Konrad,
The Wolfram site you found is indeed confused and incorrect regarding the
Allais paradox. But the Mannheim site has it presented
correctly. I reproduce their example below:
Consider the following choice situation (<i>A</i>) among two lotteries:
Lottery L1 promises a sure win of $30,
Lottery L2 is a 80% chance to win $45 (and zero in 20% of the cases).
Typically, L1 is strictly preferred to L2.
Now, consider another choice situation (B):
Lottery K1 promises a 25% chance of winning $30
Lottery K2 is a 20% chance to win $45.
Here, the typical choice is K2 over K1.
This pair of choices violates the independence axiom of expected
utility theory: Consider situation B', in which one has a 25% chance
of having to choose between L1 and L2 (and a 75% chance of receiving
nothing). If one prefers L1 over L2 in situation A, then presumably
one should prefer L1 over L2 should the chance arise in situation B'
(this is the independence axiom). But one can reduce situation B' to
situation B by simply multiplying probabilities:
Choosing L1 should the chance arise in B' = 25% chance at $30 =K1.
Choosing L2 should the chance arise in B' = 25% chance at 80%
chance at $45 = 20% chance at $45 = K2.
So subjects choosing L1 in situation A (perhaps due to risk
aversion) and K2 in situation B (the typical pair of choices) are
violating the independence axiom. There is no possible utility
function that will result in these choices. The paradox is
that one can reframe situation B so that it seems clear one should choose
K1 when in fact one initially chose K2
Zadeh dismisses expected utility theory in his 7/31/03 message ("We
can use, of course, the principle of maximization of expected utility,
but it is well known that the principle leads to counterintuitive
conclusions ( Allais' paradox).") But this dismissal is misguided and
depends on conflating descriptive and prescriptive purposes of
expected utility theory. It is now widely agreed (due to the Allais
paradox and other empirical results) that expected utility theory is a
poor tool for describing people's choices. But for prescribing choices
(the situation Zadeh was discussing), expected utility theory is still
the standard, in spite of the Allais and Ellsberg paradoxes, which
merely impugn its validity as a descriptive model. Indeed, abandoning
the independence axiom and expected utility theory opens one to
ridiculous prescriptive recommendations that no one would accept.
Gordon Hazen
At 06:57 AM 8/4/2003 -0700, you wrote:<br>
<blockquote type=cite class=cite cite>Dear Lotfi,<br><br>
> We can use, of course, the principle of maximization of
expected<br>
> utility, but it is well known that the principle leads to<br>
> counterintuitive conclusions ( Allais' paradox).<br><br>
Thanks for that reference - I'm always interested in paradoxes.<br>
Not having heard of Allais' paradox, I looked it up on 2 sites:<br><br>
<a href="http://mathworld.wolfram.com/AllaisParadox.html";
eudora="autourl">http://mathworld.wolfram.com/AllaisParadox.html</a><br><br>
gives an example which it claims "appears to violate" the
independance <br>
axiom (also defined on this site). I do not see how anyone could think
<br>
this: it is based on the suggestion that 0.89x is the same as 0.9x
because <br>
the value of x is the same in both expressions! So no paradox
here.<br><br>
Next, I tried:<br><br>
<a href="http://www.sfb504.uni-mannheim.de/glossary/allais.htm";
eudora="autourl">http://www.sfb504.uni-mannheim.de/glossary/allais.htm</a><br><br>
This gives a more detailed description, which amounts to an explanation
of<br>
the well known fact that many humans are risk averse in some
situations. <br>
Again, no paradox - it seems that Allais' paradox is merely a comment
on<br>
the psychological phenomenon of risk aversion. But the fact that
some<br>
instances of risk averse behaviour are contrary to the principle of<br>
maximising expected utility is not a criticism of the principle any
more<br>
than the existence of crime is a criticism of law.<br><br>
Am I missing something here? Certainly I cannot dispute that the
principle<br>
of maximisation of expected utility leads to conclusions that may
be<br>
counterintuitive to many people - after all, people's intuitions
differ.<br>
But I was hoping to see examples of conclusions that _I_ find<br>
counterintuitive.<br><br>
As an aside, one reason for people having a range of intuitions about
this<br>
issue is that it is easy to misuse the principle by maximising the
wrong<br>
utility, which can easily lead to misunderstandings. E.g. one might<br>
maximise the expected value of a payoff when one ought to be
maximising<br>
the probability of having a payoff > k, in cases where the latter may
be<br>
more important for the situation at hand - this would explain many <br>
examples of risk aversion that do not contradict the principle.<br><br>
Konrad</blockquote>
<x-sigsep><p></x-sigsep>
<br>
Gordon Hazen<br>
Department of Industrial Engineering and Management Sciences<br>
McCormick School of Engineering and Applied Science<br>
2145 Sheridan Road<br>
Northwestern University<br>
Evanston IL 60208-3119<br><br>
Fax 847-491-8005<br>
Phone 847-491-5673<br>
Web:
<a href="http://www.iems.nwu.edu/~hazen/";
eudora="autourl">www.iems.nwu.edu/~hazen/</a></html>
