#### Axioms of Expected Utility

The Axioms of Expected Utility:

• Preferences over possible outcomes are complete, reflexive, and transitive.

• Compound lotteries is able to be reduced to simple lotteries.

• Continuity: For each outcome xi between x1and xn , the consumer can name a probability, pi, such that s/he is indifferent between getting xi with certainty and playing a lottery (which involves getting xn with probability pi and x1 with probability (1− pi)). We utter that xi is the certainty equal to the lottery ~xi where:

~x= (xn with pi and x1 with (1−pi)).

• Substitutability. The lottery~xi is able to forever be substituted for its certainty equivalent xi in any other lottery.

• Monotonicity. If two lotteries with the similar two alternatives each differ only in probabilities then the lottery that gives elevated probability to the most-preferred alternative is preferred to the other lottery.

If preferences over lotteries satisfy above axioms then we is able to assign numbers U(xi) which is associated with the outcomes xi such that if we compare two lotteries L and L′ which offer probabilities (p1, L, pn) and (p1, ,pn) ′L′ of obtaining those outcomes L will be preferred to L′ if and only if:

This signifies that the rank order by expected utilities reflects the rank order of preference over the lotteries and that the rational individual will choose amongst risky alternatives as if s/he is maximizing expected utility.

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