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Handbook > Decision-Making Under Uncertainty > Basic Concepts: Section 1 Printer Friendly

Introduction to Decision-Making Under Uncertainty

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:

  1. Expected Value
  2. The St. Petersburg Paradox
  3. Von Neumann-Morgenstern Expected Utility Theory
  4. Risk-Aversion

Expected Value

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.

To illustrate:

  1. 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.

  2. 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.

Quiz:

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?

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