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