Who is (More) Rational?

Syngjoo ChoiUniversity College London

Shachar KarivUC Berkeley

Wieland MüllerUniversity of ViennaTilburg University

Dan SilvermanUniversity of Michigan

NBER

To be presented at the Norwegian School of Economics

Nov 7, 2011

Prologue

• Uncertainty is a fact of life so people’s attitudes towards risk enter everyrealm of economic decision-making.

• We must study individual behavior with respect to choice involving uncer-tainty.

• Models of decision making under uncertainty play a key role in every fieldof economics.

Experiments à la Allais

• Each theory predicts indifference curves with distinctive shapes in theMarschak-Machina probability triangle.

• By choosing alternatives that theories rank differently, each theory can betested against the others.

• The criterion typically used to evaluate a theory is the fraction of choicesit predicts correctly.

The Marschak-Machina probability triangle

1

HP

Increasing preference

LP 0

1

H, M, and L are three degenerate gambles with certain outcomes H>M>L

A test of Expected Utility Theory (EUT)

A

B

CD

1

HP

LP 0

1

EUT requires that indifference lines are parallel so one must choose either A and C, or B and D.

Results have generated the most impressive dialogue between observationand theorizing (Camerer, 1995):

...EU violations are much smaller (though still statistically significant)when subjects choose between gambles that all lie inside the triangle...

...due to nonlinear weighting of the probabilities near zero (as the rankdependent weighting theories and prospect theory predict)...

...the only theories that can explain the evidence of mixed fanning,violation of betweeness, and approximate EU maximization inside thetriangle...

Although this practice is understandable, it limits the usefulness of thedata:

[1] While these experiments reveal that violations exist, they give us littlesense of how important they are or how frequently they occur.

[2] The analysis builds on ad hoc assumptions about an error generatingprocess and lacks any substantive econometric methodology.

[3] Choice scenarios are not very important to the applications of theoriesof choice under uncertainty in economics.

Research questions

Consistency

— Is behavior under uncertainty consistent with the utility maximizationmodel?

Structure

— Is behavior consistent with a utility function with some special struc-tural properties?

Recoverability

— Can the underlying utility function be recovered from observed choices?

Extrapolation

— Given behavior in the laboratory, can we forecast behavior in otherenvironments?

A new experimental design

An experimental design that has a couple of fundamental innovations overprevious work:

— A selection of a bundle of contingent commodities from a budget set(a portfolio choice problem).

— A graphical experimental interface that allows for the collection of arich individual-level data set.

The experimental computer program dialog windows

Rationality

Let {(pi, xi)}50i=1 be some observed individual data (pi denotes the i-thobservation of the price vector and xi denotes the associated portfolio).

A utility function u(x) rationalizes the observed behavior if it achieves themaximum on the budget set at the chosen portfolio

u(xi) ≥ u(x) for all x s.t. pi · xi ≥ pi · x.

Afriat’s Theorem The following conditions are equivalent:

— The data satisfy GARP.

— There exists a non-satiated utility function that rationalizes the data.

— There exists a concave, monotonic, continuous, non-satiated utilityfunction that rationalizes the data.

Goodness-of-fit

• Verifying GARP is conceptually straightforward but it can be difficult inpractice.

• Since GARP offers an exact test, it is necessary to measure the extent ofGARP violations.

• Measures of GARP violations based on three indices: Afriat (1972), Varian(1991), and Houtman and Maks (1985).

Afriat’s critical cost efficiency index (CCEI) The amount by whicheach budget constraint must be relaxed in order to remove all violationsof GARP.

The CCEI is bounded between zero and one. The closer it is to one, thesmaller the perturbation required to remove all violations and thus thecloser the data are to satisfying GARP.

The construction of the CCEI for a simple violation of GARP

2x

1x

D

C

B A

x

y

The agent is ‘wasting' as much as A/B<C/D of his income by making inefficient choices.

Decision-making quality

• In economics, heterogeneity in choices is attributed to heterogeneity inpreferences, information, beliefs, or constraints.

• Several strands of empirical research consider heterogeneity in choicesdriven, instead, by differences in the quality of decision-making.

• The revealed preference criterion: if decisions are high quality then thereexists a utility function that the choices maximize.

Judgment about the quality of decision-making is generally made difficultby twin problems of identification and measurement:

— The identification problem Distinguishing differences in decision-making quality from unobserved differences in preferences, information,beliefs or constraints.

Identification is important because welfare conclusions and thus (con-strained) optimal policy will depend on the sources of any systematic dif-ferences in choices.

— The measurement problem Defining (and implementing) a portable,practical, autonomous, quantifiable, and economically interpretable mea-sure of decision-making quality.

The CentERpanel

• A representative sample of over 2,000 Dutch-speaking households (5,000individual members) in the Netherlands.

• A wide range of individual socio-demographic and economic informationfor the panel members.

• The subjects in the experiment were randomly recruited from the entireCentERpanel body.

Participants DropoutsNon-

participants

Female 45.43 37.89 50.00Age

16-34 18.53 3.16 26.1435-49 26.14 12.11 32.1350-64 35.62 38.42 27.5865+ 19.71 46.32 14.15

EducationLow 33.59 42.63 30.99Medium 29.70 22.63 31.61High 36.72 34.74 37.40

Household monthly income€0-2500 22.42 34.73 21.28€2500-3499 25.13 26.32 18.90€3500-4999 28.85 16.32 28.93€5000+ 23.60 22.63 30.89

OccupationPaid work 53.13 39.47 62.91House work 11.59 7.89 8.78Retired 20.90 42.63 13.95Others 14.38 10.00 14.36

Household compositionPartner 80.88 67.89 82.64# of kids 0.84 0.32 1.09

# of obs. 1182 190 968

Types of analysis

[1] A descriptive overview of the average quality of decisions (consistency witheconomic rationality).

[2] The relationship between decision-making quality and demographic andeconomic characteristics.

[3] The correspondence between wealth accumulation and behavior in the lab-oratory.

Mean CCEI scores

CCEI scores Combined

CCEI CCEI.887*** .735***(.022) (.037)

-.024*** -.011(.009) (.015)

Age-.016 -.007(.011) (.020)

-.052*** -.077***(.011) (.020)

-.051** -.081**(.020) (.032)

Education.009 .021

(.011) (.017).026** .060***(.011) (.018)

Income.026** .026(.012) (.019).020 .006

(.013) (.020).033** .017(.014) (.022)

Occupation.028 .030

(.018) (.026).047** .039(.021) (.030).037* .035(.019) (.030)

Household composition-.026** -.023(.011) (.018).001 .001

(.004) (.007)R 2 .068 .058# of obs. 1182 1182

# of children

65+

€2500-3499

€3500-4999

€5000+

Partner

High

Paid work

Medium

House work

Others

Constant

Female

35-49

50-64

 

Dominance

• Violations of monotonicity with respect to first-order stochastic dominance(FOSD) are errors, regardless of risk attitudes.

• A decision to allocate less tokens to the cheaper account violates domi-nance but need not involve a violation of GARP.

• We use expected payoff calculations (largest upward probabilistic shift) toassess how nearly choice behavior complies with dominance.

A scatterplot of CCEI and FOSD scores

Wealth differentials

=⇒ The heterogeneity in wealth is not well-explained either by standard observ-ables (income, education, family structure) or by standard unobservables(intertemporal substitution, risk tolerance).

=⇒ If consistency with utility maximization in the experiment were a goodproxy for (financial) decision-making quality then the degree to which con-sistency differ across subjects should help explain wealth differentials.

Wealth regressions

1.425** 1.348* 1.781** 1.728**

(0.565) (0.714) (0.746) (0.750)0.078 -0.091 -0.038

(0.381) (0.381) (0.384)-1.361 -1.366(0.838) (0.840)

0.103(0.072)

0.601*** 0.602*** 0.520*** 0.514***(0.127) (0.127) (0.121) (0.121)-0.228 -0.229 -0.299 -0.321(0.164) (0.164) (0.168) (0.169)-0.286 -0.284 -0.310 -0.282(0.316) (0.316) (0.319) (0.316)0.006 0.006 0.007 0.006

(0.005) (0.005) (0.005) (0.005)0.000 0.000 0.000 0.000

(0.000) (0.000) (0.000) (0.000)0.682*** 0.682*** 0.733*** 0.714***(0.183) (0.183) (0.191) (0.191)0.103 0.103 0.095 0.090

(0.092) (0.093) (0.095) (0.095)Education controls Y Y Y Y

5.932 5.888 7.797 7.371(5.862) (5.879) (5.880) (5.841)

R 2 0.1794 0.1778 0.1801 0.1819

# of obs. 517 517 494 494

# of children

Constant

Ln(hhld wealth)

Female

Age

Age2

Age3

Partner

CCEI

Combined CCEI

Risk attitude

Conscientiousness

Ln(hhld income '08)

 

Sources of correlation Checking Saving Stocks House

1.907** 1.792** -0.100* -0.179* -0.003 0.336***(0.751) (0.705) (0.057) (0.096) (0.050) (0.129)

0.686*** 0.25 -0.030** -0.062*** 0.011 0.098***(0.133) (0.184) (0.013) (0.021) (0.010) (0.023)

0.467*(0.262)0.312*(0.176)

-0.144 -0.037 0.019 0.017 0.005 -0.039(0.188) (0.184) (0.018) (0.029) (0.012) (0.039)

0.790*** 0.740*** -0.033 -0.058* -0.008 0.136***(0.222) (0.222) (0.021) (0.033) (0.014) (0.044)0.111 0.123 -0.004 -0.043*** 0 0.069***

(0.107) (0.101) (0.009) (0.013) (0.007) (0.019)4.282 -0.415 0.045 1.42 -0.287 -0.777

(5.947) (6.032) (0.750) (1.197) (0.388) (1.289)R 2 0.236 0.269 0.051 0.104 0.029 0.146# of obs. 377 377 512 502 514 479

Ln(hhld income '04)

Constant

Female

partner

# of children

Fraction of hhld wealthLn(hhld wealth)

CCEI

Ln(hhld income '08)

Ln(hhld income '06)

Risk aversion – the fraction of tokens allocated to the cheaper asset

Risk aversion

All sample CCEI ≥ 0.8 CCEI ≥ 0.9

-0.020*** -0.021*** -0.026***(0.007) (0.008) (0.010)

Age-0.010 -0.011 -0.019(0.011) (0.012) (0.014)

-0.045*** -0.048*** -0.054***(0.010) (0.011) (0.013)

-0.052*** -0.054*** -0.067***(0.014) (0.018) (0.021)

Education0.018** 0.018* 0.025**(0.008) (0.009) (0.012)0.015** 0.016 0.026**(0.008) (0.010) (0.012)

Income-0.002 -0.005 -0.007(0.008) (0.011) (0.013)-0.003 -0.008 -0.006(0.009) (0.011) (0.014)0.013 0.009 0.003

(0.011) (0.013) (0.016)Occupation

-0.006 -0.009 -0.014(0.011) (0.014) (0.017)0.011 0.007 0.006

(0.013) (0.017) (0.020)0.003 -0.006 -0.010

(0.012) (0.016) (0.020)Household composition

0.004 0.007 0.006(0.009) (0.010) (0.013)

-0.007** -0.005 -0.004(0.003) (0.004) (0.005)

0.632*** 0.644*** 0.645***(0.015) (0.019) (0.023)

R 2 0.048 0.045 0.054# of obs. 1,182 901 681

Others

Constant

Partner

# of children

Female

65+

High

€5000+

€2500-3499

€3500-4999

Paid work

House work

35-49

50-64

Medium

 

Loss/disappointment aversion

The theory of Gul (1991) implies that the utility function over portfoliostakes the form

min {αu (x1) + u (x2) , u (x1) + αu (x2)} ,

where α ≥ 1 measures loss/disappointment aversion and u(·) is the utilityof consumption in each state.

If α > 1 there is a kink at the point where x1 = x2 and if α = 1 we havethe standard EUT representation.

An illustration of the derived demand

2

1lnxx

2

1lnpp

0ln2

1 =xx

0ln2

1 =pp

Scatterplot of the CARA NLLS estimates

0.0

2.0

4.0

6.0

8.1

.12

.14

.16

.18

.2R

isk

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Loss

Risk aversion (CARA)

All sample CCEI ≥ 0.8 CCEI ≥ 0.9

0.060*** 0.086*** 0.110***(0.021) (0.026) (0.032)

Age0.020 0.025 0.047

(0.030) (0.036) (0.043)-0.007 0.011 0.026(0.029) (0.035) (0.042)0.085* 0.148** 0.187***(0.045) (0.058) (0.071)

Education-0.026 -0.049 -0.069*(0.024) (0.031) (0.038)0.002 -0.018 -0.053

(0.024) (0.030) (0.038)Income

-0.023 0.004 0.023(0.027) (0.035) (0.044)-0.010 0.009 0.010(0.027) (0.035) (0.044)0.003 0.022 0.041

(0.031) (0.039) (0.048)Occupation

0.098*** 0.141*** 0.146**(0.038) (0.050) (0.065)0.046 0.056 0.017

(0.042) (0.057) (0.072)0.091** 0.133** 0.108(0.043) (0.056) (0.072)

Household composition-0.033 -0.034 -0.031(0.026) (0.032) (0.039)0.005 0.008 0.005

(0.010) (0.013) (0.015)0.125** 0.088 0.113(0.054) (0.068) (0.088)

R 2 0.022 0.03 0.037# of obs. 1,182 901 681

House work

Others

Partner

# of children

Constant

Medium

High

€2500-3499

€3500-4999

€5000+

Paid work

Female

35-49

50-64

65+

 

Loss aversion (CARA)

All sample CCEI ≥ 0.8 CCEI ≥ 0.9

-0.021 0.024 0.151(0.166) (0.203) (0.247)

Age0.468* 0.543* 0.706**(0.241) (0.280) (0.333)0.013 0.121 0.301

(0.236) (0.277) (0.331)0.834** 1.230*** 1.413**(0.363) (0.455) (0.560)

Education0.155 0.078 0.116

(0.192) (0.239) (0.296)0.475** 0.523** 0.428(0.192) (0.235) (0.291)

Income0.396* 0.620** 0.773**(0.216) (0.270) (0.337)-0.118 0.046 -0.098(0.217) (0.272) (0.339)0.097 0.216 0.192

(0.244) (0.299) (0.369)Occupation

0.394 0.752* 0.782(0.306) (0.394) (0.508)0.293 0.752* 0.524

(0.340) (0.447) (0.561)0.750** 1.253*** 1.283**(0.347) (0.440) (0.563)

Household composition-0.565*** -0.742*** -0.799***

(0.207) (0.249) (0.305)0.180** 0.204** 0.215*(0.083) (0.100) (0.120)

1.663*** 1.316** 1.432**(0.436) (0.538) (0.692)

R 2 0.036 0.047 0.051# of obs. 1,182 901 681

House work

Others

Partner

# of children

Constant

Medium

High

€2500-3499

€3500-4999

€5000+

Paid work

Female

35-49

50-64

65+

 

Individual-level data

• A review of the full data set reveals striking regularities within and markedheterogeneity across subjects.

• A quick look at the individual-level data indicates that whatever is goingon is not consistent with the standard interpretation of EUT.

• Examples that reveal some of the unexpected features of the data andillustrate the role of heuristics.

Diagonal

Boundary

Smooth

Diagonal and boundary

Diagonal, boundary and smooth

Diagonal, boundary and smooth

Center

Vertex

Centroid (budget shares)

Edge

Bisector

Center and bisector

Edge and bisector

Center, vertex, and edge

Vertex and edge

Center and bisector

Procedural rationality

• How subjects come to make decisions that are consistent with an underlyingpreference ordering?

• Boundedly rational individuals use heuristics in their attempt to maximizean underlying preference ordering.

— There is a distinction between true underlying preferences and revealedpreferences.

— Preferences have an EU representation, even though revealed prefer-ences appear to be non-EU.

2x

1x

21 xx =

A

True preferences

Reveled preferences

Discussion

Suppose there are states of nature and associated Arrow securities andthat the agent’s behavior is represented by the decision problem

max u (x)s.t. x ∈ B (p) ∩A

where B (p) is the budget set and A is the set of portfolios correspondingto the various archetypes the agent uses to simplify his choice problem.

The only restriction we have to impose is that A is a pointed cone (closedunder multiplication by positive scalars), which is satisfied ifA is composedof any selection of archetypes except the Centroid.

We can derive the following properties of the agent’s demand:

1. Let pk denotes the k-th observation of the price vector and

xk ∈ argmaxnu (x) : x ∈ B

³pk´∩A

odenotes the associated portfolio. Then the data

npk,xk

osatisfy

GARP.

2. There exists a utility function u∗ (x) such that for any price vector p,

x∗ ∈ argmax {u (x) : x ∈ B (p) ∩A}⇔

x∗ ∈ argmax {u∗ (x) : x ∈ B (p)} .