Decision theory derives from a simple, common-sense premise: a rational person should make a decision that maximizes his expected benefit. This principle is called the principle of maximum expected utility, and its origins can be traced to Daniel Bernoulli.
Decision theory makes it possible to associate a real number between zero and one with each possible consequence of a decision. This number— termed a utility—is the value of that consequence to the decision maker. In the theory, the number is always expressed on a scale from zero to one, where zero typically means the worst possible outcome and one the best possible outcome. In practice, the scale can be other than zero to one (but, technically, it is not called a utility).
The expected utility of a decision is simply the sum of the utilities of each of the possible consequences of the decision, multiplied by their probabilities. For example, if I am considering placing a $10 wager on a horse (in the hope of receiving a $20 payout from the track) and I believe the horse has a 0.3 probability of winning, the expected utility of my decision to place the bet is the sum of 0.3 x ($20 - $10) (I win my bet and take home $10 profit) and 0.7 x -$10 (I lose my $10 wager). The expected value of a decision to place the bet is - $4 (0.3 x $10 + 0.7 x -$10=-$4), and the expected value of a decision not to bet is $0. Of course, if I ascribe to the maximum expected value principle, I would not place this bet.
The alternative of not placing the bet is $0 (I neither gain nor lose money), which is greater than -$4. In general, one should only take bets with a positive expected utility.1
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