Bounded Rationality and Socially Optimal Policy to Affect Choice Probabilities by Eytan Sheshinski Presentation at the Turkish Economic Association Meeting,
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Bounded Rationality
and
Socially Optimal Policy to Affect
Choice Probabilities
by
Eytan Sheshinski
Presentation at the Turkish Economic
Association Meeting, Ankara, September 11-13
i
page 2
The benefits of choice
Assumption: consumers are well informed and make the
right choices. Choice supports differentiated tastes and
needs, promotes competition among providers (lower
prices, improved quality).
page 3
• When individuals are 'Boundedly Rational', the probabilities of
making errors may depend on the domain of choice, in
particular, on the number of alternatives.
• Cognitive limitations: facing difficult decisions, individuals
tend to procrastinate, choose default options or use ever more
simple decision rules (cheapest-hoping for a bargain, most
expensive-highest quality).
page 4
• Examples where outcomes depend on design: Opt-in vs.
Opt-out in organ donations (Johnson and Goldstein
(2003)); 401(K) retirement plans (Choi et-al (2002)).
Default options in mandatory car insurance
(Loewenstein (2002)).
Shortsightedness (credit cards, Ausubel (1998)) and
preference for immediate gratification.
page 5
Detailed Example: Social Security (SS)
• Social Security now offers almost no choice
• Only choice for individuals is when to claim benefits
once they are eligible ('Delayed Retirement Credit’).
• Substitution of all or part of SS benefits by individual
saving accounts gives workers responsibility for managing
their funds.
page 6
Investment Choices
• One or more accounts (programs).
• Division between Stocks and Bonds (TIAA-CREF,
specific investments made by the fund).
• Single or joint accounts (dividing family income between
spouses).
• Combine retirement with other insurance.
• Life Insurance - general issue of survivors' benefits.
page 7
• Allowing early withdrawals for specific purposes such as
unemployment Insurance (Stiglitz (2002)).
• Deferred annuities (when, how much, type of annuities).
• Choice of fee structure - front loading, management fees
(combination of formulas).
• Should Add-On options be allowed?
• Early eligibility, partial retirement and delayed
retirement credit.
page 8
Annuitization Choices
• One or many annuity types (x-year certain, single-joint etc.).
• Indexation choice (nominal, CPI, Wages).
• Mandatory full or partial annuitization (programmed
withdrawal) replacement rate.
• Choice of issuer of annuities and provider of benefits (SS,
pension or insurance firms).
• Different benefit profiles over time.
Major question SS reformers face:
What trade-offs will people be offered?
How will they respond?
page 10
Conclusion:
When individuals make errors, more choice may exacerbate
mistakes; Consequently, a government whose objective is to
maximize an aggregate of expected utilities of a heterogeneous
population may find it optimal to influence the choice set
which individuals face and its associated probability measure,
the elimination of some alternatives being an extreme option.
page 11
When government considers a policy that affects individual
choice probabilities, three factors seem a-priori important:
(a) The 'individuals’ degree’ of rationality;
(b) The distribution of preferences in the population as
revealed by self-selection;
(c) Intensity of preferences.
page 12
Our major conclusions:
(a) At low degrees of rationality it is best to entirely eliminate
individual choice. The single remaining alternative depends on
the distribution of preferences and on their intensity;
(b) At high degrees of rationality all alternatives should be
assigned a positive probability;
(c) The optimal weight assigned to each alternative may not
vary monotonically with the 'degree of rationality';
page 13
(d) While policy to shift choice probabilities becomes
ineffective when individuals are perfect choosers,
substantially shifting choice probabilities is called for
even at high degrees of rationality when errors of choice
made by individuals are very small. Such policy aims at
reducing differentially the larger errors made by
individuals with less pronounced preferences and hence
prone to make mistakes.
page 14
Model of Probabilistic Choice:
Utility is deterministic but choice is probabilistic (Tversky, 1972).
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).
Sa
AA ).a(p)S(p
page 15
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}).
Comments: (ii) implies that pT(a) = 0. (i) is a path
independence property.
page 16
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
.)b(U
)a(U)a(p
Sb
S
page 17
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).
Sb
)b(u
)a(u
Se
e)a(p
)b(p)a(p
)b(p)a(p
T
T
S
S
page 18
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) = .
21
21
41
31
page 19
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) = .
21
21
41
31
page 20
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
choice probabilities.
page 21
Modelling The Degree of Rationality
Individuals are characterized by a parameter (e.g. ability,
labor disutility, health) which is private information.
They choose one among a finite number, n, of alternatives,
i = 1, 2,.., n. Individual's utility of alternative i is denoted
ui().
Following the Multinomial Logit Model, the probability that
individual chooses alternative i is
page 22
n..,,2,1i
e
e),q(p)(qu
)(qu
ij
i
q is a positive constant representing the precision of choice.
,n1),0(pi all i and .
Let
For i M(), pi(q, ) strictly increases with q, approaching
as q , where R() is the number of elements in M().
For i M(), pi(q, ) strictly decreases with q, approaching 0
as q .
)]}(u),...,(u),(umax[arg)(u|i{)(M n21i
)(R1
page 23
Henceforth q is called the ‘Degree of Rationality’.
Individuals’ welfare is represented by expected utility:
For each , V(q, ) strictly increases in q, approaching
for any i M(), as q .
n
1iii )(u),q(p),q(V
)(u),q(V i
page 24
We assume that all individuals have the same q
[with different levels of q, individuals aware of their low q
may attempt to follow choices made by those considered to
have similar preferences but higher q].
Utilitarian social welfare:
F() is the distribution function of .
)(dF),q(V)q(W
page 25
Policy to Influence Probabilities
Policy depends only on observables (not ). Consider a
tax/subsidy, ti, on alternative i:
j
n
1j
qu
iqu
n
1j
)tu(q
)tu(q
i
ge
ge
e
e),q,(pj
i
ij
ii
g
).g,...,g,g(,eg n21qt
ii g
page 26
[Other interpretations of g: multi-stage choice.
Example: Let n = 3. Expected utility of the 'package’ 2 + 3,
Probability of choosing alternative 1, when in first-round
selection is between 1 and the 'package' 2 + 3 is
We have
Another method to influence probabilities, add alternatives along
the Blue bus- Red bus paradox]
).ee/(epwhere,u)p1(upuis,u 322 quququ232222323
).ee/(ep~ 2311 quququ1
.pp~ 11
page 27
pi(g, q, ) is homogeneous of degree zero in
When all gi > o, then
independent of g. By continuity, the optimal policy for large q is
not to exclude any alternative.
n
1i
.1gg
W)(dF)(V),(W g
n
1iiiWg
n1)0,(W g
wheren,...,2,1i,)(dF)(uW ii
is social welfare when all individuals choose alternative i.
page 28
Let Wm arg max[W1, W2,…, Wn].
Optimal policy when q = 0 is gm = 1 and gi = 0, i m,
implying the elimination of individual choice.
F.O.C. for maximization of W w.r.t. g,
0)(dF)],q,(V)(u)[,q,(pg1
gWg ii
iii gg
ij
2
ggW
0)],q,(V)(u)[,q(pn
1iii
g for any g].
[The matrix is negative semi-definite since
page 29
Denote the solutions:
Sign of is ambiguous. However,
))q(g),...,q(g),q((g (q) *n
*2
*1
* g
dqdg*
i
0q),(qW q),(
dqdW **
gg
whenever there are at least two different alternatives with
positive weights.
page 30
Proposition 1.
(a) There exists a positive number, q0, and an index m, such that
for all
and hence
for all and W(g*, q) = Wm;
)n,...,2,1i,mi,0g(1g,qq0 *i
*m0
),q,(V)(u),n,...,2,1i,mi,0p(1p *mim g
(b) For q > q0, W(g*(q), q) strictly increases with q,
approaching asW
(c) For large for all 0)q(g,q *i .n,...,2,1i
;q
page 32
An Example with Two Alternatives
Let n = 2. Then where
Condition for an interior solution:
0)(dF]1g)1e[(
)(egWg 2)(q
)(q
.g1ggand)(u)(u)( 2121
Second-order condition is satisfied:
0)(dF]1g)1e[(
)1e)((e2gW
3)(q
)(q)(q
2
2
),1g)1e/((ge),q,g(pp )(q)(q1
page 33
2 2 Case: Two groups, 1, 2. Fraction of group 1 is f, 0 <f <1.
Then:
)1e(e)1e(e
eeg1221
22
q2qq2
q
2qe
2q
*
.2,1i),(u)(uwhere i2i1i
.f
)f1( 21
1
2
Assume 1 > 0, 2 < 0.
page 34
Two cases:
(I) > 1.
.qqfor1)q(g0;lnq,qq0for0g 0*
2100*
0
0
0
where
where
where
0
11
1
)q(glim)(g
21
21
21
*
q
*
(II) < 1. .lnqwhere,qq0for1g 1200*
The limit g*() is as before.
page 35
Conclusions:
(a) g*(q) need not be monotone in q.
(b) Even for large q, a substantial shift of choice
probabilities is called for.
Reason: close to 'perfect rationality', shifting relative
weight has little effect, but it is differentially important
for those who have 'weaker' preferences and hence prone
to make mistakes.
page 38
Behavior of g*(q) around q0.
By Proposition 1, all choice is eliminated at q q0, for some
positive q0. As q decreases to q0, is the elimination of alternatives
gradual or abrupt?
A continuous example where it is optimal to provide positive
probabilities to all alternatives once q is slightly higher than q0.
page 39
Suppose is continuous with density
The choice set is the real line x (- , + ). The probability that
type chooses x, p(q, x, ) is
. varianceandmean),(f 2
dx)x(ge
)x(ge) x, p(q,
-
),x(qu
),x(qu
where g(x) is a generalized function (non-negative measure).
page 40
Expected utility, V(), is
and social welfare, is maximized w.r.t. g(x).
Assume that
dx),x,q(p),x(u) V(q,-
d)(f),q(VW
.)x(),x(u 2
Definition: f() is normally concentrated if
is bounded for all .
0
dm)m(mf)(f
1
page 41
Proposition 2
if f() is normally concentrated there exists a positive q0 such that
when q< q0, W is maximized when g(x) is a singleton at
For the normal density, and for q > q0,
.x
20 41q
.
q14
xexp)x(g2
2
page 42
Note that:
(a) there is a sharp transition from no choice to full choice
(providing all options) at q = q0.
(b) as rationality becomes perfect, options are weighted
towards those that are most commonly the best, i.e. at high
q's, g(x) is not uniform [recall the outcome in the previous
discrete binary choice model].
page 43
Sketch of proof of Proposition 2
Without loss of generality, take Then, .0
.d)(fdx)x(he
dx)x(he)x2(q1d)(f
d)(fdx)x(ge
e)x2x(W
xq2
xq22
22
)x2x(q
)x2x(q22
22
22
Where (since q is given, we henceforth ommit
the dependence on q). The first term is 2, the level of W when
there is no choice, everyone getting
2qxe)x(g)x(h
.0
page 44
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).
dx)x(he)m( mx
.dx)x(hex)m('',dx)x(hxe)m(' mx2mx
Hence,
.d)(f)q2(
)q2('')q2('n2W 2
page 45
Further define
Then,
.)q2()q2(')(
).q2'2(W 22
Define Then, integrating by parts,.dm)m(mf)(G0
.d)(')(fq41)(G2W 2
page 46
If f is normally concentrated, is bounded. If q is small,
is negative for all . is non-decreasing in . Hence, for small q,
When = 0 for all , = 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,
fG
qfG
.W 2 .W 2
dm)m(mf4
)(finfq0
.4
q2
0
page 47
Now,.d)(f)
qf2'f2(W 22
Choose to maximize W:
4m
21 2
e)q2
m(f)m( Integrating
.)(qf4
)('f
for a normal distribution this corresponds to the g(x) function
given above.
page 48
Two comments:
(a) Nonuniform Degree of Rationality
What are the consequences of heterogeneous q's?
In the 22 example presented above, same conclusions
(e.g. elimination of choice at some q0) pertain (with
different formulas).
page 49
With heterogeneous q’s, ’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 q’s can be regarded as (partially) endogeneous,
individuals purchasing information to support their decision-
making: chosen q’s are then correlated with incomes (Arrow).
page 50
(b) Benevolent Government?
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: it is interesting to explore the interaction of such
limits with the degree of individual rationality.
page 51
Self-Selection and Aggregate Constraint
Let ui(xi, ) be individual ’s utility of alternative i, where xi is
some government policy which has a cost (in terms of the
numeraire) of c(xi). Correspondingly, choice probabilities are
pi = pi(q, xi, ).
Suppose there is a resource constraint
R)(dF)x(c),x,q(pn
1iiiii
page 52
Denote the policies that maximize W s.t. the resource constraint
by and the corresponding level),q(xi
)(dF)),q(x(u)),q(x,q(p)q(W iii
When where ))0(x(Wn1)0(W,0q ii
),(dF)),0(x(u))0(x(W iiii
and is the limit of)0(xi 0.q as )q(xi
page 53
Let be the feasible policy when only alternative m is permitted:
The corresponding social welfare is
Even when
it does not follow necessarily that
Eliminating choice at is now a possible but not necessary
outcome.
ix~
.R)x~(c mm
).(dF),x~(uW~
mmm
))0(x(W)),...0(x(W)),0(x(Wmaxarg),x~(W~
nn2211mm
).0(WW~
m
0qq
page 54
Controlling Only the Number of Alternatives
When government cannot fine-tune probabilities, only
eliminate some, what are the errors generated at the critical
elimination point?
In the above 22 example, if only alternative 1 is allowed,
social welfare is
)f1(ufuW 21
111
page 55
When individuals are allowed to choose between the two
alternatives, but the weights in the choice probabilities are fixed
and equal, social welfare is
)f1)(1e
uue(f)
1e
uue()q(W 2
2
1
1
q
22
21
q
q
12
11
q
Equating we obtain an implicit equation for the level
of q, at which alternative 2 is eliminated:
)q(WW1
q
2
q
q
1
2
e1
e1
where has been defined above.
page 56
What are the type II errors of each group at ?
Take, for example, and The errors are then
and respectively.
That is, 27 percent of type 1 individuals erroneously choose
alternative 2 while 38 percent of type 2 individuals erroneously
choose alternative 1 (the fact that group 2 errors are larger is
expected in view of their 'weaker' preferences).
q
21
1
2 .
21f
27.)q(p1 1 ,38.)q(p2
page 57
A Work-Retirement (Self-Selection) Model
u(ca) - = workers’ utility
v(cb) = non-workers’ utility
F() = distribution function of
Resource constraint:
R(> - 1) is the level of external resources.
0
00
ba R)(dFc)(dF)1c(
page 58
Social welfare
First-best (labor disutility observable): (ca*, cb
*, * ) satisfying:
* > 0 provided a “Poverty-Condition” is satisfied:
0
00
ba )(dF)c(v)(dF))c(u(W
)c('v)c('u *b
*a
)cc1)(c('u)c(v)c(u *b
*a
*a
*b
**a
).R(v)1R(u
page 59
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).
)]c(v)c(u,0[Maxˆba
).ˆ,c,c( ba
)y(v)x(u
ac bc
0cc1and)c('v)c('u baba
)y('v)x('u
Labor Disutility Unobserved (Self-Selection Equilibrium)
page 60
Logit Model of Self-Selection
Let the probability that individual chooses to work, Pa, be
Social welfare
Resource constraint:
Denote the optimum solution by ).c,c( ba
)c(qu))c(u(q
))c(u(q
baaba
a
ee
e),q,c,c(P
0
abaa )(df)]P1)(c(vP))c(u[(W
0
abaa R)(df)]P1(cP)1c[(
page 61
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
and),c('v)c('u ba ;1R2cc ba
.Rcb
;1Rca
or
(b) the retirement option is eliminated, setting
or
(c) the work option is eliminated, setting
page 62
Logarithmic Two-Class Example
Two types, 1 < 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*, is
If both types work, welfare is wa,
The condition that type 2 retires in the First-Best allocation is
therefore,
.f)fRln(*w 111
.ff)1Rln(w 2211a
.R
fRln
f1 1
22
page 63
In addition, the ”poverty condition” ensures that the optimum has
type 1 working:
.R
fRln
f1 1
21
Self-Selection Equilibrium
Under the ’moral-hazard condition’, the following relation holds:
From this and the resource constraint
).c(u)c(u ba
solve,Rfcf)1c( 2b1a
21
1b
21
1a
ffe
fRc,
ffe
)fR(ec
11
1
page 64
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
.ffe
fRln)(W
21
11
.f)ffe(fR1R
lnf1
112112
21
,21
R)0(c)0(c ba
).ff(21
)21
Rln()0(W 2211
page 65
Social welfare without a retirement option, Wa, exceeds iff
.f
21
R
1Rln2
f1
112
2
)0(W
The following table uses parameters:
R = 0, 1 = 0, 2 =1.5 and f1 = .5.
page 66
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