What would you rather fight? If you win you get $1,000,000

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This is an exercise in decision theory, and among the elements of decision theory are decision criteria, which are rules by which a decision may be made. Another element is payoff, which is what you get as the result of making a particular decision. The expected value of the billion dollar option is 10 million dollars, which is the average payoff you would get per trial if you were able to make a large number of trials. You usually wouldn’t get anything, but the occasional big payoff more than makes up for it. The payoff for the million dollar option is a flat million dollars per trial no matter how many trials you make.

A common decision criterion in real life is to pick the option that gives the greatest expected value. This is the bread-and-butter of gamblers and financiers, who make many such decisions all the time, and is appropriate if you are in fact going to be doing this over and over again.

If it’s a one-shot affair, or you don’t know the probabilities needed to calculate an expected value, there are several other criteria that are commonly used. One might be called pessimistic, or conservative, and it says: pick the option that gives you the best payoff if the worst should happen. If you picked the billion dollar option and the worst happens, you get nothing. If you picked the million dollar option it doesn’t matter what happens, you get a million dollars. Under the pessimistic criterion, you should take the million. Most of us would do that.

Different problems make for different criteria. Just changing the scale could change you from a pessimist to an optimist: imagine the same problem, with 1 million changed to 1 penny and 1 billion changed to $10. The proportions are the same, and the pessimistic criterion guarantees you a penny, but you would be perfectly rational to say that’s not enough to worry about, I’m going to go for the $10 and the worst that happens is I miss the opportunity to make one cent.