Lecture on

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Lecture on Bounded Rationality and Socially
Optimal Limits on Choice by Eytan
Sheshinski January, 2002
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• Is more choice always better?
• Benefits of more choice
• Differentiated tastes and needs, promotes
  competition among providers (lower prices,
  improved quality).
• Consumers must be well informed and make the
  right choices.

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• Costs of More Choice
• Three kinds of costs (1) time (2) error (3)
  psychic costs (Loewenstein, 2000).
• When decisions become difficult, consumers tend
  to procrastinate or choose default options.
• People are shortsighted when facing choices
  between immediate gratification and long-term
  gains.

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Extreme risk-aversion. When consumers make
errors, more choice may exacerbate mistakes! Wish
to explore the tradeoff between varying tastes
and bounded rationality (i.e. imperfect
maximization).
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A Model of Stochastic Decision Rules Utility is
deterministic, but the choice process is
probabilistic (Tversky, 1972). Individuals do not
necessarily select what is best for themselves
(Luce, 1959). This captures the idea of Bounded
Rationality.
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Notation the probability that an individual
chooses a ? S when confronted with the choice set
S is denoted PS(a). Probability that the
alternative chosen in a set A belongs to the
subset S is denoted PA(S), so When S a, b,
P(a, b) stands for PS(a).
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The (Luce) Choice Axiom For any S ? A and T ? A
such that S ? T, (i) if, for a given a ? S , P(a,
b) ? 0, 1 for all b ? T , then PT(a)
PT(S)PS(a) (ii) if P(a, b) 0 for some a and b ?
T, then for all S ? T PT(S) PT-a(S-a). (ii)
implies that PT(a) 0. (i) is a path
independence property.
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Theorem (Luce, 1959). Assume that P(a, b) ? 0, 1
for all a, b ? A. Part (i) of the choice axiom is
satisfied if and only if there exists a positive
real valued function u defined on A such
that This function is unique up to
multiplication by a constant. Using the
transformation U(a) ln u(a), we have
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This is the familiar Logit model. Implication of
the choice axiom for all S ? A, T ? A, such that
S ? T, and for all a, b ? S, This independence
property leads to the Blue bus/red bus paradox
(Debreu, 1960).
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Suppose P (car, bus) . Now let the choice
set become A car, blue bus, red bus. Assume
PA (red bus) PA(blue bus).We expect PA(car)
and PA(blue bus) PA(red bus) . However,
the choice axiom implies that PA(car) PA(red
bus) PA(blue bus) .
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The proof is immediate PA(car) PA(car, blue
bus). P(car, blue bus). PA(car, blue bus
PA(car) PA(blue bus). Since PA(car)PA(blue
bus) PA(red bus) 1 and PA(red bus) PA(blue
bus), it follows that PA(blue bus) (1 -
PA(car)). Hence, PA(car, blue bus) 1
PA(car). Thus, PA(car) 1 PA(car) ?
PA(car) .
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Conclusion A new alternative is seen to reduce
the probabilities of similar alternatives less
than proportionately. We shall use the blue
bus/red bus paradox to model options available
for social policy to influence individual
decisions.
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Individuals have a deterministic net (of cost)
utility function u u(x, ?), where x is a choice
variable and ? is an individual characteristic
(ability, labor disutility, etc.) x can assume n
discrete values x1, x2, ..., xn. We call these
the basic choices. The probability that a
?-individual chooses alternative xi, denoted
Pi(?), is a version of the Logit model,
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where q ? 0 and g(xi) ? 0, i 1, 2, ..., n. The
latter is a constant representing the relative
weight of alternative i can
benormalized to 1 without loss of generality).
In the spirit of the blue bus/red bus paradox,
where changing the choice set affects
disproportionately related probabilities, we can
call the extended-choices.
Note that g(xi), being policy instruments, do not
depend on ?, which is private information.
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When Thus, q can be called the degree of
rationality. (q ? is termed perfect
rationality ).
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Expected utility of individual ?, V (?), is
Let f (?)(? 0) be the density of ? (?f (?)d? 1).
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Assume that all individuals have the same q.
Utilitarian social welfare,W(q), is
Optimal social policy is a choice of (g(x1),
g(x2), ..., g(xn)) that maximize W for any given
f(?) and q.
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Viewing individuals as 'imperfect maximizers'
while governments choose optimal policies, may be
questioned. Boundedly rational or systematically
biased governments call for constitutions that
will limit their power to make decisions the
possible interaction of such limits with
individual choice will not be discussed here.
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Three observations (1) When q ?,
(perfect-rationality ), the policy that ensures
the social optimum for any f (?) is to set g (xi)
gt 0 for all i 1, 2, ..., n, because this policy
ensures that for all?. In this case,
independent of the specific configuration
(g(x1), g(x2), ..., g(xn)). This is the
First-Best allocation.
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(2) When q 0, the optimal policy is to have
some g(xj) 1 and g(xi) 0 for all i ? j. That
is, individual choice is eliminated. Denote by Wi
the level of social welfare when xi is chosen by
everyone, Wi ?u(xi, ?)f(?)d?, and let When q
0,
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Wm is attained by the policy stated above. Set
g(xm) 1 and g(xi) 0, i ? m. The specific xm
chosen depends, of course, on the density
f(?). (3) W(q) is strictly monotone increasing
in q
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Theorem There exists a unique q0 ? 0,
generically q0 gt 0, such that no choice is
optimal when q ? q0. Question How does the
optimal configuration (g(x1), ..., g(xn)) change
with the level of q? Since at q 0 it is optimal
for only one g (xm) to be positive, while at very
high values of q, all g (xj), j 1, 2, , n, are
positive, is the response to an increase in q
abrupt (i.e., at q0, a switch occurs from one
positive g (xm) to all positive g (xi), i 1,
2, ..., n), or does the set of positive optimal
gs increase gradually as q increases?
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Question At q0 (individual choice being
eliminated for lower q), to what extent are
individuals making erroneous choices? For
example, consider two alternatives, 1 and 2, and
two groups of individuals, one prefers
alternative 1 and the other alternative 2. At the
level of rationality at which individual choice
is eliminated, q0, what are the (statistical)
type I and type II errors in the choice of the
eliminated alternative?
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Consider W as a function of (g1, g2, ..., gn)
W(g1, g2, ..., gn q), where gi g(xi). For gi gt
0 and gj gt 0,
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The matrix
is negative definite (it is strictly negative
definite provided u(xi, ?) ? u(xj , ?) for all
i, j and ?). Hence, the optimum can never have a
cyclical pattern w.r.t. q (i.e. as q increases,
gk becomes positive, then gk 0 over an interval
of higher qs and becomes positive again for
still higher qs).
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Suppose that for 0 ? q ? q0, g(x1) 1 and g(x2)
g(x3) ... g(xn) 0. That is, at q below
q0, it is socially optimal to eliminate all
alternatives except the first alternative. By the
Kuhn-Tucker conditions, it is optimal to
introduce alternative 2 at q0 if
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If the introduction of alternative 3 also becomes
optimal at q0, a similar condition has to hold
and so on. Below we provide an example where
these conditions are mutually satisfied at q0
w.r.t. all alternatives. However, examples where
alternatives are introduced gradually can be
produced.
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Example Two alternatives, i 1, 2, and two
individual types, ?h, h 1, 2. Denote u(xi,
?h), Ph P1(?h), gh g(xh), i, h 1, 2, and
f1 f (0 lt f lt 1) is the relative weight of
group 1.
where ?h ½ (uh1 - uh2), h 1, 2. An interior
solution requires that ?2 lt 0 and ?1 gt 0 or ?2 gt
0 and ?1 lt 0 (i.e. each alternative is preferred
by one individual).
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Using the definition of Pi, solve for g1 g (g2
1 - g)
where
It is seen that
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It is interesting that when preferences are not
symmetric , the relative weight of the
groups, represented in ?, is irrelevant for large
q. Note also that when preferences are symmetric
and group size is equal
for all q. Note also that for large
q, 0 lt g lt1 for any finite q, that is, no
alternative is eliminated.
(a) ? gt 1. It is seen that g 0 for
where
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? gt 1
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At , alternative 1 is
eliminated.
Recall that
Hence, ? gt1 reflect a stronger relative
preference (for either alternative) by group 2
and/or a larger relative weight of group 2 in the
population. When ? 1, 0 lt g lt 1 for all q ? 0,
i.e. no alternative is eliminated.
(b) ? lt 1. It is seen that g 1 for
, where That is, alternative 2 is
eliminated for small qs.
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? lt 1
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Except the borderline case ? 1, when group
preferences are symmetric and the weight of these
groups in the population is equal, choice is
eliminated for small q. Continuous Example ? is
continuous with density f(?), mean and
variance ?2. The choice set is the real line x ?
(- ?, ?). The probability that type ? chooses
x, P(x, ?), is
where g(x) is a generalized function
(non-negative measure).
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Expected utility, V(?), is and social welfare,
W ?V(?)f(?)d ? is maximized w.r.t. g(x). Assume
that u(x, ?) - (x - ?)2 Definition f(?) is
normally concentrated if is bounded for all ?.
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Theorem if f (?) is normally concentrated
there exists a positive r such that when q ? r, W
is maximized when g(x) is a singleton at For
the normal density, and for q gt r,
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Note (a) there is a sharp transition from no
choice to full choice (providing all options) at
q r. (b) as rationality becomes perfect,
options are weighted towards those that are most
commonly the best, i.e. at high qs, g(x) is not
uniform recall this outcome in the previous
discrete binary choice model.
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Without loss of generality, take Then,
Where The first term is - ?2, the level of
W when there is no choice, everyone getting
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Define
(the denominator in the above expression). If h
is a weighting function, ? must exist for all
real m (it is the moment-generating function,
or Laplace transform, for h).
Hence,
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Further define
Then,
Define Then,
integrating by parts,
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If f is normally concentrated, is bounded.
If q is small, is negative for all ?. ? is
non-decreasing in ?. Hence, for small q, W ? -
?2. When ? 0 for all ?, W - ?2. ? 0 means
that ? is constant, i.e. only one option, x 0,
is available. For the first part of the theorem,
set for the normal distribution,
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Now, Choose ? to maximize W Integrating
for a normal distribution this corresponds to
the g(x) function given above.
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Non-Uniform Rationality Suppose individuals
differ in preferences, including the degree of
rationality, q. Take the two-types,
two-alternatives example above. Let qi, I 1, 2
be the degree of rationality of type i. Continue
to assume symmetry ?1 - ?2 gt 0. That is, type
1 (type 2) prefers alternative 1 (alternative
2). Then W is maximized when g1(g2 1) is given
by
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Suppose that q1 lt q2. The optimal relative weight
of the alternative preferred by the
less-rational type (g1) increases with q2. As
type 2 individuals become more precise in their
choice, optimal policy shifts the choice
probabilities towards the alternative preferred
by the less-accurate group. Unlike the case
with equally rational individuals, elimination of
choice at low q1 is still possible but not
necessary g1 is eliminated at low q1, provided
-(ln? q2 ?1) gt 0.
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With heterogeneous qs, deeper questions
arise (1) Are individuals aware of their own q
(ability to choose) compared to others and, if
so, are they able to identify individuals with
similar tastes, i.e. ?, and imitate their
choice? (2) The qs can be regarded as
(partially) endogeneous, individuals purchasing
information to support their decision-making
chosen qs are then correlated with incomes
(Arrow).
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Education Choice Model Two groups, i 1, 2,
with relative weight of group 1 of f (0 lt f lt
1). Income with education is wi for a group i
individual, and equal to 1 without education for
both groups. Utility of consumption ( income),
U(c), is u(c) lnc. Let w1 gt 1 gt w2( ? 0). With
perfect rationality, group 1 individuals have
utility of u(w1) lnw1, while group 2
individuals have utility u(1) 0.
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Hence, with a utilitarian social welfare, W
(lnw1) f ln(w1f). Now assume that the
probability of group i individuals selecting
education, pi, is Expected utility of group i
individuals, Vi, is and social welfare,
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where Denote social welfare when education is
mandatory by Wa, Social welfare when education
is not available, Wb, is, At q 0, either Wa
gt W(0) gt Wb 0 or Wb 0 gt W(0) gt Wa. In
either case, choice should be eliminated.
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Take the case Wa gt Wb 0. Solving for W(q0)
Wa,
Example for
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A Work-Retirement (Self-Selection) Model u(ca) -
? workers utility v(cb) non-workers
utility F(?) distribution function of
? Resource constraint R(gt - 1) is the level of
external resources.
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Social welfare First-best (labor disutility
observable) (ca, cb, ? ) satisfying ? gt
0 provided a Poverty-Condition is satisfied
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Labor Disutility Unobserved (Self-Selection
Equilibrium)
Solution Assume
(moral-hazard condition )
implies for
all x, x ? 0. Under the above assumption,
and satisfy the relation (a positive
implicit tax on labor at the optimum).
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Logit Model of Self-Selection Let the probability
that individual ? chooses to work, Pa, be
Social welfare Resource constraint Denote
the optimum solution by
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Proposition When q 0 the optimal allocation
has one of the following forms (a) consumption
levels of workers and of non-workers equate their
marginal utilities
or (b) the retirement option is eliminated,
setting or (c) the work option is
eliminated, setting
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Logarithmic Two-Class Example Two types, ?1 lt
?2, with weights f1 and f2 1 - f1. Let u(c)
v(c) lnc. In the First-Best, if type 1 works
and type 2 retires, social welfare, w, isIf
both types work, welfare is wa,The condition
that type 2 retires in the First-Best allocation
is therefore,
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In addition, the poverty condition ensures that
the optimum has type 1 working
Self-Selection Equilibrium Under the
moral-hazard condition, the following relation
holdsFrom this and the resource constraint
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With the corresponding level of social
welfare, The condition that at the
self-selection eq. type 2 does not work is In
the Logit Model, when q 0,and
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Social welfare without a retirement option, Wa,
exceeds iff
The following table uses parameters R 0, ?1
0, ?2 1.5 and f1 .5.
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