Finance| Finance |
Utility
Usually no one has a problem telling you where they'd spend an extra dollar if they had one!
But consider these two puzzles;
Puzzle #1
Is it true that
"If you add one stone to a small pile of stones you still have a small pile of stones" ?
Seems logical. But what happens if you apply that rule over and over again? What if you apply it a billion times? Exactly where/when does it break down? Why?
Puzzle #2
Lets play a game where I flip a fair coin until heads comes up. If it comes up on the very first flip, I pay you $2. If it first comes up on the second flip, I pay you $4. If on the third flip, I pay $8. And so on, doubling the payoff for each further case. How much should you be willing to pay me for the privilege of playing this game once?
Discussion
Puzzle #1 reminds us of the power of "integration"--the adding up of lots and lots of small pieces. The statement is only approximately true. The pile is LESS small after the operation. Small amounts that seem unnoticeable or not to matter can make a big difference when integrated. For instance, instead of making a mortgage payment once a month, making half that payment every two weeks will pay off the loan surprisingly faster. Of course, we all know that 28 days is shorter than the average month. The impact over many years of compounded interest is less obvious.
Puzzle #2 requires some math. In textbook fashion you should evaluate each possible outcome by multiplying the payoff times the probability of it occurring, and then add them all up. Outcome 1 has a probability of 1/2 (heads) and a payoff of $2, so it contributes $1 of expected value to the total. Outcome 2 has probability of 1/4 (tails, then heads) and a payoff of $4, so it contributes $1 of expected value to the total. Outcome 3 has probability 1/8 (tails, tails, then heads) and a payoff of $8, so it contributes $1 of expected value to the total. And so on, for all the scenarios. Every scenario has an expected payoff of $1, so when we sum them all up we get an infinite value. Oops! Suppose I offered you a real bargain, and said I'd play the game if you only gave me $1,000,000 to play it. You still wouldn't play, would you? The deep reason is that your utility for money is not linear. At SOME point your real practical value for money flattens out. Two million dollars is probably worth twice as much as one million dollars to most people; but two billion is not usually worth twice one billion (except perhaps to the government....but then again it might opt to play the game at a higher entrance fee!). We shouldn't add up the expected payoff dollars, but the expected utilities of those dollars. When this is done, those really "long shot" scenarios will contribute miniscule amounts to the total, instead of the $1. If you have spent a dollar on a lottery ticket for a one-in-fifty-million chance to win fifty million dollars, you probably didn't get you money's worth. If you spent that dollar on a raffle ticket with a one-in-a-thousand chance to win a thousand dollars, you probably DID get your money's worth!
© 2003 Michael E. Doherty