Reading about Decision Theory I have come about Allais Paradox to be an argument against expected utility theory.
One faces the following lotteries each with 100 tickest and the following payoffs per ticket group.
| Lottery | Ticket no. 1 | Ticket no. 2-11 | Ticket no. 12-100 |
|---|---|---|---|
| Gamble 1 | $\$1M$ | $\$1M$ | $\$1M$ |
| Gamble 2 | $\$0$ | $\$5M$ | $\$1M$ |
| Gamble 3 | $\$1M$ | $\$1M$ | $\$0$ |
| Gamble 4 | $\$0$ | $\$5M$ | $\$0$ |
The book I am reading says the following:
In a choice between Gamble 1 and Gamble 2 it seems reasonable to choose Gamble 1 since it gives the decision maker one million dollars for sure ($\$1M$), whereas in a choice between Gamble 3 and Gamble 4 many people would feel that it makes sense to trade a ten-in-hundred chance of getting $\$5M$, against a one-in-hundred risk of getting nothing, and consequently choose Gamble 4. Several empirical studies have confirmed that most people reason in this way.
There is simply no utility function such that the principle of maximising utility is consistent with a preference for Gamble 1 to Gamble 2 and a preference for Gamble 4 to Gamble 3.
However, since many people who have thought very hard about this example still feel it would be rational to stick to the problematic preference pattern described above, there seems to be something wrong with the expected utility principle.
I know wonder why this is an issue with expected utility theory and not just people making mistakes.
I argued before knowing his explanation that regarding Gamble 1 and Gamble 2, assuming increasing utility for money (be that risk averse or risk-loving or neither) one would rather give up 1M on one ticket and gain additional 4M on 10 other tickets. And regarding Gamble 3 and 4 I argued, well again, giving up 1M on one ticket to add 4M to 10 other tickets is the better choice. This "moving 4M to 10 other tickets an sacrificing one million on one ticket" seems to only increase Expected payoff and in turn expected utility for an increasing utility function in money...)
I think I am missing something and this is supposed to be a nobel-prize worthy paradox. So i should not think Gamble 2 is preferred over Gamble 1 (assuming increasing utility for money) and Gamble 4 to be preferred over Gamble 3 (assuming the same)....
(Quotes all from p. 79 and 80 in Peterson: Introduciton to Decision Theory)
Risk aversion is importance, but the paradox is not that risk aversions exists. It is that if gamble 1 is prefered to gamble2 by expected utility, then gamble3 should be prefered to gamble 4. Gamble1 is prefered to gamble2 by assumption so : $$1/100u(1M)+1/10U(1M)+89/100u(1M)>1/100u(0)+1/10u(5M)+89/100u(1M)$$ cancelling the common terms, $$1/100u(1M)+1/10(1M)>1/100u(0)+1/10u(5M).$$ But that last equation is just the expected utility comparison between gamble3 and gamble4. So an expected utility maximizer who prefers gamble1 to gamble2 should prefer gamble3 to gamble4, but experiments show people prefer 1 to 2 but 4 to 3. So that is the paradox: expected utility does not predict how people behave.
This also nnicely illustrates which axiom of expected utility breaks down in this case: the independence of irrelavant alternatives. In each comparison,the prize for tickets 12-100 is the same across alternatives, but it seems to make a difference in how people rate the gambles.