Decision Analysis I 2010

BAMS 517 Decision Analysis: A Dynamic Programming Perspective Martin L. Puterman UBC Sauder School of Business Winter Term 2010

Introduction to Decision Analysis -outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Some dynamic decision problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],In each case there is a trade-off between immediate reward and uncertain long term gain

Common ingredients of these dynamic decision problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Decision Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Decision Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Simple decision problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Assessing Probabilities Through Decision Trees

The election stock market problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The election problem on September 8 Payoff (Gain) Obama wins Obama loses Buy 1 share 1 (+.44) 0 (-.56) Do not 0 0

A decision tree for the election problem Do not invest $0 Buy 1 share $0.44 Obama wins Obama loses 1-q q -$0.56

Valuing gambles ,[object Object],[object Object],[object Object]

Solving the election problem - a reduced problem Do not invest $0 Buy 1 share $( q-.56) We replace the gamble by its expectation – latter we use expected utility of the gamble

The election problem solution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Odds ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Odds revisited ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

A decision tree for the Super Bowl problem Do not bet $0 Bet on Arizona $40 Arizona wins Arizona loses 1-q q -$1

Solving the Super Bowl problem Do not bet $0 Bet on Arizona $ 41q -1 You would be indifferent between the two decisions if q = 1/41 or 1-q =40/41

Odds and Bookmaker’s Odds ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Games of Chance and Odds The payout on a successful bet on a single number is 35 to 1 plus the amount bet. The true bookmaker’s odds are 37 to 1 on an American roulette wheel (with 0 and 00). (assuming a fair wheel)

A decision tree for a single number bet in roulette Do not bet $0 Bet $1 on 7 $35 Ball stops on 7 Lose 37/38 1/38 -$1

Solving the roulette problem Do not $0 Bet $1 on 7 -$ 0.0526

Bet name Winning spaces Payout Odds against winning Expected value (on a $1 bet) 0 0 35 to 1 37 to 1 − $0.053 00 00 35 to 1 37 to 1 − $0.053 Straight up Any single number 35 to 1 37 to 1 − $0.053 Row 00 0, 00 17 to 1 18 to 1 − $0.053 Split any two adjoining numbers vertical or horizontal 17 to 1 18 to 1 − $0.053 Trio 0, 1, 2 or 00, 2, 3 11 to 1 11.667 to 1 − $0.053 Street any three numbers horizontal (1, 2, 3 or 4, 5, 6 etc.) 11 to 1 11.667 to 1 − $0.053 Corner any four adjoining numbers in a block (1, 2, 4, 5 or 17, 18, 20, 21 etc. ) 8 to 1 8.5 to 1 − $0.053 Five Number Bet 0, 00, 1, 2, 3 6 to 1 6.6 to 1 − $0.079 Six Line any six numbers from two horizontal rows (1, 2, 3, 4, 5, 6 or 28, 29, 30, 31, 32, 33 etc.) 5 to 1 5.33 to 1 − $0.053 1st Column 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 2 to 1 2.167 to 1 − $0.053 2nd Column 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35 2 to 1 2.167 to 1 − $0.053 3rd Column 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 2 to 1 2.167 to 1 − $0.053 1st Dozen 1 through 12 2 to 1 2.167 to 1 − $0.053 2nd Dozen 13 through 24 2 to 1 2.167 to 1 − $0.053 3rd Dozen 25 through 36 2 to 1 2.167 to 1 − $0.053 Odd 1, 3, 5, ..., 35 1 to 1 1.111 to 1 − $0.053 Even 2, 4, 6, ..., 36 1 to 1 1.111 to 1 − $0.053 Red 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 1 to 1 1.111 to 1 − $0.053 Black 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 1 to 1 1.111 to 1 − $0.053 1 to 18 1, 2, 3, ..., 18 1 to 1 1.111 to 1 − $0.053 19 to 36 19, 20, 21, ..., 36 1 to 1 1.111 to 1 − $0.053

Roulette ,[object Object],[object Object],[object Object],[object Object],[object Object]

Money Lines or “odds sets” ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Assigning probabilities to events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Assigning probabilities to events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Assigning probabilities to events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The implied decision tree Thumbtack land up Thumbtack lands on “ tip down” Spinner blue Spinner red +1 0 0 +1 Choice A Choice B

Implications of using reference lottery ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Another option – acquire information ,[object Object],[object Object],[object Object],[object Object],[object Object]

Axioms of Probability ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The law of total probabilities ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Bayes’ rule ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Updating probability assessments ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Updating probability assessments ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Probabilities for single events ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The “Monty Hall” problem ,[object Object],[object Object],[object Object],[object Object],[object Object]

The “Monty Hall” problem ,[object Object],[object Object],[object Object]

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object]

Decision Analysis I 2010

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Decision Analysis I 2010