PowerPoint Slides © Michael R. Ward, UTA 2014

St. Petersburg Paradox I• I offer you the following gamble:

• I will continue to flip a coin until I get a heads.• If I get a heads with the first flip, I will pay $2.• If I get my first heads the second flip, I will pay $4.• If I get my first heads the third flip, I will pay $8.• If I get my first heads the Nth flip, I will pay $2N.

• How much is this gamble worth?• How much would you be willing to pay in order to get in

on this gamble?

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St. Petersburg Paradox II• This problem illustrates issues when dealing with

uncertainty• This problem is contrived but it helps us to think about the

problems that managers face• Eventually, it helps us to look at ways to help deal with

uncertainty and arrive at decisions that will best profit your firm

• By modeling uncertainty carefully, you can: • Learn to make better decisions• Identify the source(s) of risk in a decisions• Compute the value of collecting more information

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Random Variables• Thinking about random variables• To model uncertainty we use random variables to

compute the expected costs and benefits of a decision • A Random Variable represents outcomes that occur with

different probabilities. • When different outcomes can occur we define the

different possibilities as States of the World (SOW)

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States of the World• Before I flip a coin, we know that it could land as heads or

tails • The random variable “coin side” has different values in the SOW

“heads” and “tails” • After I flip the coin, one of these SOWs is realized and the world

proceeds along that “path”

• We attach probabilities to each potential SOW before they are realized: 50% “heads” and 50% “tails”• Where do probabilities come from?• Use past experience, reasoning ability, etc. – pretty objective • But must decide if this experience is similar enough to references

– ultimately subjective

• After SOW is realized, probabilities are 0% and 100%

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More Complicated Uncertainty• Before I roll a die, we know that it could land as 1, 2, 3, 4,

5 or 6. These are the possible SOWs (assuming six-sided)• We usually assume each has probability of 1/6

• Is the die “loaded?”

• After SOW is realized, probabilities for five SOWs are 0% and one is 100%

• Can build up to much more complicated gambles:• Two dice or more• 52 cards• XYZ Corp. meets earnings expectations• Calculating probabilities & outcomes is “just” math

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Modeling Uncertainty• Identify the possible States of the World

• Mutually Exclusive (can’t have two SOWs at once)• Exhaustive (must include all possible SOWs)

• Assign probabilities to each event • Prob(SOW) ≥ 0 for all SOWs• Sum Prob(SOW) = 100%

• To represent values that are uncertain:• List the possible values the variable could take• Assign a probability to each value• Compute the Expected Value

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Expected Value• The Expected Value (EV) of a gamble is the average

outcome that will occur if the gamble were repeated many times

• Example Gamble on coin flip with $10 payout on “heads” and $5 cost on “tails.” Two SOWs with probability of 50% each. EV = $10×50% + (-$5)×50% = $2.50

• Example Gamble on roll of die with $9 payout on “1” and $2 cost on all others. Six SOWs with probability of 1/6 each. EV = $9×1/6 + (-$2)×1/6 + (-$2)×1/6 + (-$2)×1/6 + (-$2)×1/6 + (-$2)×1/6 ≈ -$0.167

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Back to St. Pete’s I• What are the probabilities?

• Heads on first roll = ½• Heads on second roll means tails on first one= ½ × ½ = ¼.• Heads on third roll means tails on first two = ½ × ½ × ½ = 1/8.• Heads on Nth roll means tails on first N-1 = (½)N.

• What is Expected Value?• EV = Prob(1st) × Payout1 + Prob(2nd) × Payout2 + Prob(3rd) × Payout3

+ …

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Back to St. Pete’s IIEcon 5313

First Heads Prob Payout1 0.5 22 0.25 43 0.125 84 0.0625 165 0.03125 326 0.01563 647 0.00781 1288 0.00391 2569 0.00195 512

10 0.00098 102411 0.00049 204812 0.00024 409613 0.00012 819214 0.000061 1638415 0.000031 3276816 0.000015 6553617 7.6E-06 13107218 3.8E-06 26214419 1.9E-06 52428820 9.5E-07 1048576

Back to St. Pete’s IIIEcon 5313

First Heads Prob Payout EV(SOW)1 0.5 2 12 0.25 4 13 0.125 8 14 0.0625 16 15 0.03125 32 16 0.01563 64 17 0.00781 128 18 0.00391 256 19 0.00195 512 1

10 0.00098 1024 111 0.00049 2048 112 0.00024 4096 113 0.00012 8192 114 0.000061 16384 115 0.000031 32768 116 0.000015 65536 117 7.6E-06 131072 118 3.8E-06 262144 119 1.9E-06 524288 120 9.5E-07 1048576 1

Back to St. Pete’s IV• Paradox resolved• EV = Sum EV(SOW)

• Infinite sum of $1• EV = $Infinity

• Why aren’t we willing to pay $Infinity to play this gamble?• Two possible explanations:

• How much do you value $2billion over $1billion? • Need value of the payout not just payout

• Do you really trust that I can payoff if it ever hit $1million? • Calculated probabilities not likely accurate

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Inferring Probabilities• “Wheel of Cash” example:

• The carnival game wheel is divided like a pie into thirds, with values of $100, $75, and $5 on each of the slices

• The cost to play is $50.00

• Should you play the game? • Three possible outcomes: $100, $75, and $5 with equal

probability of occurring (assuming the wheel is “fair”)• Expected value of playing the game is: 1/3 ($100) + 1/3 ($75) +

1/3 ($5) = $60• But, if the wheel is biased toward the $5 outcome, the expected

value is: 1/6 ($100) + 1/6 ($75) + 2/3 ($5) = $32.50

• If a deal seems to good to be true, it probably is

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European Expansion I• For expansion into Europe, your market research has

identified three potential market opportunities: UK, France, and Germany

• You can only enter one market and your entry costs are $250,000 regardless of which you enter

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UK France GermanyProbability

Great Success 0.5 0.4 0.2Moderate Success 0.3 0.4 0.5Failure 0.2 0.2 0.3

Gross ValueGreat Success $800,000 $1,000,000 $1,500,000 Moderate Success $360,000 $300,000 $420,000 Failure $0 $0 $0

European Expansion II• Should you enter?• If so, where?• How much profit should you expect?

• Calculate the expected value of each option• Net out the cost of entry• Compare results

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European Expansion III• EV[option] = Pr1 × Value1 + Pr2 × Value2 + Pr3 × Value3

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UK France GermanyProbability

Great Success 0.5 0.4 0.2Moderate Success 0.3 0.4 0.5Failure 0.2 0.2 0.3

Gross ValueGreat Success $800,000 $1,000,000 $1,500,000 Moderate Success $360,000 $300,000 $420,000 Failure $0 $0 $0

Expected Gross ValueGreat Success $400,000 $400,000 $300,000 Moderate Success $108,000 $120,000 $210,000 Failure $0 $0 $0

Expected Net Value $258,000 $270,000 $260,000

European Expansion IV• Should you enter?

• In each case, expected net value > 0, so yes

• If so, where?• Highest expected net value is in France• Or is it? Can the calculation be more precise than the values

going into it?• How precise are our probabilities?• One significant digit in implies 0.3 could be within 0.25 and 0.34• Or, calculate the size of the probability would have to be to

overturn the decision. • Is the difference within your “margin of error?”• This is enough variation to overturn the decision

• Do not be lulled into false precision

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Reducing Uncertainty• Reduce uncertainty by gathering information• To gather information about the benefits and costs of a

decision you can run natural experiments • Natural experiment - A restaurant chain example:

• A regional manager wanted to test the profitability of a 10% price increase

• To do this, the menu was introduced in the Dallas restaurants but not the Fort Worth restaurants

• In comparing sales between the Dallas locations (the treated group) and the Fort Worth locations (the control group) the manager hoped to isolate the effect of the price change on demand (and profit)

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Diff-in-Diff• This is a difference-in-difference estimator• The first difference is before vs. after the price change; the

second difference is the treatment vs. control groups• Difference-in-difference controls for unobserved factors

that can influence changes• Count number of customers:

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Pre Post Diff

Fort Worth Stores 550 625 75

Dallas Stores 560 525 -35

Diff-in-Diff -110

Understanding Diff-in-Diff• The manager found that sales fell a little from the price

increase – but there was an increase for the treatment group too • This is likely due to factors affecting both groups

• The raw difference would underestimate the effect of the price change. The diff-in-diff is likely to more accurately measure the effect of the price change

• Possible problems with diff-in-diff:• Leakage: Dallasites travel to Fort Worth due to price change • Representativeness: Is Fort Worth a good enough control for

Dallas behaviors?

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Understanding Biases• Leakage: If Dallasites travel to Fort Worth due to price

change how does this affect our experiment?• Implies Fort Worth increase is biased upward• The ‘control’ is not controlling for only ‘other factors’• Implies the estimate is biased and we can tell the direction

• Representativeness: What does it mean for Fort Worth to be a poor control for Dallas behaviors?• Are the ‘other factors’ adequately captured by Fort Worth?• If not, change could be due to unmeasured factor

• Could choose Oklahoma City as control• Less leakage but also less representative• Tradeoffs

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Data Driven Decision Making• Collecting and analyzing the data from pre-price change to

post-price change is good• Comparing the ‘treated’ group to a ‘control’ group is even

better• Understanding how biases might still creep into the

estimate is better still

• Ex Elevator conversation with CEO• Huge increase in “Data Driven Decision Making”

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Minimizing Expected Error Costs• Sometimes, when faced with a decision, instead of

focusing on maximizing expected profits it can be useful to think about minimizing expected “error costs” • Want to make decisions that accept true hypotheses and reject

false ones• But we are not perfect and we make errors• Want to minimize costs of these errors

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TRUE FALSE

Accept OK Type II

Reject Type I OK

Product Launch• VP for new product introductions hypothesizes that the

product launch would be profitable• Collect cost studies, market research studies and any

other pertinent reports to assess the probability, p, that this hypothesis is true

• Choose the row with the smaller expected decision error• Accept if (1-p)×(Type II Error Cost) < p×(Type I Error Cost)

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TruthProfitable

(Hypothesis True)Not Profitable

(Hypothesis False)

Decision

Launch Product (Accept Hypothesis) No Error Cost Type II Error Cost

Do Not Launch Product (Reject Hypothesis) Type I Error Cost No Error Cost

Decision Maker Incentives• More information on error costs• Can calculate = (Type II Error Cost)/(Type I Error Cost +

Type II Error Cost) • Is your estimate of p > ?• But will the VP be too cautious?• Will her boss know if a launch fails?• Will anyone know if a killed launch would have been

successful?• Implies an incentive to be too cautious on decisions• Ex FDA’s drug approval

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Too much Information?• Alternatively, the VP can seek more studies to make more

certain that p > or p < • Cost of more studies is both expense and delay• Will VP balance value of precision with these costs?• Or will the VP be too cautious again and gather too much

information?• Ex FCC on cellphones

• Need to encourage people to take chances• “If you never miss an airplane, then you spend too much time in

airports”

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From the Blog• Chapter 17• Facebook Trolls• Never Punt• Experiments to Fight Poverty• Selling Lottery Tickets• Estimating via Regressions

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Summary of Main Points• Uncertainty means you must replace actual values with

expected values.• Identify the SOWs and assign probabilities to each.• Be wary of the information sources for probability

estimates.• Don’t be lulled into false precision.• Run experiments (e.g., diff-in-diff ) to uncover uncertain

parameter values. Know how to interpret results.• Some decisions lend themselves to minimizing decision

errors. Understand the incentives of the decision error minimizer.

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