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Unformatted text preview: Probability, Expected Payoffs and Expected Utility • In thinking about mixed strategies, we will need to make use of probabilities . We will therefore review the basic rules of probability and then derive the notion of expected value. • We will also develop the notion of expected utility as an alternative to expected payoffs . • Probabilistic analysis arises when we face uncertainty. • In situations where outcomes (or “states of the world”) are uncertain, a probability measures the likelihood that a particular outcome (or set of outcomes) occurs. – e.g. The probability that the roll of a die comes up 6. (1/6). – The probability that two randomly chosen cards in a 52 card deck add up to 21 (Natural Blackjack) (<5%). Sample Space or Universe • Let S denote a set (collection or listing) of all possible outcomes or “states” of the environment known as the sample space or universe; a typical state is denoted as s. For example: • S={s 1 , s 2 }; success/failure, or low/high price. • S={s 1 , s 2 ,...,s n1 ,s n }; number of n units sold or n offers received. • S=[0, ); stock price or salary offer. (continuous positive sample space). States and Events • A state is the result of an experiment or other situation involving uncertainty. • An event is a collection of those states, s, that result in the occurrence of the event. • An event can be that state s occurs or that multiple states occur, or that one of several states occurs (there are other possibilities). • Event A is a subset of S, denoted as A S. • Event A occurs if the true state s is an element of the set A, written as s A. Venn Diagrams S A 1 A 2 • Illustrates the sample space and events. • S is the sample space and A 1 and A 2 are events within S. • “Event A 1 does not occur.” Denoted A 1 c (Complement of A 1 ) • “Event A 1 or A 2 occurs.” Denoted A 1 A 2 (For probability use Addition Rules) • “Event A 1 and A 2 both occur”, denoted A 1 A 2 (For probability use Multiplication Rules). Probability • To each uncertain event A, or set of events, e.g., A 1 or A 2 , we would like to assign weights which measure the likelihood or importance of the events in a proportionate manner. • Let P(A i ) be the probability of A i . • We further assume that: . ) ( 1 ) ( a ll a ll i i i i i A P A P S A Addition Rules • The probability of event A or event B: P(A B) • If the events do not overlap, i.e. the events are disjoint subsets of S, so that A B= , then the probability of A or B is simply the sum of the two probabilities. P(A B) = P(A) + P(B). • If the events overlap, (are not disjoint) A B use the modified addition rule : P(A B) = P(A) + P(B) – P(A B) Example Using the Addition Rule • Suppose you throw two dice. There are 6x6=36 possible ways in which both can land....
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This note was uploaded on 03/04/2012 for the course ECON 1200 taught by Professor Staff during the Fall '08 term at Pittsburgh.
 Fall '08
 Staff
 Utility

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