Most of the basic ideas in the theory of decision-making under uncertainty stem from a rather unlikely source - gambling. This becomes increasingly evident as one notices the literature is dotted with phrases like 'expected value', and of course, 'lotteries'. This section covers the following topics:
- Expected Value
- The St. Petersburg Paradox
- Von Neumann-Morgenstern Expected Utility Theory
The term "expected value" provides one possible answer to the question: How much is a gamble, or any risky decision, worth? It is simply the sum of all the possible outcomes of a gamble, multiplied by their respective probabilities.
- Say you're feeling lucky one day, so you join your office betting pool as they follow the Kentucky Derby and place $10 on Santa's Little Helper, at 25/1 odds. You know that in the unlikely event of Santa's Little Helper winning the race, you'll be richer by 10 * 25 = $250.
What this means is that, according to the bookmaker of the betting pool, Santa's Little Helper has a one in 25 chance of winning and a 24 in 25 chance of losing, or, to phrase it mathematically, the probability that Santa's Little Helper will win the race is 1/25.
So what's the expected value of your bet?
Well, there are two possible outcomes - either Santa's Little Helper wins the race, or he doesn't. If he wins, you get $250; otherwise, you get nothing. So the expected value of the gamble is:
(250 * 1/25) + (0 * 24/25) = 10 + 0 = $10
And $10 is exactly what you would pay to participate in the gamble.
- Another example:
A pharmaceutical company faced with the opportunity to buy a patent on a new technology for $200 million, might know that there would be a 20% chance that it would enable them to develop a life-saving drug that might earn them, say $500 million; a 40% chance that they might earn $200 million from it; and a 40% chance that it would turn out worthless.
The expected value of this patent would then be:
(500,000,000 * 0.2) + (200,000,000 * 0.4) + (0 * 0.4) = $180 million
So of course, it would not make sense for the firm to take the risk and buy the patent.
Now that we've established that when people gamble, they should be willing to pay the expected value of the gamble in order to participate in it, ask yourself this question:
Suppose you were made an offer. A fair coin would be tossed continuously until it turned up tails. If the coin came up tails on the nth toss, you would receive $2n, i.e. if it came up tails on the 5th toss, you would receive $25 = $32.
How much would you be willing to pay, to participate in this gamble?