Utility theory Fred Wenstp 11212020 Fred Wenstp Utility

Utility theory Fred Wenstøp 11/21/2020 Fred Wenstøp: Utility 1

Decisions under risk Terminology • The likelihoods of the different states of nature are known in terms of – probabilities • subjective or objective • Choices are between prices – x 1, x 2, x 3. . . – lotteries • prices with certain probabilities – The choice will depend on the decisions maker's kind of rationality – and attitude towards risk 11/21/2020 Fred Wenstøp: Utility 2

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? 11/21/2020 Fred Wenstøp: Utility 3

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 11/21/2020 Fred Wenstøp: Utility 4

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. 11/21/2020 Fred Wenstøp: Utility 5

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 • Controversial in many situations where z – for instance irreversible environmental dam – Then there must exist a value of p which makes you indifferent between th 11/21/2020 Fred Wenstøp: Utility 6

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 11/21/2020 Fred Wenstøp: Utility 7

v. N-M axioms for preferential. 8 A consistency V 400000 . 2 • 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 11/21/2020 . 2. 8 B . 25 C 0. 75 Fred Wenstøp: Utility 0 300000 400000 0 300000 . 75. 8 0 400000 . 2 0 300000 0 8

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 11/21/2020 Fred Wenstøp: Utility 9

The Allais paradox 11/21/2020 Fred Wenstøp: Utility 10

Measuring a utility function • Change xc until it is equivalent to the lot • Then U(xc)=0. 5 • Select the working domain • Repeat the process as many times as n Use the xc'sall asvalues new end – As narrow as possible, still wide • enough to new contain youpoints might wa • 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 11/21/2020 Fred Wenstøp: Utility 11

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 11/21/2020 Fred Wenstøp: Utility 12

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? 11/21/2020 Fred Wenstøp: Utility 13