The Allais Paradox involves the choice between two alternatives:
decision/allais Added material
Problem
The Allais Paradox involves the choice between two alternatives:
A. 89% chance of an unknown amount
10% chance of $1 million
1% chance of $1 million
B. 89% chance of an unknown amount (the same amount as in A)
10% chance of $2.5 million
1% chance of nothing
What is the rational choice? Does this choice remain the same if the
unknown amount is $1 million? If it is nothing?
Show solution
This is "Allais' Paradox".
Which choice is rational depends upon the subjective value of money.
Many people are risk averse, and prefer the better chance of $1
million of option A. This choice is firm when the unknown amount is
$1 million, but seems to waver as the amount falls to nothing. In the
latter case, the risk averse person favors B because there is not much
difference between 10% and 11%, but there is a big difference between
$1 million and $2.5 million.
Thus the choice between A and B depends upon the unknown amount, even
though it is the same unknown amount independent of the choice. This
violates the "independence axiom" that rational choice between two
alternatives should depend only upon how those two alternatives
differ.
However, if the amounts involved in the problem are reduced to tens of
dollars instead of millions of dollars, people's behavior tends to
fall back in line with the axioms of rational choice. People tend to
choose option B regardless of the unknown amount. Perhaps when
presented with such huge numbers, people begin to calculate
qualitatively. For example, if the unknown amount is $1 million the
options are:
A. a fortune, guaranteed
B. a fortune, almost guaranteed
a tiny chance of nothing
Then the choice of A is rational. However, if the unknown amount is
nothing, the options are:
A. small chance of a fortune ($1 million)
large chance of nothing
B. small chance of a larger fortune ($2.5 million)
large chance of nothing
In this case, the choice of B is rational. The Allais Paradox then
results from the limited ability to rationally calculate with such
unusual quantities. The brain is not a calculator and rational
calculations may rely on things like training, experience, and
analogy, none of which would help in this case. This hypothesis
could be tested by studying the correlation between paradoxical
behavior and "unusualness" of the amounts involved.
If this explanation is correct, then the Paradox amounts to little
more than the observation that the brain is an imperfect rational
engine.
Additional material
[ Editor: from a December 1995 thread "Utilities; problems" on the Allais preferences and the axioms of utility theory. ] arromdee@jyusenkyou.cs.jhu.edu (Ken Arromdee) wrote: > Actually, there is no inconsistency in this example if you don't assume that > utility is proportional to monetary value. Yes, there is, Ken. Try assigning utilities, you'll see. A couple of people have mentioned the assumption of monotonicity. It is not clear to me that even that assumption is needed. (Can someone post a coherent assignment of utilities for the Allais preferences, dropping the assumption of monotonicity?) karr@cs.cornell.edu (David Karr) wrote: > A college friend of mine once gave a different counter-example to > utilities. She proposed to make the following choices: > > When I go to dinner at the cafeteria, if the choice of main courses > is between pasta salad and chicken, I'll take pasta salad because > it's healthier. > > If the choice is between chicken and roast beef, I'll take chicken > because it's healthier. > > But if the choice is between pasta salad and roast beef, I'll take > the roast beef because it tasts so much better than pasta salad. > > I can't make sense of this if the entrees (or any of their properties) > have to be converted into utilities. But it can make sense if "I could > have been eating X" is part of the outcome of a choice. Great! Great answer. Actually, I think the distinction here is more like this. We can't make sense of the preferences if the outcomes are individuated in the usual way; but we can make sense out of them if they are partly differentiated by "what I could have done instead", as you say. In the latter case, we can even assign coherent utilities. [e.g., u(chicken when the alternative was pasta) = 1; u(pasta when the alternative was chicken) = 2; well, you get the idea.] John Broome gives a similar example in his _Weighing Goods_. Maurice prefers skiing in the alps to staying home, *because it would be cowardly to stay home* if one had a chance to ski in the alps. Maurice prefers going to some museums to skiing in the alps (because going to a museum instead of skiing isn't cowardly, it's cultured). But he prefers staying home to going to the museums, because museums are boring. There are other nice examples available. These examples are, on the surface, counterexamples to an axiom of decision theory that is almost never questioned, namely, transitivity. But, as David Karr points out, they can be rescued by redescribing the outcomes. And, as David suggests (I think), the Allais preferences, which look like they violate the Independence axiom, can also be saved, in a similar way. We have to assign utilities, not to money amounts, but to outcomes like "I don't get anything when I could have had a sure thing half million dollars". There are some potential technical difficulties in the formal construction of an expectational utility function, if we allow these alternative-sensitive outcomes into our fields. (The 'Rectangular Field Assumption' is threatened.) I am very interested in this particular problem, but it's not very intuitive and probably not of general interest in this forum. So instead let me give another, related example. Suppose you have a kitten to give away. Horace and Talia each want the kitten. You think that Talia would make a very slightly better kitten-owner than Horace. But you want to be fair. So you consider these three distribution methods: i) Give the kitten to Horace. ii) Give the kitten to Talia. iii) Flip a coin, giving the kitten to Horace if the coin comes heads, and to Talia if it comes Tails. Your preferences: (iii) > (ii) > (i). But this preference schedule appears to violate the monotonicity axiom. (A probability mixture of two outcomes must be ranked between the two outcomes.) This example is based on one invented by Peter Diamond. -Jamie (I still think the Allais preferences are irrational, myself.) -- Jamie Dreier <pl436000@brownvm.brown.edu>