Utility Axioms Axiom: something obvious, cannot be proven Utility axioms (rules for clear thinking)

The seven axioms (rules of clear thinking) 1.Ordering and transitivity 2.Reduction of compound uncertain events 3.Continuity 4.Substitutability 5.Monotonicity 6.Invariance 7.Finiteness

3 1. Ordering and transitivity You can order consequences for all alternatives in preference level from best to worst; some may be at the same level. A 1  A 2  means you prefer A 1 to A 2 A 1  A 2  means you are indifferent between A 1 to A 2 If A 1  A 2  and A 2  A 3 then A 1  A 3

2. Reduction of compound uncertain events Decision-maker indifferent between compound uncertain event and simple uncertain event determined by reduction of compound uncertain event using standard probability manipulations

5 3. Continuity For any three consequences, A 1  A  A 2, the decision-maker can set a probability p (preference probability) of receiving A 1 versus A 2 that will make you indifferent to receiving A for sure. A1A1 A2A2 A p 1-p 

6 4. Substitutability If the decision-maker faces the choice presented in the continuity axiom between receiving A for sure and a lottery that may produce A 1 (with probability, p, equal to his/her preference probability) or A 2 (with probability 1-p), then the decision-maker remains indifferent.

7 5. Monotonicity If you have to choose between two lotteries with the same two consequences, you will choose the lottery with the higher probability of the better consequence. A1A1 A2A2 q 1-q A1A1 A2A2 p 1-p I: II: I>II if A 1  A 2 and p>q

6. Invariance All needed to determine decision maker’s preferences among uncertain events are the consequences (payoffs) and the associated probabilities

7. Finiteness No consequence is infinitely good or bad

Theorem If a decision-maker accepts the seven rules of clear thinking then It is possible to find a utility function to evaluate consequences The decision-maker should select the alternative course of action with highest expected utility

Paradoxes Framing effect: preferences depended on the way the decision problem is framed Allais paradox: certainty effect Epistemic uncertainty aversion

Some rules of clear thinking for decision-making: a broader perspective Know your objectives Evaluate alternatives only on the basis of their consequences Structure a decision so that all issues, and only those issues, that matter are accounted for