The essence of this series is to provide tools to help make decisions in real life and in programming an automatic computer response to sensory input. Most often the decision of what to do in a given situation is determined by evaluating to probabilities of given outcomes of the possible choices. We do this all the time. Because we make decisions continuously without thinking about it, we tend to become complacent and think we know how to do it. Therein is the charm of paradoxes. They kick us in the head and force us to learn how to make better decisions.

Consider this example of how to estimate probabilities in a simple betting game. This example is an old one, but I found a nice description.

Like most old puzzles, this one has many faces. Assume three people are sitting at a table in a bar. One of the friends is a math professor. He is tired of hearing his two colleagues argue about the wisdom of invading Iraq, so he interrupts and suggests that if they are really good at analyzing probabilities, they can play a game. Alan and Ben look up. The Prof continues, “My proposed game is simple. Both of you are carrying some money. I don’t know how much. If you both agree to play the game, we will count your money individually. Whoever has the most must give it all to the other. It’s simple. Now, do you want to play?”

The common way of looking at this game is to take Alan’s role. The most he can lose is the money in his wallet, but if he wins, by the terms of the game, he will win more than he can possibly lose. He assumes the odds of winning are fifty-fifty, but the expected return is greater than one. That is, if he plays the game many times, he will leave a net winner. Therefore he should play the game.

The problem is that Ben uses the exact same reasoning and comes to the same conclusion. Yet both players cannot be favored. So what is going on?

The value of considering these kinds of paradoxes is that by resolving them, we gain a better understanding of the routine assumptions we make in life. In this game, something is obviously wrong. Both players cannot have an advantage over the other. Our minds rebel at the thought. Something deep inside says that neither player has an advantage, but the logic indicates otherwise. Try to resolve it before looking up the answer.

The history of this game is quoted from the curiouser site given above: “This paradox was originated by Maurice Kraitchik in his book ‘Mathematical Recreations.’ He describes the paradox with neckties instead of wallets. Unfortunately he offers no explanation of what is wrong with the players’ reasoning. In ‘Aha! Gotcha,’ Martin Gardner writes: “We have been unable to make this clear in any simple manner. Kraitchick is no help, and so far as we know, there is no other reference on the game.”

In fact, there are several analyses available, and one of them is given later on the same site. I will give my version is a later article.
What is the relevance of this puzzle other than as a mathematical recreation? Suppose instead of gambling for money, the two barflies were heads of state and each was considering the value and potential cost of invading the other. Each one thinks his country has the advantage in provoking an armed conflict. Are you ready to bet that all the guys running the show in all the important nations of the world can unravel the logical conundrum presented by this choice and make the right decision? If so, I have a nice bridge I would like to sell you.

For those who wish to go further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.