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

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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.

• A1A2means you prefer A1 to A2

• A1A2means you are indifferent between A1 to A2

• If A1A2and A2A3 then A1A3

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

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3. Continuity

• For any three consequences, A1AA2, the decision-maker can set a probability p (preference probability) of receiving A1 versus A2 that will make you indifferent to receiving A for sure.

A1

A2

A

p

1-p

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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 A1 (with probability, p, equal to his/her preference probability) or A2 (with probability 1-p), then the decision-maker remains indifferent.

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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.

A1

A2

q

1-q

A1

A2

p

1-p

I: II:

I>II if A1A2 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