Allais' Paradox

Imagine you have to participate in two lotteries, Lottery 1 and Lottery 2. In each lottery, you have two choices. Here are the choices for Lottery 1:

• Option A:  Win $1,000,000 for certain (100% probability)
• Option A*: A 10% chance of winning $5,000,000, an 89% chance of winning $1,000,000, and a 1% chance of winning nothing.

Which option would you choose?

Now let’s go to Lottery 2:

• Option B: A 11% chance of winning $1,000,000 and an 89% chance of winning nothing.
• Option B*: A 10% chance of winning $5,000,000 and a 90% chance of winning nothing.

Which option would you choose?

From a probabilistic viewpoint, if you choose option A you should also have chosen option B and if you have chosen option A* you should have chosen option B* because these two options are identical, at least in relation to options A and B. Before your head explodes, suffice it to say that a rational investor would have chosen either options A and B or options A* and B*, but not A and B* or A* and B.

Yet, a large minority of people will choose option A (the safe gain) and option B*. The reason is that when compared to option B, option B* seems like you got roughly the same odds of winning, but if you win you win five times as much as in option B. So option B* looks more interesting. Meanwhile, in the first lottery, the chance of winning nothing is a mere 1%, yet even such a little chance of missing out on a certain gain of $1,000,000 is enough to tempt people to forego the chance of winning $5,000,000 and take the safe $1,000,000.

Obviously, investors make similar bets all the time in financial markets. Option A is essentially a bond investment, while option A* is a typical stock market investment. Option B is an income stock with steady dividends while option B* is a growth stock with no dividends but a bright future.