1 Subjective probability Often, we estimate likelihood of outcomes of uncertain events using judgment Examples: –Likelihood of major earthquake (7.5-8 on Richter scale) in Southern California over next 30 years? –Flipped a coin that has landed on floor. Have not seen coin. What is the likelihood that it is heads?

2 Interpretations of probability Probability Long-term relative frequency Subjective: Degree of decision-maker’s belief that outcome of uncertain will of occur

3 Premise We can always assess a decision-maker uncertainty using probabilities. To assess a decision-makers probability observe his/her attitude toward accepting bets about the outcome in question.

4 Assessing subjective probability Assume risk neutral decision maker; his/her utility of an amount of money is proportional to the amount

5 Lottery ticket

6 Assessing probability of A If a decision-makers subjective probability of A is p, then he/she is willing to pay p of the ticket AND He/she is willing to sell a ticket for p.

7 Practical ways to assess subjective probabilities 1.Ask directly decision-maker 2.Ask decision-maker what bets she/he is willing to place 3.Compare lottery involving the outcome whose probability you want to estimate with reference lottery whose probability mechanism is known

8 Asking decision-maker what bets she/he is willing to place Bet for A Bet against A A occurs A does not occur X -Y-Y A occurs A does not occur -X-X Y Find amounts X and Y that make the two bets equivalent. Then

9 Asking decision-maker what bets she/he is willing to place Procedure Let X+Y=$1,000. Start with X=$ Then the decision-maker would prefer betting for A than against it. Reduce incrementally X each time asking decision maker which bet he/she prefers. Stop when the decision maker becomes indifferent for the two bets. Find probability of A Example: X=$100 and Y=$900 makes decision maker indifferent for two bets. Then P(A)=0.9.

10 Compare lottery involving the outcome whose probability you want to estimate with reference lottery whose probability mechanism is known Lottery Reference lottery A does not occur $1000 Free hamburger A occurs Find probability p that makes reference lottery equivalent to first lottery. A occurs (p) $1000 (1-p) Free hamburger p 1-p

11 Assessing continuous probabilities Assess few values of cumulative probability distribution function, for different values of the random variable. Connect with smooth curve Grade, z P(grade  80) Cumulative probability distribution of grade

12 Decomposition for assessing subjective probabilities Break task of assessing probability into smaller tasks. Outcomes whose probabilities are easier to assess. Example: probability of accident? Human error No human error Accident No accident Accident No accident

13 Decomposition for assessing subjective probabilities P(Accident) = P(Accident/Human Error)  P(Human Error)+ P(Accident/No Human Error)  [1-P(Human Error)] It could be easier to assess probabilities on right hand side of equation than probability of an accident

14 Coherence Probabilities (subjective or long term relative frequencies) must cohere –Must be between 0 and 1 –P(A  B)=P(A)+P(B) if A and B are disjoint –P(A)+P(A C )=1 A ACAC

15 Why coherence Because if a decision makers subjective probabilities do not cohere he/she may incur sure loss; a competitor can set us a Dutch book to drain up his/her account Dutch book: heads, competitor wins; tails, decision maker losses.