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See [definition | http://pespmc1.vub.ac.be/ASC/DECISI_THEOR.html] at PrincipiaCybernetica.

Here is a fairly simple example of analyzing a decision under uncertainty. Say you are given the opportunity to play a street game. The cost of playing is $10. You draw a card from a deck and the payoff depends on what card you draw. If you draw a black card you get nothing. If you draw a heart, you get $10 (your money back). If you draw diamond you get $40 (quadruple your money back). The decision is whether or not to play.

To apply decision theory to the question of whether or not to play, you calculate the expected payoffs of both the alternatives. The expected payoff of playing the game is equal to the payoffs of the possibilities multiplied by the probability of the outcome. There are 3 possible outcomes of playing the game:
* draw a black card (probability = 0.5, payoff = 0, expected payoff = 0.5 x 0 = $0.00)
* draw a heart (probability = 0.25, payoff = 10, expected payoff = 0.25 x 10 = $2.50)
* draw a diamond (probability = 0.25, payoff = 40, expected payoff = 0.25 x 40 = $10.00)

The expected payoff of the game is the sum of the calculations above, minus the cost of the game, i.e. ($0.00 + $2.50 + $10.00 - $10.00 = $2.50). Therefore you can expect to win $2.50 if you play the game. Notice that the ''expected'' payoff is not among the possible outcomes.

Now compare the expected payoff of not playing the game ($0, no risk, no reward), and the expected payoff playing the game ($2.50). If you want to maximize expected payoff you __should__ play the game.

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