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Imperfect Information / Utility
Scott MatthewsCourses: 12-706 / 19-702
12-706 and 73-359 2
Willingness to Pay = EVPI
We’re interested in knowing our WTP for (perfect) information about our decision.
The book shows this as Bayesian probabilities, but think of it this way.. We consider the advice of “an expert who is always
right”. If they say it will happen, it will. If they say it will not happen, it will not. They are never wrong.
Bottom line - receiving their advice means we have eliminated the uncertainty about the event.
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Is EVPI Additive? Pair group exercise Let’s look at handout for simple “2 parts
uncertainty problem” considering the choice of where to go for a date, and the utility associated with whether it is fun or not, and whether weather is good or not.
What is Expected value in this case? What is EVPI for “fun?”; EVPI for “weather?”
What do the revised decision trees look like? What is EVPI for “fun and Weather?” Is EVPIfun+ EVPIweather = EVPIfun+weather?
12-706 and 73-359 4
Is EVPI Additive? Pair group exercise Let’s look at handout for simple “2 parts
uncertainty problem” considering the choice of where to go for a date, and the utility associated with whether it is fun or not, and whether weather is good or not.
What is Expected value in this case? What is EVPI for “fun?”; EVPI for “weather?”
What do the revised decision trees look like? What is EVPI for “fun and Weather?” Is EVPIfun+ EVPIweather = EVPIfun+weather?
12-706 and 73-359 5
Additivity, cont.
Now look at p,q labels on handout for the decision problem (top values in tree)
Is it additive if instead p=0.3, q = 0.8?What if p=0.2 and q=0.2?Should make us think about sensitivity
analysis - i.e., how much do answers/outcomes change if we change inputs..
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EVPI - Why Care?
For information to “have value” it has to affect our decision
Just like doing Tornado diagrams showed us which were the most sensitive variables
EVPI analysis shows us which of our uncertainties is the most important, and thus which to focus further effort on If we can spend some time/money to further
understand or reduce the uncertainty, it is worth it when EVPI is relatively high.
12-706 and 73-359 7
Similar: EVII
Imperfect, rather than perfect, information (because it is rarely perfect)
Example: expert admits not always right Use conditional probability (rather than assumption
of 100% correct all the time) to solve trees.
Ideally, they are “almost always right” and “almost never wrong”. In our stock example.. e.g.. P(Up Predicted | Up) is less than but close to 1. P(Up Predicted | Down) is greater than but close to
0
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12-706 and 73-359 9
Assessing the Expert
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Expert side of EVII tree
This is more complicated than EVPI because we do not know whether the expert is right or not. We have to decide whether to believe her.
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Use Bayes’ Theorem
“Flip” the probabilities with Bayes’ rule
We know P(“Up”|Up) but instead need P(Up | “Up”).
P(Up|”Up”) == =0.825
P(“Up”|Up)*P(Up)
P(“Up”|Up)*P(Up)+ .. P(“Up”|Down)P(Down)0.8*0.5
(0.8*0.5) + (0.15*0.3) +(0.2*0.2)
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EVII Tree Excerpt
12-706 and 73-359 13
Rolling Back to the Top
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Sens. Analysis for Decision Trees (see Clemen p.189)
Back to “original stock problem” 3 alternatives.. Interesting results visually
Probabilities: market up, down, samet = Pr(market up), v = P(same)
Thus P(down) = 1 - t - v (must sum to 1!) Or, (t+v must be less than, equal to 1) Know we have a line on our graph
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Sens. Analysis Graph - on board
t
v1
0 1
Friday
How to use Precision Tree software Makes solving decision trees easier, Make sensitivity analysis easier
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12-706 and 73-359 18
Risk Attitudes (Clemen Chap 13)
Our discussions and exercises have focused on EMV (and assumed expected-value maximizing decision makers) Not always the case. Some people love the thrill of making tough decisions
regardless of the outcome (not me)
A major problem with Expected Value analysis is that it assumes long-term frequency (i.e., over “many plays of the game”)
12-706 and 73-359 19
Example from Book0.5
30Game 1 30 30
0 14.5 0.5
-1-1 -1
250 0.5
2000Game 2 2000 2000
0 50 0.5
-1900-1900 -1900
Exp. value (playing many times) says we would expect to win $50 by playing game 2 many times. What’s chance to lose $1900 in Game 2?
12-706 and 73-359 20
Utility Functions
We might care about utility function for wealth (earning money). Are typically: Upward sloping - want more. Concave (opens downward) - preferences for
wealth are limited by your concern for risk. Not constant across all decisions! Recall “how much beer to drink” example
Risk-neutral (what is relation to EMV?)Risk-averseRisk-seeking
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Individuals
May be risk-neutral across a (limited) range of monetary values But risk-seeking/averse more broadly
May be generally risk averse, but risk-seeking to play the lottery Cost $1, Expected Value much less than $1
Decision makers might be risk averse at home but risk-seeking in Las Vegas
Such people are dangerous and should be treated with extreme caution. If you see them, notify the authorities.
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0.5Up
15001700 1500
0.3High-Risk Same
100-200 580 300 100
0.2Down
-1000-800 -1000
0.5Up
10001200 1000
1580 0.3
Low-Risk Same200
-200 540 400 200
0.2Down
-100100 -100
Savings Acct500
500 500
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(Discrete) Utility Function
Dollar Value
Utility Value
$1500 1.00
$1000 0.86
$500 0.65
$200 0.52
$100 0.46
$-100 0.33
$-1000 0.00
Recall: utility function is a “map” from benefit to value - here (0,1)
Try this yourselves before we go further..
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0.5Up
15001700 1500
0.3High-Risk Same
100-200 580 300 100
0.2Down
-1000-800 -1000
0.5Up
10001200 1000
1580 0.3
Low-Risk Same200
-200 540 400 200
0.2Down
-100100 -100
Savings Acct500
500 500
EU(high)=0.5*1+0.3*.46+0.2*0 = 0.638
EU(low)0.652
EU(save)=0.65
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Certainty Equivalent (CE)
Amount of money you would trade equally in exchange for an uncertain lottery
What can we infer in terms of CE about our stock investor? EU(low-risk) - his most preferred option maps
to what on his utility function? Thus his CE must be what?
EU(high-risk) -> what is his CE?
We could use CE to rank his decision orders and get the exact same results.
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Risk Premium
Is difference between EMV and CE. The risk premium is the amount you are
willing to pay to avoid the risk (like an opportunity cost).
Risk averse: Risk Premium > 0 Risk-seeking: Premium < 0 (would have
to pay them to give it up!) Risk-neutral: = 0.
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Utility Function Assessment
Basically, requires comparison of lotteries with risk-less payoffs
Different people -> different risk attitudes -> willing to accept different level of risk.
Is a matter of subjective judgment, just like assessing subjective probability.
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Utility Function Assessment
Two utility-Assessment approaches: Assessment using Certainty Equivalents
Requires the decision maker to assess several certainty equivalents
Assessment using Probabilities This approach use the probability-equivalent (PE) for
assessment technique
Exponential Utility Function: U(x) = 1-e-x/R
R is called risk tolerance
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Exponential Utility - What is R?
Consider the following lottery: Pr(Win $Y) = 0.5 Pr(Lose $Y/2) = 0.5
R = largest value of $Y where you try the lottery (versus not try it and get $0).
Sample the class - what are your R values? Again, corporate risk values can/will be higher
Show how to do in PrecisionTree (do: Use Utility Function, Exponential, R, Expected Utility)
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Next time:
Deal or No Dealhttp://www.nbc.com/Deal_or_No_Deal/game/flash.shtml