Economics 219DExperimental economics

(Spring 2014)

Risk preferencesLectures III & IV

Preferences

Let X be some set of alternatives (consumption set). Formally, we rep-resent the consumer’s preferences by a binary relation % defined on theconsumption set X.

For any pair of baskets (or bundles) x and y, if the consumer says that xis at least as good as y, we write x % y and say that x is weakly preferredto y.

Bear in mind: economic theory often seeks to convince you with simpleexamples and then gets you to extrapolate. This simple construction worksin wider (and wilder circumstances).

The basic assumptions about preferences

The theory begins with two (not three!) assumptions about preferences

[1] Completeness

x % y or y % xfor any pair of bundles x and y.

[2] Transitivity

if x % y and y % z then x % zor any three bundles x, y and z.

Together, completeness and transitivity constitute the formal definition ofrationality as the term is used in economics.

Rational economic agents are ones who [1] have the ability to make choices,and [2] whose choices display a logical consistency.

The preferences of a rational agent can be represented, or summarized, bya utility function (more later).

 The paternity of decision theory and game theory (1944) 

Preferences toward risk

The standard model of decisions under risk (known probabilities) is basedon von Neumann and Morgenstern Expected Utility Theory.

Let X be a set of lotteries, or gambles, (outcomes and probabilities). Afundamental assumption about preferences toward risk is independence:

For any lotteries x, y, z and 0 < α < 1

x  y implies αx+ (1− α)r  αy + (1− α)r.

Experiments à la Allais

Allais (1953) I

— Choose between the two gambles:

$25, 000.33%

A :=.66−→ $24, 000 B := 1−→ $24, 000&.01

$0

Allais (1953) II

— Choose between the two gambles:

$25 000 $24 00033%

34%

:= :=&67

&66

$0 $0

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.

Contributions

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

— Violations of EUT raise criticisms about the status of the Savage axiomsas the touchstone of rationality.

— These criticisms have generated the development of various alternativesto EUT, such as Prospect Theory.

Limitations

Choice scenarios narrowly tailored to reveal “anomalies” limits the useful-ness of data for other purposes:

— Subjects face “extreme” rather than “typical” decision problems de-signed to encourage violations of specific axioms.

— Small data sets force experimenters to pool data and to ignore individ-ual heterogeneity.

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.

 Foundations of Economic Analysis (1947) 

  

  

 

Paul A. Samuelson (1915‐2009) – the first American Nobel laureate in economics and the  foremost  (academic)  economist  of  the  20th  century  (and  the  uncle  of  Larry Summers…). 

   

Revealed preference

A portfolio xi is directly revealed preferred to a portfolio xj if pi · xi ≥pi · xj, and xi is strictly directly revealed preferred to xj if the inequalityis strict.

The relation indirectly revealed preferred is the transitive closure of thedirectly revealed preferred relation.

Generalized Axiom of Revealed Preference (GARP) If xi is indirectlyrevealed preferred to xj, then xj is not strictly directly revealed preferred(i.e. pj · xj ≤ pj · xi) to xi.

GARP is tied to utility representation through a theorem, which was firstproved by Afriat (1967).

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.

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

A benchmark level of consistency

A random sample of hypothetical subjects who implement the power utilityfunction

() =1−

1−

commonly employed in the empirical analysis of choice under uncertainty,with error.

The likelihood of error is assumed to be a decreasing function of the utilitycost of an error.

More precisely, we assume an idiosyncratic preference shock that has alogistic distribution

Pr(∗) =·(

∗)R:·=1

·()

where the precision parameter reflects sensitivity to differences in utility.

If utility maximization is not the correct model, is our experiment suffi-ciently powerful to detect it?

The distributions of GARP violations – ρ=1/2 and different γ

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

CCEI

γ=1/4

γ=1/2

γ=1

γ=5

γ=10

Bronnars’ (1987) test (γ=0)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

CCEI

n=5

n=15n=50

n=10

n=20

Recoverability

• Revealed preference relations in the data contain the information that isnecessary for recovering preferences.

• Varian (ECMA, 1982) uses GARP to generate an algorithm that can re-cover preferences from choices.

• This approach is purely nonparametric making no assumptions about theparametric form of the underlying utility function.

Risk neutrality

Infinite risk aversion

Loss / disappointment aversion

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.

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

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)

 

Risk aversion and loss aversion

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

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 ≥ 0.90.060*** 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+

 

Ambiguity aversion

Ambiguity aversion

• The distinction between settings with risk and ambiguity dates back to atleast the work of Knight (1921).

• Ellsberg (1961) countered the reduction of subjective uncertainty to riskwith several thought experiments.

• A large theoretical literature (axioms over preferences) has developed mod-els to accommodate this behavior.

Experiments à la Ellsberg

Consider the following four two-color Ellsberg-type urns (Halevy, 2007):

I. 5 red balls and 5 black balls

II. an unknown number of red and black balls

III. a bag containing 11 tickets with the numbers 0-10; the number writtenon the drawn ticket determines the number of red balls

IV. a bag containing 2 tickets with the numbers 0 and 10; the numberwritten on the drawn ticket determines the number of red balls

• A cleverly designed experiment that allows distinguishing between fourmodels of ambiguity aversion – SEU, MEU, REU and RNEU.

• The different models of ambiguity aversion generate different predictionsabout how the urns will be ordered.

• For each subject, there will be a unique model that predicts (is consistentwith) the subject’s reservation values.

• No single model predicts all the observed behaviors, and all models arerepresented in the pool of subjects.

The distribution of relative demands

01

23

Ker

nel d

ensi

ty

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Relative demands

)/( 323 xxx

)/( 311 xxx

)/( 211 xxx

The number of diagonal portfolios by subject (ambiguity - vertical axis / loss - horizontal axis)

05

1015

2025

3035

40Y

0 5 10 15 20 25 30 35 40X

Parametric analysis

There is a variety of theoretical models of attitudes toward risk and ambi-guity, but they all give rise to one of two main specifications:

[1] kinked specification rationalized by -Maxmin Expected Utility (-MEU), Choquet Expected Utility, or Contraction Expected Utility.

[2] smooth specification, based on the class of Recursive Expected Utility(REU) models.

The kinked specification has the form

h23(min{1 3}) +

13(2)

i+ (1− )

h23(max{1 3}) +

13(2)

i

where is the ambiguity parameter.

The indifference curves have a “kink” at all unambiguous portfolios where1 = 3.

The general form of the smooth specification isZ∆

µZ

() ()

¶()

where ∈ ∆(∆ ()) is a (second-order) distribution over possible priors on and : (R+) → R is a possibly nonlinear transformation overexpected utility levels.

The REU models are based a cardinal utility indicator — the preferencesgenerated are not independent of a change in the scale of utility.

To clarify, suppose risk preferences are represented by a von Neumann-Morgenstern utility function () with constant absolute risk aversion(CARA)

() = −−

where is the number of tokens, a scale parameter, and is the coef-ficient of absolute risk aversion.

The concavity of the transformation implies that the ranking of uncertainprospects will not be invariant to changes in (cannot identify and separately).

Smooth (left) and kinked (right) specifications

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(p1/p3)

x 1/(x

1+x 3

)The Simulated x1/(x1+x3) and log(p1/p3) in the REU Model when alpha = 0.1

rho = 0.05rho = 0.10rho = 0.25rho = 0.5rho = 1rho = 2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(p1/p3)

x 1/(x

1+x 3

)

The Simulated x1/(x1+x3) and log(p1/p3) in the Alpha-MEU Model when alpha = 0.55

rho = 0.05rho = 0.10rho = 0.25rho = 0.5rho = 1rho = 2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(p1/p3)

x 1/(x

1+x 3

)

The Simulated x1/(x1+x3) and log(p1/p3) in the REU Model when alpha = 1

rho = 0.05rho = 0.10rho = 0.25rho = 0.5rho = 1rho = 2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log(p1/p3)

x 1/(x

1+x 3

)

The Simulated x1/(x1+x3) and log(p1/p3) in the Alpha-MEU Model when alpha = 0.6

rho = 0.05rho = 0.10rho = 0.25rho = 0.5rho = 1rho = 2

Estimation

→ Our first parametric assumption is that risk preferences are represented bya von Neumann-Morgenstern utility function with CARA,

() = −−

→ We restrict, WLOG, the parameters so that preferences are always riskand ambiguity averse — ≥ 0 in both specifications, 12 ≤ ≤ 1 in thekinked specification and 0 ≤ in the smooth specification.

→ For each subject and for each specification, we generate estimates of theambiguity and risk aversion parameters, ̂ and ̂, using nonlinear leastsquares (NLLS).

Econometric results

Kinked specification

56 subjects (38.9%) have non-kinky preferences ̂ ≈ 12. We cannotreject the hypothesis that = 12 for a total of 109 subjects (75.7%)at the 5% significance level.

Smooth specification

44 subjects (30.6%) have ̂ ≈ 0, indicating ambiguity neutrality. Wecannot reject the hypothesis that = 0 for a total of 97 subjects(67.4%) at the 5% significance level .

Scatterplot of the estimated parameters – kinked specification

0.0

5.1

.15

.2.2

5.3

.35

.4.4

5.5

Rho

.5 .55 .6 .65 .7 .75 .8 .85 .9 .95 1Alpha

Scatterplot of the estimated parameters – smooth specification

0.0

5.1

.15

.2.2

5.3

.35

.4.4

5.5

Rho

0 .05 .1 .15 .2 .25 .3 .35 .4 .45 .5Alpha

• There is the strong tendency for subjects to equate the demand for thesecurities that pay off in the ambiguous states, 1 and 3.

• This feature of the data is consistent with the kinked specification but notwith the smooth specification.

• But the tendency to equate 1 and 3 cannot be attributed solely toambiguity aversion.

• In the first place, there is also strong tendency to equate the demand forthe securities that pay off in any pair of states.

A model of ambiguity aversion and loss/disappointment aversion

• If both loss aversion and ambiguity aversion are present in the data, weneed a structural model in order to disentangle the two effects.

• In order to allow for kinks at portfolios where = 0 for any 6= 0, wemake use of the rank-dependent utility (RDU) model of Quiggin (1982).

• This is a generalization of the SEU model that replaces probabilities withdecision weights when calculating the value of expected utility.

• In Quiggin (1982), the decision weight of each payout depends only on its(known) probability and its ranking position.

Following -MEU, we assume that the unknown probabilities 1 and 3are skewed using the weights and 1− :

min = min{1 3}

is given a probability weight 23 and

max = max{1 3}

is given probability weight 23 (1− ) where the parameter12 ≤ ≤ 1

measures the degree of ambiguity aversion.

The utility of a portfolio x = (1 2 3) takes the form

I. 2 ≤ min1 (2) + 2 (min) + (1− 1 − 2) (max)

II. min ≤ 2 ≤ max

3 (min) + (1 + 2 − 3) (2) + (1− 1 − 2) (max)

III. max ≤ 2

3 (min) + 4 (max) + (1− 3 − 4) (2)

where

1 = ³13

´

2 = ³23+

13

´−

³13

´

3 = ³23´

4 = ³23

´−

³23´

and the mapping from the four parameters 1 4 to two parameters and is as follows:

1 =13 +

2 =13 +

3 =13 + +

4 =13 −

The parameter measures the degree of ambiguity aversion and the para-meter measures the degree of loss aversion:

— ≥ 0 and = 0 — kinked specification

— = 0 and ≥ 0 — loss/disappointment aversion (Gul, 1991)

— = 0 and = 0 — standard SEU representation.

The indifference curves will have kinks where = 0 and agents willchoose portfolios that satisfy = 0 for a non-negligible set of prices.

Econometric results

The vast majority of the subjects are well described by the loss- andambiguity-neutral SEU model. The remainder appear to have a signifi-cant degree of loss and/or ambiguity aversion

AmbiguityNeutral Averse Total

Loss Neutral 604 167 771Averse 181 49 229Total 785 215

There is considerable heterogeneity in both ̂ or ̂ and that their valuesare not correlated (2 = 0029).

Scatterplot of the estimated parameters – generalized kinked specification (ambiguity - horizontal axis / loss - vertical axis)

0.0

3.0

6.0

9.1

2.1

5.1

8.2

1.2

4.2

7.3

.33

Y

0 .03 .06 .09 .12 .15 .18 .21 .24 .27 .3 .33X

Procedural rationality

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.

Archetypes and polytypes

• We identify a finite number of stylized behaviors, which collectively posea challenge to decision theory.

• We call these basic behaviors archetypes. We also find mixtures of archetypalbehaviors, which we call polytypes.

• The archetypes account for a large proportion of the data set and play arole in the behavior of most subjects.

• The combinations of types defy any of the standard models of risk aversion.

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)} .

2x

1x

21 xx =

A

True preferences

Reveled preferences

Takeaways

[1] Classical economics assumes that decisions are based on substantive ratio-nality, and has little to say about the procedures by which decisions arereached.

[2] Rather than focusing on the consistency of behavior with non-EUT theo-ries, we study the fine-grained details of individual behaviors in search ofclues to procedural rationality.

[3] The “switching” behavior that is evident in the data leads us to preferan “alternative” approach — one that emphasizes standard preferences andprocedural rationality.

Individual-level data

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