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Understand Utility Theory: Decision Making under Risk

Discover the principles of utility theory for decision-making under risk, including Von Neumann-Morgenstern's axioms and practical applications. Learn about preferences, lotteries, and utility functions.

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Understand Utility Theory: Decision Making under Risk

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  1. Utility theory Fred Wenstøp Fred Wenstøp: Utility

  2. Decisions under riskTerminology • The likelihoods of the different states of nature are known in terms of • probabilities • subjective or objective • Choices are between prices • x1, x2, x3... • lotteries • prices with certain probabilities • The choice will depend on the decisions maker's kind of rationality • and attitude towards risk Fred Wenstøp: Utility

  3. The St. Petersburg Paradox • Daniel Bernoulli 1738 • Suppose you are offered the following lottery • A coin is tossed until tails turn up the first time • If it happens at toss #1, you get kr 2 • If it happens at toss #2, you get kr 4 • If it happens at toss #3, you get kr 8 • etc..... • How much are you willing to pay to participate? Fred Wenstøp: Utility

  4. von Neumann-Morgenstern's axioms for preferential consistency I • Axiom 1. Complete ordering • All prices and lotteries can be ordered by the decision maker according to his preferences • No prices or lotteries can be incomparable • This is an uncontroversial axiom • To be able to speak about decision making, one must be able to decide Fred Wenstøp: Utility

  5. von Neumann-Morgenstern's axioms for preferential consistency II • Axiom 2: Transitivity • If the decision maker prefers • x to y • and • y to z • He must prefer x to z • Uncontroversial • Does not hold for football teams etc. Fred Wenstøp: Utility

  6. von Neumann-Morgenstern's axioms for preferential consistency III • Axiom 3: Continuity • Suppose that • x is preferred to y • y is preferred to z • You get a choice between • y for certain • or • x with probability p • z with probability 1-p • Then there must exist a value of p which makes you indifferent between the choices • Controversial in many situations where z is very bad • for instance irreversible environmental damages Fred Wenstøp: Utility

  7. von Neumann-Morgenstern's axioms for preferential consistency IV • Axiom 4: Reduction of compound lotteries • The only things that matter for the decision maker are the final prices and their probabilities • Therefore compound lotteries are identical to reduced lotteries • This means that fun of gambling is ruled out • The process whereby we arrive at the final prices is irrelevant Fred Wenstøp: Utility

  8. .8 A 400000 .2 0 300000 400000 .2 0 .8 B 300000 .25 0 .75 .25 .8 400000 .2 C 0 300000 0.75 0 vN-M axioms for preferential consistency V • Axiom 5: Substitutability • Suppose that you are indifferent between a price b and a lottery c • Then c and b can substitute each other in any compound lottery without affecting its attractiveness Fred Wenstøp: Utility

  9. The expected utility theorem • If a decision maker's preferences conforms to von Neumann-Morgenstern's axioms • Then it is possible to represent his preferences with a utility function • When the decision maker makes decisions according to his preferences, he is maximising expected utility • Utility function • Suppose x is a numerical price, u(x) is a function of x • xo is the worst possible price, u(xo) = 0 • x* is the best possible price, u(x*) = 1 • u(x) is monotone • Then u(x) is a utility function Fred Wenstøp: Utility

  10. The Allais paradox Fred Wenstøp: Utility

  11. Measuring a utility function • Select the working domain • As narrow as possible, still wide enough to contain all values you might want to analyse • Let the utility of the worst point be 0, and of the best point 1.0 • Offer a choice: • Either a 50/50 lottery between the end points • Or a certain outcome xc • Change xc until it is equivalent to the lottery • Then U(xc)=0.5 • Repeat the process as many times as needed • Use the new xc'sas new end points Fred Wenstøp: Utility

  12. Terminology • Certainty equivalent • Certain price which is equally attractive as a lottery • Risk premium • The difference between the certainty equivalent and the expected price • Risk averse utility functions are concave • Risk prone utility functions are convex • Risk neutral utility functions are linear Fred Wenstøp: Utility

  13. Insurance vs. betting • Insurance companies make their living from people who pay a risk premium to avoid uncertainty • Gambling organisations make their living from people who pay to achieve uncertainty • These are the same people! • How can it be explained? Fred Wenstøp: Utility

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