Sunk Costs and Escalation

Transcription

1 Sunk Costs and Escalation The sunk cost fallacy occurs when we take into consideration costs we have already paid when making decisions about what we should do in the future. I ve been with this boyfriend (girlfriend) for a long time, I shouldn t break up with him (her). I ve taken several classes toward my degree, I shouldn t change now.

2 Sunk Costs and Escalation In addition to being susceptible to the sunk cost fallacy, people tend to react to negative results by paying even larger costs! There s an additional cover charge to see the band? Well, I already bought the tickets You want to get married and move to Somalia? Well, I did put up with your family last Christmas

3 Sunk Costs Arkes and Blumer (1985) presented participants with the following: Suppose you spend $100 on a ticket for a ski trip to Michigan. A few weeks later, you spend $50 on a ticket for a ski trip to Wisconsin. After making your purchase, you notice that the trips are for the same weekend.

4 Sunk Costs Suppose that you think you will enjoy the trip to Wisconsin (for which you paid $50) more than you will enjoy the trip to Michigan (for which you paid $100). And suppose that you cannot sell or return either ticket. Which trip would you take?

5 The rational thing to do here is take the trip you think will be more enjoyable. Sunk Costs

6 Sunk Costs The rational thing to do here is take the trip you think will be more enjoyable. But many people don t actually do that!

7 Sunk Costs What Trip Participants Chose The rational thing to do here is take the trip you think will be more enjoyable. But many people don t actually do that! Michigan ($100) Wisconsin ($50)

8 Sunk Costs Sunk costs have already been paid. So, you should ignore them when deciding what to do in the future. Instead, you should always take your best current course of action. Learn from the past, but don t be a slave to it.

9 Escalation of Commitment Closely related to sunk costs is escalation of commitment. When things start to go wrong, people often double down. Bidding wars, as well as regular wars, are classic examples of escalating commitment. As the saying goes, Good money follows bad.

10 Escalation of Commitment In July of 1965, Under-Secretary of State George Ball sent a memo to President Johnson. George Ball ( ) Under-Secretary of State for JFK and LBJ.

11 Escalation of Commitment Once large numbers of U.S. troops are committed to direct combat, they will begin to take heavy casualties George Ball ( ) Under-Secretary of State for JFK and LBJ.

12 Escalation of Commitment Once we suffer large casualties, we will have started a well-nigh irreversible process. George Ball ( ) Under-Secretary of State for JFK and LBJ.

13 Escalation of Commitment Our involvement will be so great that we cannot without national humiliation stop short of achieving our complete objectives. George Ball ( ) Under-Secretary of State for JFK and LBJ.

14 Escalation of Commitment Of the two possibilities, I think humiliation would be more likely than the achievement of our objectives even after we have paid terrible costs. George Ball ( ) Under-Secretary of State for JFK and LBJ.

15 Escalation of Commitment From 1966 to 1972, around 55,000 U.S. soldiers died in Vietnam. Three times that many South Vietnamese soldiers were killed. George Ball ( ) Under-Secretary of State for JFK and LBJ.

16 Escalation of Commitment Escalation also happens with investment decisions. In the first paper on escalation of commitment, Staw (1976) presented results of an investment experiment he conducted on undergraduates in UIUC s College of Commerce and Business Administration.

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18 Moral Take your best current course of action, and learn to cut your losses. Giving up on a project in which you have invested considerable time, money, and energy is sometimes the most rational thing to do.

19 Probability Theory Introduction

20 Review At the beginning of this course, we said that an argument is a collection of sentences where some of the sentences (the premisses) allegedly evidentially support one of the sentences (the conclusion).

21 Review We then set out to construct an artificial language to help us study one species of the evidential support relation: validity. However, an argument need not be valid in order to be good. An argument might make its conclusion more likely without guaranteeing its truth.

22 Preview In the rest of the course, we are going to build some different formal machinery for expanding our account of evidential support. Our account of ampliative inference will be framed within probability theory.

23 Introducing Probability Probability is a measure.

24 Introducing Probability Probability is a measure. But what is it a measure of?

25 Introducing Probability Probability is a measure. Relative Frequency Degree of Entailment Degree of Belief?

26 Introducing Probability For now, we are going to set aside the question of what is being measured and focus on the mathematical framework. This is not because interpretive puzzles are unimportant but because the mathematical account is (or might be) shared by all interpretations.

27 Introducing Probability We are going to treat the probability function as an alternative valuation function. Implicitly, we have applied a function v(φ) to assign either True or False to each sentence φ in our language. We now introduce a function Pr(φ) that assigns a probability value p to each sentence φ in our language.

28 Introducing Probability Consequence: Probability operates on sentences of our language. What is the probability that the Cardinals win the 2015 World Series? What is the probability that Hillary Clinton is elected President in 2016?

29 Probability As a place-holder, let s assume probability is: The ratio of the number of models to the number of all (possible) worlds.

30 Probability We assign probability values to all of the sentences so that three constraints are satisfied. Non-negativity. Normality. Finite Additivity.

31 Probability We assign probability values to all of the sentences so that three constraints are satisfied. Non-negativity. The probability of a sentence must be equal to or greater than zero.

32 Probability We assign probability values to all of the sentences so that three constraints are satisfied. If { } φ, then Pr(φ) = 1. Normality.

33 Probability We assign probability values to all of the sentences so that three constraints are satisfied. Finite Additivity. If { } ~(φ ᴧ ψ), then Pr(φ v ψ) = Pr(φ) + Pr(ψ).

34 Probability In thinking about probability, we will often help ourselves to an equivalent formulation of our constraints in terms of set theory. One way to make sense of the connection is by treating a sentence as a collection of possible worlds (its models).

35 Probability In the set theoretic approach, we begin with a universe of discourse. The universe of discourse is a set. Also called sample space or population. In order to simplify our lives, we will only consider finite universes.

36 Universes and Events Consider an experiment, like picking someone at random from among the students in this class and asking what his or her age is.

37 Universes and Events Suppose that none of you lies, and suppose that we only care about your age in years. We want to represent all of the possible outcomes that we might observe for our experiment.

38 Universes and Events Suppose that none of you lies, and suppose that we only care about your age in years. We want to represent all of the possible outcomes that we might observe for our experiment. How?

39 Universes and Events Suppose that none of you lies, and suppose that we only care about your age in years. We want to represent all of the possible outcomes that we might observe for our experiment. Use a set!

40 Universes and Events Suppose that the ages of the students in the class range from 18 to 22 and all of the intermediate ages are present. Then our universe of discourse might be the set: U = {18, 19, 20, 21, 22}

41 Universes and Events The universe is a set. Subsets of the universe are called events. The set of all events relative to a universe U is the power set of U.

42 Universes and Events How many events are there relative to our universe U? U = {18, 19, 20, 21, 22}

43 Universes and Events How many events are there relative to our universe U? U = {18, 19, 20, 21, 22} P (U) has 2 5 = 32 elements.

44 Universes and Events The events are the subsets of U: E 0 = { } E 1 = {18} E 2 = {19} E 3 = {20} E 4 = {21} E 5 = {22} E 6 = {18, 19} E 7 = {18, 20} : E 15 = {21, 22} E 16 = {18, 19, 20} E 17 = {18, 19, 21} : E 25 = {20, 21, 22} E 26 = {18, 19, 20, 21} E 27 = {18, 19, 20, 22} E 28 = {18, 19, 21, 22} E 29 = {18, 20, 21, 22} E 30 = {19, 20, 21, 22} E 31 = {18, 19, 20, 21, 22}

45 Universes and Events Consider the subset E drink = {21, 22}. This set corresponds to the event of picking a student who is legally allowed to drink alcohol. Probability is defined over events.

46 Universes and Events Consider the subset E drink = {21, 22}. This set corresponds to the event of picking a student who is legally allowed to drink alcohol. Probability is defined over events. For example, we will be able to talk about the probability of picking a student who is legally allowed to drink alcohol.

47 Probability Again We will now think of the probability function as a mapping from events into real numbers. Suppose A U. The expression Pr(A) = p says that the probability of the event A is equal to the real number p.

48 Probability Again How do we decide which real number the probability function assigns to an event? We start with the simple events.

49 Probability Again How do we decide which real number the probability function assigns to an event? We start with the simple events. A simple event is an event having exactly one element, i.e. a singleton subset of U.

50 Probability Again How do we decide which real number the probability function assigns to an event? We start with the simple events. A simple event is an event having exactly one element, i.e. a singleton subset of U. This only works for finite universes!

51 Probability Again Consider again the subsets of U: E 0 = { } E 1 = {18} E 2 = {19} E 3 = {20} E 4 = {21} E 5 = {22} E 6 = {18, 19} E 7 = {18, 20} : E 15 = {21, 22} E 16 = {18, 19, 20} E 17 = {18, 19, 21} : E 25 = {20, 21, 22} E 26 = {18, 19, 20, 21} E 27 = {18, 19, 20, 22} E 28 = {18, 19, 21, 22} E 29 = {18, 20, 21, 22} E 30 = {19, 20, 21, 22} E 31 = {18, 19, 20, 21, 22}

52 Probability Again Consider again the subsets of U: E 0 = { } E 1 = {18} E 2 = {19} E 3 = {20} E 4 = {21} E 5 = {22} E 6 = {18, 19} E 7 = {18, 20} : E 15 = {21, 22} E 16 = {18, 19, 20} E 17 = {18, 19, 21} : E 25 = {20, 21, 22} E 26 = {18, 19, 20, 21} E 27 = {18, 19, 20, 22} E 28 = {18, 19, 21, 22} E 29 = {18, 20, 21, 22} E 30 = {19, 20, 21, 22} E 31 = {18, 19, 20, 21, 22} These are the simple events!

53 Probability Again We assign probability values to all of the simple events so that three constraints are satisfied. Non-negativity. Finite Additivity. Normality.

54 Probability Again We assign probability values to all of the simple events so that three constraints are satisfied. Non-negativity. The probability of a simple event must be equal to or greater than zero.

55 Probability Again We assign probability values to all of the simple events so that three constraints are satisfied. Finite Additivity. The probability of an arbitrary event E is the sum of the probabilities of the simple events that are subsets of E.

56 Probability Again We assign probability values to all of the simple events so that three constraints are satisfied. Normality. The probability of the universe of discourse is one. Formally: Pr(U) = 1.

57 Probability Again Our three constraints are called the Kolmogorov axioms of probability theory.

58 Probability Again As long as the constraints are satisfied, we are free to assign whatever values we like to the sentences of our language. However, assignments might be constrained by the interpretation of probability one adopts as well.

59 Probability Again As long as the constraints are satisfied, we are free to assign whatever values we like to the sentences of our language. However, assignments might be constrained by the interpretation of probability one adopts as well. Example?

60 An Illustration In 2009, John Oliver visited the Large Hadron Collider. And he talked to Walter Wagner from Citizens Against the LHC

61 An Illustration Walter Wagner s assignment of probabilities satisfies the Kolmogorov axioms. Is that enough to make the assignment reasonable?

62 Another Illustration As long as the constraints are satisfied, we are free to assign whatever values we like to the simple events. So, if we want to find the probability of E drink, we need to start by assigning probabilities to the simple events.

63 Another Illustration Suppose the simple events in U are all equally likely. Then Pr(E 1 ) = Pr({18}) = 1/5 Pr(E 2 ) = Pr({19}) = 1/5 Pr(E 3 ) = Pr({20}) = 1/5 Pr(E 4 ) = Pr({21}) = 1/5 Pr(E 5 ) = Pr({22}) = 1/5

64 Another Illustration Suppose the simple events in U are all equally likely. Then Pr(E 1 ) = Pr({18}) = 1/5 Pr(E 2 ) = Pr({19}) = 1/5 Pr(E 3 ) = Pr({20}) = 1/5 Pr(E 4 ) = Pr({21}) = 1/5 Pr(E 5 ) = Pr({22}) = 1/5

65 Another Illustration Suppose the simple events in U are all equally likely. Then Pr(E 1 ) = Pr({18}) = 1/5 Pr(E 2 ) = Pr({19}) = 1/5 Pr(E 3 ) = Pr({20}) = 1/5 Pr(E 4 ) = Pr({21}) = 1/5 Pr(E 5 ) = Pr({22}) = 1/5 Pr(E drink ) = Pr({21, 22}) = Pr({21}) + Pr({22}) = 1/5 + 1/5 = 2/5

66 Another Illustration Suppose the simple events in U are all equally likely. Then Pr(E 1 ) = Pr({18}) = 1/5 Pr(E 2 ) = Pr({19}) = 1/5 Pr(E 3 ) = Pr({20}) = 1/5 Pr(E 4 ) = Pr({21}) = 1/5 Pr(E 5 ) = Pr({22}) = 1/5 Pr(E drink ) = Pr({21, 22}) = Pr({21}) + Pr({22}) = 1/5 + 1/5 = 2/5 This assignment satisfies the constraints, but is it reasonable?

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68 Probability Redux Suppose I have a collection of 21 different hats.

69 j n m v b t h d s a u o q r e i g p c k f

70 Probability Redux We will often assume that all possibilities are equally likely in order to make our lives a bit easier. This is an idealization that is not generally true.

71 xkcd #669: Experiment

72 Probability Redux Okay. Now, suppose that I draw one hat at random from my collection. And suppose that we make our equal likelihood simplifying assumption. What is the probability that I draw a plastic hat?

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75 j n m v b Plastic Hats t h d s a u o q r e i g p c k f

76 Probability Redux In four worlds, I draw a plastic hat, and there are 21 worlds altogether. So, the probability that I draw a plastic hat is equal to 4/21. What is the probability that I draw a straw hat?

77 j n m v b t h d s a u o q r e i g p c k f

78 j n m v b Straw Hats t h d s a u o q r e i g p c k f

79 Probability Redux In seven worlds, I draw a straw hat, and there are 21 worlds altogether. So, the probability that I draw a straw hat is equal to 7/21 = 1/3. What is the probability that I draw either a plastic hat or a straw hat?

80 j n m v b t h d s a u o q r e i g p c k f

81 Probability Redux There are no worlds where I draw both a plastic hat and a straw hat, so by finite additivity, the probability of the disjunction is the sum of the probabilities of the disjuncts. Hence, the probability that I draw either a plastic hat or a straw hat is 4/21 + 7/21 = 11/21.

82 Probability Redux We can put the point a different way by framing the question in terms of sets. In set theory, disjunction corresponds to union. Hence, we are looking for the probability of the union of plastic hats and straw hats.

83 Probability Redux What is the probability of the union of two sets? Special Rule for Unions: Let A and B be events, and suppose that (A B) = Ø. Pr(A B) = Pr(A) + Pr(B).

84 Probability Redux The probability of picking a plastic hat is 4/21, and the probability of picking a straw hat is 7/21. Since these events are disjoint The probability of picking either a plastic hat or a straw hat or both is 4/21 + 7/21 = 11/21.

85 Probability Redux Again suppose that I draw one hat at random from my collection with the equal likelihood simplifying assumption. What is the probability that I draw a red or straw hat?

86 Probability Redux Since the sets overlap, we need a new rule. General Rule for Unions: Let A and B be arbitrary events. Pr(A B) = Pr(A) + Pr(B) Pr(A B).

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89 j n m v b t h d s a u o Red Hats q r e i g p c k f

90 Probability Redux In five worlds, I draw a red hat, and there are 21 worlds altogether. So, the probability that I draw a red hat is equal to 5/21.

91 j n m v b t h d s a u o q r e i g p c k f

92 j n m v b Straw Hats t h d s a u o q r e i g p c k f

93 j n m v b t h d s a u o q r e i g p c k f

94 Probability Redux In seven worlds, I draw a straw hat, and there are 21 worlds altogether. So, the probability that I draw a straw hat is equal to 7/21 = 1/3. What is the probability that I draw a red and straw hat?

95 j n m v b t h d s a u o q r e i g p c k f

96 Probability Redux The probability of picking a red hat is 5/21, the probability of picking a straw hat is 7/21, and the probability of picking a red straw hat is 2/21. Hence, the probability of picking either a red hat or a straw hat is 5/21 + 7/21 2/21 = 10/21.

97 Probability Redux To solve more complicated problems, we will often have recourse to derived rules, which we can prove from the Kolmogorov axioms.

98 Probability Redux Derived Rules: If { } ~φ, then Pr(φ) = 0. If { } (φ ψ), then Pr(φ) = Pr(ψ). Pr(~φ) = 1 Pr(φ).

99 Next Time Conditional probability and independence.