Here are some quickies to see if you are looking at the right things.
A constant flow rate pump is hooked to a balloon. At t = 0, the diameter of the balloon is 1 foot. The pump will run for a random time inflating the balloon, but a switch will stop it if the diameter gets to be 3 feet. Assume the tensile strength of the balloon is sufficiently small that no pressure difference exists between the inside of the balloon and the atmosphere. The game is to bet on the range of the final diameter. You can bet it is between 1 and 2 feet or between 2 and 3 feet. The odds offered are even. How do you bet?
Three candidates are running for mayor. You don’t know anything about their relative strengths, but you can bet on the winner. You pick candidate A and reason that since you don’t know anything about the election, A can win or not win, therefore the odds are 50/50. An even money bet on A would be fair. But of course you can say the same thing about candidates B or C. Now we all know the winner will be decided in the courts in real life, but in the spirit of the mathematical problem, what is wrong with this analysis? (Hint: misuse of the principle of indifference.)
Then there is one of my personal favorites: Pascal’s wager. The original is rather wordy (see Pascal’s Pensees, Thought 233), but can be recast more succinctly as a mathematical proof of why everyone should become practicing Christians.
Here goes: (1) Either Christianity is correct or it is not. (2) I can believe or not. Those two statements can have four combinations. If Christianity is correct and I believe, then my reward is infinite (in heaven, harps, etc.) and my cost is minor (restrictions on fun things in this life). If Christianity is correct, and I don’t believe, then my loss is infinite (burning sulphur, pitchforks, etc.) while my cost is still minor. On the other hand, if Christianity is bogus, and I believe, then I have wasted the opportunity to take in relatively small and meaningless enjoyments for nothing. Finally, if Christianity is bogus and I don’t believe, then I can lead a finitely happier life by doing those things that give me pleasure but might offend Christians. I can choose whether to believe or not, but the probability of Christianity being correct is not in my control. However, I can assign any finite probability I wish to its being true and find from the matrix implied above that my reward for believing is always more than the cost of not believing. So strictly on the basis of rational mathematics, the best wager is to believe.
This argument might have persuaded more people to convert if another Frenchman, Diderot, had not pointed out that we can substitute “Islam” for “Christianity” and come to a similar conclusion about becoming Muslims. In fact, we can substitute any religion that posits a similar system of eternal rewards and punishments. So what is wrong with Pascal’s wager? BTW, these issues do not arise from the properties of infinity – suitable collections of large numbers work. In fact, the introduction of infinite rewards and punishments introduces some complications since normal infinity times anything positive is still infinity, and infinity plus anything is still infinity. These properties can complicate things unless handled correctly.
Pascal’s wager is linked to the other two puzzles by a similar underlying assumption. Learning to ferret out those hidden assumptions that we all suffer from is what this series is all about.
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.