(Advanced) Decision Theory
Miguel A. Ballester
Universitat Autònoma de Barcelona (BGSE, MOVE)
2011-2012, IDEA program
Course Evaluation
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Problem Set
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Report
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Paper
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Take-Home Exam
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Participation, including paper presentation
Problem Set
Exercises appear in the slides. Some of them (if not all !!) require
conceptual discussion. Each exercise must be written in at most
one page (one side).
Due Time: December 7th, 8 pm (box or e-mailed)
Report
List of possible papers to report in the slides. There is a group of
six papers with close discussions. The first five must be reported
by two students that can discuss the paper but have to report
individually. The last paper will be reported by a single student
(notice however that the author of this paper is available for
discussion at the department). Use a maximum of three pages
(one side) for your reports.
Paper Assignment: November 7th, 11.10 am (students will
come with a proposal to class. If the assignment fails to satisfy the
conditions above, I will impose an assignment)
Due Time: November 28th, 8 pm (box or e-mailed)
Paper
Students will divide in groups of 3/4 to write a paper proposal on
the topics of the course. Group Formation: November 7th,
11.10 am (students will come with a proposal to class. If the
assignment fails to satisfy the conditions above, I will impose an
assignment). Each group will use a maximum of four pages (one
side) for the paper. The paper has to contain:
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A novel question in decision theory.
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The model to address it.
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Conjectures or Hypothetical results (credible!).
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Interpretation and Conclusions.
First Proposal Limit: November 30th, during the day (office)
Due time: December 21st, before presentations that will
start at 10 am
Take-Home Exam
Students will be given few questions involving between half a day
and one day of work. They will be e-mailed December 19th, 8
pm, and after solved individually, they must be handed back (box
or e-mailed) before December 21st, 9 am (please notice this
coincides with the presentations day).
Participation
Participate, you are here because you want !
Introduction
What is Decision Theory? What is Behavioral Economics?
Introduction
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A theoretical/descriptive research field aiming to help
understanding and driving economic phenomena through the
understanding of individual decisions?
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A theoretical/descriptive research field in the service of
(individual and collective) normative economics by helping us
to understand how individuals make decisions?
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A purely normative enterprise whose goal is to help individual
decision makers pursue their own goals?
How people make/should make decisions?
Introduction
Paraphrasing Gilboa:
Decision theory appears to be at a crossroad, in more sense than
one. Over half a century after the defining and, for many years,
definitive works of the founding fathers, the field seems to be
asking very basic questions regarding its goal and purpose, its main
questions and their answers, as well as its methods of research.
This soul searching is partly due to empirical and experimental
failures of the classical theory. Partly, it is the natural development
of a successful discipline, which attempts to refine its answers and
finds that it also has to define its questions better. In any event,
there is no denying that since the 1950s decision theory has not
been as active as it is now.
Introduction
Homo Oeconomicus
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Rationality as Maximization: max U(·)
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P
Consistency in dealing with uncertainty:
p(z)U(z)
P t
Consistency in dealing with time:
ρ U(xt )
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Choice from Menus. maxx∈A U(x)
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Non-Altruistic Preferences: Ui (xi )
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Game Theoretical considerations
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Advanced Decision Theory and Economics: Experimental
Economics, Game Theory, I.O., Macroeconomics...
ADVANCED DECISION THEORY
I Riskless Choice**
II Choice and Uncertainty*
III Menu Choice*
Complement with Time Preferences (Temporal inconsistencies,
hyperbolic discounting...)
I RISKLESS CHOICE
1
2
3
4
5
6
7
8
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Standard Individual Decision-Making
Basic Violations of Consistency
Reference-dependent models
Sequential models
Other Multicriteria models
Search models
A classification of cyclical models
Measuring Consistency
Welfare analysis
1 STANDARD INDIVIDUAL DECISION-MAKING
1.1
1.2
1.3
1.4
Preference and Utility
Choice Behavior
Rationalizability in a consumer framework
Rationalizability in a general framework
1.1 Preference and Utility
Preferences/Maximization
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Preference over alternatives, as binary relations R ⊆ X × X .
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Greatest Alternatives {y ∈ A : yRx for all x ∈ A}
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Undominated Alternatives
{y ∈ A : for all x ∈ A, not (xPy )} = max(A, R)
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Consistency on preferences: xRy , yRz ⇒ xRz (transitivity),
absence of cycles with some strict relation (acyclicity),
comparability (completeness), . . .
1.1 Preference and Utility
Utility/Maximization
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Utility Function u : X → R
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Maximization: max u(·) and arg max u(·)
1.1 Preference and Utility
A utility function u : X → R represents the preference R if, for all
x, y ∈ X : xRy ⇔ u(x) ≥ u(y )
Representation problem: Which preferences can be represented?
The finite and countable case. Transitivity (and completeness) as
the only requirement. Hint: measure the lower contour sets of
alternatives by counting or by weighting-counting each of the
alternatives which is inferior
X
U(xj ) =
1k
k:xj Pxk
U(xj ) =
X
k:xj Pxk
1
( )k
2
1.1 Preference and Utility
The general case: Transitivity (and completeness) is not enough.
Let X be the unit square, that is, X = [0, 1] × [0, 1]. Let x %k y if
xk ≥ yk . The lexicographic preferences %L induced from %1 and
%2 are:
(a1, a2) %L (b1, b2) if a1 > b1 or both a1 = b1 and a2 ≥ b2
1.1 Preference and Utility
Proposition The lexicographic preference relation %L on
[0, 1] × [0, 1] does not have a utility representation.
Proof: Assume, by contradiction, that u : X → R represents %L .
For any a ∈ [0, 1], it is (a, 1) L (a, 0). Hence, it must be
u(a, 1) > u(a, 0) and there exists a rational number q(a) in the
interval (u(a, 0), u(a, 1)). The function q : [0, 1] → Q should be a
one-to-one function because if b > a, then (b, 0) L (a, 1) and
hence, u(b, 0) > u(a, 1) and q(b) > q(a). But the cardinality of
the rational numbers is lower than that of the continuum, a
contradiction.
The general case: A preference can be represented if and only if
there exists a countable dense set
Continuous representations: Debreu (1954), Eilenberg (1941),
Rader (1963)
1.2 Choice Behavior
Samuelson (1938): I propose, therefore, that we start anew in
direct attack upon the problem, dropping off the last vestiges of
the utility analysis. This does not preclude the introduction of
utility by any who may care to do so, nor will it contradict the
results attained by use of related constructs. It is merely that the
analysis can be carried on more directly, and from a different set of
postulates. All that follows shall relate to an idealised individual
not necessarily, however, the rational homo-economicus.
I assume in the beginning as known, i.e., empirically determinable
under ideal conditions, the amounts of n economic goods which
will be purchased per unit time by an individual faced with the
prices of these goods and with a given total expenditure. It is
assumed that prices are taken as given parameters not subject to
influence by the individual.
1.2 Choice Behavior
Samuelson identifies choice as a more fundamental notion than
preference mostly because the latter is not observable while the
former is. A natural question thus arises:
Rationalizability problem: Can we uncover the utility/preferences
of a decision maker by observing the choices she makes when
choosing a commodity bundle from her budget set?
1.2 Choice Behavior
Choice Behavior Let X be a set of alternatives. A choice behavior
is a mapping c : Ω ⊆ 2X \ ∅ → X such that c(A) ∈ A. The
elements of Ω are called menus.
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Consumer Choice: X ≡ Rn , Ω is a set of compact sets defined
n
X
pi xi ≤ m.
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Universal Domain: X is a general set of alternatives,
Ω = 2X \ ∅
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Binary Domain: X is a general set of alternatives, Ω subsets
of two alternatives.
i=1
Choice correspondences versus choice functions.
1.3 Rationalizability in a consumer framework
Rationalizability: Existence of a preference relation over bundles
such that xt is the maximal/greatest bundle achievable at prices pt .
Key observation by Samuelson: . . . if an individual selects batch
one over batch two, he does not at the same time select two over
one.
A preference for the first batch is said to be revealed. The logic
behind this terminology is straightforward: A decision maker who
views an alternative as superior to another will never choose the
latter when the former is available (Stochastic Choice !)
Revealed Preference Given some vectors of prices and chosen
bundles (p t , x t ) for t = 1, . . . , T we say x t is directly revealed
preferred to a bundle x (written x t RD x) if p t x t ≥ p t x.
Weak Axiom of Revealed Preference If x t RD x s then it is not
the case that x s RD x t . Algebraically, p t x t ≥ p t x s implies
ps x s < ps x t .
1.3 Rationalizability in a consumer framework
Samuelson (1948) offered a graphical proof of rationalizability with
two commodities under the Weak Axiom. Houthakker (1950)
offered a formal proof for any dimension by using the Strong
Axiom.
Indirect Revealed Preference We say x t is (indirectly) revealed
preferred to x (written x t RI x) if there is some sequence
r , s, t, . . . , u such that p r x r ≥ p r x s , p s x s ≥ p s x t , . . . , p u x u ≥ p u x
The relation RI is the transitive closure of the relation RD .
Strong Axiom of Revealed Preference If x t RI x s then it is not
the case that x s RI x t .
Generalized Axiom of Revealed Preference If x t RI x s then
ps x s ≤ ps x t
Exercise 1: We can just rewrite SARP in the form...if x t RI x s then
it is not the case that x s RD x t . Why? Can you think of other
alternative ways to write this axiom?
1.3 Rationalizability in a consumer framework
Rose (1958) offered a formal argument that the Strong Axiom and
the Weak Axiom were equivalent in two dimensions, providing a
rigorous proof for Samuelson’s earlier graphical exposition.
Afriat (1967) contributed by showing a general version admitting
indifferences (GARP) and a constructive method for the utility
function.
1.3 Rationalizability in a consumer framework
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Varian (2005)
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Rubinstein (2007)
1.4 Rationalizability in a general framework
Rationalizability A choice function c is rationalizable if there
exists a well-behaved preference P such that c(A) = max(A, P).
1.4 Rationalizability in a general framework
Revealed Preference (choice functions) Alternative x is
revealed better than alternative y if y ∈ A and c(A) = x.
Weak Axiom of Revealed Preference (choice functions) If
{x, y } ∈ A ∩ B and x = c(A) then it is not the case that y = c(B).
Chernoff Condition (α condition or IIA). If x ∈ S ⊆ T and
x = c(T ) then it must be the case that x = c(S).
See Chernoff (1954) and Sen (1969, 1971). For a simple survey on
the classical ideas, see Moulin (1984).
1.4 Rationalizability in a general framework
Proposition: The following statements are equivalent (in the
universal domain):
1. A choice behavior c is rationalizable
2. c satisfies WARP
3. c satisfies the α property.
1.4 Rationalizability in a general framework
Exercise 2: Check whether WARP is sufficient to guarantee
rationalizability in a domain Ω which is not universal (a domain
that contains only some but not all menus). Formulate the Strong
Axiom of Revealed Preference in this general framework and check
whether it is sufficient to guarantee rationalizability.
1.4 Rationalizability in a general framework
Proof: 1 ⇒ 2. Let P the preference that rationalizes c. Consider
two menus A and B and two alternatives {x, y } ⊆ A ∩ B. If
x = c(A) then, x = max(A, P). In particular, xPy . Hence,
y 6= max(B, P) = c(B).
2 ⇒ 3. Let x ∈ S ⊆ T and x = c(T ). If y = c(S) with y 6= x, we
would have a violation of WARP involving menus S and T and
alternatives x, y .
3 ⇒ 1. Define the binary revealed preference xPB y ⇔ x = c(xy ).
We show that PB is transitive. Suppose xPB y and yPB z and
consider the set {x, y , z}. It cannot be c(xyz) = y as this is a
contradiction with c(xy ) = x. It cannot be c(xyz) = z as this is a
contradiction with c(yz) = y . Hence, c(xyz) = x. By the α
property, c(xz) = x and thus xPB z. Clearly, PB rationalizes c.
1.4 Rationalizability in a general framework
Expansion x = c(A) and x = c(B) imply x = c(A ∪ B).
Always Chosen x = c(xy1 ), . . . , x = c(xyk ) imply
x = c(xy1 . . . yk ).
1.4 Rationalizability in a general framework
Proposition: A choice behavior is rationalizable if and only if: (i)
the Binary Revealed Preference is acyclical and (ii) it satisfies the
always chosen property.
Proof: We only need to prove the sufficiency part. Since the
choice behavior does not contain any binary cycle, finiteness
guarantees that PB is well-behaved. Let A be a subset of
alternatives, and consider max(A, PB ). This element dominates
any other alternative in A and hence, it is chosen in all binary
problems. By the always chosen property, we are done.
See Tyson (2008) for extra discussions on the Binary Revealed
Preference and refinements of rationalizability.
2. BASIC VIOLATIONS OF CONSISTENCY
2.1
2.2
2.3
2.4
2.5
Existence of Binary Cycles
Status Quo and Endowment Effect
Attraction Effect
Compromise Effect
Other Effects
2.1 Existence of Binary Cycles
The cycle of order three is well-known in voting, being called
Condorcet Cycle. In essence, majority voting over the pairs of
alternatives in a triple does not entail necessarily a transitive
relation.
This natural violation of transitivity may appear in many choice
situations in which aggregation of criteria sounds reasonable. No
cyclical choice is observed when choosing from amounts of cash !
Consider an agent that studies a collection of items by focusing on
a group of characteristics/components common to all these items.
Circularities thus may arise if items are ordered in conflicting ways
according to each component.
2.1 Existence of Binary Cycles
May (Econometrica 1954). College students face the choice of
three hypothetical marriage partners diferring in intelligence, looks
and wealth.
y
z
Very Intelligent
Intelligent
Fairly Intelligent
Plain Looking
Very Good Looking
Good Looking
Well-off
Poor
Rich
x
2.1 Existence of Binary Cycles
In a very simple presentation, 27 percent of students presented
cyclical preferences of the type xPB yPB zPB x.
The intransitive pattern is easily explained as the result of choosing
the alternative that is superior in two out of three criteria.
Other cyclical experiments: Tversky (1969), Loomes, Starmer and
Sugden (1991), Roelofsma and Read (2000), 20-50 percent of
cyclical choice.
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Waite (2001). Birds prefer to take 1 raisin from a 28cm tube
to 2 raisins from a 42cm tube, and the latter to 3 raisins from
a 56cm tube. However, they choose the last confronted to the
first.
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Female Lions always choose the biggest male lion (compatible
with transitivity of choice in cash environments).
2.2 Status Quo and Endowment Effect
Experimental evidence has revealed that Individuals are reluctant
to make changes in their current status (often called Status Quo
Bias). Consider the following hypothetical description of three
professional positions.
x
y
z
Very Stimulating Work
Fairly Stimulating Work
Stimulating Work
Pleasant Life
Very Pleasant Life
Fairly Pleasant Life
2.2 Status Quo and Endowment Effect
While having no particular work (studying), the agent declares an
informal preference for position y over position x.
The agent gets position z. After some time working, she receives
an offer to work in position y . It comprises a much better quality
of life but a less stimulating work. Doubting on it, the agent
rejects the offer.
The agent receives an offer to work in position x. The offer, being
better than the original position in all dimensions, is accepted.
The previous example conforms a cycle among alternatives x, y , z.
2.2 Status Quo and Endowment Effect
Samuelson and Zeckhauser (1988) constitutes the first and widest
analysis of SQB.
Each question begins with a brief description of an individual, a
manager, or a government policymaker, followed by a set of
mutually exclusive alternative actions from which to choose. The
subject plays the role of the decision maker and in many of the
decisions, one alternative occupies the status quo position
(externally framed or self-selected in a prior stage).
Presence of (statistically significant) status quo bias.
2.2 Status Quo and Endowment Effect
Knetsch (1989) provides the simplest experiment on SQB.
Mug, 76
Candy, 87
choice, 55
Mug
89 percent
10 percent
56 percent
Candy
11 percent
90 percent
44 percent
2.2 Status Quo and Endowment Effect
See Masatlioglu and Uler (2011) for a modern experiment that also
compares explanatory theories.
These phenomena are also observed in real markets by means of
field studies:
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Insurance: Madrian and Shea (2003)
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Organ donations: Johnson and Goldstein (2003)
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Residential Electric Service: Hartman, Doane and Woo (1991)
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Religious Choices: Chaves and Montgomery (1996)
2.2 Status Quo and Endowment Effect
Individuals are likely to value an alternative they possess more than
one that they do not (often called the Endowment Effect). The
usual consequence of the endowment effect is the willingness to
accept (WTA)/willingness to pay (WTP) gap, i.e., the fact that
WTA is larger than WTP. Consider the following hypothetical
situation
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A person moves to a new apartment and finds there an old
drawing. She is offered 100 euros for the drawing and rejects
the offer, thus revealing that x = (1, I )PB y = (0, I + 100).
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If the person had moved to the new apartment and found 100
euros, she would have wished to buy the drawing for no more
than 50 euros in the paintings shop, thus revealing that
y = (0, I + 100)PB z = (1, I + 30).
The previous example conforms a cycle among alternatives x, y , z.
2.2 Status Quo and Endowment Effect
Thaler (1980) provides evidence of the Endowment Effect, mainly
on lottery choices. Kahneman, Knetsch and Thaler (1990) show
similar results for a trading experiment. For the most basic
consumption setup on bundles, see for instance Bateman et al.
(1997, 2005).
2.2 Status Quo and Endowment Effect
Suggested explanations for these effects:
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Experimental Misconceptions: Plott and Zeiler (2005, 2007,
2011).
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Rational explanations: Transaction costs, uncertainty.
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Cognitive Misperceptions: Perceived losses are more relevant
than perceived gains, switching from the SQ may generate a
loss in some attribute/value, or a mere loss. Similar to
anchoring process where we adapt views of the world relative
to the current position but in an insufficient amount.
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Psychological Commitment: sunk costs as relevant even if this
is not rational, to justify previous commitments or
investments, i.e., people like to think their decisions were
correct or avoid regret in future moments about past decisions
or feel in control of their actions and thus, avoid difficult
choices of any kind.
2.2 Status Quo and Endowment Effect
Exercise 3: Read in detail one of the recent suggested
empirical/experimental papers (last 20 years) and describe in detail
one of the cycles that could appear as a consequence of the
irrational behavior analyzed there. Might you suggest a rational
explanation for the cycle? Alternatively, might you suggest a
plausible cognitive failure that explains the cycle?
Exercise 4: Think of other real life example fitting these effects
and look for empirical evidence or literature supporting the
alternative example.
2.3 Attraction Effect
The attraction effect refers to the ability of an asymmetrically
dominated or relatively inferior alternative, when added to a set, to
increase the choice probability of the dominating alternative.
For a first observation, see Huber, Payne and Puto (1983).
Students are asked to choose products (different categories), in
sets of two or three. The alternatives in each category represent a
target, a competitor and a decoy, that differ in several dimensions.
The target and the competitor do not dominate each other, but
the target dominates the decoy, and thus, a stimulus for the target.
The effect of the decoy is tested by checking the percentages in
which target and competitor are chosen, with and without decoy.
A greater percentage of target under the decoy treatment would
suggest that some students fall in the attraction effect. They
would choose the competitor in the two alternative case but the
target in the three alternative case.
2.3 Attraction Effect
It is observed an increase in Target Demand from 50 percent to 59
percent.
The effect is stronger when the decoy shares with the target the
dimension in which the target is better, and less important if the
decoy is mainly dominated in terms of the dimension in which the
target is worse.
2.3 Attraction Effect
The attraction effect constitutes a violation of always chosen. Let
x, y and z, where the target y clearly dominates the decoy z and
hence yPB z. The competitor x is perceived as better than the
target and the decoy, xPB y and xPB z. Hence, no cyclical choice is
present.
However, the target is chosen when both the competitor and the
decoy are present, i.e., c(xyz) = y .
See also Shafir, Simonson and Tversky (1993).
2.4 Compromise Effect
The compromise effect refers to the ability of an extreme (but not
inferior) alternative, when added to a set, to increase the choice
probability of an intermediate alternative.
For a first observation, see Simonson (1989). The following is an
example from Herne (1997).
The alternatives were imaginary policy proposals concerning
various topics from everyday politics (such as economic conditions,
social security, taxation, environmental questions). Respondents
were asked to make their choices as a member of parliament, as a
member of the local council, as a voter in a referendum, etc.
The amount of people who made a choice for the compromise
option is smaller in the 3 alternative set versus the 2 alternative
set. However, the percentage of people who selected the
compromise alternative among those who did not select the new
alternative increased from 64 to 75 percent.
2.4 Compromise Effect
The compromise effect can operate as the attraction effect or in a
more crude way. Consider x, y and z, where the intermediate
alternative y is never chosen against the extreme alternatives x
and z. Hence, no cyclical choice is present. However, the
intermediate alternative is chosen when both extremes are present,
i.e., c(xyz) = y . This is also called a difficult choice.
2.5 Other Effects
Several empirical papers have reported on order effects in panel
decisions in contests such as the World Figure Skating Competition
(Bruine de Bruin 2005), the International Synchronized Swimming
Competition (Wilson 1977), the Eurovision Song Contest (Bruine
de Bruin 2005) and the Queen Elisabeth Contest for violin and
piano (Glejser and Heyndels 2001).
Other empirical observations on how the alternatives are
considered in an structured way, and the models that explain this
can be seen in Caplin, Dean and Martin (Forthcoming). Consider
that, in general, Presentation/Description of the alternatives
(framing effect) has been deeply analyzed.
3. REFERENCE DEPENDENT MODELS
3.1
3.2
3.3
3.4
3.5
Loss Aversion and Constant Loss Aversion models
Anchored Preferences
Status Quo Bias
Endogenous version: Default Choice
Other Models
3.1 Loss Aversion
Tversy and Kahneman (1991) introduce the first
Reference-dependent model where losses and gains are defined
relative to a reference-point.
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Primitives are preferences over consumption bundles with n
dimensions.
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For each reference point r, which is a consumption bundle,
the decision-maker has preferences %r , and they assume the
existence of Ur
P
Ur (x) = i gi (ui (xi ) − ui (ri )).
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Concavity of gi for a > 0. Diminishing sensitivity for gains.
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Convexity of gi for a < 0. Diminishing sensitivity for losses.
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gi (0) = 0 and gi (a) < gi (−a) captures loss aversion.
3.1 Loss Aversion
The theory is useful in a context of uncertainty (Prospect Theory),
where there is one dimension, money. In a multidimensional
general setting, however, it allows for very controversial behaviors.
In particular, Reference-dependent choice cycles are permitted.
(Munro and Sugden, 2003): An agent may have preferences of the
form y %x x, z %y y and x %z z, allowing for money pumps.
3.1 Constant Loss Aversion
Munro and Sugden (2003): Constant Loss-Aversion model
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gi is linear both on gains and losses.
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gi (a) < gi (−a) captures loss aversion.
Exercise 5: Exemplify with a two-dimensional setup why loss
aversion models may present reference-dependent cycles. Show
that, if g1 and g2 are linear, this is not possible.
3.2 Anchored preferences
Sagi (2006): A model of Reference-dependent preferences for
lotteries, alternative to prospect theory. It can be adapted to
riskless choice.
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Primitives are preferences.
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Ur (x) = inf u∈∆ u(x) − u(r ), where ∆ is a set of real,
continuous and bounded utility functions on X .
3.2 Anchored preferences
A graphical comparison of models.
3.3 Status Quo Bias
Masatlioglu and Ok (2005) present a model of choice with SQB.
They differentiate decisions where the SQ is present and decisions
where there is not a SQ.
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The latter are solved in a standard way, by maximizing a
standard preference/utility over the set of alternatives.
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The former are solved by considering the set of alternatives
that dominate the SQ in all attributes. If this set is empty, the
agent sticks at the SQ. If the set is non-empty, the agent
maximizes the standard preference/utility over such a set.
The model assumes the possibility of observing the Status Quo of
the agent, and thus, deals with data of the form c(A, x) where A is
the menu, x is the status quo (if exists) and c(A, x) is the choice.
3.3 Status Quo Bias
SQ WARP If {x, y } ∈ A ∩ B and x = c(A, z) then it is not the
case that y = c(B, z).
SQ Independence If x 6= c(T , x) for any {x} ⊂ T ⊆ S, then
c(S, x) = c(S, ).
SQ Bias If y = c(S, x) then y = c(S, y ).
3.3 Status Quo Bias
Theorem. A Choice Model with Status Quo satisfies WARP, SQI
and SQB if and only if there exists a positive integer q, an injective
function u : X → Rq and a strictly increasing map f : u(X ) → R
such that:
(
c(S, x) =
x
arg
max
if {y ∈ S : u(y ) > u(x)} = ∅,
f (u(z)) otherwise
z∈{y ∈S:u(y )>u(x)}
3.3 Status Quo Bias
Proof:
Step 1: Define the binary revealed preference when the status quo
is discarded (thus, clearly inferior even with the status quo plus).
That is, x % y ⇔ x = c(xy , y ). This relation is transitive.
Let x % y and y % z. Then x = c(xy , y ) and y = c(yz, z). By
WARP, the second equality guarantees that z cannot be the
chosen alternative in (xyz, z). By WARP, the first equality
guarantees that y cannot be the chosen alternative in (xyz, y ). By
SQB, y cannot be the chosen alternative in (xyz, z). Hence,
c(xyz, z) = x and by WARP, c(xz, z) = z. Thus, x % z.
Step 2. Consider all linear orders extending %, P1 , . . . , Pn . We can
represent them by a vector of utility functions u = (u1 , . . . , un ).
3.3 Status Quo Bias
Step 3. Define a transitive and complete preference P over all the
alternatives by using the choices without status quo, c(·, ).
Represent them by a utility function v .
Step 4. v is monotone on u.
Suppose u(x) > u(y ). Then, it must be x % y and hence,
x = c(xy , y ). By SQB, it must be x = c(xy , ). Hence
v (x) > v (y ). We can thus write v = f (u) in a monotone way.
3.3 Status Quo Bias
Step 5. Final proof.
Suppose first that {y ∈ S : u(y ) > u(x)} = ∅. Then, by definition,
x = c(xy , x) for all y ∈ S, and by WARP, it must be x = c(S, x),
as desired.
Suppose now that {y ∈ S : u(y ) > u(x)} = A 6= ∅. Consider the
set A ∪ {x}. For any subset T , it must be c(T , x) 6= x since x is
dominated by all elements in A. Hence, by SQI, it must be
c(A ∪ {x}, x) = c(A ∪ {x}, ) = arg maxz∈{y ∈S:u(y )>u(x)} f (u(z)),
as desired.
3.4 Endogenous version: Default Choice
Reference-Dependent models are usually exogenous, in the sense
that the reference point is an exogenous parameter.
Testing these models require thus to obtain data both on the set of
available options and the reference point at the moment of choice.
The validity of this assumption depends on the particular
interpretation of the reference point (status quo, default option,
etc)
3.4 Endogenous version: Default Choice
The simplest endogenous version of Masatlioglu and Ok’s model
can be seen in Apesteguia and Ballester (2011a):
Default Choice: A choice function c is a default choice, DC, if
there exists an element d ∈ X , a positive integer q, an injective
function u : X → Rq and a strictly increasing map f : u(X ) → R
such that:
I
For all S ⊆ X with d 6∈ S, it is c(S) = argmaxx∈S f (u(x)).
I
For all S ⊆ X with d ∈ S it is:
d
c(S) =
arg maxy ∈S∩X d f (u(y ))
with X d = {x ∈ X : u(x) > u(d)}
if S ∩ X d = ∅,
if S ∩ X d 6= ∅.
3.4 Endogenous version: Default Choice
DC satisfy Expansion
Proof: Let x be chosen in sets S and T . If x = d, then it
dominates each of the alternatives in at least one attribute. Hence,
it would be chosen in S ∪ T . If x is not the Default, and
d 6∈ S ∪ T , standard maximization satisfies Expansion. Finally, if x
is not the Default but d ∈ S ∪ T , then the chosen element in
S ∪ T is the alternative maximizing f (u) in (S ∪ T ) ∩ X d , call it,
z. Since x ∈ (S ∪ T ) ∩ X d it must be f (u(z)) ≥ f (u(x)). Let
wlog, z ∈ S. Then {x, z} ∈ S ∩ X d and hence, f (u(x)) ≥ f (u(z).
It can only be z = x, as desired.
3.4 Endogenous version: Default Choice
DC satisfy Cyclical Choice Consistency If we observe two
cyclical chains xPB yPB zPB x and aPB bPB cPB a, then there must
exist t ∈ {x, y , z} ∩ {a, b, c} such that c(xyz) and c(abc) are the
alternatives that dominate t.
Proposition A choice function c is a DC if and only if it satisfies
Expansion and Cyclical Choice Consistency.
3.4 Endogenous version: Default Choice
Proof: If there are no binary cycles, since c satisfies AC, it must
be rationalizable. Hence, we can simply find a preference P and a
utility representation u for which the result holds.
If there are binary cycles, cyclical consistency guarantees that there
exists one alternative d present in all of them and such that the
choice in the triple is the alternative dominating d.
Consider the sets A = {x ∈ X : xPB d}, B = {x ∈ X : dPB x}.
3.4 Endogenous version: Default Choice
Proof: Define P1 on X \ d as xPi y ⇔ xPB y . Complete it by
placing d on the bottom of the preference.
Place AP2 dP2 B, with P2 on A as xPi y ⇔ xPB y and P2 on B as
xPi y ⇔ xPB y .
Represent Pi with the lower contour sets utility function ui . Finally
define f (u(x)) = u1 (x) + u2 (x). Also notice that A = X d for the
given representation u.
3.4 Endogenous version: Default Choice
If d 6∈ S, S presents no cycle and AC guarantees that the
alternative selected is the maximal one according to PB or P1 .
Given the structure of f , this is the element that maximizes f , as
desired.
If d ∈ S and dPB x for all x ∈ S, then by AC, d = c(S). This is
merely what we desire as A ∩ S = ∅.
If d ∈ S and A ∩ S 6= ∅, we need to prove that c(A) is the maximal
alternative in A = X d according to f (u). To prove it, consider that
maximal alternative z and any other y ∈ S. If zPB y , given that we
also have zPB d we can use AC to derive z = c(zyd). If yPB z it
must be y 6∈ A, because in the set A, f (u) is merely equivalent to
u1 or u2 and z was the maximizer. Hence, we know that
zPB dPB yPB z and by cyclical consistency, it must be z = c(zyd).
The union of all triples zyd is merely S and using expansion, we
have c(S) = z, as desired.
3.4 Endogenous version: Default Choice
SQB
Cycles
SQ cyles
Attraction
Difficult
NO
YES
NO
NO
Exercise 6: Consider three alternatives x, y and z, and a Status
Quo Model with two utility functions u1 and u2 (two attributes or
two selves). Prove formally that the attraction and the difficult
choice effects cannot happen. Hint: Notice that this has been
already proved in a more general form before, I am just asking to
prove it in a different way using the model directly!
3.5 Other models
I
A Model of Reference-Dependent Preferences (Koszegi and
Rabin, 2006): lotteries and expectations.
I
A theory of reference-dependent behavior (Apesteguia and
Ballester, 2009): Bridge between choice and preference.
Rationalizability.
I
Money matters: an axiomatic theory of the endowment effect
(Giraud, 2011): A monetary model to analyze axiomatically
the endowment effect (WTA-WTP).
I
Extended SQB: Extension of Rational SQB paper, not only
Pareto domination (Masatlioglu and Ok, 2011). Endogenous
extended version in Ok, Ortoleva and Riella (2011).
I
Aspiration levels: External Reference-Dependent behavior.
Guney, Richter and Tsur (2011).
4. SEQUENTIAL MODELS
4.1
4.2
4.3
4.4
4.5
Elimination by Aspects and Lexicographic Semiorders
Sequential Rationalizability
Consideration Sets
Rationalization and Categorize Then Choose
Other Models
4.1 Elimination by Aspects and Lexicographic Semiorders
I
Elimination by Aspects (Tversky 1972): The set of
alternatives is narrowed down sequentially by dropping inferior
alternatives with respect to some aspect/attribute. Random
model of choice.
I
Lexicographic Semiorders (Tversky 1969): Preference is
generated by the sequential application of numerical criteria,
by declaring an alternative x better than an alternative y if
the first criterion that distinguishes between x and y ranks x
higher than y by an amount exceeding a fixed threshold.
Model of preferences based on the semiorder concept (see
Luce 1956).
4.2 Sequential Rationalizability
I
The DM applies a number of criteria (incomplete binary
relations) in a fixed order of priority, gradually narrowing down
the set of alternatives, until one is identified as the choice
I
Same criteria applied in the same fixed order to every choice
problem
I
Individual choice: multiple criteria or selves, orderly applied
I
Collective choice: refining the set of efficient allocations with
a fairness notion
4.2 Sequential Rationalizability
Manzini and Mariotti (2007):
Sequential Rationalizability by Asymmetric Rationales
(SR(As)): A choice function c is sequentially rationalizable by
asymmetric rationales or simply SR(As) whenever there exists a
non-empty ordered list {P1 , . . . , PK } of asymmetric rationales on
X such that c(A) = M1K (A) for all A ∈ P(X ), where
M1K (A) = M(M(. . . M(M(A, P1 ), P2 ), . . . , PK −1 ), PK )
4.2 Sequential Rationalizability
SR satisfy Always Chosen
Proof: Let x be chosen among alternatives y1 , . . . , yk . We prove
that x ∈ M1j ({x, y1 , . . . , yk }) for every j inductively. It is obvious
for j = 1, since otherwise yi P1 x for some i and hence, x 6= c(xyi ).
If x ∈ M1t ({x, y1 , . . . , yk }), we prove that
x ∈ M1t+1 ({x, y1 , . . . , yk }). Suppose that there exists
yi ∈ {x, y1 , . . . , yk } such that yi Pt+1 x. Then, since x = c(xyi ),
there must exist p < t + 1 such that xPp yi and hence,
yi 6∈ M1t ({x, y1 , . . . , yk }). This proves that
x ∈ M1t+1 ({x, y1 , . . . , yk }).
4.2 Sequential Rationalizability
The case of two rationales.
Weak WARP If {x, y } ∈ B ⊆ A and x = c(xy ) = c(A) then it is
not the case that y = c(B).
Proposition A choice function c is a 2-SR if and only if it satisfies
Expansion and Weak WARP.
4.2 Sequential Rationalizability
Proof: xP1 y if not yPD x and xP2 y if xPB y .
Let c(S) = x. Clearly, there does not exist w ∈ S such that wP1 x,
since xPD w thanks to set S. Then x survives the first round.
We now prove that for any y ∈ S such that yP2 x, y is eliminated
in the first round. Otherwise, for all z ∈ S \ y , there exists Tyz
such that c(Tyz ) = y . By Expansion, y = c(∪Tyz ). Since
{x, y } ⊂ S ⊂ ∪Tyz , Weak WARP guarantees that x 6= c(S),
absurd. Hence, y survives the second round too.
If some alternative y survives the first round, we know that
x = c(xy ) and thus, xP2 y . Hence, x is the only alternative in
M12 (S), as desired.
4.2 Sequential Rationalizability
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
NO
YES
YES
YES
NO
NO
NO
NO
Exercise 7: Consider three alternatives x, y and z, and a Rational
Shortlist Method (two attributes or two selves). Prove formally
that the attraction and the difficult choice effects cannot happen.
Hint: Notice that, again, this has been already proved in a more
general form before !
4.2 Sequential Rationalizability
I
Choice by Sequential Procedures (Apesteguia and Ballester
2011a). All Sequential procedures are equivalent if they
impose acyclicity to each attribute. Manzini and Mariotti’s
assymetric version is more general. Is it credible?
I
Lexicographic compositions of two criteria for decision making
(Houy and Tadenuma, 2009). Other Sequential Compositions.
I
Manipulation of Choice Behavior (Manzini, Mariotti and
Tyson, 2011). How can this behavior be manipulated?
4.3 Consideration Sets
Choice with Limited Attention (Masatlioglu, Nakajima and Ozbay,
forthcoming)
I The revealed argument relies on the implicit assumption that
a DM considers all feasible alternatives. Without the full
consideration assumption, the standard revealed preference
can be misleading. It is possible that the DM prefers x to y
but she chooses y when x is present simply because she does
not realize that x is also available.
I The marketing literature calls the set of alternatives to which
a DM pays attention in her choice process consideration set.
Due to cognitive limitations, DMs cannot pay attention to all
the available alternatives.
I The common property in the formation of consideration sets is
that it is unaffected when an alternative she does not pay
attention to becomes unavailable (attention filters).
I The agent proceeds in two-stages, by maximizing a
well-behaved preference over the alternatives in her
consideration set, and only these.
4.3 Consideration Sets
Attention Filter A consideration set mapping Γ : 2X \ ∅ → 2X \ ∅,
with Γ(A) ⊆ A is an attention filter if for any menu S,
Γ(S) = Γ(S \ x) whenever x 6∈ Γ(S).
Choice with Limited Attention A choice function c is a choice
with limited attention if there exists a preference P and an
attention filter Γ such that c(A) = M(P, Γ(A)).
4.3 Consideration Sets
Choice with limited attention are not necessarily SR choices. There
are attention filters that do not correspond to a maximizing
process.
I
Elements that are the best for one particular characteristic.
I
Elements that are the most popular in the set according to
some characteristics.
Exercise 8: Consider a set of four alternatives {a, b, c, d}. Define a
choice function in the universal domain that: (1) it is not a
rational shortlist method, (2) it is a Limited Attention model.
Which of the properties of a Rational Shortlist Method is not
satisfied? Give an intuitive interpretation to the Attention Filter
that you have defined.
4.3 Consideration Sets
I
Limited Attention model allows for cycles. Let xPyPz and
Γ(xz) = z is the only problem with partial attention. Clearly
xPB yPB zPB x.
I
Limited Attention model allows for the attraction effect. Let
xPyPz and Γ(xyz) = yz the only problem with partial
attention. Clearly c(xyz) = y and the binary revealed
preference is consistent with P.
I
Limited Attention model allows for Difficult Choices. Let
xPyPz and Γ(xy ) = y , Γ(xz) = z the only problems with
partial attention. Clearly, c(xyz) = x and the binary revealed
preference is yPB zPB x as in the compromise effect.
4.3 Consideration Sets
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
NO
YES
YES
YES
YES
YES
NO
NO
YES
NO
NO
YES
4.3 Consideration Sets
I
When More is Less: Choice by Limited Consideration (J.
Lleras, Y. Masatlioglu, D. Nakajima and E. Ozbay, 2011).
Variant.
I
Limited information and advertising in the US personal
computer industry, Goeree (2008). Awareness of a fraction of
products.
I
Consideration sets and competitive Marketing (K. Eliaz and
R. Spiegler, forthcoming). I.O. market implications, series of
papers.
I
A good application for Neuroeconomics ?! Neuroeconomics:
How neuroscience can inform economics (Camerer,
Loewenstein and Prelec, 2011)
4.4 Rationalization and Categorize Then Choose
Rationalization (Cherepanov, Feddersen and Sandroni, 2011)
I
In a first stage, a decision maker uses a set of
rationales/norms to determine a subset of alternatives, the
ones she can rationalize i.e., those that are optimal according
to at least one of her rationales.
I
Among the alternatives that she can justify according to some
rationale/norm, the agent maximizes a well-behaved
preference.
4.4 Rationalization and Categorize Then Choose
Categorize Then Choose (Manzini and Mariotti, forthcoming)
I
The first stage involves a coarse form of maximization, using a
binary relation (interpreted as a psychological shading
relation) defined on categories, namely sets of alternatives.
For instance, the presence of salad dishes in the menu shades
pasta dishes, or the presence of hamburgers shades other
types of sandwiches.
I
Then in the second stage the agent picks an alternative which
is preferred to all surviving alternatives.
4.4 Rationalization and Categorize Then Choose
I
Both models are equivalent in terms of explained behavior
(choice functions).
I
They can explain cycles and the attraction effect.
I
They cannot explain difficult choices. Suppose x = c(xz) and
y = c(yz) and reason according to rationalization. If zPx,
then z is dominated by x according to all
norms/rationalizations. Hence, it cannot be maximal in xyz
for any norm and z 6= c(xyz). It must be then xPz. Using the
same reasoning, yPz. Thus, in order to choose z from xyz we
should discard x and y . This means x and y are not maximal
for any norm and thus, z is maximal for all the norms. But
then z would be maximal for all the norms in xz and
z = c(xz) absurd.
4.4 Rationalization and Categorize Then Choose
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
R-CTC
NO
YES
YES
YES
YES
YES
YES
YES
NO
NO
YES
YES
NO
NO
YES
NO
5. OTHER MULTICRITERIA MODELS
5.1
5.2
5.3
5.4
5.5
Additive Difference Model
Rationalization by Multiple Rationales
Rationalization by Game Trees
Aggregation and Dual Aggregation
Other Models
5.1 Additive Difference Model
The Additive Model:
I
Preferences. Alternatives can be represented in an attribute
space Rn P
and there exist
P real functions f1 , . . . , fn such that
x % y ⇔ i fi (xi ) ≥ i fi (yi ).
I
Choices. Alternatives can be represented in an attribute space
[0, 1]n and thereP
exist real functions f1 , . . . , fn such that
c(A) = arg max i fi (x).
That model is equivalent to a rational individual.
5.1 Additive Difference Model
The Additive Difference Model: Alternatives can be represented in
an attribute space [0, 1]n and there exist
P functions g1 , . . . , gn with
gi (−a) = −gi (a) such that x % y ⇔ i gi (xi − yi ) ≥ 0.
This model guarantees transitivity only if gi are linear. To see this,
just notice that
X
gi (xi − yi ) ≥ 0 ⇔
i
X
λi (xi − yi ) ≥ 0 ⇔
i
⇔
X
λi xi ≥
i
which clearly leads to transitivity.
X
i
λi yi
5.1 Additive Difference Model
The general version may explain violations of rationality. However,
it can only be defined for pairs of alternatives. How to write a
reasonable choice version of this model to apply revealed analysis?
I
Reference-Dependence version. All alternatives are
P compared
to some alternative r in the set, receiving value i gi (xi − ri )
I
Aspiration level. All alternatives are compared to some
non-feasible ideal. For instance, ai = maxx∈A xi . That
resembles a regret model, though regret is a concept mostly
analyzed in decisions under uncertainty like Sugden (1993),
Sarver (2008) or Hayashi (2008).
5.2 Rationalization by Multiple Rationales
Kalai, Rubinstein and Spiegler (2002)
I
The choice set conveys information about its constituent
elements and given this information, the DM chooses what he
thinks is the best alternative.
I
The DM has in mind a partition of the set of menus and she
applies one ordering to each cell in the partition.
I
A cell is like a state of the world. The DM’s behavior is
rationalized after the state of the world is added to the
description of the alternatives.
5.2 Rationalization by Multiple Rationales
Rationalization by Multiple Rationales: A choice function c is
Rationalizable by Multiple Rationales whenever there exists a
non-empty collection {P1 , . . . , PK } of linear orders X such that for
all A ∈ P(X ), there exists i with c(A) = M(A, Pi ).
5.2 Rationalization by Multiple Rationales
Every Choice Function is Rationalizable by Multiple
Rationales
Define {P1 , . . . , Pn } as a collection of |X | = n linear orders, each
of them placing an alternative x ∈ X on top.
The authors propose to focus on the minimal explanation in terms
of number of rationales. The larger such a number, the less
meaningful is the rationalization by multiple rationales that can be
given to the choice behavior. For an application, see Crawford and
Pendakur (2010).
5.2 Rationalization by Multiple Rationales
Every Choice Function is Rationalizable by at most |X | − 1
Rationales
Let x an alternative with c(X ) 6= x and define {P1 , . . . , Pn−1 } as a
collection of |X | − 1 linear orders, each of them placing an
alternative y ∈ X \ {x} on top, and x as the second best
alternative. Let A be any menu. If c(A) 6= x, assign A to the
corresponding rationale with c(A) on top. If c(A) = x, it must be
A 6= X and hence, there exists y ∈ X \ A. Assign A to the
rationale with y on top.
The proportion of choice functions that can be rationalized
by less than |X | − 1 orderings tends to 0 as |X | tends to
infinity.
5.2 Rationalization by Multiple Rationales
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
R-CTC
RMR
NO
YES
YES
YES
YES
YES
YES
YES
YES
YES
NO
NO
YES
YES
YES
NO
NO
YES
NO
YES
Exercise 9: Consider the set of alternatives X = {1, 2, . . . , n} and
define a choice function implicitly by considering one of the
previous models of choice (please define the specification of the
model formally!). Construct the explanation of this choice function
with a minimal number of rationales using the idea of Kalai,
Rubinstein and Spiegler.
5.2 Rationalization by Multiple Rationales
Computational difficulties in explaining behavior. RMR model is
computationally complex in general.
I
Apesteguia and Ballester (2010)
I
DeMuynck (Forthcoming)
5.3 Rationalization by Game Trees
Xu and Zhou (2007). For an extension, see Horan (2011a).
I
The choices of the DM are the equilibrium outcome of an
extensive game with perfect information. The tree has
alternatives of X as terminal nodes, each alternative
appearing once and only once. Every node of the tree
represents one criterion.
I
Multiple selves competing between them
I
Collective choice; hierarchical decision-making
5.3 Rationalization by Game Trees
Rationalizability by Game Trees: A choice function c is
rationalizable by game trees whenever there exists an extensive
game with perfect information (G , P) where alternatives of X are
terminal nodes (each alternative appearing once and only once),
every node of the tree represents the decision of some agent, Pi ,
and c(A) = SPNE (G |A; P) for all A ⊆ X
5.3 Rationalization by Game Trees
I
I
RGT allows for cycles. Let agent 1 choose between x or yz,
while agent 2 chooses for yz. If zP1 xP1 y and yP2 z, we have
xPB yPB zPB x.
RGT does not explain the attraction effect or the compromise
effect. Suppose PB is well-behaved on a triple, with x over y
over z and consider c(xyz). If one agent is decisive over the
entire triple, then this agent has preferences xPi yPi z and
c(xyz) = x. Hence, there must exist one agent i splitting
{x, y , z} into a single option or a pair (for which another
agent will decide later). If the alternative splitted is x, we
know that xPi y and xPi z and hence this agent would choose
x as equilibrium. If the alternative splitted is z, we know that
xPi z and yPi z and hence this agent will choose xy . The
decision on xy we know is x and hence, x will be the
equilibrium. If the alternative splitted is y , we know that the
agent deciding on xz prefers x. Hence i will go for xz for a
resulting equilibrium of x.
5.3 Rationalization by Game Trees
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
R-CTC
RMR
RGT
NO
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
NO
NO
YES
YES
YES
NO
NO
NO
YES
NO
YES
NO
Exercise 10: Prove formally (Please be very precise in defining the
game theoretical tools that you need) that any individual or
collectivity following an RGT model satisfies Always Chosen.
5.4 Aggregation Models
Green and Hojman (2011) formulate an additive model with a
voting procedure.
I
The choices of the DM are the result of a voting/aggregation
of multiple criteria.
I
Multiple selves aggregated.
I
Collective choice; voting.
5.4 Aggregation Models
An aggregation-explanation of a choice function c is a pair (λ, v )
consisting of a distribution of probabilities of the different
preferences λ and a aggregation rule v such that c(A) = v (A, λ).
Any choice function can be explained using a monotonic
aggregator.
Dual Aggregation: A Dual aggregation/explanation of a choice
function c is a pair (λ = (λ1 , λ2 ), v ) consisting of a distribution of
probabilities of two possibly different preferences and an
aggregation rule v such that c(A) = v (A, λ). This can explain the
attraction effect but not the other violations.
5.4 Aggregation Models
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
R-CTC
RMR
RGT
DAGG
NO
YES
YES
YES
YES
YES
NO
YES
YES
YES
YES
YES
YES
NO
NO
NO
YES
YES
YES
NO
YES
NO
NO
YES
NO
YES
NO
NO
5.5 Other Models
Ambrus and Rozen (2011): An aggregation model, based on
preferences/utilities.
DeClipel and Eliaz (forthcoming): A bargaining model of two
selves.
6. SEARCH MODELS
6.1 Search and Report papers
6.2 Choice by sequential elimination
6.1 Search and Report papers
Simon (1955) considered a model, satisficing, having the following
two characteristics:
I
The examination of alternatives comes in an ordered,
structured way.
I
The individual chooses the first alternative over certain
threshold.
Papers to report analyze or extend the structure of the choice set
or the existence of a satisficing idea in models similar to the ones
analyzed in this course.
6.1 Search and Report papers
1. Search, Choice and Revealed Preference (Caplin and Dean
2011)
2. Choice by Iterative Search (Masatlioglu and Nakajima, 2011)
3. The satisficing Choice Procedure (Papi 2011)
4. Sequential Choice and Choice from Lists (Horan, 2011)
5. Choice over Lists (Rubinstein and Salant, 2006)
6. Choice in Ordered-Tree-Based Decision Problems (Mukherjee,
2011)
6.2 Choice by sequential elimination
I
The final choice is the surviving alternative of an elimination
process. The supermarket example, google lists, etc.
I
Individual choice: models of choice by ordered elimination.
The election in any menu of alternatives is determined, as in
the classical model, by binary comparisons. Instead of a
maximization process, there is an elimination heuristics.
Salant (2003), Rubinstein and Salant (2006), Apesteguia and
Ballester (2011a).
I
Collective choice and Political Environment: Voting by
successive elimination as in Dutta, Jackson and LeBreton
(2001, 2002) or Apesteguia, Ballester and Masatlioglu (2011).
Elimination of alternatives is done by majority voting.
6.2 Choice by Sequential Elimination
Apesteguia and Ballester (2011a), Apesteguia, Ballester and
Masatlioglu (2011)
Choice by sequential elimination: A choice function c is Choice
by Sequential Elimination whenever there exists a profile of
preferences P and a linear order < over the set of alternatives (an
agenda-list) such that for every A ∈ P(X ), c(A) is the alternative
surviving the majority process over the agenda-list <.
6.2 Choice by Sequential Elimination
I
Choice by sequential elimination satisfies Always Chosen
I
Choice by sequential elimination allows cycles (consider the
classical Condorcet Cycle).
6.3 Choice by Sequential Elimination
Cycles
SQ cyles
Attraction
Difficult
SQB
SR
LAt
R-CTC
RMR
RGT
DAGG
CSE
NO
YES
YES
YES
YES
YES
NO
YES
YES
YES
YES
YES
YES
YES
NO
YES
NO
NO
YES
YES
YES
NO
YES
NO
NO
NO
YES
NO
YES
NO
NO
NO
7. Classification of Cyclical Models
7.1 Examples
7.2 A Nested Classification of Cyclical Models
7.1 Examples
We have seen that the following models explain some sort of
cyclical behavior, but satisfy Always Chosen (and thus, do not
explain Attraction Effect or Compromise Effect).
I
Reference Dependent Model: DC
I
Sequential Model: SR
I
Multicriteria Model: RGT
I
Search Model: CSE
7.1 Examples
I
Default: Alternative x
I
Attribute 1: u 1 = (1, 2, 4, 9)
I
Attribute 2: u 2 = (7, 8, 8, 5)
Average: ū = (4, 5, 6, 7)
I
I
I
Base Relation
x `o
> yO
w
/z
c1 (x, y , w ) = y , c1 (x, z, w ) = c1 (x, y , z, w ) = z
7.1 Examples
I
Criterion 1: wP1 z and zP1 y
I
Criterion 2: zP2 yP2 x
Criterion 3: xP3 wP3 y
I
I
I
Base Relation
x `o
> yO
w
/z
c4 (x, y , w ) = w , c4 (x, z, w ) = c4 (x, y , z, w ) = x
7.1 Examples
P1
P2
x
I
I
I
P3
y
z
'
w
Agent 1: zP1 xP1 wP1 y
Agent 2: yP2 xP2 wP2 z
Agent 3: yP3 xP3 wP3 z
I
I
Base Relation
x `o
> yO
w
/z
c3 (x, y , w ) = c3 (x, y , z, w ) = w , c3 (x, z, w ) = x
7.1 Examples
I
Supermarket Products Ordered:
x
I
Criterion 1: wP1 yP1 zP1 x
I
Criterion 2: xP2 wP2 zP2 y
Criterion 3: zP3 yP3 xP3 w
I
I
I
Base Relation
/y
/z
x `o
> yO
w
/z
/w
c2 (x, y , w ) = c2 (x, z, w ) = c2 (x, y , z, w ) = w
7.2 A Nested Classification of Cyclical Models
Proposition 1
C DC ⊂ C CSE
7.2 A Nested Classification of Cyclical Models
Proof (inclusion):
I
Default: x
I
Base Relation
x `o
w
I
> yO
/z
c1 (x, y , w ) = y , c1 (x, z, w ) = c1 (x, y , z, w ) = z
I
I
Construct P to explain the Base Relation (McKelvey 195..)
To explain the first choice, we need to put alternative y after
alternatives x and w . Similarly, for the second choice we need
to put alternative z after alternatives x and w . For instance,
agenda x < w < y < z. General Agenda: default < Worse
than default < Better than default
7.2 A Nested Classification of Cyclical Models
Proof (strict):
I
Base Relation
x `o
w
I
> yO
/z
c2 (x, y , w ) = c2 (x, z, w ) = c2 (x, y , z, w ) = w
I
Cycle x, y , w determines as default alternative y . However,
this is incompatible with the existence of the Cycle x, z, w .
7.2 A Nested Classification of Cyclical Models
Proposition 2
C CSE ⊂ C GT
7.2 A Nested Classification of Cyclical Models
Proof (inclusion):
I
Political Agenda:
I
Base Relation
x `o
w
I
/z
/y
x
/w
> yO
/z
c2 (x, y , w ) = c2 (x, z, w ) = c2 (x, y , z, w ) = w
7.2 A Nested Classification of Cyclical Models
Proof (inclusion):
I
P1
P2
P3
x
w
z
y
General Game with the same structure
I
Pi consistent with the base relation. In the example, for
instance xP1 wP2 (y , z)
7.2 A Nested Classification of Cyclical Models
Proof (strict):
I
Base Relation
x `o
w
I
> yO
/z
c3 (x, y , w ) = c3 (x, y , z, w ) = w , c3 (x, z, w ) = x
I
Cycle x, y , w determines that alternative w comes after x in
the agenda. However, Cycle x, z, w determines that alternative
x comes after w in the agenda.
7.2 A Nested Classification of Cyclical Models
Proposition 3
C GT ⊂ C SR
7.2 A Nested Classification of Cyclical Models
Proof (inclusion):
I
P1
P2
P3
x
I
Base Relation
x `o
w
I
y
z
'
w
> yO
/z
c3 (x, y , w ) = c3 (x, y , z, w ) = w , c3 (x, z, w ) = x
7.2 A Nested Classification of Cyclical Models
Proof (inclusion):
First, decisions where players 2 and 3 will be decisive: yP1 x, wP2 z.
Then, decisions where player 1 is decisive in any order, since only
two alternatives will remain at this point.
I
Order linearly the players in the game, starting from the lower
part of the hierarchy
I
Associate to each player all binary comparisons for which they
are decisive
I
Order linearly the binary comparisons respecting the order of
players and respecting some specific order regarding the
preference of the player involved
I
Construct a rationale where only one binary comparison is
executed, according to the base relation
7.2 A Nested Classification of Cyclical Models
Proof (strict):
I Base Relation
x `o
> yO
I
/z
w
c4 (x, y , w ) = w , c4 (x, z, w ) = c4 (x, y , z, w ) = x
I
Cycle x, y , w determines that alternative w appears in a
different subgame than x, y . Cycle x, z, w determines that
alternative x appears in a different subgame than x, y . Then,
it must be
P1
P2
P3
x
y
z
'
w
However, since yP2 x, c4 (x, y , z, w ) cannot be explained
7.2 A Nested Classification of Cyclical Models
C DC ⊂ C CSE ⊂ C GT ⊂ C SR
Exercise 11: Prove formally that every choice function generated
by a Rational Shortlist Method is indeed a Default Choice. Find a
Default Choice function that cannot be explained sequentially by
only two rationales. Given that C DC ⊂ C SR , you can explain it with
more. Can you do it with only three? Can you explain any Default
Choice function with a maximum of three rationales? How?
Exercise 12: Pick up one of the other models and analyze how
general it is in relation to the cyclical models. Which of the
cyclical models are just particular cases of the one you have
selected? Why?
8. MEASUREMENT OF CONSISTENCY
8.1 Classical Measures
8.2 Minimal Indices
8.3 A characterization of minimal indices
8.1 Classical Measures
Two main features of the maximization principle in the classical
theory of choice are:
I
it provides a simple and versatile account of individual
behavior, and
I
a tool for individual welfare analysis
However, how severe the observed deviations are from that simple
description?
8.1 Classical Measures
Applied work has tried to applied the idea of rationalizability since
long ago. See Koo (1963) for an interesting analysis of household
consumption. To just mention some experimental works that use
these classical measures, see
I
Sippel (1997): Laboratory experiment.
I
Harbaugh, Krase and Berry (2001): Rationality of children
I
Andreoni and Miller (2002, 2008): Consistency on Altruist
behavior.
I
Chen, Lakshminarayanan, and Santos (2006): Rationality of
tufted capuchin monkeys.
I
Choi et al. (2007): Portfolio Decisions
I
Choi et al. (2011): Linking Rationality and Sociodemographic
conditions.
8.1 Classical Measures
A common framework for all measures by just allowing the same
menu observed several times.
I
An observation (A, a) consists of a non-empty menu of
alternatives A ⊆ X and an element a that is chosen from A.
Observations: O
I
A collection of observations is a mapping f : O → Z+ .
Data: F
I
An inconsistency index is a mapping I : F → R+ that
measures how inconsistent any collection of observations is.
8.1 Classical Measures
Afriat (1973): To measure the amount of relative wealth
adjustment required in each budget constraint to avoid violations
of the maximization principle.
Revealed preference at e: xRfφ,e y ⇔ ∃A : f (A, x) >
0 and y can be bought with a proportion of income e
8.1 Classical Measures
Afriat can be reinterpreted as an attention concept, in a general
framework:
I
An attention mapping φ assigns to every menu A the
alternatives in A that are considered at the attention level
e ∈ [0, 1].
1. φ(A, 0) = ∅
2. φ(A, 1) = A, and
3. φ(A, e) is increasing in e.
I
Revealed preference at e:
xRfφ,e y ⇔ ∃A : f (A, x) > 0 and y ∈ φ(A, e)
8.1 Classical Measures
IAf (f ) =
e:Rfφ,e
(1 − e).
inf
is acyclic
1. Allows for exogenous information on the plausibility of
violations.
2. Does not consider number of violations.
3. Does not allow for endogenous information (preferences) on
the plausibility of violations.
8.1 Classical Measures
I
Varian (1982) proposes a nonparametric approach for the
estimation of demand functions and later in this line of
research, Varian (1990) refines Afriat’s measure of
inconsistency by considering potentially different levels of
attention in the different observations e = {e(A,a) }
xRfφ,e y ⇔ ∃A : f (A, x) > 0 and y ∈ φ(A, e(A,x) )
I
Varian is interested in the vector of attention levels e that is
closest to 1 and respects rationalizability
IV (f ) =
e:Rfφ,e
inf
is acyclic
X
(A,a)
f (A, a)(1 − e(A,a) ).
8.1 Classical Measures
Houtman and Maks (1985) suggest to compute the maximal
subset of observations that is consistent with revealed preference.
In other words, inconsistencies are measured by the minimal
subset of observations that needs to be eliminated from the data
in order to make the remainder rationalizable.
IM (f ) =
g ≤f :f −g
min
is rationalizable
X
g (A, a).
(A,a)
1. Does not allow for exogenous information on the plausibility
of violations.
2. Considers the number of violations.
3. Does not allow for endogenous information (preferences) on
the plausibility of violations.
Banker and Maindiratta (1988) extend on this approach and Dean
and Caplin (2011) discuss a quicker algorithm for computing that
maximal subset.
8.1 Classical Measures
I
Rationality has also been measured by counting the number of
times in the data a consistency property is violated (see, e.g.,
Swofford and Whitney, 1985 or Famulari 1995).
I
Echenique, Lee and Shum (2010) make use of the monetary
structure of budget sets to suggest a version of this notion,
the money pump index, that captures also the severity of each
violation.
8.2 Minimal Indices
Apesteguia and Ballester (2011b)
A weighting function is a mapping w : P × O → R+ such that
w (P, A, a) = 0 if and only if a = m(P, A). That is, observations
that are explained by the preference P receive a null
weight/inconsistency, while any other observation receives a
positive one.
Iw (f ) = min
P∈P
X
(A,a)
f (A, a)w (P, A, a).
8.2 Minimal Indices
I
Proposition 1: IM is a minimal index.
I
Proposition 2: IV is a minimal index.
I
Minimal SwapsP
Index:
IS (f ) = minP∈P (A,a) f (A, a)|{x ∈ A : xPa}|.
I
Minimal Loss Index:
P
IL (f ) = minP∈P (A,a) f (A, a)(um(P,A)
− uba(P) ).
b
Exercise 13: Can you describe which are the weights that we need
to consider for understanding Houtman-Maks and Varian as
minimal indices? Can you explain with a simple example why
counting the number of cycles is NOT a minimal index?
8.2 Minimal Indices
Exercise 14: Consider a set X = {1, 2, . . . , n} of cake pieces,
ordered from biggest to smallest, where n is an even positive
integer. Define, on the universal domain, the glutton-educated
choice behavior as pick up the second largest piece of cake. Is this
rationalizable? Which of the effects described in chapter 2 are
present in this behavior? Analyze whether this choice behavior
satisfies each of the properties defined in these notes.
Exercise 15: Compute the inconsistency of the glutton-educated
choice behavior for Houtman-Maks, the number of violations of
WARP and the swaps index.
8.3 A characterization of minimal indices
Denote by r ∈ R a rationalizable collection of observations where
all binary menus are observed. Given f and r , we say that f is
r -invariant if I (f ) = I (f + r ).
8.3 A characterization of minimal indices
Denote by r ∈ R a rationalizable collection of observations where
all binary menus are observed. Given f and r , we say that f is
r -invariant if I (f ) = I (f + r ).
1. Rationality (RAT): I (f ) = 0 if and only if f is rationalizable.
8.3 A characterization of minimal indices
Denote by r ∈ R a rationalizable collection of observations where
all binary menus are observed. Given f and r , we say that f is
r -invariant if I (f ) = I (f + r ).
1. Rationality (RAT): I (f ) = 0 if and only if f is rationalizable.
2. Invariance (INV): For every f , there exists r such that f is
r -invariant.
8.3 A characterization of minimal indices
Denote by r ∈ R a rationalizable collection of observations where
all binary menus are observed. Given f and r , we say that f is
r -invariant if I (f ) = I (f + r ).
1. Rationality (RAT): I (f ) = 0 if and only if f is rationalizable.
2. Invariance (INV): For every f , there exists r such that f is
r -invariant.
3. Attraction (ATTR): For every f and every r , there exists a
positive integer z such that f + zr is r -invariant.
8.3 A characterization of minimal indices
Denote by r ∈ R a rationalizable collection of observations where
all binary menus are observed. Given f and r , we say that f is
r -invariant if I (f ) = I (f + r ).
1. Rationality (RAT): I (f ) = 0 if and only if f is rationalizable.
2. Invariance (INV): For every f , there exists r such that f is
r -invariant.
3. Attraction (ATTR): For every f and every r , there exists a
positive integer z such that f + zr is r -invariant.
4. Separability (SEP): For any two collections of observations f
and g , I (f + g ) ≥ I (f ) + I (g ), with equality if and only f and
g are r -invariant for some r .
8.3 A characterization of minimal indices
Theorem 1: An inconsistency index I satisfies (RAT), (INV),
(ATT) and (SEP) if and only if it is a minimal index.
8.3 A characterization of minimal indices
1. Categorization: Relate f with the corresponding r (and P r ).
2. Separation: For certain constant znr that forces all collections
with at most n observations to bePr -invariant collections,
proceed to I (f ) = I (f + nznr r ) = (A,a) f (A, a)I (1(A,a) + znr r )
3. Representation: Let w (P r , A, a) = 0 whenever P r explains
(A, a), and w (P r , A, a) = I (1(A,a) + z1r r ) otherwise. Then
show that I (1(A,a) + z1r r ) = I (1(A,a) + znr r ).
0 consider g = f + zr 0 with z
4. Minimality: For any P 0 and rP
0
large enough. We know
P that (A,a) gr(A, a)w (P , A, a) =
I (g ) ≥ I (f ) + 0 = (A,a) f (A, a)w (P , A, a).
8.3 A characterization of minimal indices
Extra properties characterize the specific families/indices described
above.
I
Corollary 1: I satisfies (RAT), (INV), (ATTR), (SEP), (UC)
and (PI) if and only if I is a scalar transformation of a
Varian’s index.
I
Corollary 2: I satisfies (RAT), (INV), (ATTR), (SEP), (UC),
(PI) and (NEU) if and only if I is a scalar transformation of
the minimal inconsistent subset index.
I
Corollary 3: I satisfies (RAT), (INV), (ATTR), (SEP), (UC),
(NEU) and (DC) if and only if it is a scalar transformation of
the minimal swaps index.
I
Corollary 4: I satisfies (RAT), (INV), (ATTR), (SEP),
(NEU) and (COM) if and only if it is a minimal loss index.
9. WELFARE ANALYSIS
9.1 A neutral approach using Pareto
9.2 The non-neutral approach
9.3 A neutral approach using measures of rationality
9.1 A neutral approach using Pareto
Two main features of the maximization principle in the classical
theory of choice are:
I
it provides a simple and versatile account of individual
behavior, and
I
a tool for individual welfare analysis
how to extract relevant information from the choices of the
individual to do welfare analysis?
9.1 A neutral approach using Pareto
Accepting the welfare information involved in individual
preferences, we may approach the second question in two different
ways:
I
The neutral approach: We have limited information about the
behavioral violations of the consistency conditions and hence,
we cannot adopt one specific model as reflecting individual
behavior. We can only assume there are several
framings/ancillary conditions that determine different
preferences/choices of the individual, or the preferences of the
individual are subject to some variability across problems.
From that, we can try to extract welfare implications.
I
The non-neutral approach: We have specific information that
the agent proceeds in a specific behavioral way. Under some
behavioral processes, there is an identifiable preference that
we can intuitively use as the preference of the individual.
9.1 A neutral approach using Pareto
Bernheim and Rangel (2010): Choice with ancillary conditions.
Individuals can be inconsistent across choice problems and also in
the same problem if operating under different framings/ancillary
conditions.
We can declare x better than y if for every choice problem and for
every ancillary condition, y is not directly revealed preferred to x.
9.1 A neutral approach using Pareto
Limitation of the approach:
We make very limited judgments on individual welfare, and reject
many information available (as the likelihood of different ancillary
conditions or the consistency of the individual within ancillary
conditions, which is treated as a yes/no binary issue merely.
9.2 The non-neutral approach
Koszegi and Rabin (2007), Rubinstein and Salant (2011): The use
of the underlying behavioral process is inexcusable.
The real preference of the agent can contradict the revealed
preference.
9.2 The non-neutral approach
An example of non-neutral analysis. Limited Attention.
Remember that Limited Attention model allows for cycles. Let
xPyPz and Γ(xz) = z is the only problem with partial attention.
Clearly xPB yPB zPB x.
However, this another agent zP 0 xP 0 y overlooking z in xyz and yz
makes the same choices.
But these are the only two possible cases. Thus, we know for sure
that x is preferred to y !
9.2 The non-neutral approach
Proposition Under the limited attention model, we can claim xPy
if and only if there exists T such that c(T ) = x 6= c(T \ y ).
Proof: Since c(T ) 6= c(T \ y ), we know for sure that y was
considered in T . Since c(T ) = x, it must be xPy .
Now we show that we can complete such P in any way. Let be
any completion of P. Define Γ(S) = {x ∈ S : c(S) x} ∪ c(S).
Clearly, c(S) is the maximal element according to in Γ(S). Also,
Γ is an attention filter, since y 6 inΓ(S) means y x and by
construction, not xPy . Hence, c(S) = c(S \ y ) and hence
Γ(S) = Γ(S \ y ).
Exercise 16: Check whether the educated-glutton behavior can be
explained by some of the models in these notes. If it can be,
describe the welfare implications of accepting the corresponding
model (which would be the real preference of the agent).
9.3 A neutral (?!) approach using measures of rationality
Apesteguia and Ballester (2011b), minimal indices.
Wonder about which preference relation approximates best the
choices, and the extent of such an approximation:
I
the latter provides a measure of inconsistency/rationality as
described before, and
I
the former a tool for the welfare analysis of possibly
inconsistent individuals
9.3 A neutral (?!) approach using measures of rationality
Computation of P ∗ is a complex matter in general. Strategies:
1. Identification of the optimal solution through well-known
quick algorithms/techniques. Dynamic programming, branch
and bound methods, etc.
2. Identification of a suboptimal solution.
3. Specific domains that allow easy computations.
9.3 A neutral (?!) approach using measures of rationality
Algorithms and Suboptimality:
I
Dean and Martin (2011) connect Houtman-Maks to the
Minimum Set Covering Problem, and using this relation, they
provide a quick algorithm for computing the optimal
preference.
I
Apesteguia and Ballester (2011b) connect the Minimal Swaps
index to the (integer) Linear Order Problem. For a study and
a series of applications of the LOP problem, see Brusco, Kohn
and Stahl (2008). The most relevant application to economics
is related to the triangularization of input-output matrices
(see for instance Fukui, 1986). Connecting LOP and the
swaps index allows for quick algorithms and also, identification
of suboptimal solutions.
9.3 A neutral (?!) approach using measures of rationality
The integer LOP problem over the set of vertices X and directed
weighted edges that connect all vertices x and y with (integer)
cost cxy consists of finding the linear order relation over the set of
vertices that maximizes or minimizes the total aggregated cost.
That is, if we denote by Π the set of all mappings from X to
{1, 2, . . . , k}, the LOP involves solving
arg min
π∈Π
X
π(x)<π(y )
cxy
9.3 A neutral (?!) approach using measures of rationality
Theorem:
1. For any collection of observations f one can define a LOP
with vertices in X , the solution of which provides the optimal
preference for the minimal swaps index.
2. For any LOP with vertices in X one can define a collection of
observations f , its optimal preference being the solution to
the LOP.
Exercise 17: Consider again the glutton-educated behavior.
Imagine that each menu has a different ancilliary condition. Which
is the Bernheim-Rangel incomplete preference over the
alternatives? Which is the optimal preference for the
Houtman-Maks and the Swaps indices?
9.3 A neutral (?!) approach using measures of rationality
Specific Domains:
We say that a collection of observations f is balanced if all the
menus of alternatives of the same cardinality are observed the
same number of P
times. That is, |A| = |B| ⇒
P
f
(A,
x)
=
x∈A
y ∈B f (B, y ) (universal domain, binary
domain...)
Given a collection f , a basic preference relation P B (f ) is any
preference
relation such
P
P that
B
(A,a):a=x f (A, a) >
(A,a):a=y f (A, a) ⇒ xP (f )y . Clearly, such
a preference relation is extremely easy to compute. Hence, the
following result is of substantial interest.
9.3 A neutral (?!) approach using measures of rationality
Theorem: For any balanced collection of observations f , and for
any vector u, P is an optimal preference relation for the minimal
loss index if and only if P is a basic preference relation for f .
II Choice under Uncertainty
10.
11.
12.
13.
14.
Objective and Subjective Uncertainty
Experimental Observations
Objective Uncertainty and Sophisticated Behavior
Choquet EU Model
Multiple Priors
10. Objective and Subjective Uncertainty
10.1 VonNeumann and Morgenstern
10.2 Anscombe and Aumann
10.3 Savage
10.1 VonNeumann and Morgenstern
Von Neumann and Morgenstern (1947)
Let L the set of all probability measures on a (finite) prize set Z . A
binary relation % on L satisfies
I
independence/substitution, i.e., for any p, q, r ∈ L and any
α ∈ (0, 1), p % q ⇔ αp ⊕ (1 − α)r % αq ⊕ (1 − α)r .
I
archimedean/continuity, i.e., if p q then there are
neighborhoods B(p), B(q) such that for all
p 0 ∈ B(p), q 0 ∈ B(q), we have p 0 q 0 .
if and only if there are numbers v (z)z∈Z such that
p % q ⇔ U(p)
X
z∈Z
p(z)v (z) ≥ U(q) =
X
q(z)v (z)
z∈Z
See for instance Rubinstein (2007) or any basic textbook!
10.2 Anscombe and Aumann
Anscombe and Aumann (1963)
I
When the preference is defined over roulettes, P, preference
satisfies the VNM axioms.
I
Horse lotteries (H) are defined by [p1 , . . . , ps ], with each state
providing a roulette lottery.
I
Let P ∗ be the set of roulette lotteries with prizes being horse
lotteries in H.
I
When the preference is defined over these compounded
roulette lotteries, the preference satisfies the VNM axioms.
I
Monotonicity and Reversal connect the two systems of
preferences.
10.2 Anscombe and Aumann
THEOREM is a binary relation on P ∗ satisfying the axioms if
and only if there exists a unique set of s non-negative numbers
π1 , . . . , πs summing to 1, such that
u ∗ [p1 , ., ps ] = π1 u(p1 ) + . . . πs u(ps )
10.3 Savage
Savage (1954)
Let S be the set of states and X be the set of consequences. Acts
are the collection F of mappings from S to X and is a binary
relation on F .
Ordering is a preference relation.
Sure-Thing Principle For all f , f 0 , g , g 0 and A ⊆ S, if f = f 0 ,
g = g 0 on A, and f = g , f 0 = g 0 on Ac , then f g if and only if
f 0 g 0.
Conditional Preferences make sense thanks to Axiom 2, by
equating the outcomes in the complementary event. An event A is
null if f ∼ g whenever f = g on Ac .
10.3 Savage
Eventwise Monotonicity For all f , g ∈ F , x, y ∈ X and non-null
A ⊆ S, if f = x and g = y on A, then f g given A if and only if
x y.
Conditional Preferences respect preferences over constant acts.
Weak Comparative Probability For all f , f 0 , g , g 0 ∈ F ,
x, y , x 0 , y 0 ∈ X , A, B ⊆ X , if
I
x y, x0 y0
I
f = x on A, f = y on Ac , g = x on B, g = y on B c , and
I
f 0 = x 0 on A, f 0 = y 0 on Ac , g 0 = x 0 on B, g 0 = y 0 on B c
then f g if and only if f 0 g 0 .
Thanks to this axiom, we are able to define a qualitative
probability relation on events. An event A is more likely than an
event B if we prefer the good prize on A.
10.3 Savage
Nondegeneracy There exist x, y ∈ X such that x y .
Small Event Continuity For all f ∈ F , x ∈ X , there exists a finite
set of events {A1 , . . . , An } forming a partition of S such that: (1)
for all i, f g 0 with g 0 = x on Ai and g 0 = g otherwise, and (2)
for all j, f 0 g , with f 0 = x on Aj and f 0 = f otherwise.
10.3 Savage
THEOREM A binary relation on F satisfies (Ord), (STP), (EM),
(WCP), (Non) and (SEC) if and only if there exists a unique,
finitely additive (non-atomic) probability measure µ on S and a
state independent utility u on X such that finite outcome acts are
ranked according to:
Z
v (f ) =
u(f (s))dµ(s) =
n
X
u(xi )µ(f −1 (xi ))
i=1
This result characterizes probabilistically sophisticated
non-expected utility maximizers. These preferences are often
referred to as yielding a separation of preferences from beliefs.
11. Experimental Observations 11.1 Allais’ Paradox
11.2 Ellsberg’s Paradox
11.1 Allais’ Paradox
f1
f2
f3
f4
1
1M
0
1M
0
2 − 11
1M
5M
1M
5M
12 − 100
1M
1M
0
0
For most decision makers, f1 f2 but f4 f3 .
11.1 Allais’ Paradox
The paradox can be easily discussed for any model of uncertainty.
Under any reasonable assumption, Allais is describing roulettes. No
sensible individual would derive subjective probabilities other than
the objective ones. Hence, the paradox seems to rely on the
non-expected utility maximizer nature of agents.
This paradox would be resolved just by considering individuals who:
I
Face objective uncertainty and then rank the lotteries not in
an expected utility form, or
I
Even if they face subjective uncertainty, form sophisticated
beliefs (assign probabilities to states) and then rank the
lotteries induced from the acts, but not in an expected utility
form.
Exercise 18. Prove formally that a VNM individual facing the
objective urn, or any Savage type of individual cannot behave
according to the Allais paradox. In the latter case, discuss which
axioms in Savage’s theory are key for this observation.
11.1 Allais’ Paradox
I
Camerer (1995): Survey on Experimental Observations.
I
Starmer (2000): Survey on Experimental/Theoretical research.
11.2 Ellsberg’s Paradox
g1
g2
g3
g4
Red(1/3)
100
0
100
0
Black
0
100
0
100
Yellow
0
0
100
100
For most decision makers, g1 g2 but g4 g3 .
11.2 Ellsberg’s Paradox
Difference with Allais.
I
Comparisons between the acts in the Ellsberg example involve
only exchanging a pair of outcomes, thus concern only beliefs
and not the risk preferences !
Comparisons between acts in the Allais paradox concern also
the risk preferences of the agent. Specically, it requires the
agent to compare getting 1M with sometimes getting 0 and
sometimes getting 5M. Probabilistic Sophisticated
Non-Maximizers may therefore incur in Allais paradox, but not
in Ellsberg, as they build well-behaved beliefs.
Exercise 19. Prove formally that a Savage type of individual with
any beliefs cannot behave according to the Ellsberg paradox.
Discuss which axioms in Savage’s theory are key for this
observation.
11.2 Ellsberg’s Paradox
I
Camerer (1995): Survey on Experimental Observations.
I
Halevy (2007): Comparisons of Models explaining Ellsberg’s
paradox.
I
Eliaz and Ortoleva (2011): Uncertainty on probabilities and
prizes.
12. Objective Uncertainty and Sophisticated Behavior
12.1 Prospect Theory
12.2 Sophistication
12.1 Prospect Theory
Prospect Theory (Kahneman and Tversky, 1979)
I
Value is assigned to gains and losses instead of final assets, in
a reference-dependent sort of way. Concave for wins, convex
for losses.
I
Probabilities are transformed. Overweighting of small
probabilities.
12.2 Sophistication
Probabilistic Sophistication (Machina and Schmeidler, 1992)
It contains a general behavior regarding subjective probability in
which agents are fully sophisticated (they form probabilities over
states in a consistent way) but are not utility maximizers. That is,
the agent transforms acts into roulettes
(u(x1 ), µ(f −1 (x1 ), . . . , u(xn ), µ(f −1 (xn )) with a subjective
probability µ but does not apply EU.
A different functional w on roulettes is used
v (f ) = w (u(x1 ), µ(f −1 (x1 ), . . . , u(xn ), µ(f −1 (xn )). Linearity on
probabilities is discarded. The authors preserve some mixture
continuity properties and monotonicity properties (related to
stochastic dominance) in the functional w .
12.2 Sophistication
Probabilistically Sophisticated Non-Maximizers satisfy (EM) and
(WCP). To see the first, consider that the individual forms
well-behaved probabilities and thus we can define conditional
probabilities. In this case, a dominating act will define stochastic
dominant outcomes and thus (EM) will hold thanks to the
monotonicity condition. To see the second, f g implies
w (x, µ(A), y , 1 − µ(A)) w (x, µ(B), y , 1 − µ(B)). Given the
dominance in between x and y , it must be µ(A)µ(B) and then we
can apply stochastic dominance on
w (x 0 , µ(A), y 0 , 1 − µ(A)) w (x 0 , µ(B), y 0 , 1 − µ(B)) to conclude
f 0 g 0.
The key difference can be seen to be (STP). However, we cannot
just dispense completely with this axiom or the individual may fail
to be sophisticated.
12.2 Sophistication
Strong Comparative Property.
For all f , f 0 , g , g 0 ∈ F , x, y , x 0 , y 0 ∈ X , A, B ⊆ X A ∩ B = ∅, if
I
x y, x0 y0
I
f = x on A, f = y on B, g = y on A, g = x on B, and f = g
on (A ∪ B)c .
I
f 0 = x 0 on A, f 0 = y 0 on B, g 0 = y 0 on A, g 0 = x 0 on B and
f 0 = g 0 on (A ∪ B)c .
then f g if and only if f 0 g 0 .
Exercise 20. Prove formally that SCP implies SWP but they are
not equivalent. Prove also that any Probabilistically Sophisticated
Non-Maximizer can neither behave according to the paradox, even
if we have eliminated the Sure-Thing Principle. How is this
possible?
13. Choquet EU Model
13. Choquet EU model
Ellsbergs examples capture the idea that ones condence in the
probability assignment is relevant. Following Schmeidler (1989)
and Gilboa (1992) we discuss Choquet Expected Utility (CEU).
They need of subjectivity and hence, the former is in the
Anscombe-Aumann framework, and the second in the Savage’s
framework, but share the same idea. They capture the notion of
uncertainty or ambiguity aversion. The model combines a capacity
(instead of an additive probability) and a utility function.
13. Choquet EU Model
I
I
A real valued set function v on events (subsets of S) is a
capacity if v (∅) = 0, v (S) = 1 and v is monotonic, i.e.,
v (E ) ≤ v (F ) if E ⊆ F .
P
Let a be a finite step function where a = ki=1 αi Ei and
α1 > α2 > · · · > αk and {Ei }ki=1 is a partition of S and υ be
a capacity. The
R Choquet integral of a with respect to υ is
denoted by adυ and defined by
Z
adυ =
k
X
i=1
where αk+1 = 0.
(αi − αi+1 ) υ ∪ij=1 Ej
13. Choquet EU Model
If the capacity is additive, it allows to define a probability over
states by considering the value of v in each state. In this case, the
integral coincides with the classical notion.
Axioms in the two setups characterizing behaviors of the form
Z
Z
f g ⇔ u (f (s)) dυ ≥ u (g (s)) dυ
Exercise 21. Describe the condition for the capacity v to be
additive. Does any capacity function generate the ambiguity
aversion present in the Ellsberg Paradox? Which capacities would
create the paradox?
14. Multiple Priors
14. Multiple Priors
A natural story behind the Ellsberg paradox can be described as
follows. Given that the individual has not information about how
many black-yellow balls are, the individual forms a set of possible
priors. Being uncertainty averse, considers the minimal expected
payoff across these payoffs.
Z
v (f ) = min u(f (s))dµ(s)
µin∆
This model, MMEU, is analyzed in Gilboa and Schmeidler (1989)
and Casadesus-Masanell, Klibanoff and Ozdenoren (2000) for the
Anscombe-Aumann and Savage’s frameworks.
14. Multiple Priors
Other multiple prior models:
I Variational preferences (Maccheroni, Marinacci and Rustichini,
R
2006): v (f ) = minµin∆ ( u(f (s))dµ(s) + c(p)) where c
describes ambiguity aversion. The MMEU case corresponds to
c(p) = 0 for all p, while SEU corresponds to c(p) = 0 for a
particular p = q and c(p) = ∞ otherwise.
I Multiplier Preferences (Hansen and Sargent, 2001; Strzalecki,
R
2010): v (f ) = minµin∆ ( u(f (s))dµ(s) + θR(p||q) where θ is
a trust parameter and R(p||q) describes the (entropy)
distance from p to the true subjective probability q.
I Knightian Uncertainty (Bewley, 2002). Incomplete preferences
(Pareto domination over priors).
I Sophisticated MMEU (Marinacci, 2002). Shows how the
intersection between MMEU and sophistication looks like.
Exercise 22. Describe a set of plausible priors for the Ellsberg
experiment, explaining the intuition and the consequences, that
allows to resolve the paradox when considering the MMEU model
III Menu Preferences
15. Indirect Utility Theory
16. Preference for Flexibility
17. Temptation and Self-Control
15. Indirect Utility Theory
15. Indirect Utility Theory
In the classical approach, a menu is valued as much as the best
alternative on it. The indirect utility function describes this idea.
The value of a menu A is equal to v (A) = maxx∈A u(x).
Some axiomatic exercises in Nehring and Puppe (1996) and
Ballester, DeMiguel and Nieto (2003).
Exercise 23. Let X be a finite set of alternatives and % a
preference on X (all non-empty subsets of X , or menus). We say
that the preference over menus is perfectly rational if for all
A, B ∈ X , A % B =⇒ A ∼ A ∪ B % B. Prove that % is perfectly
rational if and only if there exists u : X → R such that
A % B ⇔ maxx∈A u(x) ≥ maxx∈B u(x).
16. Preference for Flexibility
16. Preference for Flexibility
Consider an individual who must decide in the morning a menu
(restaurant) from which she will have to make a choice (lunch).
With regards to this interpretation, Kreps (1979) discusses how the
individual may like the flexibility of having more options to choose
from later in the day. The reason might be that she is not sure
about which her actual tastes will be at that moment. A key
property analyzed to represent this idea is the following:
Kreps (1979) and Dekel, Lipman and Rustichini (2001) deeply
analyze this idea describing a model where states of nature are
completely subjective (the possible different preferences of the
individual), providing finite or continuous versions of the following
model:
A%B⇔
X
s∈S
π(s) max U(x, s) ≥
x∈A
X
s∈S
π(s) max U(x, s)
x∈B
16. Preference for Flexibility
The key axiom to characterize this preference is:
Monotonicity. A ⊆ B ⇒ B % A.
An alternative explanation on monotonic preferences on menus is
freedom of choice. See for instance Pattanaik and Xu (1990),
Dutta and Sen (1996), or Alcalde-Unzu and Ballester (2004).
17. Temptation and Self-Control
17. Temptation and Self-Control
Gul and Pesendorfer (2001)
Consider an individual who must decide what to eat for lunch. She
may choose a vegetarian dish or a hamburger. In the morning,
when she feels no hunger, she prefers the healthy, vegetarian dish
but at lunchtime, she experi- ences a craving for a hamburger
Consider an individual who must decide what to study for the day.
She may choose an ambitious (and costly) plan of work or a
minimal effort one. In the morning, when she feels no pressure, she
prefers the ambitious plan, but as soon as she starts working she
experiences a craving for the minimal effort one.
17. Temptation and Self-Control
To lessen the impact of her lunchtime craving, the decision-maker
may seek to limit the options at lunchtime. For example, she may
choose a vegetarian restaurant. When this is not possible and the
individual is confronted with a menu that includes both the
vegetarian meal and the ham- burger, she may exercise
self-control, that is, resist the craving for the hamburger and
choose the vegetarian meal.
To lessen the impact of her daily craving, the decision-maker may
seek to limit the options at daily work. For example, she may study
in a place where the benefits of the minimal effort plan are not
perceived. When this is not possible and the individual is
confronted with a menu that includes both plans of work, she may
exercise self-control, that is, resist the craving for the minimal
effort plan.
17. Temptation and Self-Control
Recognizing Temptation: When there is no choice, the agent is not
tempted and prefers the long-run menu {x} to the tempting {y }.
However, when confronted with the menus {x} and {x, y }, the
agent is not indifferent, and prefers to commit, avoiding
temptation. That is, {x} {x, y }.
Recognizing Self-Control. When there is no choice, the agent is
not tempted and prefers the long-run menu {x} to the tempting
{y }. When confronted with the menus {y } and {x, y }, the agent
prefers the latter because she will self-control (maybe with some
cost that does not exceed the benefit of x over y ) and choose the
better option x. That is, {x, y } {y }.
Set-Betweeness: If A % B then A % A ∪ B % B.
15. Temptation and Self-Control
Temptation Preferences with Self-Control:
U(A) = max u(x)−[max v (y )−v (x)] = max[u(x)+v (x)]−max v (y )
x∈A
y ∈A
y ∈A
x∈A
The individual considers the cost of temptation. Cost can be
measured by the difference in short-run value to the optimal
alternative.
Limiting case when cost is sufficiently high, Overwhelming
Temptation:
U(A) =
max
for all
x∈A:v (x)≥v (y )
u(x)
y ∈A
This model corresponds to Strotz’s idea (Strotz, 1955). An agent
commits/chooses in the first period knowing that a different
self/agent will choose in the second period.
17. Temptation and Self-Control
Dekel and Lipman (2011) show a very interesting result on Gul and
Pesendorfer’s model of temptation. It can be represented as a
random Strotz model.
U(A) =
X
i
max
for all
x∈A:vi (x)≥vi (y )
u(x)
y ∈A
Hence {x} x, y {y } can be interpreted as: (1) The agent
prefers to commit because there are chances of falling in
temptation but (2) the agent prefers the good alternative available
because there are chances of self-controlling.
Dekel, Lipman and Rustichini (2009) provide a complete overview
of Flexibility and Temptation.
17. Temptation and Self-Control
Exercise 24. Consider a dieting agent who wishes to commit
herself to eating only broccoli. There are two kinds of snacks
available: chocolate cake and high-fat potato chips. Let b denote
the broccoli, c the chocolate cake and p the potato chips. How
could you interpret the following ranking over menus, {b} {b, c}
and {b, p} {b, c, p}? Prove whether this is possible under Gul
and Peserdorfer’s model of temptation.
Exercise 25. Consider again the dieting agent facing multiple
temptations, but now suppose the two snacks available are high-fat
chocolate ice cream i and low-fat chocolate frozen yogurt y . How
could you interpret the following ranking over menus,
{b, y } {y }, {b} {b, i} and {b, i, y } {b, i}? Prove whether
this is possible under Gul and Peserdorfer’s model of temptation.