Decisions Under Uncertainity

# Decisions Under Uncertainity - Prof Dr Reinhard H Schmidt...

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1 Decision Theory SS 2009 4th lecture, May 5, 2009 Decisions under Uncertainty (Part II) Prof. Dr. Reinhard H. Schmidt Copyright © Reinhard H. Schmidt

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2 Content and Structure of Today’s Lecture Once again Bernoulli principle and its axiomatic foundations – Summing up of the last lecture – Critique of the axioms – Determining or selection of an utility function Risk attitude , forms of utility functions , certainty equivalent and deduction for risk Traditional decision criteria in the light of the Bernoulli principle Measures of absolute and relative risk aversion Various utility functions and their attitudes towards risk aversion
3 The Axioms Needed to Derive the Bernoulli Principle (A1) Ordering axiom (over outcomes) (A2) Transitivity axiom (A3) Independence axiom, which can be subdivided into substitution axiom and reduction axiom (A4) Continuity axiom (also called consistency axiom) (A5) Dominance axiom or rule (A1) and (A2) are defined in the same manner as under certainty, but here they only refer to the preference ordering of results and not of alternatives (otherwise we wouldn’t need Bernoulli’s principle).

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4 Axioms of Rational Behaviour and (Cardinal) Utility Functions Axioms are statements that are considered evidently true or evidently reasonable or evidently rational Together, the five axioms of the Bern. Principle lead to – a rule that “rational” decision makers would (and should) follow – A definition for rationality (in decision making) – A foundation for the complex statement “ there is a (cardinal) utility function” The content of the B Principle is: „Maximize the expected value of the utility of the results that are possible when you make a choice, and thereby evaluate the results using your own utility function that expresses your personal attitude towards risk. A utility function can be determined/found by using – Interviews, introspection and mathematical approximation – This yields a (cardinal) utility function, however only up to a positive linear transformation
(A3) Independence Axiom Three probability distributions are given: p , q and r (including the special case that e.g. q or r are certain or risk-free values) as well as a number α from the open interval (0,1) . Furthermore assume p > q (or p ~ q or p < q). The relative preference for r is discretionary. (A3) indicates that If p ~ q then α p + (1 - α)r ~ αq + (1 - α)r and if p>q then αp + (1 - α)r > αq + (1 - α)r How p is valued in comparison to q is independent of other events (namely r) and the probability of the occurrence of q or p (thus α). See transformation concept under certainty and multiple goals. Criticism (?): Ellsberg paradox

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## This note was uploaded on 06/12/2011 for the course ECON 101 taught by Professor Schmidt during the Spring '09 term at Uni Frankfurt.

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Decisions Under Uncertainity - Prof Dr Reinhard H Schmidt...

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