Even Odds? Want To Bet?

In the previous article in this series, I introduced a gambling game in which two players evaluate the payoff and after reasoning it out logically, each believes the other is at a disadvantage. That is, each player independently has decided the odds favor him.

To recap, the game goes like this: three people are sitting at a table in a bar. One 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.

Alan reasons: The most I can lose is the money in my wallet, but if I win, by the terms of the game, I will win more than I 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 since he always wins more than he loses. 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?

We feel intuitively that something is wrong here. Both players in a two-person game cannot have the advantage over the other. In fact, the odds of either winning are fifty-fifty, just as you might guess. Proving it takes a bit of theoretical work, which we are not going to do here.

Instead of plowing through equations, the easiest way to show the odds are even is by simply working our way through all possible combinations and then seeing how many of them are winners and how many are losers. This game is simple enough that such an approach of proving by exhausting all possibilities is practical. In fact, this approach is quite practical for similar situations with somewhat more realistic decision trees. We often try for an elegant closed form solution of the type that earned us an “A” in high school honors classes, but elegance in not called for here.

To start our analysis, let’s agree to ignore tie scores. The conditions of the game implicitly assume that one or the other would win. Create a payoff matrix with players A and B having either High or Low. If you wish, you can assign arbitrary amounts to High and Low. For instance, Low could be $1.00, and High could be $5.00. It doesn’t really matter as long as one is higher than the other.

The elements in the matrix show the net to each player:

Where I put in $5.00, but each time the game is played, the High amount can change. Presenting the game in this way shows that on the average each player will come out even, regardless of the changing amounts. The point is that the value of the Low amount never appears. All that matters is that the winning amount is passed from one to the other.

The faulty thinking here was similar to that of some of the other examples in this series and the tutorial. It has to do with thinking about the wrong thing. For instance, Affirming the Consequent, the Monty Hall Paradox, and the Prosecutor’s Fallacy all involve concentrating on the wrong parameters.

But once we are over that hurdle, we can relax a bit and let our minds roam. The way the game was presented built in a number of assumptions. For instance, we assumed that each player had a non-zero positive amount of money. Suppose one of them was broke. Should he play anyway? If we allow that either or both could be broke, but holding negative money in the form of IOUs, would that change?

And finally, the conditions of the game assume that each player has a finite amount on money. Suppose we relax that condition and assume either or both could have an infinite amount of positive or negative money. What happens to the game then? Is that different than if either or both could have a very large, but finite wad of money?

While this is fun and games, remember that we are looking at puzzles and problems that have analogs in the real world with the idea that some readers are actively engaged in programming computers to make decisions based on data input. If the coding matches the mental model the programmer has of the problem, then the computer will happily compute the wrong answer.

If two gangs meet in an alley and both think they have the advantage, great harm could be done. That harm is unlikely to be avoided just because the alley also contains a math professor armed with a computer and chalkboard, and he explains the real situation to the antagonistic parties. But I would like to think that at the highest levels of government, decisions on whether to risk lives by attacking other countries are considered using sound reasoning, not the fallacious analysis that the players of this simple gambling game made.

For those who wish to delve 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.