Lecture 4 PPE 110

In most situations when people make choices, the outcomes are random. For example, if you buy a stock, it may go up or down. If you decide to support a soccer team, it may finish the season in a number of positions. If you woo another person, your efforts may lead to a number of different outcomes. We have had a quick acquaintanceship with probability theory, which is a way of quantifying uncertainty. Now we proceed to modeling choice by adding in decisions

It is very convenient to have such situations of uncertainty depicted by probability trees. In the rest of this note, we will write down some rules that we claim should govern these probability trees from the perspective of a rational decision-maker. At this point in time, we will not bother to investigate how the probabilities are arrived at – we will assume they are there for the decision maker. Either someone provided them to her/him (a bookie, from a lottery, empirical observation) or s/he came up with them on her/his own. The ultimate aim is to present a succinct way to capture rational human behavior when faced with situations of uncertainty. This theory will not be perfect – we will point out many shortcomings. However, it is the best we have, and in some sense, incorporates some desirable properties of decision making, such as consistency. Further, in experiments, this theory seems to do better than other theories. This of course does not mean that it cannot be improved.

In most real life choices the probabilities are not given, but rather are a subjective embodiment of the nature of the uncertainty. Thus when you took this class, you did not know what fraction of students had received what grade, but at the same time used what information you did have, about the class and yourself, to come up with something that rational choice theory claims looks like a probability distribution -- we will not cover this part of the theory, but this is known as rational choice with subjective probabilities. Do people always manage to come up with probabilities that make sense when they are faced with uncertainty?

The following exercise illustrates this issue in very stark terms. Subjects were asked about their choices in the following context. Part 1: An urn contains 300 colored marbles. 100 of the marbles are red and the remaining are some mixture of blue and green, but the number of blue and green marbles cannot be equal. You reach into the urn and select a marble at random. You will receive $1000 if the marble you select is of a specific color, which you have specified before drawing the marble. Would you rather that marble be: (R) Red (B) Blue Part 2 Now, suppose you were to again draw a marble at random from the same urn, and win $1000 if the marble is not of a specific color. That color will be announced by you before drawing the marble. Would you rather that color be: (B) Blue (R) Red A majority of people choose red in part 1 and red in part 2. Let us analyze their set of choices. Are they consistent with rational choice? Why or why not? Why do you think they acted the way they did?

Since the # of marbles cannot be equal, there are 2 possibilities, written in decreasing order: GRB and BGR The fact that people chose R in part 1 immediately implies that they believe R>B, so the order is in effect GRB. In that case, in part 2, they want to choose the color which is least likely to be picked. So, they should pick B.

What are these rules? We will first assume that the individual is aware of the set of outcomes that can occur after the uncertainty resolves (stock goes up, soccer team wins championship, wooing results in rude rebuffal etc.) Then for any situation, we will label the set of outcomes associated with that outcome X, and the set of all probability trees that end with outcomes in X as ∆(X). Question: If X has only two elements, how many elements does ∆(X) have? We will also assume that the individual is a consequentalist. Who is a consequentalist? It is someone who cares only about the consequences. This is a well defined term in Philosophy ( Roughly speaking, a consequentalist believes that the end result is what matters, not the path that got there. The validity/worthwhileness of a path is to be judged according to its outcome. This may seem amoral – but a consequentalist would say that if there are particular values attached to the paths themselves, then the utility of the outcomes should be modified to reflect the worth of travelling by a particular path. In other words, describe the set of outcomes, X and then put a probability distribution over it – if two situations result in the same probability distribution, then they are identical, no matter in what other aspects they differ.

This philosophy may seem very amoral. Loosely translated, it seems to suggest that the end is what matters not the means. For instance, should the two situations depicted in the next slide be seen as the same, just because the outcomes are the same? In that slide an individual has a choice between stealing and gambling $10,000. The punishment for stealing is a fine of $10,000, and the potential for loss in gambling is also $10,000. The probability of ‘success’ and `failure’ is the same in both situations.

You might say: the consequences of being caught stealing and of losing at gambling are different. Gambling is a private vice, while stealing is a crime. But a consequentalist would then say: the set of consequences have not been described properly. One needs to augment the set by describing not just the dollar losses but also the emotions associated with such an outcome. This is shown next

In other words, the consequences were not the same after all. But if the consequences are unambiguously the same, and do not depend on the interpretation of context, are people consequentalists? The next couple of slides ask you to choose between a pair of options in different situations

Experiment 1 Part 1 Imagine that the US is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: A. If program A is adopted, 200 people will be saved. B. If program B is adopted, there is one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved.

Experiment 1 part 2 Imagine that the US is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows: C. Program C is adopted, 400 people will die. D. If program D is adopted, there is a one-third probability nobody will die and a two-thirds probability that 600 people will die. In the next slide, we see what people chose and the way this tree can be drawn.

Here is another example related to consequentalism: Experiment 2, part 1 Imagine that you have decided to see a play and paid the admission price of $10 per ticket. As you enter the theater, you discover that you have lost the ticket. The seat was not marked and the ticket cannot be recovered. Would you pay $10 for another ticket? A: Yes B: No

Experiment 2, part 2 Imagine that you have decided to see a play. As you enter the theater, you discover that you have lost a $10 bill, before buying the ticket. Would you still pay $10 for a ticket to see the play? A: Yes B: No

There are problems with assuming consequentalism – yet it is not clear what those problems are due to. Further, if people are pointed out their inconsistencies, will they continue to behave as before, or will they become more consequentalist? Even if they continue to behave as before, we want the theory we are developing to serve as a benchmark level of reference, a theory that embeds some notion of consistency, so that deviations from it may be clearly apparent. We now proceed to some of the more concrete rules. We write these rules down one by one. The first category of rules are what may be called “simplifying axioms.” The second category of rules may be called “weak order axioms” The third category of rules may be called “Substitution axiom and Archimedean axiom”

Simplifying rules I (i) Zero probability limbs in a tree may be pruned. This rule means that if a state may be conceived of, and yet be assigned zero probability, then that state may safely be ignored in the analysis of the decision. Example: You are thinking of joining a multi-national company. Your expected salary is (a) less than <$200,00 with probability 0.8 (b) between $200,000 and $500,000 with probability 0.2 (c ) equal to the salary of the CEO. The last state is possible but so unlikely that it may be assigned probability zero.

0 c b a a b This axiom is relatively uncontroversial.

I(ii): If there is a situation of uncertainty which leads to another, then the probability of any outcome that results at the end may be reached by multiplying the probabilities. Example: You are thinking of joining a multi- national company. You could join and find your expected salary is (i) is $200,00 with prob 0.8 (b) or with probability 0.2 you find your salary random: with probability 0.5 you may make $300,000 or with probability 0.5 you may make $100,000

=> ,

I (iii) If an outcome occurs more than once in a situation of uncertainty, then the probability of that outcome is equal to the sum of the probabilities associated with that outcome. Example: In the multinational company you are thinking of joining, you could be stationed in San Francisco with probability 0.3, New York with probability 0.4, San Francisco with probability 0.1 and London with probability 0.2.

This situation, shown on the left, is equivalent to the one on the right SF NY => SF NY LO

I(iv) Numerical outcomes that accrue along different stages of a probability tree may be added along each path Example: Again you find yourself thinking about joining that MNC. You could join and find your expected salary is: (i) $80,000 with prob 0.8, but if this happens then with probability 0.4 you will be allowed to try and get an additional project where you may get a bonus of $40,000 Or with probability 0.2 you may make $300,000 with probability 0.5 or $100,000 with probability 0.5

=> ,

Do people behave as these axioms suggest? Lets do some more experiments Experiment 3 part 1 Which of the following options do you prefer? A: 25% chance to win $30 B: 20% chance to win $45

Experiment 3 part 2 Consider the following 2 stage game. In the first stage, there is a 75% chance to end the game without winning anything and a 25% chance to move into the second stage. If you move into the second stage you have a choice between: C: A sure win of $30 D: 80% chance to win $45

C Violation of compounding and coalescing as A=C and B=D

Experiment 4 part 1 Choose between: E: 25% chance to win $240 and 75% chance to lose $760 F. 25% chance to win $250 and 75% chance to lose $750

Experiment 4, part 2 Two problems: Choose between A: A sure gain of $240 B: 25% chance to gain $1000 and 75% chance to gain nothing Also, choose between C:A sure loss of $750 D: 75% chance to lose $1000 and 25% chance to lose nothing