STAR
Forest - I read fpdk's post as an implicit argument for cardinal voting
(which was why it was relevant to STAR). Each friend states the utility
of each topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is
greatest. I don't think there's a better answer.
CJC
On 18/08/2023 21:50, Forest Simmons wrote:
He posed a pizza choice among friends pronlem.. a problem of consensus
as opposed to "tyranny of the majority" ... how to find the best
consensus decision when a simple majority first place preference would
not be ideal.
On Fri, Aug 18, 2023, 10:41 AM Colin Champion
<colin.champion@routemaster.app
mailto:colin.champion@routemaster.app> wrote:
Forest – I may be being slow, but... what problem are you trying
to solve? The problem which fpdk (quite plausibly, to my mind)
said was optimally solved by cardinal voting? Or the problem which
I claimed was optimally solved by decision theory? Or something to
do with tactical voting?
CJC
On 18/08/2023 18:32, Forest Simmons wrote:
It's been a while since I thought about this but here's something
that somebody with some number crunching resources should
experiment with ... a lottery method that I used to call "the
ultimate lottery" back before Jobst invented MaxParC, which
arguably has at least an equal claim to ultimateness:
Ballots are positive homogeneous functions of the candidate
probability variables. The homogeneity degree doesn't matter as
long as all of the ballots are of the same degree.
The candidate probabilities are chosen to maximize the product of
the ballots.
This candidate probability distribution can be realized as a
spinner. The spinner is spun to determine the winner.
How would this work for our pizza example?
For example, each voter's ballot could be her pizza desirability
[score] expectation as a function of the lottery probabilities.
Then each A faction voter would submit the same ballot ... namely
the function given by the expression
100pA+80pC, while each B faction voter would submit the expression
100pB+80pC.
When these ballots are multiplied together, we get the product
(100pA+80pC)^60×(100pB+80pC)^40.
The p values that maximize this product (subject to the
constraint that they are non-negative and sum to 100 percent) are
pA=pB=0, and pC=100%.
The lottery that maximizes the expectation product is called the
Nash lottery after John Nash who first used this idea for
efficient allocation of limited resources.
Since expectations are linear combinations of the probabilities,
they are homogeneous of degree one ... one person, on vote. Their
product is homogeneous of degree n ... so n people, n votes.
Instead of using voter expectations for their ballots, the voters
could have used other homogeneous expressions ... for example, by
simply replacing each sum of products by a max of the same products.
The product of these modified ballots would be ...
[max(100pA,80pC)]^60
×[max(100pB,80pC)]^40.
Maximization of this product with the same constraints as before,
yields the same consensus distribution ... pC=100%.
This information is new in the sense that it has never been
submitted for official publication ... it's an exclusive bonus of
Rob Lanphier's EM list archive... first posted to this list back
in 2011 after Jobst and I published our 2010 paper on the use of
mixed strategies for achieving consensus.
Anyway, it turns out that using the Max operator in place of the
Sum operator yields a distribution with less entropy whenever the
two distributions are not identical.
Less entropy means less randomness, which means less chance,
which in this context, means more consensus.
In our example, the candidate distribution turned out to be 100
percent candidate C ... zero randomness ... zero entropy ... 100
percent consensus.
Now you can see why I mentioned the need for number crunching
capability ... experimenting with these ballot product
maximizations requires some serious number crunching.
The field is wide open. Is the Ultimate Lottery Method strongly
monotonic? For that matter, how about even the Nash Lottery?
Can MaxParC be formulated in terms of the Ultimate Lottery?
Somebody with some grad students should get them going on this!
fws
On Thu, Aug 17, 2023, 11:10 AM Forest Simmons
<forest.simmons21@gmail.com <mailto:forest.simmons21@gmail.com>>
wrote:
Suppose voter utilities for three kinds of pizza are
60 A[100]>C[80]>>B[0]
40 B[100]>C[80]>>A[0]
Suppose the voters must choose by majority choice between
pizza C and the favorite pizza of a voter to be determined by
randomly drawing a voter name from a hat.
The random drawing method would give voter utility
expectations of
60%100+40%0 for each A groupie, and
40%100+60%0 for each B groupie.
The max utility expectation would be 60.
On the other hand, if voters decide to go with the sure deal
C, the assured utility fo every voter will be 80.
Every rational voter faced with this choice will choose C.
Here we have an ostensibly random method that is sure to
yield a consensus decision when voters vote ratkonally.
More on this topic at
https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971
<https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971>
fws
On Thu, Aug 17, 2023, 1:18 AM Forest Simmons
<forest.simmons21@gmail.com
<mailto:forest.simmons21@gmail.com>> wrote:
The best methods that I know of for the friends context
are minimum entropy lottery methods characterized by max
possible consensus (min entropy) consistent with a
proportional lottery method with higher entropy fallback
to disincentivize gratuitous defection.
Jobst's MaxParC (Max Partial Consensus) is the best example.
Too late to elaborate tonight.
fws
I'll
On Wed, Aug 16, 2023, 10:01 AM <fdpk69p6uq@snkmail.com
<mailto:fdpk69p6uq@snkmail.com>> wrote:
On Mon, Aug 14, 2023 at 12:09 AM C.Benham wrote:
I think this is an interesting point. We can
ask at a philosophical level what makes a good
voting method. Is it just one that ticks the most
boxes, or is it one that most reliably gets the
"best" result?
The one that most reliably gets the best result in
the real world. The difficulty with this approach is
accurately modeling human voting behavior and the
consequent utility experienced from the winner, but
it's still the better answer philosophically.
(Note that VSE predates Jameson Quinn by decades, and
has had several different names:
https://en.wikipedia.org/wiki/Social_utility_efficiency
<https://en.wikipedia.org/wiki/Social_utility_efficiency>)
And that's partly because the premise of
Condorcet is essentially built on a logical
fallacy - basically that if A is preferred to B
on more ballots that vice versa then electing A must
be a better result than electing B.
I'd be interested in reading your explanation of
why you think that is a
"logical fallacy". What about if there are only
two candidates?
Ranked ballots can't capture strength of preference.
It's possible for a majority-preferred candidate to
be very polarizing (loved by 51% and hated by 49%),
while the minority-preferred candidate is
broadly-liked and has a much higher overall
approval/favorability rating. Which candidate is the
rightful winner?
https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html
<https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html>
"Suppose you and a pair of friends are looking to
order a pizza. You, and one friend, really like
mushrooms, and prefer them over all other vegetable
options, but you both also really, /really/ like
pepperoni. Your other friend also really likes
mushrooms, and prefers them over all other options,
but they're also vegetarian. What one topping should
you get?
Clearly the answer is mushrooms, and there is no
group of friends worth calling themselves such who
would conclude otherwise. It's so obvious that it
hardly seems worth calling attention to. So why is
it, that if we put this decision up to a vote, do so
many election methods, which are otherwise seen as
perfectly reasonable methods, fail? Plurality,
top-two runoffs
<http://en.wikipedia.org/wiki/Two-round_system>,
instant runoff voting
<http://en.wikipedia.org/wiki/Instant-runoff_voting>,
all variations of Condorcet's method
<http://en.wikipedia.org/wiki/Condorcet_method>, even
Bucklin voting
<http://en.wikipedia.org/wiki/Bucklin_voting>; all of
them, incorrectly, choose pepperoni."
(And strength of preference is clearly a real thing
in our brains. If you prefer A > B > C, and are
given the choice between Box 1, which contains B, and
Box 2, which has a 50/50 chance of containing A or C,
which do you choose? What if the probability were 1
in a million of Box 2 containing C? By varying the
probability until it's impossible to decide, you can
measure the relative strength of preference for B > C
vs A > C.)
----
Election-Methods mailing list - see
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Hello,
Le vendredi 18 août 2023 à 15:59:02 UTC−5, Colin Champion colin.champion@routemaster.app a écrit :
Forest - I read fpdk's post as an implicit argument for cardinal voting
(which was why it was relevant to STAR). Each friend states the utility of
each topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is greatest.
I don't think there's a better answer.
CJC
On 18/08/2023 21:50, Forest Simmons wrote:
He posed a pizza choice among friends pronlem.. a problem of consensus as
opposed to "tyranny of the majority" ... how to find the best consensus
decision when a simple majority first place preference would not be ideal.
If we're discussing a group of friends as the voters, who possess a higher
goal, not of getting the best outcome for themselves individually, but to
ensure some global happiness with the result, I guess there would be some
room to play around with what that happiness goal is.
What occurs to me in particular is that a group might want to minimize the
number of voters who are "really unhappy" with the result. So you might have
some option which maximizes utility, but the group would opt against that
one if one friend is particularly harmed by it.
To be honest "max utility wins" doesn't strike me as a rule friends would
use, unless the issues voted on are not really that critical to anyone.
Kevin
votingmethods.net
That's pretty good ... not too bad for a deterministic method, if in a good
faith, no pressure context.
Some people would resort to sortition, especially if the decision is one
that arises frequently ... every Friday, say.
Other lottery methods exist that reduce the amount of randomness (the
entropy) but remain proportional (like random ballot).
During a ten year period winding down in 2015, Jobst and I intensively
explored how much you can limit entropy (maximize the degree of consensus)
under the constraint of minimal acceptable fairness ... which for us was
the same kind of long term proportionality that random ballot favorite
would ensure ... so min randomness required for proportional fairness ...
or expressed differently ... max consensus with zero sacrifice of strict
statistically proportional representation.
Our exploration culminated in Jobst's invention of MaxParC , a Maximum
Partial Consensus method based on ballots where the voters specify (for
each alternative) how many other voters would have to be on board before
they would be willing to join with them in support of that alternative.
From that information you can easily figure out which alternative has the
potential for the most partial consensus... and that option will be awarded
a lottery probability in proportion to that max potential support ... etc.
That's the best known solution for maximising consensus without
compromising proportional fairness.
But in any context where MaxParC would give 100 percent consensus, there is
always a simpler lottery method that would ensure the same result among
rational voters who are already aware of the other voters' preferences ...
so they don't need to fill out MaxParC ballots.
If rational, well informed voters already know among themselves that at
least one candidate is a shoo-in for neing a 100 percent implicit approval
winner (or a 100 percent consensus MaxParC winner), then a score ballot is
all we need ...
... among the candidates with 100 percent implicit approval, elect the one
with the greatest score total.
The projected MaxParC winner C should be among those 100 percent Implici
Approval candidates.
However, if (by some rare fluke) neither C nor any other candidate
amassed100 percent implicit approval, then the winner must be chosen by
random ballot favorite, in order to ensure fair proportionality.
Does that seem to make sense?
fws
On Fri, Aug 18, 2023, 1:58 PM Colin Champion colin.champion@routemaster.app
wrote:
Forest - I read fpdk's post as an implicit argument for cardinal voting
(which was why it was relevant to STAR). Each friend states the utility of
each topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is greatest.
I don't think there's a better answer.
CJC
On 18/08/2023 21:50, Forest Simmons wrote:
He posed a pizza choice among friends pronlem.. a problem of consensus as
opposed to "tyranny of the majority" ... how to find the best consensus
decision when a simple majority first place preference would not be ideal.
On Fri, Aug 18, 2023, 10:41 AM Colin Champion <
colin.champion@routemaster.app> wrote:
Forest – I may be being slow, but... what problem are you trying to
solve? The problem which fpdk (quite plausibly, to my mind) said was
optimally solved by cardinal voting? Or the problem which I claimed was
optimally solved by decision theory? Or something to do with tactical
voting?
CJC
On 18/08/2023 18:32, Forest Simmons wrote:
It's been a while since I thought about this but here's something that
somebody with some number crunching resources should experiment with ... a
lottery method that I used to call "the ultimate lottery" back before Jobst
invented MaxParC, which arguably has at least an equal claim to
ultimateness:
Ballots are positive homogeneous functions of the candidate probability
variables. The homogeneity degree doesn't matter as long as all of the
ballots are of the same degree.
The candidate probabilities are chosen to maximize the product of the
ballots.
This candidate probability distribution can be realized as a spinner. The
spinner is spun to determine the winner.
How would this work for our pizza example?
For example, each voter's ballot could be her pizza desirability [score]
expectation as a function of the lottery probabilities.
Then each A faction voter would submit the same ballot ... namely the
function given by the expression
100pA+80pC, while each B faction voter would submit the expression
100pB+80pC.
When these ballots are multiplied together, we get the product
(100pA+80pC)^60×(100pB+80pC)^40.
The p values that maximize this product (subject to the constraint that
they are non-negative and sum to 100 percent) are pA=pB=0, and pC=100%.
The lottery that maximizes the expectation product is called the Nash
lottery after John Nash who first used this idea for efficient allocation
of limited resources.
Since expectations are linear combinations of the probabilities, they are
homogeneous of degree one ... one person, on vote. Their product is
homogeneous of degree n ... so n people, n votes.
Instead of using voter expectations for their ballots, the voters could
have used other homogeneous expressions ... for example, by simply
replacing each sum of products by a max of the same products.
The product of these modified ballots would be ...
[max(100pA,80pC)]^60
×[max(100pB,80pC)]^40.
Maximization of this product with the same constraints as before, yields
the same consensus distribution ... pC=100%.
This information is new in the sense that it has never been submitted for
official publication ... it's an exclusive bonus of Rob Lanphier's EM list
archive... first posted to this list back in 2011 after Jobst and I
published our 2010 paper on the use of mixed strategies for achieving
consensus.
Anyway, it turns out that using the Max operator in place of the Sum
operator yields a distribution with less entropy whenever the two
distributions are not identical.
Less entropy means less randomness, which means less chance, which in
this context, means more consensus.
In our example, the candidate distribution turned out to be 100 percent
candidate C ... zero randomness ... zero entropy ... 100 percent consensus.
Now you can see why I mentioned the need for number crunching capability
... experimenting with these ballot product maximizations requires some
serious number crunching.
The field is wide open. Is the Ultimate Lottery Method strongly
monotonic? For that matter, how about even the Nash Lottery?
Can MaxParC be formulated in terms of the Ultimate Lottery?
Somebody with some grad students should get them going on this!
fws
On Thu, Aug 17, 2023, 11:10 AM Forest Simmons forest.simmons21@gmail.com
wrote:
Suppose voter utilities for three kinds of pizza are
60 A[100]>C[80]>>B[0]
40 B[100]>C[80]>>A[0]
Suppose the voters must choose by majority choice between pizza C and
the favorite pizza of a voter to be determined by randomly drawing a voter
name from a hat.
The random drawing method would give voter utility expectations of
60%100+40%0 for each A groupie, and
40%100+60%0 for each B groupie.
The max utility expectation would be 60.
On the other hand, if voters decide to go with the sure deal C, the
assured utility fo every voter will be 80.
Every rational voter faced with this choice will choose C.
Here we have an ostensibly random method that is sure to yield a
consensus decision when voters vote ratkonally.
More on this topic at
fws
On Thu, Aug 17, 2023, 1:18 AM Forest Simmons forest.simmons21@gmail.com
wrote:
The best methods that I know of for the friends context are minimum
entropy lottery methods characterized by max possible consensus (min
entropy) consistent with a proportional lottery method with higher entropy
fallback to disincentivize gratuitous defection.
Jobst's MaxParC (Max Partial Consensus) is the best example.
Too late to elaborate tonight.
fws
I'll
On Wed, Aug 16, 2023, 10:01 AM fdpk69p6uq@snkmail.com wrote:
On Mon, Aug 14, 2023 at 12:09 AM C.Benham wrote:
I think this is an interesting point. We can ask at a
philosophical level what makes a good voting method. Is it just one that
ticks the most boxes, or is it one that most reliably gets the "best"
result?
The one that most reliably gets the best result in the real world. The
difficulty with this approach is accurately modeling human voting behavior
and the consequent utility experienced from the winner, but it's still the
better answer philosophically.
(Note that VSE predates Jameson Quinn by decades, and has had several
different names:
https://en.wikipedia.org/wiki/Social_utility_efficiency)
And that's partly because the premise of Condorcet is essentially
built on a logical fallacy - basically that if A is preferred to B on more
ballots that vice versa then electing A must
be a better result than electing B.
I'd be interested in reading your explanation of why you think that
is a
"logical fallacy". What about if there are only two candidates?
Ranked ballots can't capture strength of preference. It's possible for
a majority-preferred candidate to be very polarizing (loved by 51% and
hated by 49%), while the minority-preferred candidate is broadly-liked and
has a much higher overall approval/favorability rating. Which candidate is
the rightful winner?
https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html
"Suppose you and a pair of friends are looking to order a pizza. You,
and one friend, really like mushrooms, and prefer them over all other
vegetable options, but you both also really, really like pepperoni.
Your other friend also really likes mushrooms, and prefers them over all
other options, but they're also vegetarian. What one topping should you
get?
Clearly the answer is mushrooms, and there is no group of friends
worth calling themselves such who would conclude otherwise. It's so obvious
that it hardly seems worth calling attention to. So why is it, that if we
put this decision up to a vote, do so many election methods, which are
otherwise seen as perfectly reasonable methods, fail? Plurality, top-two
runoffs http://en.wikipedia.org/wiki/Two-round_system, instant
runoff voting http://en.wikipedia.org/wiki/Instant-runoff_voting,
all variations of Condorcet's method
http://en.wikipedia.org/wiki/Condorcet_method, even Bucklin voting
http://en.wikipedia.org/wiki/Bucklin_voting; all of them,
incorrectly, choose pepperoni."
(And strength of preference is clearly a real thing in our brains. If
you prefer A > B > C, and are given the choice between Box 1, which
contains B, and Box 2, which has a 50/50 chance of containing A or C, which
do you choose? What if the probability were 1 in a million of Box 2
containing C? By varying the probability until it's impossible to decide,
you can measure the relative strength of preference for B > C vs A > C.)
Election-Methods mailing list - see https://electorama.com/em for
list info
Election-Methods mailing list - see https://electorama.com/em for list info
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info
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info
Kevin - I think I agree really. It's a debate about utilitarianism.
There's a shallow objection to adding utilities, which is that they may
not be suitably scaled, and a more serious one that considerations of
justice etc may enter into people's moral judgements, and that these
don't lend themselves to a utilitarian calculus. Different objective
functions will have different optima. But it's one thing to cricitise
fpdk's model, and another to criticise the conclusion drawn from it.
And this is why I'm still puzzled by what Forest is saying. If
fpdk's model is what I assume it to be, I don't see how there's room to
improve on cardinal voting, or how cardinal voting can be only 'not too
bad'. Perhaps Forest has another objective function in mind, based on
consensus rather than utilities, but I'm not sure what it is or whether
it's really an improvement.
As I recall, Amartya Sen goes to town in distancing himself from
utilitarianism. In an extreme case (which I'm not sure Sen would
reject), collective utility might be an increasing function of the
utility of the worst-off member of society, and a decreasing function of
everyone else's. Obviously no monotonic transformation of individual
utilities can reconcile this with utilitarianism.
CJC
On 18/08/2023 23:23, Kevin Venzke wrote:
Hello,
Le vendredi 18 août 2023 à 15:59:02 UTC−5, Colin Champion colin.champion@routemaster.app a écrit :
Forest - I read fpdk's post as an implicit argument for cardinal voting
(which was why it was relevant to STAR). Each friend states the utility of
each topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is greatest.
I don't think there's a better answer.
CJC
On 18/08/2023 21:50, Forest Simmons wrote:
He posed a pizza choice among friends pronlem.. a problem of consensus as
opposed to "tyranny of the majority" ... how to find the best consensus
decision when a simple majority first place preference would not be ideal.
If we're discussing a group of friends as the voters, who possess a higher
goal, not of getting the best outcome for themselves individually, but to
ensure some global happiness with the result, I guess there would be some
room to play around with what that happiness goal is.
What occurs to me in particular is that a group might want to minimize the
number of voters who are "really unhappy" with the result. So you might have
some option which maximizes utility, but the group would opt against that
one if one friend is particularly harmed by it.
To be honest "max utility wins" doesn't strike me as a rule friends would
use, unless the issues voted on are not really that critical to anyone.
Kevin
votingmethods.net
Election-Methods mailing list - see https://electorama.com/em for list info
On 8/19/23 11:11, Colin Champion wrote:
Kevin - I think I agree really. It's a debate about utilitarianism.
There's a shallow objection to adding utilities, which is that they may
not be suitably scaled, and a more serious one that considerations of
justice etc may enter into people's moral judgements, and that these
don't lend themselves to a utilitarian calculus. Different objective
functions will have different optima. But it's one thing to cricitise
fpdk's model, and another to criticise the conclusion drawn from it.
And this is why I'm still puzzled by what Forest is saying. If
fpdk's model is what I assume it to be, I don't see how there's room to
improve on cardinal voting, or how cardinal voting can be only 'not too
bad'. Perhaps Forest has another objective function in mind, based on
consensus rather than utilities, but I'm not sure what it is or whether
it's really an improvement.
Forest's model is a sort of generalization of the friendly pizza setting
to where people are strategic. That is, he considers a situation where
the honest utilities are
60: A (100) > B (80) > C (0)
40: C (100) > B (80) > A (0)
and then he observes that with any sort of cardinal method passing
InfMC, the majority can force their outcome, so a large electorate might
well do:
60: A (100) > B (0) > C (0)
and then the 40% minority has no way to respond that doesn't end up with
their hated outcome.
So the whole motivation for the setting is, given that the honest
utilities are as above, how do we incentivize strategic voters to
express that information? And ideally without leaving too much to
chance, or having awful expected utility.
In a pizza setting, presumably the friends know each other well enough
and don't want to ruin their friendship over pizza, so the best outcome
will happen anyway.
As I recall, Amartya Sen goes to town in distancing himself from
utilitarianism. In an extreme case (which I'm not sure Sen would
reject), collective utility might be an increasing function of the
utility of the worst-off member of society, and a decreasing function of
everyone else's. Obviously no monotonic transformation of individual
utilities can reconcile this with utilitarianism.
I know that he objected to Homo Economicus (there's that post office
quote from Rational Fools), but I didn't know he distanced himself from
utilitarianism in general. But then there's much of his that I haven't read.
-km
Kristofer - that makes perfect sense. So Forest's method is a proposed
solution to the problem in which the objective function is as assumed by
fpdk, but voters are assumed to act deviously.
For all I know it may be a very good solution. At any rate,
strategic voting is so hard to analyse that I wouldn't dream of saying
that I knew how to characterise the true optimum, even at the vaguest
conceptual level.
CJC
On 19/08/2023 12:07, Kristofer Munsterhjelm wrote:
On 8/19/23 11:11, Colin Champion wrote:
Kevin - I think I agree really. It's a debate about utilitarianism.
There's a shallow objection to adding utilities, which is that they
may not be suitably scaled, and a more serious one that
considerations of justice etc may enter into people's moral
judgements, and that these don't lend themselves to a utilitarian
calculus. Different objective functions will have different optima.
But it's one thing to cricitise fpdk's model, and another to
criticise the conclusion drawn from it.
And this is why I'm still puzzled by what Forest is saying. If
fpdk's model is what I assume it to be, I don't see how there's room
to improve on cardinal voting, or how cardinal voting can be only
'not too bad'. Perhaps Forest has another objective function in mind,
based on consensus rather than utilities, but I'm not sure what it is
or whether it's really an improvement.
Forest's model is a sort of generalization of the friendly pizza
setting to where people are strategic. That is, he considers a
situation where the honest utilities are
60: A (100) > B (80) > C (0)
40: C (100) > B (80) > A (0)
and then he observes that with any sort of cardinal method passing
InfMC, the majority can force their outcome, so a large electorate
might well do:
60: A (100) > B (0) > C (0)
and then the 40% minority has no way to respond that doesn't end up
with their hated outcome.
So the whole motivation for the setting is, given that the honest
utilities are as above, how do we incentivize strategic voters to
express that information? And ideally without leaving too much to
chance, or having awful expected utility.
In a pizza setting, presumably the friends know each other well enough
and don't want to ruin their friendship over pizza, so the best
outcome will happen anyway.
As I recall, Amartya Sen goes to town in distancing himself from
utilitarianism. In an extreme case (which I'm not sure Sen would
reject), collective utility might be an increasing function of the
utility of the worst-off member of society, and a decreasing function
of everyone else's. Obviously no monotonic transformation of
individual utilities can reconcile this with utilitarianism.
I know that he objected to Homo Economicus (there's that post office
quote from Rational Fools), but I didn't know he distanced himself
from utilitarianism in general. But then there's much of his that I
haven't read.
-km