Economics of Uncertainty:Behavior, Perceptions and Policy-Making

PART II

Bergen, NHH, intensive PhD courseApril 11-15, 2011

Nicolas TreichLERNA, Toulouse School of Economics

ntreich@toulouse.inra.frwww.toulouse.inra.fr/lerna/treich/indextreichd.htm

4. Experimental Evidence

Outline- Early tests of Expected Utility- Probability triangle- Framing effects- Preference reversal- The debate about incentives- Are subjects risk-averse?- Field experiments

Experimental Economics

• Hundreds of experiments on risky decisions in psychology and economics

• Will select a small sample of them. Mostly on the tests of foundations, i.e., tests of EU, and of risk-aversion.

• Focus on the experimental results. (Not so much on the experimental design)

Test of Expected Utility • Early experiments suggested by economists, especially Allais (1953 Etca),

• Extended by psychologists, especially Kahneman and Tversky (1979 Etca) ⇒ Huge impact on economics

Good literature surveys: Machina (1989 JEP), Conlisk (1996 JEL), Camerer (1995 Handbook of EP), Starmer (2000 JEL)

Early Experiments

Four main effects:1. Certainty effect (or common ratio effect)2. Allais paradox (or common consequence

effect)3. Isolation effect4. Reflection effect

Certainty Effect

X1= (4000, 80%; 0, 20%)X2= (3000, 100%)

X3= (4000, 20%; 0, 80%)X4= (3000, 25%; 0, 75%)

Most subjects prefer X2 to X1 and X3 to X4

A Variant

X1’= (6000, 45%; 0, 55%)X2’= (3000, 90%; 0; 10%)

X3’= (6000, 0.1%; 0, 99.9%)X4’= (3000, 0.2%; 0, 99.8%)

Most subjects prefer X2’ to X1’ and X3’ to X4’

Allais Paradox

X1*= (2500, 33%; 2400, 66%; 0, 1%)X2*= (2400, 100%)

X3*= (2500, 33%; 0, 67%)X4*= (2400, 34%; 0, 66%)

Most subjects prefer X2* to X1* and X3* to X4*

Isolation Effect X1= (4000, 80%; 0, 20%)X2= (3000, 100%)X3= (4000, 20%; 0, 80%)X4= (3000, 25%; 0, 75%)

Two-stage gamble:X5= (X1, 25%; 0, 75%)X6= (X2, 25%; 0, 75%)

Most subjects prefer X3 to X4 and X6 to X5

Reflection Effect

X1= (4000, 80%; 0, 20%)X2= (3000, 100%)

Y1= (-4000, 80%; 0, 20%)Y2= (-3000, 100%)

Most subjects prefer X2 to X1 and Y1 to Y2Does not violate EU but need a « specific »utility function to accomodate for this choice

Three-outcome Lotteries

• Difficult to « organize » these anomalies

• Observe that these experiments involve lotteries with at most three outcomes

• Can be represented using a triangle of probabilities - Machina (1982, Etca)

Machina Triangle

p1

p3

Any lottery with the same 3 outcomes Only the vector (p1, 1-p1-p3, p3) matters

Any lottery can thus be represented on the following triangle

(0,0,1)

(0,1,0)(1,0,0)

(0.5,0, 0.5)

(2/3,1/3,0)

Risk Preferences

K=p1u(x1)+(1-p1-p3)u(x2)+p3u(x3)

Indifference curves over (p1,p3)Slope: [u(x2)-u(x1)] / [u(x3)-u(x2)]

Without loss of generality, x1<x2<x3

Machina Triangle

p1

p3

Increasing Preferences

Upwards movements increase p3 (at the expense of p2)Leftwards movements reduce p1 (to the benefit of p2)

Risk Preferences

p1

p3

Increasing Preferences

Risk Neutrality

Risk Aversion

Certainty Effect

p1

p3

Increasing Preferences

X1= (4000, 80%; 0, 20%)X2= (3000, 100%)

X3= (4000, 20%; 0, 80%)X4= (3000, 25%; 0, 75%)

X1

X2

X3

X4

Fanning Out

p1

p3

Increasing preferences; but preferences fan out within the triangle: Machina’s hypothesis

X1= (4000, 80%; 0, 20%)X2= (3000, 100%)

X3= (4000, 20%; 0, 80%)X4= (3000, 25%; 0, 75%)

X1

X2

X3

X4

Allais Paradox

p1

p3

Increasing Preferences

X1*= (2500, 33%; 2400, 66%; 0, 1%)X2*= (2400, 100%)

X3*= (2500, 33%; 0, 67%)X4*= (2400, 34%; 0, 66%)

X1*

X2*’

X3*

X4*

Test of Fanning Out

• Machina’s hypothesis: more risk aversion in the region of more preferred lotteries

• This ingenious hypothesis holds at the extremes of the triangle but does not hold in general (Conlisk 1989 AER, Camerer 1995)

• Raises the question of the appropriate alternative theory (see later)

Further Results

• Importance of framing• Intransitivity• Preferences reversal• Uncertainty effect• WTP vs WTA

Two-stage gambles

0.4

0.6

0.4

0.6

0.5

0.5

0.5

0.5

$40

$0

-$40

$0

$40

$0

-$40

$0

Gamble 1

Gamble 2

Which gamble you prefer, 1 or 2?

Two-stage gambles

0.4

0.6

0.4

0.6

0.5

0.5

0.5

0.5

$40

$0

-$40

$0

$40

$0

-$40

$0

Gamble 1

Gamble 2

Gambles are identical

People are not indifferentbetween both gambles.Usually prefer gamble 1.

Choices affected by the probability of winningin the « gain wheel »compared to probabilityof loosing in the « losswheel ».

Framing• Preferences among lotteries should be invariant to the way the question is asked

• Isolation effect or two-stage gamble are evidence of pure framing effect

• Another example. Kahneman and Tversky (1979 Etca):You have been given 1000. You are now asked to choose between 50% of a gain of 1000 or a sure gain of 500. (84% of subjects choose the sure gain)

You have been given 2000. You are now asked to choose between 50% of a loss of 1000 or a sure loss of 500. (69% of subjects choose the gamble)

Intransitivity

X1= (8£, 60%; 0, 40%)X2= (18£, 30%; 0, 70%)X3= (4£, 100%)

About 17% of subjects prefer X1 to X2 and X3 to X1 but prefer X2 to X3 (Loomes et al 1991 JRU)

Possible explanation: when payoffs are « close »prefer higher probas

Preference Reversal Let X1 and X2 two lotteries and C(X1) and C(X2) their certainty equivalent

In EU (and virtually all) models, X1 prefered to X2 is equivalent to C(X1)>C(X2)

Lichtenstein and Slovic (1971 JExpP) show that this equivalence may fail in experiments => preference reversal

Debates about whether this phenomenon disappears with monetary incentives (Grether and Plott 1979 AER)

WTP vs WTA• Typical design: some subjects are endowed with a good X, others

are not – the experimentalist asks sell price (WTA) and buying price (WTP). Results :

X = lottery: WTA = 4WTP (Knetsch et Sinden 1984 QJE)X = mug: WTP=$2.2, WTA = $5.78 (Kahneman et al. 1990)

• Another design (Kahneman et al.): endowment either mug, chocolate or nothing

27

WTP/WTA and Robustness

• Does WTP/WTA disparity disappear with:– repetition and feedback?

yes : Shogren et al. (1994), Knez et al., (1985), Coursey et al. (1987)no : Kahneman et al. (1990), Morrison (1997)

– market? yes : List (2003), Plott and Zeiler (2005)no : Frey et Pommerehne (1987), Kahneman et al. (1990), Knetsch (1995)

28

(In)Sensitivity to Risk Change (a) (b) Risk reduction 4/100,000 7/100,000 Mean willingness to pay £137 £155 Value of a statistical life £3.4 million £2.2 million Median willingness to pay £50 £50 Value of a statistical life £1.2 million £0.7 million 42% of respondents would pay same amount for each good (8% would pay more for the smaller good) Source: Jones-Lee, Hammerton and Philips (1985, EJ)

• See Hammitt and Graham (1999 JRU) for a meta-analysis confirming robustness of insensitivity to risk changes• Possible to retrieve sensitivity under some specific visual aids (Corso et al 2001 JRU)

Uncertainty Effect

• Evidence that a risky project is valued less than its worst outcome (Gneezy et al 2006)

• WTP=$38 for a gift certificate of $50, but only $28 for either a $50 or $100 certificate with 50%

• Basic failure of first-order stochastic dominance• Remark: between subjects experiment (as

opposed to within subjects)• Not replicated under a different design providing

more information (Rydval et al 2010 EE)

Incentives• Most early experiments: absence of monetary incentives (i.e., when subjects are paid in real money)

• What about the experimental tests of EU under appropriate monetary incentives?

• With incentives, and when lotteries are in the interior of the triangle, violations of EU are rare!(Camerer 1989, JRU 1995)

Are People Risk-averse? • Next step. Are subjects risk-averse in experiments with monetary incentives?

• Traditionally small incentives in experiments (a few dollars) => Individuals should be nearly risk-neutral

• Relate to Rabin (2000 Etca)’s critique

Rabin’s Calibration Theorem • Under EU: « Suppose that, from any initial wealth, a person turns down a gamble where she loses $100 or gains $110, each with 50% probability. Then she will turn down 50-50 bets of losing $1000 or gaining any sum of money ».

• The Theorem: Risk-aversion needed to explain low-stakes situations implies an absurd amount of risk-aversion in high-stakes lotteries

• Intuition: Means that marginal utility of wealth must diminish very rapidly

• Rabin: « if we think that subjects in experiments are risk averse, then we know they are not expected utility maximizers.»

Wealth vs. Income

• Yet the « from any initial wealth » in the theorem is important, implicitly means CARA utility. Nevertheless the critique can be adapted for more sensible utility functions.

• A possible explanation: difference between (experimental) income and wealth (Cox and Sadiraj 2006 GEB, Johansson-Stenmann 2008 GEB)

Debate about Incentives • Previous discussions suggest that the choice of incentives (small vs. realworld stakes) matters in experiments on risk preferences. But difficult to induce strong incentives (costly!).

• Kahneman and Tversky (1979 Etca)’s approach: « By default, the method of hypothetical choices emerges as the simplest procedure »

• Rely on the assumptions that i) people know how they would behave with high stakes and ii) do not disguise their true preferences

Risk-aversion and incentives Holt and Laury (2002 AER):

Hypothetical

Incentives Strong incentives effects!

Field Experiments • Binswanger (1980 AJAE) show that most farmers in rural India exhibit significant risk aversion

• Kachelmeier and Shehata (1992 AER) in their study in China report significant increase in risk aversion as incentives increase

• List (2003 QJE, 2004 Etca)’s results tend to show that inexperienced subjects display inconsistencies, like those identified by Kahneman and Tversky. However experience and learning significantly eliminate these inconsistencies

Animal Experiments • Presumably, one might induce salient rewards with animals: food.

• Rats may be taught about the returns (food) of a specific lever. Then, they face a « safe lever » vs. a «random lever »

• Rats display inconsistencies, like the certainty effect Battalio et al. (1985 AER), Kagel et al. (1990) – Same with monkeys (Chen et al. JPE 2004)

• Further research needed to examine the persistence of these inconsistencies among animals

Eliminate Risk-Aversion • Some experimenters may not want to elicit risk-aversion, but instead to «eliminate» it(e.g., in order to discriminate between two hypotheses, one being risk aversion)• Assuming EU, it is possible in experiments to eliminate, in principle, risk-aversion. • Idea: subjects are paid with « probability points»

Eliminate Risk-Aversion • Remark: 10% of $100 is not twice better than 10% of $50. But under EU, it is twice better than 5% of $100. And it is the same as ½ of 15% of $100 and ½of 5% of $100. • Under EU, the objective function is linear in probability points. If subjects are paid with probability points, they behave « like » risk-neutral agents. • Called the lottery-ticket procedure (Roth and Malouf PR 1979) • But the lottery-ticket procedure does not perform well experimentally (Millner and Pratt PC 1989)

5. Non-EU models

Outline- Decision weights- Rank-dependent expected utility- Prospect theory, loss aversion- Regret and disappointment- Anticipatory feelings

Non-Expected Utility Models• These models were mainly influenced by early

experimental failures of EU and, more recently, by psychologists’ work

• Over time models have become more “psychologically realistic”

• Remark: Many models simply encompass EU, so necessarily perform better

• I will present a portfolio of models, but just a few applications (there does not exist so many)

Outcomes and Probabilities• Main feature of EU: separation between outcomes and

probabilities + linearity in probabilities• Let pi and xi be resp. probabilities and outcomes• EU is a special case of Σif(pi,xi) in which f(p,x)=pu(x)• Observe that we can be more general:- f(p,x)=w(p)x : « distortion » of probabilities alone; see e.g. Handa

(1977 JPE) or Yaari (1987 Etca)’s dual theory- f(p, x) =w(p)u(x) : « distortion » of both probabilities and

outcomes; see e.g., Viscusi (1989 JRU)’s prospective theory, or weighted utility (Starmer 2000)

Outcomes and Probabilities (Cont’d)

• w(p) is usually called “decision weight” function• Remark: decision weights w(p) are different from

misperception of probabilities (which are pure judgments distortions, and often coined “probability weights”, see later)

• Non-linear w(p) can explain most of the experimental anomalies presented earlier

• Problem: a non-linear w(p) leads to unappealing normative properties like dominance violations

Dominance Violation

• Suppose w(p+q)>w(p)+w(q)• Take X=($x,p+q; $0,1-p-q)• Take Y =($x+k,p; $x,q; $0,1-p-q)• Note that Y (first-order) stochastically

dominates X• However X may induce a higher utility with

such preferences for k sufficiently small

Rank-Dependent EU

• One way to avoid dominance violation is to let decision weights transform the entire probability distribution of outcomes

with ranked outcomes x1≤x2≤…≤xn and with g increasing and g(0)=0 and g(1)=1.

• See Quiggin (1982 JEBO), Chew (1983 Etca)

1 1RDEU [ ( ) ( )] ( )

n n n

j j ii j i j i

g p g p u x= = = +

= −∑ ∑ ∑

Examples• RDEU = (1-g(p2))u(x1) + g(p2) u(x2) with two

outcomes

• Similar as « weighted » utility• Note that decision weights are now respectively

w(x1)=1-g(p2)) and w(x2)=g(p2) (new notation)

• RDEU = (1-g(p2+p3))u(x1) + (g(p2+p3)-g(p3))u(x2) + g(p3)u(x3) with three outcomes

• Different from « weighted » utility

Remarks• RDEU = (1-g(p2+p3))u(x1) + (g(p2+p3)-g(p3))u(x2) + g(p3)u(x3)

• Transforms cumulative probabilities; the decision weight varies according to how good or bad it is compared to other outcomes= u(x1) + [u(x2)-u(x1)]g(p2+p3) + [u(x3)-u(x2)]g(p3))

• Interpretation: Obtaining the minimum satisfaction u(x1) withcertainty, and evaluates successive additional utility differencesusing the decision weights

• How to characterize risk aversion under RDEU?– lottery dominated by its mean, coined « weak risk aversion », is

equivalent to u concave and g(p)<p (Chateauneuf and Cohen 1994 JRU)

– aversion to mean-preserving spreads, coined « strong riskaversion », is equivalent to u concave u and g convex (Chew, Karni and Safra 1987 JET)

Exercises• Exercise 1: Show in the three-outcome case that RDEU

is consistent with first-order stochastic dominance. Then show in the general case.

• Exercise 2: Assume RDEU with u(x)=x² and g(p)=p². What is the certainty equivalent of (100,50%; 20,50%)? Show that risk aversion holds despite u convex.

• Exercise 3: Assume we find that the certainty equivalent C of (100,p;0,1-p) equals C=100p2. Then i) assume EU, find u(x); ii) Assume RDEU with u(x)=x, find g(p). What do you conclude?

Pessimism vs. Optimism• Let X=(1, ¼; 2, ¼; 3; ¼, 4, ¼), we have w(4)=g(¼) ,

w(3)=g(½)-g(¼), w(2)=g(¾)-g(½) and w(1)=1-g(¾).

1/4 1/2 3/4W(4)W(3)

W(2)

g(p)=p2 generates pessimism g(p)=p1/2 generates optimism

W(1)

1/4 1/2 3/4

W(4)

W(3)W(2)W(1)

Inverted S-shape

• Classical experimental finding: g(.) is inverted S-shape• Axiomatic foundation for a specific class of inverted S-shape: see Prelec (1998 Etca)• Gonzalez and Wu (1999 CP) discuss two parameters for the S-shape: the « elevation » and « curvature » parameters

Prospect Theory

See Kahneman and Tversky (Etca 1979)’s seminal paper>17,000 citations on GoogleScholar (to be compared to e.g. Ellsberg (1961): 3,000 citations; Pratt (1964): 4,000; Krugman’s most citer paper: 6,700; Nash’s bargaining paper: 4,283; Tirole’s IO book: 9,000)

Prospect theory differs in three ways from EU:• People « edit » lotteries• The utility function is replaced by a psychogical value

function • Decision weights w(p)

Prospect Theory: Edition Stage

« Edit » lotteries:• Coding: e.g., people perceive outcomes as

gains or losses rather than final outcomes• Gains and losses are defined w/r to a

reference point• Other aspects: combination, segregation,

cancellation, rounding probabilities…• Implication: importance of framing

Prospect Theory: v(.)

Psychological value function v(.)• Diminishing sensitivity principle (concave

for gains convex for losses)• Loss aversion (steeper for losses)

Losses Gains

Value

Status quo wealth

Prospect Theory: w(.)

Decision weights w(.)• Should be interpreted as beliefs, not

probabilities• Discontinuities around certainty 1 and

impossibility 0 (1979, Etca)• Diminishing sensitivity principle from 0 and 1

Initial KT (1979)

InvertedS-shape

0

1

1

Prospect Theory=> A simple form for Prospect Theory (PT):

with:• r: reference point (e.g., status quo wealth)• v(z): convex (concave) if z<(>) 0 (and steeper for

losses than for gains)• w(.): inverted S-shaped (with possible discontinuities

around 0 and 1)

1PT ( ) ( )

n

i ii

w p v x r=

= −∑

Cumulative Prospect Theory

Kahneman and Tversky (1992 JRU):• Cumulative weighting of probabilities in

order to satisfy first-order stochastic dominance (similar to RDEU)

• Any number of outcomes (not only two)• Two different continuous inverted S-shaped

w(.), one for losses, one for gains• Extensions to subjective probabilities

Loss Aversion: A Few Possible Implications

• People hold losing stocks too long, and sell winning stocks too early

• Disparity WTA/WTP (Knetsch and Sinden 1994 QJE, Kahneman et al. 1990)

• New York city cab drivers quit around daily targets (Camerer et al. 1997 QJE)

• High base + penalty contracts superior to low base + bonus contracts (Hossain and List 2009)

• High « risk aversion » in experiments

Myopic Loss Aversion (MLA)Benartzi and Thaler (1995 QJE)• MLA is the combination of two factors, loss aversion

and « myopia »• Depends on how often people evaluate their portfolio• Idea: if evaluation every day, stock investments not

attractive because stock prices fall too often• Benartzi and Thaler take v(x)=xa if x>0 and bxa if x<0

using KT (1992)’s estimates a=0.88 and b=2.25• They can explain the equity premium puzzle for an

evaluation period of 13 months

Reference Points• How to select reference points? Not clear. Based on

the previous examples, they could be:– Wealth at the beginning of the period– Endowment– Purchase price– Target performance– Expected salary

• Kahneman and Tversky (1991 QJE): « The question of the origin and the determinants of the reference state lies beyond the scope of the present paper. »

• Koszegi and Rabin (2007 AER): « Unfortunately, relatively little evidence on the determinants of reference points currently exists »

Reference-Dependent Preferences

• Koszegi and Rabin (2007 AER)• More general model (« unifying prospect theory

and disappointment model »)• RD= Σipiv(xi/r)= Σipi[u(xi) + f(u(xi)-u(r))] in which r

is the reference point, and f(.) displays loss aversion and diminishing sensitivity

• Equates the reference point with « recent beliefs about outcomes »

• The reference point can be a random variable, and endogeneous

Disappointment• Let X=($1000, 99%; $0, 1%) or Y=($0, 99%;

$1000, 1%) • Suppose you win 0. Do you feel the same if

played X versus Y?• May feel disappointment if played X • Idea: capture the psychological feeling that

an outcome falls short or exceeds some «expectation»

Disappointment (cont’d)• Bell (1985 OR), Loomes and Sugden (1986 RES) • Let u*=Σipiu(xi) be the « expectation », can be

understood as a reference point here• Let v(xi)=u(xi) + D(u(xi)-u*) in which D(.) is «

disappointment » or « elation »• DET= Σipiv(xi)= Σipi[u(xi) + D(u(xi)-u*)] • D linear: EU• Bell assumes D(.) linear but kinked at the origin with

different slopes• Loomes and Sugden assume D(x) convex (concave) if

x is positive (negative)

Regret

• Loomes and Sugden (1982 EJ), Bell (1982 OR)• Let d1 or d2 two decisions and let x1s and x2s be resp.

outcomes in states of the world s• Let v(d1,d2, s)=u(x1s)+R(u(x1s)-u(x2s)) with u and R

increasing and concave• d1 « chosen », and d2 « foregone »• R captures « regret » or « rejoicing »• RT=Σipiv(d1, d2, i)• The « reference point » here is the foregone utility

Some other Models

• Similarity theory• Case-based decision theory• Anticipatory feelings• Endogenous beliefs

Similarity Theory

• Rubinstein (1988 JET)• Idea: the magnitudes which are sufficiently similar

are treated equally (related to the editing stage of Prospect theory, Tversky 1969, 1977 PR)

• Let two lotteries, X1=(x1, p1; 0, 1-p1) and X2=(x2, p2; 0, 1- p2) with xi>0.

• Based on two heuristics:1- Dominance: if p1>p2 and x1>x2 then X1>X22- If heuristics 1 fails, then if p1Sp2 and x1>x2 or p1>p2

and x1Sx2 then X1>X2 in which S means ”similar” i.e ”=”• Implies nontransitivity

Case-based Decision Theory• Gilboa and Schmeider (1995 QJE), an outlier model• In many real decision problems the consequences are

not fully known; this is an attempt to model this• The agent stores in memory a ”case” which is a triple (a

”problem”, and ”act” and a ”result”)• When an agent faces a new problem he knows the

”feasible acts”, and the only remaining information is ”memory” ie past cases

• Then there is a ”similarity function”: the agent scans his memory for similar problems, and identify acts that generated good results

• Agents may be ”locked-in” choosing suboptimal decisions

Anticipatory Feelings• People experience feelings of anticipation prior to the

resolution of uncertainty• Hopefulness, anxiety...• Recursive preferences (Kreps and Porteus, 1978

Etca, Epstein and Zin 1989 Etca) allow for a preference w/r to uncertainty resolution, and can somehow capture anticipatory feelings

• A more recent model: Caplin and Leahy (2001 QJE)• See also psychological games (i.e. belief dependent

games) like models of trust game using guilt aversion

Endogeneous Beliefs• Idea: choose beliefs in models in which there are

anticipatory feelings • Brunnermeir and Parker (2005 AER)’s optimal

expectations• Trade-off between the gain of having « rosy » beliefs (and

be happy) and the cost of poor choices due to mistakenbeliefs: systematic but limited optimism

• An example of how it works: – Let U(d,p) the subjective utility which depends on decision d and

subjective belief p, leading to optimal d(p)=arg maxd U(d,p). – The objective function could then be: U(d(p),p)+U(d(p),q) which is

optimized over subjective beliefs p and where q is objective beliefs

6. Risk Perceptions and Ambiguity

Outline- Subjective Probabilities- Risk perceptions- Mistakes, heuristics and biases- Probabilities elicitation- Ellsberg Paradox- Ambiguity Aversion

Introductory Remark

Here: no more objective probabilities (or « risk », contrary to previous chapters)

=> Calls for a distinction:- Judgments: How do people perceive

uncertainty?- Choices: How do they behave in face of

uncertainty?

What is a Probability?• Traditionnally three views of a probability:

– frequentist, – logical and – subjectivist

• Subjectivist’s view can be summarized by the famous De Finetti (1928)’s sentence: « probability does not exist »

• This sentence means that an « objective » probabilitydoes not exist, a probability only exists in the mind of individuals, it is a « degree of belief »

• De Finetti (1928) and then Savage (1954) essentiallypropose the same « behavioral » definition of probabilityof an event, namely it is a rate at which an individual iswilling to bet of this event

Subjective Probabilities in Economics

• vNM is based on objective probabilities, so it is implicitly« frequentist » or « logical »

• Instead Savage (1954) generalize EU to subjective probabilities

• In Savage’s framework, there is essentially no differencebetween risk and uncertainty

• Moreover this approach suggests that probabilities are not directly observable (one observe only choices) and can be anything (since they are subjective)

• It is no suprise then that subjective probabilities have not been traditionnally an object of research for economists

• This contrasts with the long tradition of research in the psychology of perception

• In particular, this field in cognitive psychology has emphasized systematic biases of perception of risk

Biases and Mistakes

• Milton Friedman: people make mistakes. Just not systematic.

• Tversky and Kahneman (1974, Science): many systematic mistakes

• Confronted with uncertainty, all individuals use similar heuristics, that are useful, but that lead to systematic biases

• These heuristics are instrumental to understand individual risk perceptions

• Since systemantic, they can have macro effects

Mortality Risk Perceptions

Probability Perceptions

Objective Probabilities

Subjective Probabilities

The Inverted S-shape Pattern. Most common shape in probability judgments.

Two-Card Game

• Two cards• Each has the two faces colored• One card has red/red• The other card has red/green• You observe red• What is probability that the other side is

red?

Risk Heuristics• Detection test of a disease (women’s breast

cancer)• Breast cancer prevalence at 40 years old: 1%• The test is imperfect:- 90% with a breast cancer are tested “positive”- 10% with no breast cancer are also tested

“positive”

• What is the probability that a person tested positive has actually the disease?

Small Sample

• Hospital A : 45 babies/day • Hospital B : 15 babies/day• Nb of days for which >60% of babies are

male: larger for A or B?

• Answers: 21% A, 53% the same• The law of large numbers does not

belong to the repertory of intuitions

Law of Small Numbers

• Coin: Tail (T) or Head (H) ; proba = 1/2• Results: TTTT• Probability that H comes out next flip? • Many subjects say >1/2• “gambler’s fallacy”• Misperception of correlations

Repeated Sequences

• Gneezy (1996 AP)• A financial asset follows a random walk• +$1 or -$1 with p and 1- p• Assume p=0.7• Stop if the asset reaches either 0 or 10• Probability to reach 10?• If starting value $1, proba=0.57; if $5

proba=0.9857• It has been shown that exponential growth is

considerably underestimated (Wagenaar and Timmers 1979 AP)

Compound Events• Suppose you were able to build a ballistic missile

defense that depended on the successful performance of 500 independent parts or subsystems and suppose that each part or subsystem were 99% reliable. What are the chances that a system would work in a first attempt? (p<1%)

• Let a group composed of N people. What is the value of N necessary to make it more that 50% the chances of 2 people having the same birthday? (N=23)

Sampling and Bayes’ Rule• You can bet on three gambles:- 1) Drawing a red ball from a bag containing 50%

red balls - 2) Drawing a red ball for seven successive times,

with replacement, from a bag containing 90% red balls

- 3) Drawing a red ball at least once in seven successive tries with replacement, from a bag containing 10% red balls

Usually people prefer 2 (p=0.48) to 1 (p=0.5)and 1 to 3 (p=0.52)

Sampling and Bayes’ Rule• Imagine an urn filled with balls, of which 2/3 are of

one color and 1/3 of another. One individual has drawn 5 balls from the urn, and found that 4 were red and 1 was white. Another individual has drawn 20 balls and found that 12 were red and 8 were white. Which of the two individuals should be more confident that the urn contains 2/3 red balls and 1/3 white balls?

• Correct posterior odds are resp. 8 to 1 and 16 to 1• “Conservative estimates”: usually estimates are

far less extreme than the correct values

Heuristics

• Need to explain and organize patterns of probability (mis-)perceptions

• Tversky and Kahneman (1974, Science): People use heuristics when they have to assess probabilities

• Three main heuristics: “Representativeness”, “Availability” and “Anchoring”

Representativeness

• Heuristic: Probabilities are evaluated by the degree something resembles something else

• Typical induced bias: Detailed scenarios (who seem more realistic) are judged more likely; But more details in fact lessen likelihood

• Implications: May include preconception of chance, misconception of regression to the mean

Availability

• Heuristic: Probabilities evaluation depends on the ease in which instances or occurrences can be brought to mind

• Typical induced bias: More salient (e.g., sensational, recent) events are judged more likely

• Implications: May include overestimation of small probabilities, illusory correlation

Anchoring

• Heuristic: People make probability estimates by starting from an initial value that is adjusted insufficiently

• Typical induced bias: Estimates end up around a “natural” anchor

• Implications: May include conjunctive vs disjunctive events, anchoring to the mean, insufficient adjustment

Hindsight Bias• Limited memory• Hindsight bias or “I knew it all along”• Systemic bias in how people recall their

own past judgments of probabilities• People believe that an event was

inevitable; Determinism• After an event occurs, people

overestimate their own past probabilities predictions that this event would occur (Fischhoff, 1975, JExp.Psych.)

Overconfidence• Calibration: a judge is calibrated if, over the long

run, for all propositions assigned a given probability, the probability that it is true equals the probability assigned

• Example. “Absinthe is a a) precious stone, b) a liqueur, c) a Caribbean island”. The assessor provides an answer and then gives a probability that an answer is correct

• Most pervasive finding in miscalibration is overconfidence (Fischhoff and al, 1977, J.Exp.Psych)

• Overconfidence is more extreme with tasks of great difficulty; Underconfidence may be found for easy tasks

Illusion of Control• People behave as if chance events are subject

to control• Dice players play softly (strongly) if they want a

small (large) number• Illusion of control: Expectancy of a personal

success probability is inappropriately higher than the objective probability under conditions of perceived control (Langer 1975 JPSP)

• Example: Car accidents vs. plane accidents

Many other Biases• Projection bias• Self-serving bias• Attribution bias• Attribution bias• Confirmation bias• Cognitive dissonance • Distinction bias• Probability neglect• Outcome bias• Availability cascade• False consensus effect

Remark: Debiasing

• There exists a literature on how to reduceor eliminate the biases

• See especially Gerd Girgerenzer in psychology (mostly using framing effects), and John List in economics (mostly marketeffects)

Implication for Economics?• What are the implications of this psychological

research on beliefs for economists?• Even if economists may not be interested in

beliefs per se, there are interested in choices• Yet, choices presumably depend on beliefs• Misattribution problem: a cautious behavior may

be attributed to risk aversion, while it may be in fact due to pessimistic beliefs

• In other words, misrepresentation of beliefs maylead to a misestimation of risk preferencesparameters

• How to solve this empirical difficulty?

Identification Problem • Choice data are consistent with many combinations

of choice/beliefs models; see Manski (Etca, 2004) • Strategy 1 (most common): Make strong

assumptions about beliefs (e.g., rational, naive etc.). Problem: erroneous if people hold different beliefsthat those assumed

• Strategy 2: Use stated data on beliefs (e.g. surveys, opinion polls). Problem: how reliable are these data? See the debate about hypothetical vs real stakes

• Strategy 3 (in experiments): use incentivized beliefs elicitation techniques, that is, “truth serums” like scoring rules

Scoring Rule

• Suppose a subject believes that the probability of an event (e.g., no rain tomorrow) is q

• Objective: obtain truthful subjects’ estimates of the probability she assigns to this event by giving rewards contingent of the realization or not of this observable event

• Essentially, De Finetti and Savage’s idea

Scoring Rule• A scoring rule is characterized by contingent

payments S(p) and SN(p) when reported probability is p

• Namely S(p) is the monetary amount the subject is paid if the event occurs, and SN(p) if the event does not occur

• Under risk neutrality, the subject thus maximizesMaxp qS(p)+(1-q)SN(p)FOC: qS’(p)+(1-q)SN’(p)=0SOC: qS’’(p)+(1-q)SN’’(p)<0• Let consider a quadratic scoring rule: S(p)=2p-p2

and SN(p)=1-p2

• FOC: q(1-p)-(1-q)p=0 => p*=q (truthful revelation)

Quadratic Scoring Rule• It is said that the quadratic scoring rule is

proper (i.e., incentive compatible)• Statisticians have shown that other scoring

rules are proper (Savage 1971 JASA), e.g. the logarithmic scoring rule (Good 1952 JRSS)

• Selten (1998 EE) provides an axiomatic foundation for the quadratic scoring rule (based on four axioms: symmetry, elongation invariance, incentive compatibility and neutrality)

Efficiency of Scoring Rules• Allow beliefs to be elicited even without mentioning the

word “probability”, or defining it• Yet, the efficiency of a scoring rule in experiments is

not clear (Camerer 1995). It is often too complex (Hogarth 1987). It may alter the behavior of the subject during the course of an experiment (Croson2000 JEBO).

• Predictions are too extreme (Palfrey and Wang 2009 JEBO), and erratic (Nyarko and Schotter 2002 Etca)

• Theoretically, not incentive compatible under risk-aversion and if the subject has “stakes” in the predicted events (Armantier and Treich 2010)

Models of Ambiguity

• Ellsberg paradox• Modelling ambiguity aversion• Different choice models

Ambiguity and climate sensitivitySavage’s EU predicts that individuals make choices « as if » there exists a unique subjective pdf. Is it reasonable?

Source: Millner et al. (2010)

Ambiguity: Ellsberg’s Experiment

• Ellsberg (1961, QJE)’s experiment is one the most popular experiment ever

• A decision-maker chooses from an urn that contains 30 red balls and 60 balls in some combination of black and yellow.

• The probability of winning is thus unknown, or ambiguous.

Ellsberg’s ExperimentAct 30 blue balls white balls red balls

X $W 0 0

Y 0 $W 0

X’ $W 0 $W

Y’ 0 $W $W

Many people choose X>Y and Y’>X’

Inconsistency with Savage axiom

• Contradicts the « sure-thing principle »: let f, f’, g and g’ be lotteries and S an event. If f=g and f’=g’ on S, and on ~S (not S), then f is preferred to f’ iff g is preferred to g’ (Let S equals red, f=X, f’=Y, g=X’ and g’=Y’)

Ambiguity

• Defining ambiguity is a popular pastime..• Ambiguity is “known-to-be-missing

information” (Camerer 1995, Handbook Exp. Econ.)

• Ambiguity aversion is a robust finding in experiments (e.g., Slovic and Tversky BS)

Ambiguity Aversion Literature• Experimental evidence: Subjects are usually often found

to be averse to ambiguous probabilities (e.g., Camerer and Weber 1992 JRU)

• Survey evidence (Hogarth and Kunreuther 1985; Viscusi et al. 1991; Chesson and Viscusi 2003; Burks et al. 2008)

• Ambiguity aversion may explain some empirical puzzle (e.g.) in finance (Dow and Werlang 1992; Mukerji and Tallon 2001; Chen and Epstein 2002; Ju and Miao 2009; Epstein and Schneider 2010)

• Ambiguity in markets: Sarin and Weber (1993 MS) in double-oral and sealed-bid auctions, preference for home-country investments (French and Poterba 1991 AER) and entrepreneurship and labor contracting (Bewley 1986)

Causes to Ambiguity Aversion?

• Hyper-rationality: Rational account of incomplete information: “Agents can have different models (..) and be aware of the possibility that their model is misspecified” Maccheroni et al. (1995)

• Paranoia: the decision-maker behaves as if a malevolent Nature changes the odds against him(Cerreia et al. 2008)

• Fear: Using brain imaging, Camerer et al. (2007) findevidence that the amygdala is more active underambiguity conditions, and notice that the “amygdalahas been specifically implicated in processinginformation related to fear”

Modeling Ambiguity Aversion• Let θ=0,1,…,n the number of white balls in the Ellsberg’s

urn, leading to the associated subjective probability p(θ) of drawing a white ball

• Savage’s subjective expected utility: essentially assumes that only the subjective expectation Ep(θ) matters

• Maxmin: Gilboa and Schmeidler (1989 JME)’s proposes an axiomatics so that the decision maker maximizesexpected utility under the « worst » probability p(θ) withina set

• Many variant versions of Maxmin theory (e.g., Epstein 1999; Ghirardato et al. 2003; Maccheroni 2006; Gajdos et al. 2009)

Smooth Ambiguity Aversion

• See Klibanoff, Marinacci and Mukerji (Etca, 2005)• Suppose θ is known. Let U(θ)= (1-p(θ))u(w) +

p(θ)u(w+100) the expected utility reached ex post for a specific θ (e.g., when choosing act Y in the Ellsberg urn)

• The ambiguity averse agent then evaluates his welfare ex ante by the certainty equivalent of the random EU:Φ-1(EΦ(U(θ)) = Φ-1(∑θq(θ) Φ(U(θ))

q(θ) is the subjective probability that the parameter value is θ;Also coined the « second order belief »

Ø reprensents ambiguity aversion

KMM preferences• Interpretation: two-stage lottery (first-stage determining probability

distribution, and second-stage determining outcome)• Axiomatics: unique second order belief giving rise to an expectation of

an expected utility• Permit to distinguish ambiguity, ambiguity aversion (a concave ø), and

Arrow-Pratt risk aversion – moreover defines an « index » of ambiguity aversion

• Two benchmarks: expected utility for ø linear, and MaxMin of Gilboa-Schmeidler (1989) for ø CARA with infinite absolute risk aversion

• Open to a violation of the axiom of reduction of compound lotteries for ø non-linear (see Segal 1987 IER; see also Kreps and Porteus 1978 Etca)

• Preferences are not « kinked » (as with MaxMin) but are « smooth »when ø is differentiable, thus improving tractability

Difficulties with Ambiguity Theories• Portfolio of ambiguity (aversion) models – which one to

choose?• Few empirical evidence based on market data – can we

test these models? What is a « proxy » for ambiguity? How to estimate ambiguity aversion?

• Robust to reflection? Consider the following argument (Raiffa 1961): Assume in the Ellsberg urn that you are offered two options A (you get either X or Y’ with 50%) or B (you get either Y or X’ with 50%). A should bepreferred to B based on the Ellsberg paradox. Yet A and B are objectively identical.

A Time Inconsistency Problem• See, e.g., Al-Najjar and Weinstein (2009) for arguments

that all ambiguity (aversion) models are time-consistent, and if not, lead to other anomalies:– Sophistication (Siniscalchi 2006), but then information aversion– Distortion of the updating rule (Hanany and Klibanoff 2007,

2009), but then not fact-based updating (« Dynamic consistent beliefs must be bayesian », Epstein & Le Breton 1993)

– Restriction of information structures (Sarin & Wakker 1998, Epstein & Schneider 2003, Maccheroni et al. 2006), which is not sensible (prevents choices that are problematic for the theory!)

Hyper-rationality or Mistake?• Halevy (2007 Etca)’s experimental evidence

• Four (two-colour) urns1: 5 blue balls and 5 white balls

2: unknown proportion

3: number of blue balls uniform in {0,1,…,10}

4. either 10 blue balls or 10 white balls

• Urns 1 and 2 are Ellsberg’s urns; Urns 1, 3 and 4 equivalentusing reduction of objective compound lotteries (ROCL)

• 80% of subjects are ambiguity averse, and 86% fail ROCL• But conditional on ROCL, 96% are neutral to ambiguity; and

conditional on failing ROCL, 95% display Ellsberg’s paradox

Status of Ambiguity Models?

• Either describe « rational behavior », but they induce other anomalies

• Or describe behavioral choices (people use heuristics leading to mistakes), but there exist much simpler behavioral economics models

• No clear interpretation of « beliefs » (and as a result almost no applications to information economics and game theory)