[EM] Being serious about the limits to utilitarian methods

Kristofer Munsterhjelm km_elmet at t-online.de
Fri Sep 17 16:21:41 PDT 2021

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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