

--
Martin's Generalised Strategy-Free Criterion:
If, for some X and Y, a majority prefer X to Y, then Y should not win, 
provided the following hold:
1) That majority votes a Naive vote.
2) Everyone votes a Safe vote.
3) X is in the Sincere Smith Set.
Further, a method fails GSFC if there are no safe votes for some set of 
utilities. {otherwise it would pass by default}

: Note that this criterion doesn't help us at all if the minority do 
risky votes. I'd like to quote from Mike's text here: "In theory, that 
minority could cause the other candidate to win by falsifying their 
preferences, but that would be a very risky offensive strategy that is 
more likely to backfire than to succeed". It's the riskiness of 
order-reversal that makes it something we don't expect to see too much 
of, I'd argue - not the reversal of preferences per se.

--
Martin's Strong Defensive Strategy Criterion:
If a majority prefer some candidate (not necessarily the same candidate) 
to Y, then Y cannot win, provided that that majority votes a Naive vote.

: Approval still fails SDSC, but now it fails it for a *reason*! 
Previously it failed it by default - because it is impossible to avoid 
falsely voting two alternatives equal in general in Approval. Now it 
fails it because the Naive voting method won't always keep out Y.

--
Martin's Weak Defensive Strategy Criterion:
If a majority prefer some candidate (not neccessarily the same 
candidate) to Y, then there should exist a Safe means for the majority 
to vote, which will ensure that Y cannot win.

: In other words, if I want to keep someone I hate out of office, I 
don't need to take any risks. Again, I'd argue that whereas all methods 
have *some* defensive strategy, the key to good methods is that their 
defensive strategy is safe. The key to even better methods is that heir 
defensive strategy is naive.
--

IRV has no safe votes, which is where its problems come. Plurality has 
only one safe vote: a vote for your favourite, which is also its only 
naive vote. Condorcet has safe votes where you don't reverse any 
preference, which is also the Case for Approval, and indeed for CR and 
Dyadic Approval and Approval Completed Condorcet.

The Approval Completed Condorcet Naive vote would presumably be to rank 
all candidates with utilities above some number, but I'm not sure what 
that number should be. Since the method mixes Approval and Condorcet, 
I'd expect it would be between 0.5 and 0 (in 0-info), but where exactly 
the line goes is something of a mystery...