One of the anonymati, in the course of a lengthy comment on
yesterday’s post, made this request: “I
would love to see you take a step back from the mathematical details and give a
more general discussion of what your mathematical exposition means to those of
us who don't really want to get down into the technical details.” I am more than happy to do so.
It is a long story, and it begins with Jeremy Bentham’s
statement of Utilitarianism in his Introduction
to the Principles of Morals and Legislation, published in 1789. In Utilitarianism,
John Stuart Mill cites Bentham’s dictum as “everybody to count for one, nobody
for more than one.” Bentham himself does
not write these words in the Principles,
but since he was little John Stuart’s godfather, and inasmuch as John’s father,
James, raised Mill virtually from birth to be a crusader for Utilitarianism, I
think we can take Mill’s account as accurate.
Two centuries later, when Utilitarianism is as familiar and
unthreatening as bagels and lox, it is difficult to remember just how
revolutionary and dangerous Bentham’s dictum was and was perceived to be. The problem was that there were so many
peasants and workers and so few lords and ladies. No matter how you looked at it, the pains of
the masses were going to outweigh the pleasures of the classes, and that implied
frightening economic policies that were clearly not on, as the British would
say.
Oh, they tried. It was
widely believed in the drawing rooms and men’s clubs of London that the toffs
were so much more sensitive than their servants that a place by the fire and a
mess of gruel, which was more than enough to make a scullery maid happy, would
cause excruciating anguish to her mistress [hence Hans Christian Andersen’s
princess and the pea.] Mill himself had
a go at this defense [slipping a shiv between the ribs of his godfather all the
while,] distinguishing between higher and lower pleasures, and delivering the
immortal line “it is better to be a human being dissatisfied than a pig
satisfied; better to be Socrates dissatisfied than a fool satisfied.”
The problem was that there were so bloody many pigs [which
is to say, the unwashed] and fools. No
matter how you twisted and turned, the Greatest Happiness Principle was going
to require some serious redistribution of wealth.
What to do? As is so
often the case in such crises, Philosophers stepped up to protect their
employers. They discovered the Problem
of Other Minds. Each of us knows the
magnitude of his or her pleasures and pains by an act of introspection, and
each of us has access to the results of such introspection for him or herself
alone. If I cannot know with certainty the
contents of any mind save my own, then I cannot give to the pleasures and pains
of others a weight commensurate with that of my own pleasures and pains, so it
is simply impossible for me to let everyone count for one as I go about
legislating in Parliament.
Buoyed by the assurances of their philosopher brethren, who surely
could not be accused of any covert ideological parti pris, the economists relaxed and devoted themselves to the
arcana of unanimity quasi-orderings, Pareto optima, and indifference
curves. To the rebels who complained
that the masses were miserable and the masters gleeful, serious students of
capitalism could reply, “That is not a matter of science, and Economics is,
after all, a Science. It even has Nobel
Prizes.”
This defense was given canonical imprimatur in 1932 by Lionel Lord Robbins, in his classic work An Essay on the Nature and Significance of
Economic Science: “There is no means of testing the magnitude
of A’s satisfaction as compared with B’s” he wrote, italicizing for
emphasis.
And that was that.
Enter John von Neumann.
There was, von Neumann demonstrated, one tiny but theoretically
interesting exception to Robbins’ dictate. In the very special case of two individuals
with strictly opposed preferences for probability combinations of prizes,
preferences obeying a set of extremely powerful axioms, it could be
demonstrated that one could carry out an arithmetic sum of A’s satisfaction with
B’s. Indeed, with suitable permissible
transformations, it could be shown that their satisfactions from the prizes
associated with the outcome of a game added up to zero, despite the general impossibility,
pace Robbins, of testing the magnitude of A’s satisfaction as compared with
B’s.
That is the general significance of the mathematical details
to which I devoted yesterday’s post.
11 comments:
Thanks for the background.
I was with you up until "Enter John von Neumann".
Excuse my ignorance but a "science" by my understanding is only possible where the object of that knowledge is in fact observable and measurable and reproducible and formalizable in a theory. While I trust von Neumann's logic and math, I'm left at the starting gate wondering whether the labour of formalizing probabilities and preferences and satisfactions is a proper ground for science (not to speak of morality based on utilitarianism).
I'm afraid that the math leaves me cold until I can be convinced that the subject matter of this inquiry is in fact proper and reasonable.
It seems to me that utilitarianism is a 19th century idealization of "measuring people". Back then physicists were convinced that all of science was done except obtaining a few more decimal places. But what the 20th century demonstrated is that beneath the deterministic classical physics lies a strange world of quantum physics where the only "objective truth" is about collections and their statistical behaviour. The cute little hard balls of physics with their deterministic mass, direction, momentum, and energy dissolve when we peer deep down. Even math in the 20th century demonstrated that the classical view that you could formally demonstrate completeness and consistency of a formalism that captures all that we understand to be mathematical "truth" is elusive.
So I need a further story that justifies dipping my toes into the details of logic and math and measuring things about humans -- real humans, not models of human behaviour -- to convince me that the logic and math details are worth pursuing.
My prejudice is that humans are ultimately opaque. Opaque to each other and even opaque to ourselves. We can't even understand our own pleasure or pain let alone truly "control" decisions about it since science has demonstrated that the conscious mind is the tip of the iceberg, a busy rationalizer working after the fact while subsystems in the brain analyze, choose, and decide with only minimal input and "oversight" from the conscious part.
I appreciate your effort. And I really enjoyed the backgound and the post up to the bit starting "Enter John von Neumann".
I'd love to know, Bob, what it is about the zero-sum game that you find so compelling. You have repeatedly mentioned it over the years. I am no game theorist, but as an economist I know enough to know that nothing you say here or in your previous post is at all original - what is it about the idea that inspires you so? Or is it just to admonish that we should use the term "correctly"?
I don't quite see how you could test the magnitude of A's satisfaction as compared to B's or why you would.
We know that eating well satisfies everybody (that's our biological mechanism) and it seems that everyone has a human right to that satisfaction whether or not their satisfaction after eating well is greater or less than the next person. Ditto with the satisfaction of seeing a dentist if you have a toothache. The list is long.
I know that the utilitarians or at least Bentham didn't believe in human rights, but maybe they should have.
I don't understand Wallerstein's claim that "We know that eating well satisfies everybody...". Yes it is a biological mechanism, but not all people eat with the same gusto. Some are picky eaters. Some over-eat. What we find pleasure in eating is clearly not always what we "biologically" need. There is the well known mismatch between our desires for sugars and salts and the body's actual needs.
As one outside the academic world, I'm very suspicious of people putting things in neat and tidy boxes, of people making sweeping generalization, and of conceptualizations that claim to wrap up the world with a tidy bow on top.
Utilitarianism claimed to make measurements and moral claims based on pleasures and satisfactions. To me that is part of the primitive "science" of the 19th century. Just as Freud claimed to find stories about an id and ego and super-ego with lurid tales of an Oedipus complex and penis envy. Thankfully 20th century science moved beyond such primitive "science".
As for game theory and von Neumann, this is wonderful math. But when you venture to claim applicability in discerning human activity, then you are acting much like the primitive scientists of the 19th century. The 20th century proved math and real science are much more complex, complicated, and mysterious.
I enjoy your stories. I enjoy the fact that you find game theory intellectually interesting, but your delving into proofs leaves me cold when I look for meaning beyond the math. Intellectual puzzles like zero-sum games and prisoner's dilemma are fun and interesting, and may shed some light on human activities. But my problem is that I need more motivation to believe that game theory truly applies to something as complicated as a human being.
Even the idea that there is such a thing as a "human being" boggles my mind because I've never really found "two of the same" so to declare something about the "set of human beings" and their "pleasure" and "pain" and their "rational decisions" as a conceptual maneuver in writing up a mathematical theory of the world that I live in leaves me breathless. I guess I was spoiled by my college years spent in philosophy. I see a panorama of cultures and a long history of philosophical concerns and find it hard to box them up in a "science" and develop a mathematics which is more than a simplistic model of something far more complex. What was the "utility" or the rational choice behind the 1864-1870 Paraguayan war against the triple alliance of Brazil, Argentina, and Uruguay? 70% of the adult males of Paraguay died. Just how does that measure up as a "choice" in game theory?
I guess when the physicists have truly solved solved the math of turbulent flow and nailed down predictions of climate for hundreds of years into the future, then I would be ready to sit down and learn how the "exactness" of mathematics fits the woolly reality of an irrational, fuzzy, uncertain, and chaotic world that we live in and the even more complex and complicated world of humans and their "choices".
I don't quite see how you could test the magnitude of A's satisfaction as compared to B's or why you would.
Utilitarianism claimed to make measurements and moral claims based on pleasures and satisfactions.
Interpersonal comparisons of utility turns out to be a real and significant problem. (For those interested in the issue, there's a good, if pretty difficult, volume edited by Jon Elster and John Roemer, _Interpersonal Comparisons of Well-Being_, from Cambridge University press, with articles by really good people working in philosophy, economics, and psychology. I found most of it fairly hard going, though.) This has led most economists, and many philosophers, away from the hedonistic utilitarianism of Bentham, Mill, and Sidgwick to other variants, mostly to "preference utilitarianism", where it's the satisfaction of preferences, and not "happiness" that is maximized. This sounds better, but in practice there are a lot of problems, because we mostly end up dealing with revealed preferences, and, especially when the economists are doing the talking, with "intensity of preference" being measured by "willingness to pay" for the item in question, with no real account of ability to pay. It's soon no longer clear what good the theory serves, I think - it gives us something we can measure, and sometimes it's worth measuring this, but not for most important questions. For these reasons, and others, some utilitarians, in particular Peter Singer, have been moving back to the hedonism of Sidgwick, though of course the original problems remain. To my mind, it's all part of a strong argument that utilitarianism can't be right as a full moral theory, but that would be a much longer argument.
What about those of us who have no preferences?
Not a problem, Dean. You are indifferent among every pair of alternatives. We have you covered. However, if your point is that some alternatives are incommensurable, that is actually a very important point. I wrote about it in my review of Nozick. See box.net.
a thought provoked by the Bentham" quote--"everyone to count for one, nobody for more than one." I am interested in the increasingly widespread use of "they" as the gender-neutral pronoun of choice for many people. I suppose, after reading Freud, we are all properly "they," fractured selves. And, while i assume all people should be called by whatever name or pronoun they choose, "they" seems to violate the political principle Bentham espouses. No? (I get it that "you" can both singular and plural, but "they" is grammatically and politically different.) Pick another pronoun
I was riffing, both in jest and seriously, on at least those two takes on "having no preference." I can imagine also a third: I prefer A (and I express as much by voting for A), but even when my preference prevails, I get A- or B or X. It isn't a question of whether A is commensurate with A-, B, or X; it's a question of how a nominally successful preference for A nevertheless results in X. This is far afield from von Neumann, I know. But at some point my "preference" for A becomes emptied of meaning, because I know that the A that I prefer will never in fact be A. I never had a preference in the first place.
By the way, here's part of the OED etymology of "preference":
< (i) Middle French preference, French préférence superiority (1377), advantage for one person over others (1559), action of preferring (1626), (in card games) trump suit (1764), and its etymon (ii) post-classical Latin praeferentia precedence (c1266, c1330 in British sources), favour, benevolence (15th cent.) < classical Latin praeferent-, praeferēns, present participle of praeferre prefer v. + -ia...
That "trump suit" stings.
(I get it that "you" can both singular and plural, but "they" is grammatically and politically different.) Pick another pronoun
For what it's worth, singular "they" has been used in English at least since 14th Century. See here: https://public.oed.com/blog/a-brief-history-of-singular-they/ That link also has some good history about the mistaken grammatical arguments against singular "they". For other good discussion, see this list of Language Log posts: http://languagelog.ldc.upenn.edu/nll/?cat=27
Note that they go well beyond the first page.
There's no need to invoke any sort of "Freudian" reading, and it's a perfectly fine word to use. Fighting it is just silly.
But note the OED entry itself, which reads, "In anaphoric reference to a singular noun or pronoun of undetermined gender: he or she. Especially in relation to a noun phrase involving one of the indefinite determiners or pronouns any, each, every, no, some, anybody, anyone, etc. This use has sometimes been considered erroneous." Two comments: first, I think David's point, in part, was that the politics of "undetermined gender" invoked by some uses of "they" disturbs the equally gender- (but not number-) indeterminate use of "one." Second, fighting about how words are used is precisely how "they" has come to signify indeterminate or unspecified gender in our day. Not silly at all.
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