Examined Rationality

As illustrated (left), you are faced with three urns, each containing 2000 balls. A has 2 reds and the rest black; B has 20 blues and the rest black; C contains 1 red, 10 blues and the rest black. You may reach into the urn of your choice and remove a ball at random. If you draw red, you get \$1000; if you draw blue, you get \$100; if you draw black, you get nothing.

Which urn do you pick and why?

Source: The Big Questions (from a paper by Armen Alchian, with the exercise attributed to Harry Markowitz)

As you can see, the urns are arranged such that on average you’ll win exactly \$1 regardless of which urn you pick. Therefore, a wholly rational person  — in a vacuum — should be indifferent to which urn s/he picks but should never choose C. However, none of us is ever wholly rational and our preferences have various components, some rational and some not.  For example, if I have a pressing and immediate need for \$1000 — \$100 won’t do it — I would (quite rationally) choose A.

But for now, let’s stick to the scenario we’ve been given without adding anything to it.

Choosing B offers the lowest variance and provides the best chance to collect something, so the risk averse should look there.  But since the likelihood of winning anything isn’t all that high, the bigger prize might be more attractive — A offers the best chance to win \$1000, so gamblers should look there. C makes no sense to utility maximizers (combining A and B provides the same odds as C) but it does split the difference. So C could work for those who hate making decisions (think Sophie’s Choice for a particularly excruciating example) or perhaps for a — fairly typical — couple with a gambling husband and a risk averse wife (the other way around doesn’t happen much in real life).

In general, our biases are such that we want both to win big and to avoid losing.  That’s why lottery drawings and slot machines are designed the way they are.  They typically feature a single very large prize, many small prizes, and not too many intermediate prizes (plus a very significant cut for the house). Last year, in an attempt to bring in more players, the Powerball lottery increased both the size of the grand prize (from \$20 to \$40 million minimum) and the odds of winning something (from 1/35 to 1/32).

As an aside, note that our inherent biases — as usual — have more and deeper impacts than we understand or expect.  The Wason selection tasks show that even the symbols contained within choices offered create variance in how rational (as typically defined) the responses are.

If nothing else, this exercise demonstrates how difficult is can be to translate a mathematical/probability problem into a set of human actions. The probabilities are what they are, but the farther you remove the problem description from the realm of probability and more toward real human concerns, the less well it matches and the less useful it seems.

If some of you would like to explain which urn you’d pick and why in the comments, I’d appreciate it.