Lecture 2 Hastie & Dawes: Changing Our Minds: Bayes’ Theorem. In Rational Choice in an UncertainWorld, 2nd ed., 2010, pp. 178-188

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

Expected value and utility 23.9.2018

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

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, 108-113.

…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 23.9.2018

What is utility? Basically, anything you care about 23.9.2018

Pascal’s wager Revised Wager with three Gods Existence Live according to God 1 God 2 God 3 23.9.2018

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 23.9.2018

Bayes’ Theorem 23.9.2018

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? 23.9.2018

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) 23.9.2018

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? 23.9.2018

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 23.9.2018

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 23.9.2018

Bayes Theorem example 23.9.2018

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? 23.9.2018

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? 23.9.2018

Arguments for expected utility theory 23.9.2018

Why expected utility theory is normative? Long-run argument Argument from principles Decreasing marginal benefit 23.9.2018

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) 23.9.2018

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? 23.9.2018

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. 23.9.2018

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) 23.9.2018

Next lecture Axioms of expected utility theory Elicitation of probabilities, value & utilities 23.9.2018