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.
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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.
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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.
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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.
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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).
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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 ; y1 = (x, y; p1 , 1 − p1 ) ; y2 = (x, y; p2 , 1 − p2 ) → y1 y2 ↔ p1 > p2
(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 y1 and y2 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.
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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
A
B
(1 to 33)
2500
2400
(34)
0
2400
(35 to 100)
2400
2400
Now consider the choice between lotteries C and D, again with the prizes
for each ticket listed below:
Lottery
C
D
(1 to 33)
2500
2400
(34)
0
2400
(35 to 100)
0
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
A
B
Red (30)
100
0
And here are lotteries C and D:
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Black (?)
0
100
Yellow (?)
0
0
Lottery
A
B
Red (30)
100
0
Black (?)
0
100
Yellow (?)
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.
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