1. Decision Analysis Lecture 3
Tony Cox My
Course web site:

2. Agenda Problem set 2 solutions Assignment 3
Probability calculations and interpretation
Sensitivity analysis, convex dominance, Prisoner’s Dilemma
Skill: Calculating certainty equivalents
Decision psychology: Endowment effect
Wrap-up on DA via risk profiles
Review and wrap-up on decision trees and normal form

3. Problem Set 2: Options vs. stock
Investor has $500 to invest.
Stock price is now $ It will be either
$33.50 with probability 25% (if Apricot wins)
$25.75 with probability 75% (if Apricot loses)
Choice A: Buy $500 of the stock at $28.50
Choice B: Pay $500 for option to buy 1000 shares of stock for $30,000 (= $30/share)
Choice C: Buy neither, make 8% on $500

4. An unexpectedly tricky problem!
Not meant to be about how stock options work
Assume dm can get the money ($30k) to exercise the option if needed
No interest rates, opportunity costs, future resale values, or detailed accounting
Drawing the (discrete) cumulative distribution function does not use punif!
provides more information on
“Final value” vs. “change in value” accounting

5. Problem #1: Options vs. stock
Draw cumulative distribution functions (“risk profiles”) for A, B, and C, assuming 25% probability that stock price will increase to $33.50 (else will fall to $25.75)
Which choice should the investor make?
Please submit 3 numbers: EMV(A), EMV(B), EMV(C)
How large must the probability be that stock price will increase to $33.50 to make buying the option optimal (EMV-maximizing)?
Please submit one number

6. Problem #1: Options vs. stock
Stock price is now $ It will be either
$33.50 with probability 25% (if Apricot wins)
$25.75 with probability 75% (if Apricot loses)
Choice A: Buy $500 of the stock at $28.50
EMV(A) = (500/28.50)*33.50* (500/28.50)*25.75*0.75
Only subtract the $500 if you are going to evaluate EMV of change in wealth.
To quantify EMV of final wealth, just quantify final wealth for each state, multiply by state probability, sum for all states.
Either way works, provided we use it consistently.

7. Solving problem systematically List possible outcomes of choices
For Choice A, Buy $500 worth of stock:
If stock price increases, final wealth is initial wealth ($500/$28.50) shares at $33.50 per share = initial wealth (500/28.50)*33.50 = initial wealth $ = initial wealth + $87.72 (= if initial wealth is assumed to be $500)
If stock price drops, final wealth = initial wealth (500/28.50)*25.75 = initial wealth = initial wealth (= if initial wealth = 500)
Risk profile for change in wealth: Pr(X < – 48.25) = 0, Pr(X = – 48.25) = 0.75, Pr(X = $87.72) = 0.25
EMV(A) = initial wealth * * = initial wealth = if initial wealth = 500.
Since we are using EMV, we can disregard initial wealth level and maximize EMV(change in wealth).

8. List possible outcomes of choices
For Choice B, Buy $500 worth of options:
If stock price increases, final wealth = initial wealth *( ) = initial wealth
If price drops, final wealth = initial wealth - 500
Since using EMV, can disregard initial wealth level and evaluate changes
Risk profile for change in wealth: Pr(X < -500) = 0, Pr(X = -500) = 0.75, Pr(X = $3000) = 0.25
EMV(B) = initial wealth * *0 = initial wealth + $ = $875 if initial wealth is $500
$375 = change in wealth accounting, $875 = final wealth accounting

9. List possible outcomes of choices
For Choice C, Buy an 8% return on $500:
Final wealth = 1.08*500 = $540
Risk profile: Pr(X = 540) = 1
Change in wealth = $40
Risk profile for change: Pr(X = 40) = 1
EMV(C) = initial wealth + $40 = $ if the initial wealth is assumed to be $500

10. Solving problem via normal form analysis: Final wealth view
Stock price up
Stock price down
EMV
A. Buy $500 of stock
587.72
451.75
0.25* * = $485.74
B. Buy $500 option
( ) * 1000 = $3500
0.25* *0 = $875.00
(not $375!)
C. Buy CD
1.08*500 = $540
$540
Prob:
0.25
0.75

11. Change in wealth view Stock price up Stock price down EMV
A. Buy $500 of stock
= $87.72 =
0.25* *(-48.25) = $
B. Buy $500 option
( ) * = $3000
-500
0.25* *-500 = $375.00
C. Buy CD
1.08* = $40
$40
Prob:
0.25
0.75

12. Any deterministically dominated strategies/acts?
Stock price up
Stock price down
EMV
A. Buy $500 of stock
$587.72
$451.75
$485.74
B. Buy $500 option
$3500
$875.00
C. Neither
$540
Prob:
0.25
0.75

13. Any deterministically dominated strategies/acts?
Stock price up
Stock price down
EMV
A. Buy $500 of stock
$587.72
$451.75
$485.74
B. Buy $500 option
$3500
$875.00
C. Buy CD
$540
Prob:
0.25
0.75
No – none of the rows has numbers that are all higher or all lower than the corresponding numbers in some other row, so there is no deterministic dominance. There is also no stochastic dominance.
(But, there are other forms of dominance, as we shall see.)

14. Risk profile view

15. For what range of price increase probabilities is option optimal?
Stock price up
Stock price down
EMV
A. Buy $500 of stock
$587.72
$451.75
$485.74
B. Buy $500 option
$3500
$875.00
C. Buy CD
$540
Prob: p
1 - p
EMV(B) > EMV(C) if 3500p > 540, or p > 540/3500, or p > 0.154
EMV(B) > EMV(A) if 3500p > *p *(1-p)
3500p > ( )p
p >
p > 0.13
So, B is optimal for p >

16. A modified problem (by unintentional popular demand!)
Stock price up
Stock price down
EMV
A. Buy $500 of stock
$587.72
$451.75
$485.74
B. Buy $500 option
$3000
-500
$375.00
C. Buy CD
$540
Prob: p
1 - p
Obtained by mixing final-wealth calculations for A and C with change-in-wealth calculations for B.

17. Solving modified problem via decision tree analysis

18. Risk profiles (CDFs): No stochastic dominance

19. The art of risk profiles

20. Homework #3 (Due by 4:00 PM, February 7)
Problems
Readings
Required: Russo & Schoemaker, 1989, Chapter 5 (improving intelligence-gathering and estimation),
Optional (but recommended): Schoemaker, 1982, ( on risk aversion, cardinal utility; on subjective probability and subjective expected utility, SEU),
Helpful background on discrete CDFs:
Software: Download Netica, bring it next time
Skills: Review binomial distribution calculations (e.g.,

21. Assignment 3, Problem 1 (ungraded)
A fair coin is tossed once. Draw the risk profile (cumulative distribution function) for the number of heads.
Purpose: Be able to draw, interpret risk profiles
Practice!
You do not have to turn this in, but we will go over the solution next class
Helpful background on discrete CDFs:

22. Assignment 3, Problem 2 Joe’s medicine
Joe takes pills to reduce his risk of heart attack
Pharmacist can prescribe for him either 1 pill per day at full strength, or 2 pills per day, each at half strength
The probability that Joe forget to take any given pill on any occasion is p. Its value is uncertain.
Here is how pills affect daily heart attack risk:
If he takes full strength pill, multiply his risk by 0.5 (it is cut in half)
If he takes 1 half-strength pill, multiply risk by 0.7
If he takes both half-strength pills, multiply his risk by 0.5
If he takes no pill, multiply risk by 1
What should the pharmacist prescribe?
Please submit answer as two ranges (intervals) of p values for which the best choice is (A) Prescribe 1 full strength pill; (B) Prescribe 2 half-strength pills

23. Assignment 3, Problem 3 Certainty Equivalent calculation
If you buy a raffle ticket for $2.00 and win, you will get $19.00; else, you will receive nothing from the ticket.
The probability of winning is 1/3
Your utility function for final wealth x is u(x) = log(x)
Your initial wealth (before deciding whether to buy the ticket) is $10
What is your certainty equivalent (selling price) for the opportunity to buy this raffle ticket?
Please submit one number
Should you buy it?
Please answer Yes or No

24. Sensitivity Analysis and Convex Dominance (illustrated with modified problem)

25. How large must the probability now be that stock price will increase to $33.50 to make buying the option optimal?
Stock price up
Stock price down
EMV
A. Buy $500 of stock
587.72
451.75
0.25* * = $485.74
B. Buy $500 option
( ) * = $3000
-500
0.25* *(-500) = $375.00
C. Neither
1.08*500 = $540
$540
Prob: p
1-p

26. Sensitivity analysis (for p)
Stock price up
Stock price down
EMV
A. Buy $500 of stock
587.72
451.75
p* (1-p)*451.75
B. Buy $500 option
$3000
-500
p* (1-p)*(-500)
C. CD
$540
Prob: p
1 - p
For p = 1, buying the option is obviously optimal (EMV = 3000)

27. Sensitivity analysis for p
Price up
Price down
EMV
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. CD
$540 p
1 - p
EMV(A) = EMV(B) if
p* (1-p)* = p* (1-p)*(-500)
(1-p)* = p*( )
= p*( )
p = /( ) = 0.283

28. Sensitivity analysis for p
Price up
Price down
EMV
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. CD
$540 p
1 - p
EMV(A) = EMV(B) if p = 0.283
At this value of p, EMV(A) = EMV(B) = , so C is best choice
For greater values of p, EMV(B) > EMV(A)
EMV(B) = EMV(C) if p* (1-p)*(-500) = 540  3500p = 1040
EMV(B) > EMV(C) if p > 1040/3500 = 0.297

29. Sensitivity analysis for p
Price up
Price down
EMV
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. CD
$540 p
1 - p
EMV(A) = EMV(C) if p* (1-p)* = 540
p*( ) =
p = ( )/( ) = 0.649
For greater values of p, EMV(A) > EMV(C)

30. Sensitivity analysis for p
Price up
Price down
EMV
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. CD
$540 p
1 - p
EMV(A) > EMV(C) if p > 0.649
EMV(B) > EMV(C) if p > 0.297
EMV(B) > EMV(A) if p > 0.283
So, EMV(B) > EMV(A) and EMV(B) > EMV(C) if p > 0.297
EMV(C) > EMV(B) and EMV(C) > EMV(A) if p < 0.297

31. Sensitivity analysis: Summary
Sensitivity analysis shows the range of values over which an input (such as p) can vary without changing the best decision.
If p > 0.3, choose B
if p < 0.3, choose C

32. Graphical sensitivity analysis
3000 B A
587.72
540 C
451.75
-500 p
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. Neither
$540

33. Choice A is never optimal
3000 B A
587.72
540 C
451.75
-500 p
A is eliminated by “convex dominance”

34. Convex dominance A evaluates to 451.75 + 135.97p
B evaluates to p
C evaluates to 540
95% chance of doing C, else do B, evaluates to 0.95* *( p)
= p. This dominates A.
A. Buy stock
587.72
451.75
p* (1-p)*451.75
B. Buy option
$3000
-500
p* (1-p)*(-500)
C. CD
$540

35. Convex dominance
Act A is dominated by convex dominance if a probability mixture of other acts (rows of a normal form decision table) deterministically dominates act A.
Probability mixture of rows = select those rows with stated probabilities (summing to 1)

36. Generalization: Only the “upper envelope” of EU lines matters
3000 B A
587.72
540 C
451.75
-500 p

37. Comments on Dominance
In practice, identifying and avoiding dominated options is often a major value-add from DA
Several forms of dominance, including deterministic, stochastic, convex, multivariate
More objective that maximizing EU or subjective EU (SEU), requires less data (but algorithms are more complex)
However, even the principle “Don’t choose dominated strategies” does not necessarily apply if decision-makers interact

38. Example: Prisoner’s Dilemma
If each player picks the dominant strategy (Defect) to minimize jail time, each gets more jail time (5 years) than if they had each picked the dominated strategy (Cooperate). (Red = Row player, Blue = Column player)
Arms Race
Tragedy of the Commons
There is no easy “rational” solution to this dilemma.
Real people are often far more cooperative (and altruistic) than merely “rational” players.
“Altruistic punishment” helps sustain cooperation in repeated Prisoner’s Dilemma situations.

39. Further sensitivity analyses
What other input values/assumptions could affect the investor’s decision?
How would we do sensitivity analyses for them?

40. Further sensitivity analyses
What other input values/assumptions could affect the investor’s decision?
Stock price if Apricot wins
Stock price if Apricot loses
How would we do sensitivity analyses for them?
Vary each over a range, show how optimal decision changes

41. Multivariate sensitivity analysis: Spider diagram
Steepness of curves shows sensitivity of output to each input when the rest are held fixed

42. Multivariate sensitivity analysis: Spider diagram

43. Multivariate sensitivity analysis: Tornado diagram

44. Sensitivity analysis in practice
Results of important real-world decision analyses with multiple uncertain inputs should show sensitivity analysis diagrams
Spider and tornado diagrams, decision regions
Help to simplify and focus understanding of complex models
Show where additional information might change the answer/decision
Can develop probability distributions for variables which influence the decision

45. Skill 5: Calculating certainty equivalents (CEs)

46. Recall EU calculation with u(x) = log(x)
Define the utility function, as follows:
u <- function(x){
value <- log(x)
return(value) }
Application: Calculate expected utility
R: x <- c(10, 100)
R: p <- c(0.4, 0.6)
R: EU <- sum(p*u(x))
Result: EU(X) = 3.68
R: u <- function(x){ value <- log(x) return(value) }
R: x <- c(10, 100)
R: p <- c(0.4, 0.6)
x <- c(10, 100)
R: EU <- sum(p*u(x))
R: EU
p <- c(0.4, 0.6)
EU <- sum(p*u(x)) EU
[1]

47. Certainty equivalents
Problem for communicating results to client/customer/employer: Knowing that EU(X) = tells us nothing useful!

48. Certainty equivalents
Problem for communicating results to client/customer/employer: Knowing that EU(X) = tells us nothing useful!
Better approach: Answer the decision question, “What is X worth?” What is the least we should be willing to sell it for?
Answer is the selling price or certainty equivalent (CE) of X.

49. Defining certainty equivalents (CEs) of monetary r.v.s
Challenge: What is the definition of CE(X) in terms of the utility function u() and the expected utility function EU()?
Utility function u() maps outcomes (“consequences” of choice) to numbers, often scaled to run from 0 for least-preferred possible outcome to 1 for most-preferred possible outcome.
Expected utility function (or functional) maps random variables X to numbers (given u()).

50. Defining certainty equivalents (CEs) of monetary r.v.s
Challenge: What is the definition of CE(X) in terms of the utility function u() and the expected utility function EU()?
Answer: u(CE(X)) = EU(X)
Interpretation: The decision maker (d.m.) is indifferent between receiving amount CE(X) for sure and receiving the outcome of r.v. X
Both have the same EU
Mathematical implication: CE(X) = u-1(EU(X))
u-1() is the inverse function of utility function u()

51. Example: Find CE(X), given that u(x) = log(x) and EU(X) = 3.684136
Solution:
u(CE(X)) = EU(X)
log(CE(X)) =
CE(X) = u-1(EU(X)) = exp( ) = (answer)
Interpretation: A prospect that pays $100 with probability 0.6, else $10, is worth $39.81 to this d.m.
EMV is 0.6* *10 = $64

52. A solver for CEs Definition of CE(X): u(CE(X)) = EU(X)
Represent X by vectors x = (x1, x2, …, xn) and p = (p1, p2, …, pn), listing possible values of X and their respective probabilities.
CE(X) is value of C that solves u(C) = EU(x, p), or u(C) - EU(x, p) = 0
Here, EU(x, p) = sum(p*u(x))
Can use root-finding search algorithm to solve u(C) - EU(x, p) = 0.
The following R script (in CAT) solves for CE(X):
R: EU <- function(x,p){EU <- sum(p*u(x))
return(EU)}
R: f <- function(C) u(C) - EU(x, p)
R: str(xmin <- uniroot(f, c(0, 100), tol = ))

53. Theory: Risk aversion = concave (downward-bent) utility function
EMV(X) – CE(X) = Risk Premium (RP)
For a risk-averse d.m., CE(X) < EMV(X)
W = wealth, an r.v.
W1 = maximum possible, and
W0 = minimum possible wealth after uncertainty is resolved
U() = utility function
RP = risk premium
U(pW1 + (1- p)W0) < pU(W1) + (1- p)U(W0)

54. How does CE change if d.m. starts with $10,000?
With this initial wealth, CE(X) is $63.90, very close to the EMV of $64.00.
Thus, greater initial wealth reduces risk premium (and risk aversion) for a relatively small but uncertain (r.v.) change in wealth.
# Calculation in CAT *
R: x
[1]
R: x
[1]
R: EU( x, p)
[1]
R: CE <- exp(EU( x, p))
R: CE
[1]
* Recall that we defined the function EU(x, p) = p1u(x1) + p2u(x2) pnu(xn) in R as follows:
EU <- function(x,p){
EU <- sum(p*u(x))
return(EU)}

55. Important special case
Let CE(w, X) = CE of X if initial wealth is w.
Theorem: If CE(w, X) does not depend on w and the d.m. is risk-averse, then the utility function is u(x) = 1 - e-kx.
K indicates relative risk aversion
E.g., Abbas, 2007, equation 19
Theorem: If X is normally distributed and u(x) = 1 - e-kx, CE(X) = E(X) – (k/2)Var(X)
E.g., Myerson,

56. Wrap-up on certainty equivalents
Certainty equivalents (CEs) express the values of money random variables in natural units (dollars) that are more easily communicated and interpreted than expected utilities (EUs)
CE(X) = selling price of X
CEs can be calculated from same information as EUs (by solving u(CE(X)) = EU(X))
x = (x1, x2, …, xn), p = (p1, p2, …, pn), u(x)
CE(W + X) and EU(W + X) typically get closer as initial wealth W becomes larger
W = initial wealth, X = change in wealth

57. The endowment effect (Richard Thaler)

58. Is the CE well-defined for real people?
Endowment effect: CE for something jumps up when we own it.

59. Endowment effect suggests that owning something increases its utility
u(x) at time of decision  u(x) experienced after decision is made

60. Prospect theory explains endowment effect by loss aversion
Pain of loss is roughly twice pleasure of gain. Aversion to loss explains the effect.

61. Sizes of endowment effects

62. Endowment effect applications
Holding on to underperforming stocks
Clinging to opinions
Status quo bias
Adoption
Sales

63. Conclusions on endowment effect
Shows that the assumption that people have stable, well-defined preferences for outcomes does not describe some real preferences
Normative/prescriptive vs. descriptive models
Real people often care about changes relative to a reference point rather than final outcomes
This leads to logical inconsistencies and possibilities for manipulation that marketers, politicians, and others exploit

64. Wrap-up on decision analysis (DA) via risk profiles

65. DA overview Represent each alternative choice by a risk profile
Use risk models to quantify Pr(c | a)
Use addition to calculate Pr(c ≤ x | a), for all x
Use “value” as the x axis if necessary
Eliminate dominated choices (with risk profiles to the right, if more of x is better)
If risk profiles cross, compare them based on expected utility

66. Risk analysis supports DA
For most risk management decisions, all we really need to know is:
Pr(c | a) = Pr(consequence | act)
or, Pr(v | a) = Pr(value | act)
Quantitative risk assessment (QRA) builds causal models of Pr(c | a)
Learn Pr(c | a) from data if possible
Keep records, examine empirical Pr(c | a)
Use history, knowledge (theory), experience and expertise, models

67. Risk analysis supports DA
Even rough quantitative models greatly improve decisions (usually)
QRA causal modeling, although imperfect, typically improves decisions
“Improve” = make preferred outcomes more likely
Fitting simple quantitative models to one’s own judgments can improve decisions!
QRA is practical, even for complex and uncertain systems… and is too valuable not to use!

68. DA overview Decision analysis (DA) and risk analysis
Risk profiles (probability of consequence ≤ x)
Expected utility theory, stochastic dominance
Causal modeling (“knowledge representation”) + probability theory generate risk profiles
Decision trees, influence diagrams, fault trees
Bayesian networks, conditional independence
Simulation models of dynamic systems
Optimization and learning
Reinforcement learning, on-line algorithms, simulation-optimization, decision optimization