Being a geek, I sometimes, like during halftime of televised football games, find myself inventing little games of chance and then wondering about the best strategy. For example, supposing someone said that they would give me a hundred thousand dollars or, alternatively, a million dollars if I correctly call a coin flip but nothing if I call it wrong. What to do?
I find that I have an immediate emotional response that's followed by a more considered reflection nudging me toward the opposite choice, which in turn is followed by a still more considered reflection that pushes me back toward my initial inclination.
My first thought is to take the hundred grand. The prospect of losing a sum of money that large on a coin flip is too painful to contemplate. But then the lizard mind is overruled by the one that can do math. The expected return on the option involving a coin flip ($500,000) is five times greater than the option I'm choosing. Since $100,000 < $500,000, I should put aside my emotional impulses and make the obvious right choice.
But then I think: what exactly is meant by "right choice"? If my goal is to maximize my expected return, then, yes, I obviously have to try for the million. It seems, however, that I'm missing another obvious consideration relating to the concept of utility. For someone who is supposed to pay back, this coming Thursday, $80,000 that he doesn't have to a loan shark with mob connections, taking the sure hundred grand is the easy, right choice. I think it might be the right choice for me, too. But I'm not sure it is. A hundred thousand would be very useful but not really life-changing. I'd still have to go to work, still have to care what my boss thinks of me. Is that so bad though? It's kind of a perplexing problem. If you magnified by ten--I can take a million dollars or a fifty-fifty chance at $10 million--it would be easy. Put down the coin, hand over the certified check.
In other games, the perplexity is all in the math. Supposing someone proposed, for a fee, to flip a coin until it turns up heads, then pay me $2 if the first flip is a head, $4 if the first head occurs on the second flip, $8 if on the third, and so on. (If the first head occurs on the nth trial, I get 2^n dollars.) I should agree to play this game if the fee is below what price?
