1 Decision Analysis Scott Matthews 12-706 / 19-702
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Administrative Comments
Group Project 1 Back - Average 91% How graded? High level thoughts - good on NPV Some missed big picture - NPV?
HW 3 due next Wednesday
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Commentary
It is trivial to do “economics math” when demand curves, preferences, etc. are known. Without this information we have big problems.
Unfortunately, most of the ‘hard problems’ out there have unknown demand functions.
We need advanced methods to find demand
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Estimating Linear Demand Functions
As above, sometimes we don’t know demand Focus on demand (care more about CS) but can
use similar methods to estimate costs (supply) Ordinary least squares regression used
minimize the sum of squared deviations between estimated line and p,q observations: p = a + bq + e
Standard algorithms to compute parameter estimates - spreadsheets, Minitab, S, etc.
Estimates of uncertainty of estimates are obtained (based upon assumption of identically normally distributed error terms).
Can have multiple linear terms
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Also - Log-linear Function
q = a(p)b(hh)c…..
Conditions: a positive, b negative, c positive,...
If q = a(p)b : Elasticity interesting = (dq/dp)*(p/q) = abp(b-1)*(p/q) = b*(apb/apb) = b. Constant elasticity at all points.
Easiest way to estimate: linearize and use ordinary least squares regression (see Chap 12) E.g., ln q = ln a + b ln(p) + c ln(hh) ..
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Log-linear Function
q = a*pb and taking log of each side gives: ln q = ln a + b ln p which can be re-written as q’ = a’ + b p’, linear in the parameters and amenable to OLS regression.
Alternative is maximum likelihood - select parameters to max. chance of seeing obs.
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Maglev Log-Linear Function
q = a*pb - From above, b = -0.3, so if p = 1.2 and q = 20,000; so 20,000 = a*(1.2)-0.3 ; a = 21,124.
If p becomes 1.0 then q = 21,124*(1)-0.3 = 21,124. Linear model - 21,000
Remaining revenue, TWtP values similar but NOT EQUAL.
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Structuring Decisions
All about the objectives (what you want to achieve)
Decision context: setting for the decisionDecision: choice between options (there
is always an option, including status quo) Waiting for more information also an option
Uncertainty: as we’ve seen, always exists Outcomes: possible results of uncertain events Many uncertain events lead to complexity
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Structuring Decisions (2)
Can use: Fundamental objective hierarchy. Influence diagrams. Decision Trees
Risk Profiles
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Fundamental Objectives Hierarchy
Increase Lifetime Earnings
Increase Current Salary
Find New Job Get a Raise
Update Resume Network Do a Better Job
Marry Rich Go to School
Undergrad Grad School
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Influence Diagram/Decision Trees
Probably cause confusion. If one confuses you, do the other.
Important parts:
Decisions
Chance Events
Consequence/payoff
Calculation/constant
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Influence Diagram
Lifetime Earnings
Work
High Salary
Get a Raise
Find a Better Job
Marry Rich
Go to School
Undergrad
Grad School
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Other Notes
Chance node branches need to be mutually exclusive/exhaustive Only one can happen, all covered “One and only one can occur”
Timing of decisions along the way influences how trees are drawn (left to right)
As with NPV, sensitivity analysis, etc, should be able to do these by hand before resorting to software tools.
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Solving Decision Trees
We read/write them left to right, but “solve” them right to left.
Because we need to know expected values of options before choosing. Calculate values for chance nodes Picking best option at decision nodes
We typically make trees with “expected value” or NPV or profit as our consequence
Thus, as with BCA, we choose highest value.
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Texaco vs. Pennzoil
Counteroffer
$5 Billion
Texaco
Counteroffer
$3 Billion
Refuse
Settlement Amount ($ Billion)
Accept $2 Billion 2
Texaco Accept $5 Billion
5
Texaco Refuses
Counteroffer
Final Court
Decision
10.3
5
0
Final Court
Decision
10.3
5
0
Accept $3 Billion3
(0.17)
(0.5)
(0.33)
(0.2)
(0.5)
(0.3)
(0.2)
(0.5)
(0.3)
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To Solve the Tree
Solve from right to left:At chance node multiply monetary value
to probability and add them.At choice node choose highest value.
EMV for Simple Texaco vs. Pennzoil Tree:$4.63 Billion
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Risk Profiles
Risk profile shows a distribution of possible payoffs associated with particular strategies.
A strategy is what you plan to do going in to the decision. Holds your plans constant, allows chances to occur Only eliminate things YOU wouldn’t do, not things
“they” might not do.
Its not just finding the NPV of a branch.
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Risk Profiles (cont.)
Let’s think about the “subset” of the Texaco decision tree where we are only curious about the uncertainty/risk profile associated with various strategies to consider.
These represent the riskiness of each option There are only 3 “decision strategies” in the base
Texaco case:Accept the $2 billion offer (topmost branch of 1st dec.
node)Counteroffer $5 Billion, but plan to refuse counteroffer
(lower branch of 1st node, upper branch of second)Counteroffer $5B, but plan to accept counteroffer (lower
branch of both decision nodes)
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Texaco vs. Penzoil, Again
Risk Profile for "Accept to $2 Billion"
0%
100%
0% 0% 0% 0% 0% 0% 0% 0% 0%0%
20%
40%
60%
80%
100%
120%
0 1 2 3 4 5 6 7 8 9 10 11
x ($ Billion)
Chance that Settlement Equals X
Risk profile for “Accept $2 Billion” is obvious - get $2B with 100% chance.
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Risk Profile: Counteroffer $5, accept $3 billion Below is just the part of original tree to consider when
calculating the risk profile:
Counteroffer
$5 Billion
Texaco
Counteroffer
$3 Billion
Texaco Accept $5 Billion
5
Texaco Refuses
Counteroffer
Final Court
Decision
10.3
5
0
Accept $3 Billion3
(0.17)
(0.5)
(0.33)
(0.2)
(0.5)
(0.3)
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Texaco vs. Pennzoil, continued
Risk Profile for "Counteroffer $5 Billion, Accept Texaco's $3 Billion Counteroffer"
15%
0%
33%
0%
42%
0% 0% 0% 0%
10%
0%0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0 1 2 3 4 5 6 7 8 9 10 11
x ($ Billion)
Chance that Settlement Equals x
Risk Profile for "Counteroffer $5 Billion, Reject Texaco's $3 Billion Counteroffer"
25%
0% 0% 0% 0%
59%
0% 0% 0% 0%
17%
0%0%
10%
20%
30%
40%
50%
60%
70%
0 1 2 3 4 5 6 7 8 9 10 11
x ($ Billion)
Chance that Settlement Equals x
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Cumulative Risk Profiles
Graphs of cumulative distributions Percent chance that “payoff is less than
x”Cumulative Risk Profile for 3 Options in Texaco Case
0%
20%
40%
60%
80%
100%
120%
0 2 4 6 8 10 12
x ($billion)
Chance that payoff is less or equal to
x
Counteroffer $5 Billion, Accept Texaco's$3 Billion CounterofferAccept $2 Billion
Counteroffer $5 Billion, Refuse Texaco's$3 Billion Counteroffeer
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Dominance
To pick between strategies, it is useful to have rules by which to eliminate options
Let’s construct an example - assume minimum “court award” expected is $2.5B (instead of $0). Now there are no “zero endpoints” in the decision tree.
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Dominance Example
CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.
0%
20%
40%
60%
80%
100%
120%
0 2 4 6 8 10 12
x ($billion)
Chance that payoff is less or equal to x