Chapter 12: Value of Information
- Perfect Information: EVPI = | EVwPI – EVwoPI |
- If the outcome is less than the certainty or other option, Take the certainty or other
- The probability of the expert’s correct = 1 (always correct) and wrong = 0 (never wrong)
- Perfect information is rarely in reality.
- Imperfect Information: EVII = | EVwII – EVwoII |
I1= Expert says Unfavorable I =2Expert says Favorable
S1= Profit of $80,000 S2= Profit of $100,000 S3= Profit of $160,000
P(S 1 = 0.20 P(S 2 = 0.40 P(S 3 = 0.40
P(I1| S1) = 0.60 ⇒ P (I2| S1) = 0.40
P(I1| S2) = 0.80 ⇒ P (I2| S2) = 0.20
P(I1| S3) = 0.20 ⇒ P (I2| S3) = 0.80 Prior Conditional Joint Posterior
Sj P(Sj P(I |1) j P(I1∩S)j P(S|jI1) Sj P(Sj P(I2|j) P(I2∩S)j P(S|jI2)
S 1 0.20 0.60 0.12 0.2308 S 1 0.20 0.40 0.08 0.1667
S 2 0.40 0.80 0.32 0.6154 S 2 0.40 0.20 0.08 0.1667
S 3 0.40 0.20 0.08 0.1538 S 3 0.40 0.80 0.32 0.6666
1.00 P(I1) = 0.52 1.00 1.00 P(I2) = 0.48 1.00
- EVPI > EVII: Perfect Information is always better than imperfect information.
Chapter 13: Risk Attitude
- Utility is used for Risk-Avoider (concave) or Risk-Taker (convex) person. EMV and
Utility is used for a Risk Neutral (line) Person and EMV does not capture risk attitude.
- Most people are risk lover when money ($) is low, and are risk avoider when $ is high.
- Utility can be presented in 1) Utility Function, 2) graph, and 3) Table.
- Utility Function: U(x) = 1- e where R is Risk Tolerance (R make U function flatter,
smaller R more concave or more risk-avoider) To fine R(subjective): 0.5R+0.5(-R/2) = 0
- Certainty Equivalent (CE): U(CE) = 1– e
o e.g. U(CE) = 0.4, R = 100 0.4 = 1 – e ln(0.6) = -CE/100 CE = 51
o CE ≈ μ – 0.5σ / R 2
- Risk Premium = EMV – CE
- Risk Premium is amount money you are willing to give up in Expected value or willing
to pay in order to avoid the risk inherent.
- Two Way Utility Assessments: using Certainty Equivalent and using Probability. Chapter 14: Axioms for Expected Utility
- Ordering and Transitivity: A decision maker can order (establish preference or
indifference) any two alternatives, and the order is transitive. For example, if a person
prefers Amsterdam to London and London to Paris, then he would prefer Amsterdam to
Paris. I.e., U(A) > U(B), U(B) > U(C) U(A) > U(C)
- Reduction of compound uncertain events: A decision maker is indifferent between a
compound uncertain event (a complicated mixture of gambles or lotteries) and a simple
uncertain event as determined by reduction using standard probability manipulations. The
assumption says that we can perform the reduction without affecting the decision maker’s
- Continuity: A decision maker is indifferent between a consequence A (for example, win
100) and some uncertain event involving only two basic consequences A1 and A2, where
A1>A>A2. E.g. Suppose that you find yourself as the plaintiff in a court case. You believe that court will
award you either $5000 or nothing. Now, imagine that the defendant offer to pay you $1500 to drop the
charges. According to the continuity axiom, there must be some probability p of winning $5000(and the
corresponding 1-p probability of winning nothing) for which you would be indifferent between taking or
rejecting the settlement offer. Of course, if your subjective probability of winning happens to be lower than
p, then you would accept the proposal.
- Substitutability: A decision maker is indifferent between any original uncertain event
that includes outcome A and one formed by substituting for A an uncertain event that is
judged to be its equivalent. For example, you are interested in playing the lottery, and you are just
barely willing to pay 50 cents for a ticket. If I owe you 50 cents, then you should be just as willing to accept
a lottery ticket as the 50 cents in cash.
- Monotonicity: Given two reference gambles with the same possible outcome, a decision
maker prefers the one with the higher probability of winning the preferred outcome.
- Invariance: All that is needed to determine a decision maker’s preferences among
uncertain events are the payoffs ( or consequences) and the associated probabilities.
- Finiteness: No consequences are considered infinitely bad or infinitely good.
- Even though the axioms of expected utility theory appear to be compelling when we
discuss them, people do not necessarily make choices in accordance with them.
- Framing effects are among the most pervasive paradoxes in choice behavior.
- An individual’s risk attitude can change depending on the way the decision problem is
posed---that is, on the “frame” in which a problem is presented. The difficulty is that the
same decision problem usually can be expressed in different frames.
- Example: The United States is preparing for an outbreak of an unusual Asian strain of influenza. Experts
expect 600 people to die from the disease. Two programs are available that could be used to combat the
disease, but because of limited resources only one can be implemented. Which of these two programs do
o Program A (Tried and True) 400 people will be saved
o Program B ( Experimental) There is an 80% chance that 600 people will be saved and a 20%
chance that no one will be saved.
- Now consider the following two programs:
o Program C 200 people will die.
o Program D There is a 20% chance that 600 people will die and an 80% chance that no one will
- Would you prefer C or D ? You may have noticed that program A is same as C and that B is the same as D.
It all depends on whether you think in terms of deaths or lives saved. Many people prefer A on one hand, but D on the other. The reason for the inconsistent choices appear to be that different points of reference are
used to frame the problem in two different ways.
Chapter 15: Conflict Objective and Additive Utility Function
- Methods to deal with multiple conflicting objectives – Decide how best to trade off
increased value on one objective for lower value on another.
- Additive Preference Model – calculates a utility score for each objective and then add
the scores, weighting them appropriately according to the relative importance of the
various objectives. That is, it is the weighted sum of the utility of each objective
- Additive Utility Function: U(x ,……, x ) = k U (x ) + …… + k U (x )
1 m 1 1 1 m m m
Finding Utilities: To get anywhere with the construction of a quantitative model of preference,
we must assess numerical scores for each alternative, U(x), do this by:
i) Proportional Scoring: U (x ) = x − Worst Value
i i Best Value − Worst Value
U(Best) = 1 U(Worst) = 0
ii) Ratios – often use for attributes that are not naturally quantitative such as colour. E.g.
blue is twice good as red, and yellow is 2.5 times as good as red. If we assign 30 to red,
then blue = 60, yellow = 75.
o U(30) = 0 = a + b(30)
o U(75) = 1 = a + b (75)
o b = 1/45 a = -2/3
o So, U(60) = -2/3 + 60(1/45) = 2/3
iii) Standard Utility Function Assessment
o The utility functions, having been assessed in terms of your preference over
uncertain situation, are models of your preferences in which your risk attitude is
built in (Ch 13): 1) Certainty Equivalent (CE) method, and 2) Probability
o Note that Proportional Scoring Method assumes risk neutrality as the utility
function of each attribute is assumed to be linear!!! Ch 13 methods allow for the
decision maker’s risk attitude incorporated in the utility function.
Finding weights k i
i) Pricing out
o From pricing out the value of an additional of one value to get another value, e.g.,
pay $600 in price to get one more year reliable.
o Taking one decision (B) as based, create a hypothetical decision (H)
o U(C) = U (H), by using Proportional Scoring to find the k i
ii) Swing Weighting – can be used in virtually any weight-assessment situation.
1. Create a table List based on attribute (e.g. 3 attribute A, B, C):
2. 1 row = benchmark = worst possible outcome of each attribute.
3. Rank the alternatives (e.g. 3 alternatives, the extra benchmark = 4, A = 2, B=1,
C=3), then Rate the alternative (e.g. benchmark = 0, A = 60, B =100, C = 40).
Total = A + B + C = 200
4. Find the weight by k = 6A/200 = 0.3, k = 100/B00 = 0.5, k = 40/200 C 0.2 iii) Lottery Weights – Using lottery-comparison techniques to assess weights. The task is to
assess the probability p that makes you indifferent between the lottery (A) and the sure
thing (B) turns out to be the weight for the one odd attribute in the sure thing. -