introduction prospect theory or skill signaling?skilled type more likely to win, pr[winjs] >...
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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
Prospect theory or skill signaling?
Rick Harbaugh
Emory UniversityApril 2008
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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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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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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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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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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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IntroductionRisk anomaliesLoss aversionExampleFour-fold patternSkill signalingInformed DMFramingEvaluation skillRefused outcomeConclusion
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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