PowerPoint Slides © Michael R. Ward, UTA 2014
Econ 5313
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?
Econ 5313
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
Econ 5313
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)
Econ 5313
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%
Econ 5313
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
Econ 5313
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
Econ 5313
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
Econ 5313
Back to St. Pete’s I
• What are the probabilities?
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•
•
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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 + …
Econ 5313
Back to St. Pete’s II
First Heads
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Prob
0.5
0.25
0.125
0.0625
0.03125
0.01563
0.00781
0.00391
0.00195
0.00098
0.00049
0.00024
0.00012
0.000061
0.000031
0.000015
7.6E-06
3.8E-06
1.9E-06
9.5E-07
Payout
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
Econ 5313
Back to St. Pete’s III
First Heads
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Prob
0.5
0.25
0.125
0.0625
0.03125
0.01563
0.00781
0.00391
0.00195
0.00098
0.00049
0.00024
0.00012
0.000061
0.000031
0.000015
7.6E-06
3.8E-06
1.9E-06
9.5E-07
Payout EV(SOW)
2
1
4
1
8
1
16
1
32
1
64
1
128
1
256
1
512
1
1024
1
2048
1
4096
1
8192
1
16384
1
32768
1
65536
1
131072
1
262144
1
524288
1
1048576
1
Econ 5313
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
Econ 5313
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
Econ 5313
European Expansion I
• For expansion into Europe, your market research has
identified three potential market opportunities: UK,
France, and Germany
UK
Probability
Great Success
Moderate Success
Failure
Gross Value
Great Success
Moderate Success
Failure
0.5
0.3
0.2
France Germany
0.4
0.4
0.2
0.2
0.5
0.3
$800,000 $1,000,000 $1,500,000
$360,000 $300,000 $420,000
$0
$0
$0
• You can only enter one market and your entry costs are
$250,000 regardless of which you enter
Econ 5313
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
Econ 5313
European Expansion III
• EV[option] = Pr1 × Value1 + Pr2 × Value2 + Pr3 × Value3
UK
Probability
Great Success
Moderate Success
Failure
Gross Value
Great Success
Moderate Success
Failure
Expected Gross Value
Great Success
Moderate Success
Failure
Expected Net Value
0.5
0.3
0.2
France Germany
0.4
0.4
0.2
0.2
0.5
0.3
$800,000 $1,000,000 $1,500,000
$360,000 $300,000 $420,000
$0
$0
$0
$400,000 $400,000 $300,000
$108,000 $120,000 $210,000
$0
$0
$0
$258,000 $270,000 $260,000
Econ 5313
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
Econ 5313
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)
Econ 5313
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:
Pre
Post
Diff
Fort Worth Stores
550
625
75
Dallas Stores
560
525
-35
Diff-in-Diff
-110
Econ 5313
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?
Econ 5313
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
Econ 5313
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”
Econ 5313
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
TRUE
FALSE
Accept
OK
Type II
Reject
Type I
OK
Econ 5313
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)
Truth
Decision
Launch Product
(Accept Hypothesis)
Do Not Launch Product
(Reject Hypothesis)
Profitable
(Hypothesis True)
Not Profitable
(Hypothesis False)
No Error Cost
Type II Error Cost
Type I Error Cost
No Error Cost
Econ 5313
Decision Maker Incentives
• More information on error costs
• Can calculate 𝑝 = (Type II Error Cost)/(Type I Error Cost +
Type II Error Cost)
• Is your estimate of p > 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
Econ 5313
Too much Information?
• Alternatively, the VP can seek more studies to make more
certain that p > 𝑝 or p < 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”
Econ 5313
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From the Blog
Chapter 17
Facebook Trolls
Never Punt
Experiments to Fight Poverty
Selling Lottery Tickets
Estimating via Regressions
Econ 5313
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.