[EM] Being serious about the limits to utilitarian methods
Forest Simmons
forest.simmons21 at gmail.com
Fri Sep 17 18:27:11 PDT 2021
Thanks for your very cogent thoughts on how to make sense of cardinal
ratings in general and utilities in particular.
Your four tuple idea is clever, but as you say, might conceiveably lead to
a cycle.
But suppose that your range ballot B has full authority to decide between
two lotteries L1 and L2 about which you have zero prior information by
picking the one with the larger dot product ... B dot L1, or B dot L2.
[These are the expected range values for B given the respective lotteries.]
It might take some serious soul searching to arrive at a vector B, but a
"rational" player should be able to optimize B for a given ballot
resolution.
It may be impossible for these optimal range values to accurately reflect
personal "utilities" because, as has often been noted in the literature,
vectors with components limited to standard real numbers cannot adequately
cover the life and death possibilities in relation to other more mundane
possibilities. In that case it can be useful to represent utilities as
polynomials in epsilon with real coefficients ... or equivalently as
sequences of real numbers ordered lexicographically.
In any case, their use should be limited to methods that do not depend on a
particular normalization of the range vector B; any affine transformation
of B should serve equally well. This convention gets rid of the
exaggeration of joy or pain problem ... there can be no intrapersonal
comparison of utilities, nor addition or subtraction of utilities between
parties. Standard use of cardinal ratings fails this requirement miserably!
But lottery methods shine because they are based on pairwise preferences
between lotteries, which include "sure lotteries", i.e. zero entropy
lotteries that give 100 percent probability to one candidate.
In general, what Jobst and I call consensus methods have solutions that are
(at least weakly) preferred unanimously over each vying alternative
lottery, among which is usually the fallback or benchmark lottery (random
favorite).
For example, the zero entropy lottery 100%C is unanimously preferred over
the random favorite x*A + y*B when the faction sizes x and y are nearly
equal in the scenario
x: A>C>>B
y: B>C>>A
El vie., 17 de sep. de 2021 4:21 p. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:
> One problem of Range and the other utilitarian methods (/perhaps/ MJ
> notwithstanding?) is that they ask too much information. For utility
> scales, we can use lotteries to determine linear relations between
> combinations of events (under an assumption of risk neutrality). But we
> can't -- not that I know of, at least -- set a unit of utility, because
> it's impossible to say if e.g. my perception of pain is the same
> strength as yours (for a hedonist type of utilitarianism). It might be
> possible to set a zero point, e.g. where everything below zero are
> events you'd prefer not experiencing (i.e. prefer nothingness to this),
> and everything above zero are events you'd prefer experiencing; but even
> that is insufficient to construct a calibrated scale.
>
> So what does that give us? Systematic use of lotteries would give us an
> arbitrary linear scaling of the voter's utilities. However, Range (and
> similar), inasfar as they ask for a particular type of information (and
> not just "whatever you feel like"), asks for actual utility levels. Even
> honest voters may feel like normalizing is the best option, which then
> leads to de facto IIA failure (even though the method de jure passes IIA).
>
> (Another way to look at this is that the extra information that Range
> asks for is ambiguous, and that when every voter fills in this extra
> information in a way that doesn't depend on the candidates, then Range
> passes IIA. But since it's ambiguous, it's tempting to tweak the
> parameter so as to maximize the impact of one's vote, engaging in
> "manual DSV", which makes the combined method fail IIA.)
>
> Now suppose that we're frank about the limits to utilitarian methods and
> ask the voters only what they can provide. What would that be? I think
> it would be lottery data, four-tuples like this:
>
> (X, Y, a, Z): I am indifferent to:
> - even odds between X and Y winning, and
> - Z winning with probability a, some distinguished candidate W
> winning
> with probability 1-a.
>
> (W can be chosen arbitrarily as long as he's fixed throughout the
> election. W could even be a meta candidate like NOTA/run the election
> again with other candidates.)
>
> Okay, so that's the easy part. The hard part: what kind of voting method
> could we build from this? That's where I'm more stuck. Some thoughts.
>
> There seem to be analogs of the Condorcet paradox both on an individual
> and societal level. Since the tuples are inherently relative (to X, Y,
> and Z), it should be possible to set up a cycle by misreporting
> utilitarian preferences. If every voter has a well-defined internal
> utility scale, such individual cycles must be artifacts. But like
> Condorcet, society-scale cycles may not be.
>
> It's easy to naively reconstruct unbounded Range-style ballots as long
> as the tuples are transitive (no electoral tricksters of the form
> mentioned above). But there's a related honesty problem: suppose that
> we're using hedonist utilitarianism and candidate X is winning, but
> voter A says: "it will be torture to me if X wins", i.e. an extreme
> amount of disutility if X wins. Then utilitarianism would suggest that X
> shouldn't win. So taking utilitarianism seriously means that such a
> voter has an extreme impact on the result.
>
> But it seems reasonable to cap the amount of disutility that can be
> reported, like Range implicitly does. For strategic voters, it's obvious
> that otherwise, there would just be a race to tack zeroes onto the
> disutility of getting X elected, and whoever names the biggest
> (negative) number wins. Even with honest voters, though, one might
> object if there exists a single person who would be *so* happy if a
> dictator won that nobody else's opinions matter.
>
> So suppose then there's some implicit range to disutility (everything
> above this gets clamped). Then these tuples can be used to reconstruct
> Range ballots, one for each voter. But I would think we could do better.
> I just don't know how yet.
>
> Maybe the best place to start is to see what happens if there's
> transitivity (no "Condorcet" cycles). First, there's a type of unanimity
> set: if there exists a group of candidates so that everybody prefers
> lotteries between candidates in that group to lotteries between
> candidates outside of the group, it should be pretty clear that the
> winner should come from the first group. Similarly, if there exists a
> candidate who everybody prefers 100% probability of winning to any
> lottery, then that (unanimous candidate) should win.
>
> But if the method is utilitarian, we can't easily import Smith sets or
> Condorcet winners, no? Those rely on majority rule. On the other hand,
> there may be a "majority utility" analog. I would have to think more
> about it to get anywhere.
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