Introduction Prospect theory or skill signaling?Skilled type more likely to win, Pr[winjs] >...

IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion

Prospect theory or skill signaling?

Rick Harbaugh

Emory UniversityApril 2008

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Two classic literatures

Prospect theory (Kahneman & Tversky, 1979, 1992)

People often take actions that seem inconsistent withexpected utility maximizationThey are �loss averse�, like �long shots�, are a¤ected by�framing�

Career concerns (Holmstrom, 1982/1999)

Managers often take actions that are inconsistent withpro�t maximizationThey avoid some types of risks, but like others, chase badmoney with good, etc.

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Two classic literatures

Prospect theory (Kahneman & Tversky, 1979, 1992)

People often take actions that seem inconsistent withexpected utility maximizationThey are �loss averse�, like �long shots�, are a¤ected by�framing�

Career concerns (Holmstrom, 1982/1999)

Managers often take actions that are inconsistent withpro�t maximizationThey avoid some types of risks, but like others, chase badmoney with good, etc.

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Are these literatures related?

Both predict deviations from simple maximization... butfor very di¤erent reasons

Prospect theory

People have perceptual biasesExpected utility maximization asks too much

Career concerns

Managers care about looking competent (skill signaling)Expected utility maximization over monetary outcomes toonarrow

How similar are the predictions?

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Prospect theory


Career concerns



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Prospect theory


Career concerns



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Prospect theory


Career concerns



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Main argument - predictions pretty similar!

Empirical analyses could confound e¤ects

Managerial environmentsExperimental environments

Theoretical analyses might be one-sided

Behavioral economics underemphasizing social angle?Early social psychology models relevant?

Richer exchange between psychology and economics

Economics can formalize social psychology models(Benabou and Tirole, Sobel)

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Related to other literatures

Social psychology

Self-esteem (James, 1890)Achievement motivation (Atkinson, 1957)Self-handicapping (Jones and Berglas, 1978)

Violations of expected utility maximization

Regret theory (Bell, 1982; Loomes and Sugden, 1982)Rank-dependent utility (Quiggin, 1982)Disappointment aversion (Gul, 1991)

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Related to other literatures

Social psychology

Self-esteem (James, 1890)Achievement motivation (Atkinson, 1957)Self-handicapping (Jones and Berglas, 1978)

Violations of expected utility maximization

Regret theory (Bell, 1982; Loomes and Sugden, 1982)Rank-dependent utility (Quiggin, 1982)Disappointment aversion (Gul, 1991)

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Some common risk anomalies

Loss aversion �Kahneman and Tversky

Excessive risk aversion for small gambles - RabinBut also overcon�dence - Odean and Barber

Framing �Kahneman and Tversky

More likely to gamble if winning is reference pointSo how frame the status quo matters

Probability weighting �Kahneman and Tversky

Long shot bias �ThalerSimultaneous purchase of insurance, lottery tickets �Friedman and Savage

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Why so risk averse for small gambles?

Small gamble:

A. $0B. 50-50 lose $100 or win $110

Larger gamble:

A. $0B. 50-50 lose $1000 or win $718,190

Rabin (2000): For standard utility function, if avoid smallgamble then should avoid larger gamble too

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Why so risk averse for small gambles?

Small gamble:

A. $0B. 50-50 lose $100 or win $110

Larger gamble:

A. $0B. 50-50 lose $1000 or win $718,190

Rabin (2000): For standard utility function, if avoid smallgamble then should avoid larger gamble too

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Loss aversion?

Loss aversion can explainthis behavior

Marginal utility notcontinuous at status quo

So substantial riskaversion even for smallgambles

But assuming lossaversion just begs thequestion...

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Why don�t people like to lose?

Achievement motivation literature (Atkinson, 1957)

Most gambles involve some skillPeople don�t want to look unskilledSo avoid gambles that might reveal lack of skill

Career concerns literature (Holmstrom, 1982/1999)

Managers are concerned with appearing skilledSo choose investments to reduce risk of appearing unskilled

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Why don�t people like to lose?

Achievement motivation literature (Atkinson, 1957)

Most gambles involve some skillPeople don�t want to look unskilledSo avoid gambles that might reveal lack of skill

Career concerns literature (Holmstrom, 1982/1999)

Managers are concerned with appearing skilledSo choose investments to reduce risk of appearing unskilled

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Assumptions of base model

Decision maker is skilled or unskilled, q 2 fs, ugFirst assume decision maker does not know own type

Gamble has payo¤ x 2 flose,wingSkilled type more likely to win, Pr[winjs ] > Pr[winju]Utility quasilinear, U = Y + v(Pr[s j all info])Want to look skilled, v 0 > 0

Risk averse in skill estimate, v 00 < 0

And also downside risk averse, v 000 > 0

For instance, v is CRRA

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Captures �performance skill�

Some people better at a risky endeavor

Skilled manager more successful at projectSkilled athlete better at maneuver

�Career concerns� in principle-agent literature

Holmstrom (1982/1999) rat raceZwiebel (1995) corporate conformism

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Posterior skill estimate

Skill is updated based on outcome of gamble

Pr[s jwin] =Pr[win, s ]Pr[win]

= Pr[s ] +Pr[win, s ]� Pr[s ]Pr[win]

Pr[win]

= Pr[s ] +Pr[winjs ]� Pr[winju]

Pr[win]Pr[s ]Pr[u]

Pr[s jlose] = Pr[s ]� Pr[winjs ]� Pr[winju]Pr[lose]

Pr[s ]Pr[u]

Note Pr[s jwin]Pr[win] + Pr[s jlose]Pr[lose] = Pr[s ]So v(Pr[s jwin])Pr[win] + v(Pr[s jlose])Pr[lose] < v(Pr[s ])

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Example

Pr[s ] = Pr[u] = 12

Pr[winjs ] = Pr[win] + ε, Pr[winju] = Pr[win]� ε

So Pr[s jwin] = 12 +

ε2 Pr[win] and Pr[s jlose] =

12 �

ε2 Pr[win]

Suppose Pr[win] = 12 , and ε = 1

4

Then Pr[s jwin] = 34 , Pr[s jlose] =

14

So even a �friendly bet�has some real risk of making onelook unskilled

And ignoring this concern will look like loss aversion

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People are even stranger ...

Gamble or not?A. Get $10 for sureB. 10% chance win $100

Gamble or not?A. Lose $10 for sureB. 10% chance lose $100

Gamble or not?A. Get $90 for sureB. 90% chance win $100

Gamble or not?A. Lose $90 for sureB. 90% chance lose $100

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Not just �risk averse for gains�and �risk loving forlosses�

Original prospect theoryassumed risk averse ingains and risk loving inlosses

So, in addition to kink atstatus quo for lossaversion, have concavityabove and convexitybelow

But data seems moreconsistent with the�Four-Fold Pattern�

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Cumulative Prospect Theory: Four-fold pattern(FFP)

Favor long shots for gainsA. Get $10 for sureB. 10% chance win $100

Avoid small chance of lossA. Lose $10 for sureB. 10% chance lose $100

Avoid sure-things for gainsA. Get $90 for sureB. 90% chance win $100

Try to win back likely lossesA. Lose $90 for sureB. 90% chance lose $100

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Simpler if consider probability of success

Overweight small chance ofsuccess

A. Get $10 for sureB. 10% chance win $100

Underweight large chance ofsuccess

A. Lose $10 for sureB. 10% chance lose $100

Underweight large chance ofsuccess

A. Get $90 for sureB. 90% chance win $100

Overweight small chance ofsuccess

A. Lose $90 for sureB. 90% chance lose $100

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Cumulative prospect theory captures FFP withprobability weighting function

p = Pr [win]

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Is skill signaling an alternative explanation for FFP?

Probability weighting violates expected utility theory

What if people care about looking skilled?

And if expected utility maximization with this concern alsopredicts FFP?

Then the pattern could be due to either probabilityweighting or to skill signaling

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Can skill signaling �explain�apparent pattern ofprobability weighting?

Low probability of success

Success is rare and a strong signal that one is skilledFailure is common and a weak signal that one is unskilled

High probability of success

Success is common and a weak signal that one is skilledFailure is rare and a strong signal that one is unskilled

So seems low probability gambles have both more upsidepotential and less downside risk

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Are low probability gambles more attractive?

Is winning really more impressive for small Pr[win]?Is losing really more incriminating for high Pr[win]?

Pr[s jwin] = Pr[s ] +Pr[winjs ]� Pr[winju]

Pr[win]Pr[s ]Pr[u]

Pr[s jlose] = Pr[s ]� Pr[winjs ]� Pr[winju]Pr[lose]

Pr[s ]Pr[u]

Pr[s jwin] decreasing in Pr[win] if(Pr[winjs ]� Pr[winju]) /Pr[win] decreasing in Pr[win]Pr[s jlose] decreasing in Pr[win] if(Pr[winjs ]� Pr[winju]) /(1� Pr[win]) increasing inPr[win]

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Example again

Pr[s ] = Pr[u] = 12 , Pr[winjs ] = Pr[win] + ε,

Pr[winju] = Pr[win]� ε

Suppose ε = Pr[win]Pr[lose] soPr[winjs ]� Pr[winju] = 2Pr[win]Pr[lose]Then Pr[s jwin] = 1

2 +12 Pr[lose] so winning is most

impressive for small Pr[win]

And Pr[s jlose] = 12 �

12 Pr[win] so losing is most

incriminating for large Pr[win]

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Updated skill estimate as function of winningprobability

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So are long shots favored?

Low Pr[win] gambles seem attractive...

Winning is more impressive and losing less embarrassingBut winning is less common and losing is more commonSo not so clear what e¤ect dominates

Suppose utility function for skill estimate has �downsiderisk aversion�

Two gambles with equal expected values and equalvariancesThen gamble with worse bad outcome o¤ers less utilitySince low Pr[win] gamble less embarrassing, they arepreferredIf v 0 > 0, v 00 < 0, and v 000 > 0 then downside risk averse

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So are long shots favored?

Low Pr[win] gambles seem attractive...

Winning is more impressive and losing less embarrassingBut winning is less common and losing is more commonSo not so clear what e¤ect dominates

Suppose utility function for skill estimate has �downsiderisk aversion�

Two gambles with equal expected values and equalvariancesThen gamble with worse bad outcome o¤ers less utilitySince low Pr[win] gamble less embarrassing, they arepreferredIf v 0 > 0, v 00 < 0, and v 000 > 0 then downside risk averse

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Expected utility comparison for example withdownside risk aversion

Pr[win] = .2 or Pr[win] = .8, v(Pr[s ]) = 1/Pr[s ]

So higher utility and lower risk premium from �long-shot�

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General implications of this model

Losing doubly unfortunate since increases probability thatone is unskilled - loss aversion

If probability of success is high, the chance of losing issmall but the loss in skill reputation from losing issubstantial

If probability of success is low, the chance of losing is highbut the loss in skill reputation from losing is minor

Downside risk averse: Small chance of a large loss morepainful than large chance of a small loss

So there is a long-shot bias relative to other gambles - butstill a little scared of longshots

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Imputing w(p) from risk premia

Suppose (as we do) decision maker is risk neutral withrespect to money

Then risk premium π is just how much money she wouldforego to avoid expected embarrasment from gamble

And the probability weight that would be calculated ifignore embarrassment aversion is then, for p = Pr[win],

w(p) = p � π

win� lose

We have found π is smaller for p < 1/2 than p > 1/2But prospect theory says w(p) > p for small p andw(p) < p for large p, so need negative π for small p

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Imputed w(p) for the example

lose = 0, win = 10, v = �1/Pr[s ], Pr[s ] = 1/2Pr[winjs ]� Pr[winju] = 2Pr[win]Pr[lose]

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Imputed w(p) with smaller skill gap

lose = 0, win = 10, v = �1/(Pr[s ]), Pr[s ] = 1/2Pr[winjs ]� Pr[winju] = Pr[win]Pr[lose]

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Now suppose decision maker informed about ownskill

Sees noisy signal g or b where Pr[s jg ] > Pr[s jb]Accepting a gamble can show con�dence in skillRefusing a gamble can be admission that one expects tolose

Look at perfect Bayesian equilibria surviving D1

Unexpected acceptance? D1 predicts was by type gUnexpected refusal? D1 predicts was by type b

The more the decision maker knows about own skill themore negative the inference from not gambling

So might be better to risk embarrassment from losingrather than admit that one has no chance

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Overcon�dence and Loss Aversion

Theorem

Suppose v 0 > 0. In any pure strategy equilibrium the riskpremia πθ are negative for a) any given Pr[s jg ]� Pr[s jb] > 0and su¢ ciently low Pr[winjθ], or b) any given Pr[winjθ] andsu¢ ciently large Pr[s jg ]� Pr[s jb].

Theorem

Suppose v 00 < 0. In any pure strategy equilibrium the riskpremia πθ are positive for any given Pr[winjθ] and su¢ cientlysmall Pr[s jg ]� Pr[s jb].

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Overcon�dence and Loss Aversion

Theorem

Suppose v 0 > 0. In any pure strategy equilibrium the riskpremia πθ are negative for a) any given Pr[s jg ]� Pr[s jb] > 0and su¢ ciently low Pr[winjθ], or b) any given Pr[winjθ] andsu¢ ciently large Pr[s jg ]� Pr[s jb].

Theorem

Suppose v 00 < 0. In any pure strategy equilibrium the riskpremia πθ are positive for any given Pr[winjθ] and su¢ cientlysmall Pr[s jg ]� Pr[s jb].

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Long shot bias still holds

De�nition

Two gambles F and G are a symmetric pair if PrF [win]= PrG [lose], PrF [qjθ] = PrG [qjθ], and PrF [θ] = PrG [θ] = 1/2for q 2 fu, sg, θ 2 fb, gg.

Theorem

Suppose v 0 > 0, v 00 < 0, and v 000 > 0 and consider asymmetric pair of gambles F and G. For PrF [win]= PrG [lose] < 1/2, in any given pure strategy equilibrium theaverage risk premium is lower for F than for G.

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Multiple equilibria pattern same direction

Theorem

Suppose v 0 > 0, v 00 < 0, and v 000 > 0 and consider asymmetric pair of gambles F and G with EF [x ] = EG [x ]. ForPrF [win] = PrG [lose] su¢ ciently small, (i) a neither-gambleequilibrium exists for G if it exists for F ; (ii) a separating orneither-gamble equilibrium exists for G if a separatingequilibrium exists for F ; (iii) a separating or both-gambleequilibrium exists for F if a separating equilibrium exists for G;(iv) A both-gamble equilibrium exists for F if it exists for G.

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Imputed w(p) with informed decision maker

lose = 0, win = 10, v = �1/Pr[s ], Pr[s ] = 1/2Pr[winjs ]� Pr[winju] = 2Pr[win]Pr[lose]Pr[s jg ]� Pr[s jb] = 1/10, Pr[g ] = 1/2

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Multiple equilibria captures aspect of �framing�?

Subjects often behave di¤erently depending on how agamble is framed

Gamble when the no gamble outcome is portrayed as badrelative to winning reference pointDon�t gamble when the no gamble outcome is portrayed asgood relative to losing reference point

In signaling games with discrete choices multiple equilibriaeasily arise

Framing of the gamble can give some indication ofreceiver�s beliefs about which equilibrium is being played

So depending on how gamble is framed, subject can be�dared� into taking a gamble

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Suppose there is �evaluation skill�

Some people better at evaluating odds of di¤erent gambles

Skilled broker picks right stocksSkilled handicapper picks right horses

Also standard in career concerns literature

Holmstrom (1982/1999) �distorted investment decisionsPrendergast and Stole (1996) � sunk costs

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Performance skill generates same type of results

Losing doubly unfortunate since looks like one foolishlytook a bad gamble

Same tradeo¤ as with performance skill

So lower risk premium for long shot gamble thanequivalent near sure-thing gamble

Again close to prospect theory, but risk premia are alwayspositive for pure evaluation skill

If both performance and evaluation skill then can havenegative risk premia for longshots

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What if observe outcome of gamble that is refused?

Cannot escape evaluation of skill by refusing gamble

Refusing a gamble that succeeds just as embarrassing astaking a gamble that fails

Average risk premium for low Pr[win] gambles negative �just as if low Pr[win] gambles overweighted

Average risk premium for high Pr[win] gambles positive �just as if high Pr[win] gambles underweighted

Hard to distinguish from prospect theory�s weightingfunction

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Imputed w(p) with informed decision maker

lose = 0, win = 1, v = �1/Pr[s ], Pr[s ] = 12

Pr[winjs, g ]� Pr[winju, g ] =Pr[winju, b]� Pr[winjs, b] = Pr[win]Pr[lose]

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Summary - estimated probability weights if ignoreskill signaling

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Relation to Atkinson�s model of achievementmotivation

Atkinson (1957) noted di¤erent probability gamblesconveyed di¤erent skill information

Argued with reduced form model that people should bemost afraid of gambles with equal chance of success orfailure since most revealing

But data found people also favored long shots over nearsure-things

Atkinson�s model generated by initial example with skillgap proportional to Pr[win]Pr[lose] and v 0 > 0, v 00 < 0and piecewise linear

If instead v 000 < 0 then Atkinson model predicts the data

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Relevance for prospect theory and career concerns

Can skill signaling provide a foundation for prospecttheory?

Experiments done without any explicit skill component sowould seem unlikelyExcept most tests are thought experiments without realstakesSo subjects have to imagine what they would do - and inreal world almost all risk has skill component

Can prospect theory be a reduced form model of careerconcerns?

Lots of evidence that managers do engage in skill signalingAnd that their behavior is consistent with prospect theorySo why not just use prospect theory?Skill signaling is simplest career concern model - can getvery di¤erent behavior as change information andincentives

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Relevance for prospect theory and career concerns

Can skill signaling provide a foundation for prospecttheory?

Experiments done without any explicit skill component sowould seem unlikelyExcept most tests are thought experiments without realstakesSo subjects have to imagine what they would do - and inreal world almost all risk has skill component

Can prospect theory be a reduced form model of careerconcerns?

Lots of evidence that managers do engage in skill signalingAnd that their behavior is consistent with prospect theorySo why not just use prospect theory?Skill signaling is simplest career concern model - can getvery di¤erent behavior as change information andincentives

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Introduction Prospect theory or skill signaling?Skilled type more likely to win, Pr[winjs] >...

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