[EM] Strategic values, not weights

[Richard Moore]

After mentioning strategic values last night, I realized there was more
that needed to be said on that topic.

Strategic values, like utilities, reflect individual voter needs. The
strategic value is a function of a voter's utilities and the effect
that voter's choices will have on the probability of each possible
outcome. In general, for a given method, the strategic value of a
particular ballot is the expected increase in the utility of the outcome
for the voter who casts that ballot under that method, given whatever
strategic information the voter has about the other ballots cast
(typically, statistical information that can be translated into
probabilities). The voter should try to cast his ballot in the way that
has the maximum strategic value for him, and definitely should avoid
casting a ballot that has negative strategic value for him. Not voting
has a strategic value of zero, always.

The strategic value of a vote "for" a candidate in approval, plurality,
and cumulative voting is the expected value of the increase in utility
of the outcome (to that voter) as a result of his vote for that candidate.
The strategic value of a vote "against" a candidate in those methods is
the negative of that expected increase. The strategic value of a voter's
entire ballot in those methods would then be the sum of the strategic
values of all the individual "for" and "against" votes that that ballot
represents.

Richard Moore wrote:
 > Another way to look at Mike's point is that an approval ballot presents
 > exactly one vote in support of each candidate that the voter approves,
 > and one vote against each candidate that the voter disapproves. A vote
 > is a vote, whether it's for or against a candidate. A "for all 
candidates"
 > ballot or an "against all candidates" ballot may have zero effect, but
 > since the message carried by such a ballot is that the voter has no
 > preferences, then why shouldn't the effective weight of that ballot be
 > zero?

The strategic value of an optimum approval ballot is the sum of the
absolute values of the candidate strategic values. If those values are
represented as A, B, C, ..., Z, then

maximum strategic value = |A| + |B| + |C| + ... + |Z|

IRV folks like to claim approval is stacked against those who approve only
one candidate, but those voters would optimally do so only if only one
candidate had a positive strategic value. In a similar sense, it is stacked
against those who have weak central strategic choices (e.g., if your
strategic values are 10, 1, -1, and -10, your optimum ballot has a strategic
value of 22, whereas if your strategic values are 10, 9, -9, and -10, your
optimum ballot has a strategic value of 38. This isn't some big flaw in
approval voting, it's just that some voters have more choices with large
strategic significance. The fact that the first voter has nearly twice the
ballot value as the second reflects a difference in the distributions of
their strategic values.

 > Contrast lone-mark plurality: If a voter wants to vote against candidate
 > Z, and finds all other candidates reasonably acceptable, he or she must
 > split his "against" vote among N-1 candidates. The lone-mark constraint
 > makes this method asymmetrical. The result is less equitable; in terms
 > of ballot "weights", plurality favors a voter who finds only one 
candidate
 > acceptable over a voter who likes more than one candidate or who strongly
 > dislikes one or two candidates in particular, because the latter is 
forced
 > to vote against some candidates that he or she supports.

The strategic value of an optimum lone-mark ballot, if A is the candidate
receiving the "for" vote on that ballot, is

maximum strategic value = A - B - C - ... - Z

Since A + B + C + ...+Z = 0 (equivalent to a null approval ballot), we can
deduce that

maximum strategic value = 2*A

The best that can be said about plurality in this light is that the optimum
ballot never has a negative strategic value (since A cannot be negative).

So if my strategic values are 10, -2, -2, -2, -2, -2, and your strategic
values are 2, 2, 2, 2, 2, -10, then my optimal ballot has a strategic
value of 20 and yours has a strategic value of 4. This is a clear example
of the asymmetry I referred to, and illustrates the inequity of lone-mark
plurality: Even though we both have the same magnitudes of strategic value,
but with opposite signs, my ballot is worth five of yours! Your ballot's
value was reduced by your need to vote against four of your good
candidates as well as your one bad candidate; mine was strengthened by
the fact that the number of my "against" votes happens to match the number
of bad (for me) candidates one-to-one.

 > A similar thing happens in IRV: If I find two candidates acceptable, I
 > have to rank one of them second, and if I choose the wrong one to rank
 > second, I could be contributing to the early elimination of what might
 > otherwise be the winning candidate, so I am voting against that candidate
 > (quite possibly without sufficiently helping the candidate I rank first).

I don't know how to calculate strategic values for IRV in general. But
I can construct a hypothetical case with known probabilities and calculate
the strategic values of all possible ballots for that case, for a given
set of voter utilities.

Suppose a voter has preferences A>B>C; more specifically, his
utilities are A = 5, B = 3, C = 0.

Suppose he knows that the other ballots will fall into one of the
following two scenarios, each with probability 0.5 (note that
a random tie-breaker with equal outcome probabilities is assumed):

Scenario #1
===========
7 ACB
4 BAC
2 BCA
8 CAB

A wins
A still wins if the voter ranks A first
A or C wins if he ranks B first (p(A) = 0.5, p(C) = 0.5, since
he'll vote BAC not BCA)
A or C wins if he ranks C first (p(A) = 0.5, p(C) = 0.5)

Scenario #2
===========
7 BCA
4 ABC
2 ACB
8 CBA

B wins
B still wins if the voter ranks B first
B or C wins if he ranks A first (p(B) = 0.5, p(C) = 0.5, since
he'll vote ABC not ACB)
B or C wins if he ranks C first (p(A) = 0.5, p(C) = 0.5)

The possible outcome changes are as follows:

1ST CHOICE   SCENARIO 1   SCENARIO 2   EXPECTED UTILITY CHANGE
==============================================================
no vote      A            B            0
A            A            B or C       0.25*(-3) = -0.75
B            A or C       B            0.25*(-5) = -1.25
C            A or C       B or C       0.25*(-5) + 0.25*(-3) = -2

This is not just a case where a voter can do better by staying
home than by voting sincerely. Here's a situation where every
possible ballot (sincere or not) has a negative strategic value.
This voter's best choice, bar none, is to stay home.

I haven't found any other method that allows the possibility of
a situation where all possible ballots have negative value.
The mere possibility of this sort of disenfranchisement, however
arbitrary and remote, might lead to IRV being found unlawful by
the courts (IANAL so I'm not sure what laws or constitutional
provisions might apply).

  -- Richard

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