1. Lecture 2
Hastie & Dawes: Changing Our Minds: Bayes’ Theorem. In Rational Choice in an UncertainWorld, 2nd ed., 2010, pp

2. Contents About value and utility Bayes’ Theorem
Motivation for incorporating utility into decision analysis
Difference between EV and EU maximization
Bayes’ Theorem
What’s in the formula
Medical test example
Eyewitness example
Arguments for expected utility theory

3. Expected value and utility

4. Starting point: EVM evaluation
Let’s say you are offered a gamble:
Flip a coin
On heads, win x
On tails, lose y
To analyze whether you ought to take this gamble, you can check the expected monetary value (EVM)
- EVM=0,5*x+0,5*(-y)
Heads
Tails
+ x
- y

5. We’re not EV maximizers…
Samuelson[1] recalls a discussion with a colleague:
Offered 50/50 bet on +200 $ / -100 $
”I won’t take the bet, but I’ll take a hundred such bets”
To analyze whether you ought to take this gamble, you can check the expected monetary value (EVM)
EVM=0,5*200+0,5*(-100)=50
But on this analysis, the colleague should have taken the bet!
It appears he is not maximizing value
So must we give up maximizing altogether?
Heads
Tails
+ 200
- 100
[1] Samuelson, P. (1963). Risk and Uncertainty: A Fallacy of Large Numbers. Scientia, 98,

6. …but we’re EU maximizers
We can save the situation by replacing expected value maximization with expected utility maximization (EU)
Utility takes into account decreasing value for money
In fact, it turns out that EVM is a special case of EU: the case where you are risk-neutral
Note:
Utility function subject-dependent
EVM the same for all subjects!
EVM EU
0,5*200+0,5*(-100)
0,5*u(200)+0,5*u(-100)
Laitoksen nimi

7. What is utility?
Basically, anything you care about

8. Pascal’s wager Revised Wager with three Gods Existence
Live according to
God 1
God 2
God 3

9. Values and utilities – the small difference?
Value function
Under certainty
Often just ordinal
So no strength of preference
Utility function
Under uncertainty
Usually cardinal
Strength of preference
In reality almost interchangable:
Context matters
Some people prefer one word to the other
Just be consistent…
Source: von Winterfeldt, Edwards (1986) Decision Analysis and Behavioral Research. Cambridge University Press

10. Bayes’ Theorem

11. Bayes Theorem
Describes how to update a probability, when given new information 𝒑(𝑯|𝑬)= 𝒑 𝑬 𝑯 ×𝒑(𝑯) 𝒑(𝑬) Here H=hypothesis E=evidence But how do you use this?

12. Bayes’ Theorem continued
We can use conditional probability to extend the formula: Does it look any better? Maybe not… Let’s do an example!
𝒑 𝑯 𝑬 = 𝒑 𝑬 𝑯 ×𝒑 𝑯 𝒑 𝑬 = 𝒑 𝑬 𝑯 ×𝒑 𝑯 𝒑 𝑬∩𝑯 +𝒑(𝑬∩~𝑯) = 𝒑 𝑬 𝑯 ×𝒑(𝑯) 𝒑 𝑬 𝑯 ∗𝒑 𝑯 +𝒑 𝑬 ~𝑯 ∗𝒑(~𝑯)

13. Bayes’ Theorem example
A disease is present in 5% of people, and there is a test that is 90% accurate (meaning that the test produces the correct result in 90% of cases). If a person tests positive, what is the probability that this one person has the disease? What does your intuition say?

14. Bayes Theorem example
𝒑 𝑯|𝑬 = 𝒑 𝑬 𝑯 ×𝒑(𝑯) 𝒑 𝑬 𝑯 ∗𝒑 𝑯 +𝒑 𝑬 ~𝑯 ∗𝒑(~𝑯)
H= person has disease E=person tests positive 𝒑(𝑯) = 5% 𝒑 ~𝑯 =𝟏−𝒑 𝑯 =95% 𝒑(𝑬|𝑯)=90% 𝒑(𝑬|~𝑯)=10%
𝒑 𝑯|𝑬 = 𝒑 𝑬 𝑯 ×𝒑(𝑯) 𝒑 𝑬 𝑯 ∗𝒑 𝑯 +𝒑 𝑬 ~𝑯 ∗𝒑(~𝑯)
= 𝟎.𝟗×𝟎.𝟎𝟓 𝟎.𝟗∗𝟎.𝟎𝟓+𝟎.𝟏∗𝟎.𝟗𝟓 ≈𝟎.𝟑𝟐𝟏

15. Bayes Theorem example Probability of sickness, if positive:
1000 People
50 sick
0.9*50=45 disease & positive
0.1*50=5 disease & negative
950 healthy
0.1*950=95 healthy & positive
0.9*950=855 healthy & negative
Probability of sickness, if positive:
𝟒𝟓 𝟒𝟓+𝟗𝟓 = 𝟒𝟓 𝟏𝟒𝟎 ≈𝟎.𝟑𝟐𝟏

16. Bayes Theorem example

17. Another example
Two cab companies operate in River City, the Blue and the Green, named according to the colors of the cabs they run. [A total of] 85% of the cabs are Blue and the remaining 15% are Green. A cab was involved in a hit-and-run accident at night. An eyewitness later identified the cab as Green. The Court tested the witness’s ability to distinguish between Blue and Green cabs under nighttime visibility conditions. It found the witness was able to identify each color correctly about 80% of the time, but he confused it with the other color about 20% of the time. What do you think is the probability that the cab in the accident was Green, as the witness claimed?

18. Solution to the example
However, this interpretation of the problem can be debated! Perhaps the readers are answering some other problem… A conclusion: Graphical aids > formulas But what about computers?

19. Arguments for expected utility theory

20. Why expected utility theory is normative? Long-run argument
Argument from principles
Decreasing marginal benefit

21. The long-run argument
In the long run, choosing based on best expected result will:
Maximize total win
Therefore, is the optimal strategy
We can repeat the same argument for many decisions that are distributed across time, or across people (with some difficulty)

22. Argument from principles (1/2)
This rests on the axioms -> next lecture!
Suppose a well-mannered person who equally prefers apples and oranges has to choose in these cases:
Here is a (large) apple and an orange. Take your pick; I will have the other.
Here is an orange and a (small) apple. Take your pick; I will have the other.
Here is a large apple and a small apple. Take your pick; I will have the other.
Transitivity is not violated – why not?

23. Argument from principles (2/2)
The sure-thing principle: If weak ordering and sure-thing principle (plus other axioms) are adhered to, expected utility theory is the result! But more on this next time.

24. Decreasing marginal benefit
Peter tosses a coin and continues to do so until it should land ‘heads’ when it comes to the ground. He agrees to give Paul one ducat if he gets ‘heads’ on the very first throw, two ducats if he does it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul’s expectation. (Bernoulli, 1738/1954, p. 31)

25. Next lecture
Axioms of expected utility theory Elicitation of probabilities, value & utilities