[EM] [CES #4445] Re: Looking at Condorcet

[Kristofer Munsterhjelm]

On 02/04/2012 06:14 PM, robert bristow-johnson wrote:
> On 2/4/12 4:12 AM, Kristofer Munsterhjelm wrote:
>> On 02/04/2012 06:47 AM, robert bristow-johnson wrote:
>>
>>> On 2/3/12 11:06 PM, Jameson Quinn wrote:
>>>
>>>> No, he's saying that when the CW and the true, honest utility winner
>>>> differ, the latter is better. I agree, but it's not an argument worth
>>>> making, because most people who don't already agree will think it's a
>>>> stupid one.
>>>
>>> as do i. it's like saying that the Pope ain't sufficiently Catholic or
>>> something like that. or that someone is better at being Woody Allen than
>>> Woody Allen.
>>>
>>> but for the moment, would you (Jameson, Clay, whoever) tell me, in as
>>> clear (without unnecessary nor undefined jargon) and technical language
>>> as possible, what/who the "true, honest utility winner" is? how is this
>>> candidate defined, in terms the preference of the voters?
>>
>> Utilitarianism is a form of ethics that proposes that the actions to
>> be taken are the ones that produces the greatest good for the greatest
>> number.
>
> thank you. i *did* know what Utilitarianism is and suspected that the
> term "utility" referred to that. and i understand the different norms
> for combining the individual utility measures to get an aggregate
> measure of utility to the group. the "taxicab norm" and the minmax (more
> like the maxmin) norm was brought up. no one seemed to mention the
> Euclidian norm.
>
> i would say that the most fair combination is the mean magnitude
> (taxicab) because it weights every voter's franchise equally. but what
> is left unanswered is how the measure of utility for each voter is
> defined. we can say that, for each voter that voted for the eventual
> winner as their 1st choice (or most highly scored), their measure of
> utility is "1". but what measure of utility do you assign to voters that
> did not get their 1st choice?

Within a utilitarian system, each voter's utility for getting a certain 
choice is given. If you say there's a unit "utils", then a voter may get 
0.8 utils if his second choice wins - or it might be 0.2. Only the voter 
(or an omniscient entity, as in Bayesian regret calculations) know how 
many utils voter x gets if choice y wins.

> that is not well defined. given Abd's example:
>
>> 2: Pepperoni (0.61), Cheese (0.5), Mushroom (0.4)
>> 1: Cheese (0.8), Mushroom (0.7), Pepperoni (0)
>
> who says that for that 1 voter that the utility of Cheese is 0.8?

The voter does. In this thought experiment, one simply assumes the 
1-voter's utility of Cheese is 0.8 so as to show the point. The point is 
that there may be situations where utilitarian optimization and majority 
rule differs.

> how is that function defined in the "proof" that Clay repeatedly refers to
> where "it's a mathematically proven fact that Score does a better job
> picking the Condorcet winner than does Condorcet"?

It isn't. Warren's argument that "Range is more Condorcet than 
Condorcet" is quite different. I assume Clay is using that argument when 
he claims "Range does a better job", but I don't know as I haven't seen 
Clay's arguments.

The first part of the argument is that, in the real world, people will 
have access to polling data. Then, Warren argues, this means they will 
act strategically. Say that X is the honest Condorcet winner (and polls 
show that he is). Then, according strategy, "anybody-but-X" people would 
rank X last. Depending on the method and X's margin of victory, that 
could make some other candidate win, which the anybody-but-X voters prefers.

Thus, after strategy, some Y that isn't X will be elected, and the 
Condorcet method (given the assumptions of voter behavior, polling, and 
strategy) failed to elect the sincere Condorcet winner.

The second part of the argument then involves showing that if everybody 
votes strategically in Range, reducing Range to approval, the honest 
Condorcet winner wins more often than when everybody votes strategically 
in Condorcet (burying the honest CW). Warren proves that, given certain 
assumptions of Approval strategy, at http://rangevoting.org/AppCW.html .


If you want to specify the proof further, you could say it really says 
this: "if more than a certain fraction strategize in both the Condorcet 
method and Range, if Condorcet strategy involves burying the sincere CW 
if you prefer someone else, and Approval strategists know who the 
top-two are, then Range elects the sincere CW more often than does the 
Condorcet method".

What if fewer people than that fraction strategize? Then Range will 
elect the CW less often than the Condorcet method, but then Warren 
refers to utilitarian reasoning and says: if people are mostly honest, 
then the normalized Range values are based on their true utilities, and 
thus the Range winner will have greater utility than the Condorcet 
winner, since the Condorcet method knows nothing about utility.

Or, the fork of the argument, simplified: If people strategize enough, 
the sincere CW wins more often in Range. If they don't, Range's winner 
is better anyway.

> it's such a subjective thing and it can be defined in so many ways
> that i am dubious of any tight mathematical "proof" that is based on
> that. it's not subject defining the boundaries. if you get exactly
> what  you want, the utility metric is 1. if you get *nothing* of what you
> want, the utility is 0 (i.e. that pizza voter on the bottom may be a
> vegetarian and would not be eating pizza at all, if they got Pepperoni).
> there's a whole range of quantity that goes in between that is not
> objectively defined.

Utilities, as such, aren't normalized (though Range votes probably would 
be). In the pizza example above, 1 is set to some arbitrary "really 
good" level, and commensurability means that 0.8 for one voter is as 
good to him as 0.8 is to the other voter.

As another example of in-between values, consider a pizza voter who 
doesn't like pineapple. If the consensus is a Hawaiian pizza, he can 
pick the pineapple off his slices, so it's not an "I won't eat this", 
but he'd prefer ordinary ham pizza to the Hawaiian. Perhaps, though, he 
prefers pepperoni to ham. Thus, he might have a ratings order (according 
to some common standard x):

Pepperoni: 0.8 x
Ham: 0.7 x
Hawaiian: 0.4 x
Anchovy: 0 (won't eat).

If he were to submit a honest Range or MJ ballot for this, then if 
everybody doing the voting knew what the standard x referred to, he 
would vote as above. If not, he'd probably normalize between worst and 
best and give a ballot of:

Pepperoni: 1
Ham: 0.9
Hawaiian: 0.5
Anchovy: 0.

> so, i have a few questions for everyone here:
>
> 1. do we all agree that every voter's franchise is precisely equal?

Yes.

> 2. if each voter's franchise is equal, should we expect any voter
> that has an opinion regarding the candidates/choices to
> voluntarily dilute the weight or effectiveness of their vote,
> even if their preference is weak?

In a pizza scenario, voters might voluntarily dilute their weight to be 
nice to the others. In a hotly contested governmental election, not as 
likely.

> 3. so, based on the answers to 1 and 2, if there is an election or
> choice between only two alternatives (yes/no) or two candidates,
> that this election be decided any differently than, as we
> were told in elementary school, the "simple majority" with
> "one person, one vote"?

Those using utilitarian reasoning would say that if the prerequisites 
hold (people know their utilities and they're commensurable), then a 
utility-optimizing outcome is better than a majoritarian one. They might 
then further reason that if you have something like Range and it's 
contested enough that you min/max your votes, no harm done (since it'll 
pass Majority), but if it's not, then the outcome can only improve.

> if the answer to 3 is "no", on what basis would you assign non-equal
> weighting to each vote? or if "simple majority" is not the criteria for
> the collective decision, what is the alternative? award office to the
> candidate with the minority vote?

I think Warren uses a "tyranny of the majority" example in this case. 
Consider a referendum to confiscate all property of a certain minority 
to redistribute to the majority (something like the Zimbabwe land 
reform). Further, assume honest voting and the status quo is set to 
zero. You'd get something like:

Majority: confiscate (slight improvement due to the share of the 
spoils), don't (zero).
Minority: don't confiscate (zero), do confiscate (very large negative 
value).

In a majority vote, the majority wins. In Range, the minority keeps its 
property.

-

One could of course argue that in cases like tyranny-of-the-majority, 
the majority would just vote strategically to override the minority. Or 
one could argue that you can't really get commensurable utility values 
in the first place.

The former argument leads to considering the majority criterion a 
strategic one. It frees the voters from having to employ min/max 
strategy themselves, as they can honestly get what they would have to 
use strategy to otherwise get.

The latter argument was used by Arrow (of the impossibility criterion). 
He said, paraphrased, that rated methods were of little interest because 
one couldn't meaningfully compare one person's reported utilities with 
those of another. The Wikipedia article on Arrow's impossibility theorem 
uses the example of trying to get the combined ranking of performance 
within decathlon events -- how many points in a 1500 m race should 600 
points in the discus event count as equivalent to? There's no obvious 
answer.

Note that if you set the weights of the decathlon events so that, say, 
the combined output order differs by as much as possible from the output 
of any of the individual orders - i.e. that no event should dominate - 
then you get IIA violation right away, because that depends on what 
individual events you include. So trying to derive a common standard 
from relatives can reintroduce IIA in a not altogether obvious manner.