Rational Choice Theory I: The Foundations of the Theory Benjamin Ferguson Administrative Keep in mind that your second papers are due this coming Friday at midnight. They should be emailed to me, roughly 2,500 words long. Many of you indicated that you were happy with my teaching on the class surveys, so thank you! But you also gave lower marks for how satisfied you were with your own input into the class. This fact, coupled with the nature of the material for this term means that I ll be giving you some exercises to do for class in place of some of the reading questions. Mareile Drechsler will be teaching one class for me this term, but the week is not set yet. There are two teaching prizes awarded to GTAs each year, one by the Philosophy Department and one by the Student s Union. If you ve enjoyed my teaching this year, I d appreciate your nomination for these awards. Finally, there are two books you d do well to purchase for this term: 1) The Theory of Choice by Shaun Hargreaves-Heap et. al. Published in 1992 by Blackwell. 2) Economic Analysis, Moral Philosophy, and Public Policy by Hausman & McPherson. Published in 2006 by Cambridge University Press. Speak to me about where to find these and additional readings. 1 Introduction This term we will be studying Rational Choice Theory and its applications. This field covers roughly three subfields: Decision theory, which concerns itself with individual decisio ns, Game theory, which covers group decisions, and Social Choice theory, which covers social decisions as well as voting mechanisms. These three subfields will be the topic of the first half of the term. The second half is concerned with their application in Welfare economics and in the fields of normative and applied ethics. This week serves as the foundation for all three subfields. We ll be looking at what the Rational in Rational Choice Theory means. 1
2 Forms of Rationality Instrumental This form of rationality is concerned with means-ends reasoning and is the form of rationality we will be dealing with this term. Some philosophers and many economists take this to be the only form of rationality. Instrumental rationality is a non-substantive account of rationality that takes preferences as given and primative facts about individuals that are not subject to further analysis. Procedural Procedural rationality, also called, bounded rationality is really another form of instrumental rationality that takes into account what it is instrumentally rational for an agent to do, given that she is a non-ideal agent. Expressive This form of rationality is also called practical reasoning and it is a substantive view of rationality that is much broader than the instrumental rationality of choice theory. Expressive rationality concerns itself not only with means-ends reasoning but also with the value of the ends themselves. The following will help to illustrate the differences between expressive(practical) and instrumental rationality: When I found out that a famous philosopher smoked (many philosopher smoke actually) I commented to my friend Sebastian that his smoking was irrational. How can such a smart person smoke?, I asked. Sebastian replied that since As I didn t know the philosopher s preferences, I was in no position to say he was acting inconsistently and violating rationality. I said, no, the point is, he shouldn t have a desire to smoke, this desire (or preference for smoking over not smoking) itself was irrational. What was the cause of our disagreement? I had a substantive, practical account of rationality in mind, while Sebastian was speaking of purely instrumental rationality. But what would make smoking irrational, from an instrumental perspective? In order to answer this question we must examine what instrumental rationality is in more detail. 3 Choice Axioms The standard approach to instrumental rationality is axiomatic. Colloquially, we take rational choices to be consistent and preference maximizing. This means that agents act rationally if their choices do not contradict one another and when their choices are the best element in their choice set. More formally, these requirements are stated as axioms. We ll begin with the basic three axioms necessary for establishing a preference ordering in cases of choice under certainty. But first, very briefly, what do we mean choice under certainty? Well choices under certainty are choices made when we know all of the options. Contrast this with choice under risk, where we don t know the outcomes, but we know the probabilities with which they might occur. Finally, these both are different from uncertainty, where not even the probabilities are known. 2
3.1 Choice Under Certainty Reflexivity For any bundle, that bundle is at least as preferred as itself: x X x x (1) Often stated first, Reflexivity seems a bit trivial, but many relations are not reflexive. Loves, for example, is not always reflexive (taken in a romantic sense it is hardly ever reflexive). Completeness For any two bundles in the choice set, the agent can say whether one is weakly preferred to the other, or if they are indifferent between the two: xy X x y or y x (2) Note the difference between indifference and having no preference. I am indifferent between diet Coke and diet Pepsi, but I simply cannot assign a preference to the prospect of either my father or mother being killed I m not indifferent, I have no preference. In this case, I am violating completeness. Completeness is often taken to be innocuous in his Microeconomics textbook for graduates, Hal Varion says [Completeness] just says that any two bundles can be compared. This is not correct. Completeness says that all bundles in the choice set have been compared and the agent can assign either a weak preference or indifference to each. Transitivity For any three bundles x, y, and z in the choice set, if the agent weakly prefers x to y and weakly prefers y to z, then the agent weakly prefers x to z: xyz X (x y) (y z) (x z) (3) There are challenges to both completeness and transitivity that we will discuss this term and that you can read about on your own, but one of the common defenses of transitivity is that intransitive agents can be money pumped: that is, their preferences are such that they can be induced to pay a small amount to move from x to y, a small amount to move from y to z and a small amount to move from z to x; once back at x the cycle can begin again. With the addition of the continuity axiom, we can obtain an ordinal utility function that represents an agent s preferences on an ordinal (ordered) scale. Continuity Pick any bundle from your choice set. The sets of bundles that are at least as good as this bundle and no better than this bundle are closed sets. y X {x x y}, {x x y} are closed sets. (4) Continuity is merely a formal axiom and we ll not discuss challenges it may face. Thus, to be rational when making choices under certainty means that an agent s preferences can be represented by an ordinal utility function (i.e. OUF Instrumentally Rational). 3
3.2 Choice Under Risk Often, we cannot make choices under certainty. Things in life are a gamble, although often we know, at least to some degree, what kind of gamble they involve. In order to capture choices that occur under risk, the decision theory captured by (1) to (4) above must be extended. This extension is known as Expected Utility Theory (EUT) and has a number of formulations, but we ll stick pretty closely to Savage s (1954). Independence (Strong) If the agent prefers x to y regardless of whether A or A occurs, then he prefers x to y. xya X(x y A) or (y x A) (5) The independence axiom (called the sure-thing principle in Savage) is one of the most criticized axioms in decision theory and is the source of problems for a number of well-known paradoxes in decision theory as well. Savage s gives the following story as an illustration. If I would invest in a company regardless of whether the outcome of the next election was a Republican or Democratic president, then I should invest in the company. Preference Increasing with Probability If the probability of a preferred outcome within a prospect increases while the probability of the inferior falls, the prospect improves. x y ; y 1 = (x, y; p 1, 1 p 1 ) ; y 2 = (x, y; p 2, 1 p 2 ) y 1 y 2 p 1 > p 2 (6) Reduction of Compound Lotteries Agents should be indifferent between two lotteries with the same probability of winning and the same prize for winning. lotteries y 1 and y 2 and numbers 0 a, b, c, d 1, if d = ab + (1 a)c, then: L(a, L(b, x, y), L(c, x, y)) L(d, x, y)(7) This axiom introduces consequentialism to economic theory. It says that the way outcomes are arrived at should not matter to the agent, all that matters is the end result. If an agents preferences satisfy all seven axioims, then they can be represented by a cardinal utility function that represents not only the order of preferences, but their magnitudes. This utility function is unique up to positive affine (linear) transformations. That is, the zero point has no real meaning and the utility can be shifted with linear transformations. One more thing... There is an important distinction in decision theory between normative and descriptive decision theory. Normative theory is what we are talking about. It tells us what ideally rational agents should choose. Descriptive theory tells us what ordinary human beings like ourselves actually choose. Expected utility theory is a normative theory. 4
4 Challenges To The Axioms Allais Paradox The Allais paradox presents two choices between two lotteries, with Tickets numbered 1 to 100. The prizes (in dollars) won when a given ticket is drawn are presented below for lotteries A and B: Lottery (1 to 33) (34) (35 to 100) A 2500 0 2400 B 2400 2400 2400 Now consider the choice between lotteries C and D, again with the prizes for each ticket listed below: Lottery (1 to 33) (34) (35 to 100) C 2500 0 0 D 2400 2400 0 Often people choose A in lottery 1 and D in lottery 2, which violates the independence axiom. The Allais Paradox was formulated by the economist Maurice Allais in the 1950s. What it shows is that people often violate the axioms of expected utility theory. Thus, in situations that induce Allaislike choices, economic models that assume people will choose consistently may be inaccurate. To return to our distinction between normative and descriptive theory, we can say that, at least in this case, the descriptive and normative come apart. What does this mean? Well, some have said that it means that expected utility theory is incorrectthat is, the normative theory is wrong. Others have said it shows we are imperfect choosersthat is, that we sometimes make irrational choices. But if your choice was inconsistent do you think it was irrational? Take a look at our tickets again. Knowing what you now know, would you choose differently? Ellsberg Paradox A second challenge to the sure-thing principle is the Ellsberg Paradox (Ellsberg 1961). In this thought experiment an individual is again given a choice between two sets of two lotteries; however, the probabilities associated with some outcomes are unknown. Imagine an urn containing 90 balls, 30 of which are red and the remaining 60 are some assortment of black and yellow balls. Action A gives you an outcome of 100 (dollars in Ellsbergs version) if a red ball is drawn and 0 for black or yellow. B offers 100 for a black ball and 0 for red or yellow. In the second set of options, C gives an outcome of 100 for red or yellow and 0 for black; D gives 100 for black or yellow and 0 for red. The choices are summarized below for lotteries A and B. Lottery Red (30) Black (?) Yellow (?) A 100 0 0 B 0 100 0 And here are lotteries C and D: 5
Lottery Red (30) Black (?) Yellow (?) A 100 0 100 B 0 100 100 Most people choose A and D. In the first set, A is chosen because there is certainty of a 1/3 chance of payoff of 100, but in B the chance of 100 is some unknown probability between 0 and 2/3. In the second set D is chosen over C because of the certainty provided by the 2/3 probability of 100 seems less volatile than the uncertainty of the probability range of 1/3 to 1 of 100 provided by C. When we apply the sure-thing principle we can ignore the yellow column (in figure 2) since it is the same for A and B (0,0) and C and D (100,100). Looking only at red and black, we should choose the outcome we think is most likely to occur so if we believe red is most likely we should choose A, if black then B. Savages theory allows individuals to subjectively define the probability of events and this probability is revealed in choices of this kind, but when A and D are chosen a contradiction is produced (Hargreaves-Heap et al, p. 46). The choice of A in the first set shows that the individual believes red has a higher probability of occurring, while the choice of D in the second set shows they believe black to be more likely. Therefore, red is both more likely (in the first set of lotteries) and not more likely (in the second) to be drawn. This choice behavior means that you must inevitably be violating some of the Savage axioms (specificallycomplete ordering of actions or the Sure-thing Principle) (Ellsberg, p.651). The approach to probability employed by Savage starts from the notion that gambling choices are influenced by, or reflect, differing degrees of belief, this approach sets out to infer those beliefs from the actual choices, but in the case of the Ellsberg Paradox, these revealed choices cannot tell us anything about the belief of the individual (Ellsberg 645). As a result if choices are contradictory in the Ellsberg Paradox, then it would follow that there would be simply no way to infer meaningful probabilities for those events from their choices (Ellsberg, p.646). Newcomb s Paradox This is a paradox that motivated causal decision theory and it is presented quite clearly in the Hargreaves-Heap text on pp. 340 342, so I will not discuss it here, but do have a look. 6