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Prospect theory or skill signaling?
Rick Harbaugh, Indiana University
Princeton UniversityApril 2008
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Two classic literatures
Prospect theory (Kahneman & Tversky, 1979, 1992)
People often take actions that seem inconsistent withexpected utility maximization
They are �loss averse�, like �long shots�, are a¤ected by�framing�
Career concerns (Holmstrom, 1982/1999)
Managers often take actions that are inconsistent with pro�tmaximization
They avoid some types of risks, but like others, chase badmoney with good, etc.
2 / 43
Two classic literatures
Prospect theory (Kahneman & Tversky, 1979, 1992)
People often take actions that seem inconsistent withexpected utility maximization
They are �loss averse�, like �long shots�, are a¤ected by�framing�
Career concerns (Holmstrom, 1982/1999)
Managers often take actions that are inconsistent with pro�tmaximization
They avoid some types of risks, but like others, chase badmoney with good, etc.
2 / 43
Are these literatures related?
Both predict deviations from simple maximization... for verydi¤erent reasons
Prospect theory
People have perceptual biases
Expected utility maximization asks too much
Career concerns
Managers care about looking competent (skill signaling)
Expected utility maximization over monetary outcomes toonarrow
3 / 43
Are these literatures related?
Both predict deviations from simple maximization... for verydi¤erent reasons
Prospect theory
People have perceptual biases
Expected utility maximization asks too much
Career concerns
Managers care about looking competent (skill signaling)
Expected utility maximization over monetary outcomes toonarrow
3 / 43
Are these literatures related?
Both predict deviations from simple maximization... for verydi¤erent reasons
Prospect theory
People have perceptual biases
Expected utility maximization asks too much
Career concerns
Managers care about looking competent (skill signaling)
Expected utility maximization over monetary outcomes toonarrow
3 / 43
Are these literatures related?
Both predict deviations from simple maximization... for verydi¤erent reasons
Prospect theory
People have perceptual biases
Expected utility maximization asks too much
Career concerns
Managers care about looking competent (skill signaling)
Expected utility maximization over monetary outcomes toonarrow
3 / 43
How similar are the predicted behaviors?
Argument: predictions are pretty similar for simplest possiblecareer concerns model
Another example of information economics �rationalizing�apsychological phenomenon?
Signaling, herding, group think, polarization, sunk costs,option values, self-con�dence, status concerns ...
Or maybe similarity is just a coincidence
Empirical analyses could confound e¤ects
Particularly important for managerial and �nancial contextswhere skill is important
4 / 43
How similar are the predicted behaviors?
Argument: predictions are pretty similar for simplest possiblecareer concerns model
Another example of information economics �rationalizing�apsychological phenomenon?
Signaling, herding, group think, polarization, sunk costs,option values, self-con�dence, status concerns ...
Or maybe similarity is just a coincidence
Empirical analyses could confound e¤ects
Particularly important for managerial and �nancial contextswhere skill is important
4 / 43
How similar are the predicted behaviors?
Argument: predictions are pretty similar for simplest possiblecareer concerns model
Another example of information economics �rationalizing�apsychological phenomenon?
Signaling, herding, group think, polarization, sunk costs,option values, self-con�dence, status concerns ...
Or maybe similarity is just a coincidence
Empirical analyses could confound e¤ects
Particularly important for managerial and �nancial contextswhere skill is important
4 / 43
Related to other literatures
Can help formalize early social psychology models
Self-esteem (James, 1890)
Achievement motivation (Atkinson, 1957)
Self-handicapping (Jones and Berglas, 1978)
And also has similar predictions as other models that predictprospect-theory like behavior
Regret theory (Bell, 1982; Loomes and Sugden, 1982)
Rank-dependent utility (Quiggin, 1982)
Disappointment aversion (Gul, 1991)
5 / 43
Related to other literatures
Can help formalize early social psychology models
Self-esteem (James, 1890)
Achievement motivation (Atkinson, 1957)
Self-handicapping (Jones and Berglas, 1978)
And also has similar predictions as other models that predictprospect-theory like behavior
Regret theory (Bell, 1982; Loomes and Sugden, 1982)
Rank-dependent utility (Quiggin, 1982)
Disappointment aversion (Gul, 1991)
5 / 43
Some classic risk anomalies
Loss aversion �Kahneman and Tversky
Excessive risk aversion for small gambles - Rabin
But also overcon�dence - Odean and Barber
Framing �Kahneman and Tversky
More likely to gamble if winning is reference point
So how frame the status quo matters
Probability weighting �Kahneman and Tversky
Long shot bias �Thaler
Simultaneous purchase of insurance, lottery tickets �Friedman and Savage
6 / 43
Some classic risk anomalies
Loss aversion �Kahneman and Tversky
Excessive risk aversion for small gambles - Rabin
But also overcon�dence - Odean and Barber
Framing �Kahneman and Tversky
More likely to gamble if winning is reference point
So how frame the status quo matters
Probability weighting �Kahneman and Tversky
Long shot bias �Thaler
Simultaneous purchase of insurance, lottery tickets �Friedman and Savage
6 / 43
Some classic risk anomalies
Loss aversion �Kahneman and Tversky
Excessive risk aversion for small gambles - Rabin
But also overcon�dence - Odean and Barber
Framing �Kahneman and Tversky
More likely to gamble if winning is reference point
So how frame the status quo matters
Probability weighting �Kahneman and Tversky
Long shot bias �Thaler
Simultaneous purchase of insurance, lottery tickets �Friedman and Savage
6 / 43
Why so risk averse for small gambles?
Small gamble:
A. $0
B. 50-50 lose $100 or win $110
Larger gamble:
A. $0
B. 50-50 lose $1000 or win $718,190
Rabin (2000): For standard utility function, if refuse small gamblethen should refuse larger gamble too
7 / 43
Why so risk averse for small gambles?
Small gamble:
A. $0
B. 50-50 lose $100 or win $110
Larger gamble:
A. $0
B. 50-50 lose $1000 or win $718,190
Rabin (2000): For standard utility function, if refuse small gamblethen should refuse larger gamble too
7 / 43
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...
8 / 43
Why don�t people like to lose?
Achievement motivation literature (Atkinson, 1957)
Most gambles involve some skill
People don�t want to look unskilled
So avoid gambles that might reveal lack of skill
Career concerns literature (Holmstrom, 1982/1999)
Managers are concerned with appearing skilled
So choose investments to reduce risk of appearing unskilled
9 / 43
Why don�t people like to lose?
Achievement motivation literature (Atkinson, 1957)
Most gambles involve some skill
People don�t want to look unskilled
So avoid gambles that might reveal lack of skill
Career concerns literature (Holmstrom, 1982/1999)
Managers are concerned with appearing skilled
So choose investments to reduce risk of appearing unskilled
9 / 43
General structure
1 Decider o¤ered a gamble with common knowledge odds2 [Decider learns private signal re�ning odds of gamble]3 Decider chooses to take gamble or not4 Observer observes outcome, updates decider�s estimated skill5 Decider utility is function of monetary outcome and estimatedskill
10 / 43
Assumptions of base model
Decider is skilled or unskilled, q 2 fs, ugGamble has monetary payo¤ x 2 flose,wing and price zPerformance skill: Pr[winjs ] > Pr[winju]Utility quasilinear in wealth Y and skill estimate,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
CARA: v = Pr[s ]1�α/(1� α), v 0 = Pr[s ]�α,v 00 = (�α)Pr[s ]�α�1, v 000 = (�α) (�α� 1)Pr[s ]�α�2
11 / 43
Posterior skill estimate
Skill is updated based on outcome of gamble
Pr[s jwin] =Pr[s,win]Pr[win]
= Pr[s ] +Pr[s,win]� 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]
If v 00 < 0 then by Jensen�s inequality:v(Pr[s ]) > Pr[win]v(Pr[s jwin]) + Pr[lose]v(Pr[s jlose])
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Looks like loss aversion
Decider risk neutral in wealth so risk premium just:
π = v(Pr[s j don�t gamble])� E [v(Pr[s j gamble])]
Decider has no private information so simpli�es to:
π = v(Pr[s ])� (Pr[win]v(Pr[s jwin]) + Pr[lose]v(Pr[s jlose]))
Risk premium positive even for �friendly bet�
So concern for looking unskilled will look like kinked utilityfunction of loss aversion
13 / 43
Looks like loss aversion
Decider risk neutral in wealth so risk premium just:
π = v(Pr[s j don�t gamble])� E [v(Pr[s j gamble])]
Decider has no private information so simpli�es to:
π = v(Pr[s ])� (Pr[win]v(Pr[s jwin]) + Pr[lose]v(Pr[s jlose]))
Risk premium positive even for �friendly bet�
So concern for looking unskilled will look like kinked utilityfunction of loss aversion
13 / 43
Looks like loss aversion
Decider risk neutral in wealth so risk premium just:
π = v(Pr[s j don�t gamble])� E [v(Pr[s j gamble])]
Decider has no private information so simpli�es to:
π = v(Pr[s ])� (Pr[win]v(Pr[s jwin]) + Pr[lose]v(Pr[s jlose]))
Risk premium positive even for �friendly bet�
So concern for looking unskilled will look like kinked utilityfunction of loss aversion
13 / 43
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 for losses�
Original prospect theoryassumed risk averse ingains and risk loving inlosses
So, in addition to kink atstatus quo for lossaversion, have concavityabove and convexitybelow
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 with probabilityweighting function
p = Pr [win]
If w(p) > p then act likeoverweighting the trueodds - take fair gambles
If w(p) < p then act likeunderweighting the trueodds - refuse fair gambles
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Does skill signaling also generate FFP?
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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Does skill signaling also generate FFP?
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
19 / 43
Does skill signaling also generate FFP?
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
19 / 43
Are low probability gambles more attractive?
Is winning really more impressive for low 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]
If skill gap Pr[winjs ]� Pr[winju] constant:Pr[s jwin] bigger for lower Pr[win]Pr[s jlose] smaller for higher Pr[win]
Same pattern generally if skill gap does not change too fast
20 / 43
Example
Symmetric:
Pr[s ] = Pr[u] = 12
Pr[winjs ] = Pr[win] + ε, Pr[winju] = Pr[win]� ε
Probabilities bounded:
ε = Pr[win]Pr[lose]
So skill gap Pr[winjs ]� Pr[winju] = 2Pr[win]Pr[lose]Then winning is most impressive for low Pr[win]
Pr[s jwin] = 12 +
12 (1� Pr[win])
And losing is most incriminating for high Pr[win]
Pr[s jlose] = 12 �
12 Pr[win]
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Updated skill estimate as function of winning probability
22 / 43
Updated skill estimate as function of winning probability
22 / 43
Updated skill estimate as function of winning probability
22 / 43
So are long shots favored?
Low Pr[win] gambles seem attractive...
Winning is more impressive and losing less embarrassing
But winning is less common and losing is more common
So unclear what e¤ect dominates
Two gambles F and G where PrF [win] < PrG [win], same skill gap
(1) Expected skill estimate always the prior Pr[s ]
(2) PrF [s jwin]� PrF [s jlose] = PrG [s jwin]� PrG [s jlose](3) PrF [s jlose] > PrG [s jlose] so F has better downside
Together (1)-(3) imply F �TOSD GIf v 0 > 0, v 00 < 0, v 000 > 0 then F has higher expected utility
F and G have same expected skill, same variance, but F hasless �downside risk�
23 / 43
So are long shots favored?
Low Pr[win] gambles seem attractive...
Winning is more impressive and losing less embarrassing
But winning is less common and losing is more common
So unclear what e¤ect dominates
Two gambles F and G where PrF [win] < PrG [win], same skill gap
(1) Expected skill estimate always the prior Pr[s ]
(2) PrF [s jwin]� PrF [s jlose] = PrG [s jwin]� PrG [s jlose](3) PrF [s jlose] > PrG [s jlose] so F has better downside
Together (1)-(3) imply F �TOSD GIf v 0 > 0, v 00 < 0, v 000 > 0 then F has higher expected utility
F and G have same expected skill, same variance, but F hasless �downside risk�
23 / 43
So are long shots favored?
Low Pr[win] gambles seem attractive...
Winning is more impressive and losing less embarrassing
But winning is less common and losing is more common
So unclear what e¤ect dominates
Two gambles F and G where PrF [win] < PrG [win], same skill gap
(1) Expected skill estimate always the prior Pr[s ]
(2) PrF [s jwin]� PrF [s jlose] = PrG [s jwin]� PrG [s jlose](3) PrF [s jlose] > PrG [s jlose] so F has better downside
Together (1)-(3) imply F �TOSD GIf v 0 > 0, v 00 < 0, v 000 > 0 then F has higher expected utility
F and G have same expected skill, same variance, but F hasless �downside risk�
23 / 43
Downside risk aversion implies lower risk premium from�long-shot�
v(Pr[s ]) = �1/Pr[s ], Pr[win] = .2 or Pr[win] = .8
24 / 43
Downside risk aversion implies lower risk premium from�long-shot�
v(Pr[s ]) = �1/Pr[s ], Pr[win] = .2 or Pr[win] = .8
24 / 43
Overview of base model
Losing doubly unfortunate since increases probability that oneis unskilled - loss aversion?
If probability of success is high, the chance of losing is smallbut the loss in skill reputation from losing is substantial
If probability of success is low, the chance of losing is high butthe 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
25 / 43
Imputing w(p) from risk premia
Recall risk premium π is just how much money decider wouldforego to avoid skill update from gamble
Probability weight that would be calculated if ignoreembarrassment 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 and w(p) < pfor large p, so negative π for small p
26 / 43
Imputed w(p) for the example
lose = 0, win = 10, v = �1/Pr[s ]Pr[winjs ]� Pr[winju] = 2Pr[win]Pr[lose]
27 / 43
Imputed w(p) with smaller skill gap
lose = 0, win = 10, v = �1/Pr[s ]Pr[winjs ]� Pr[winju] = Pr[win]Pr[lose]
28 / 43
Now suppose decider has private information
Sees noisy signal θ 2 fg , bg where Pr[winjg ] > Pr[winjb]
Performance skill
Re�ects private info about skill
So Pr[s jg ] > Pr[s jb]Accepting a gamble can show con�dence in skill
Refusing a gamble can be admission that one expects to lose
Evaluation skill
Re�ects private info about odds of gamble
Signal more informative if decider is skilled
Pr[winjs, g ]� Pr[winjs, b] � Pr[winju, g ]� Pr[winju, b]Accepting or refusing gamble need not re�ect on skill
29 / 43
Now suppose decider has private information
Sees noisy signal θ 2 fg , bg where Pr[winjg ] > Pr[winjb]
Performance skill
Re�ects private info about skill
So Pr[s jg ] > Pr[s jb]Accepting a gamble can show con�dence in skill
Refusing a gamble can be admission that one expects to lose
Evaluation skill
Re�ects private info about odds of gamble
Signal more informative if decider is skilled
Pr[winjs, g ]� Pr[winjs, b] � Pr[winju, g ]� Pr[winju, b]Accepting or refusing gamble need not re�ect on skill
29 / 43
Now suppose decider has private information
Sees noisy signal θ 2 fg , bg where Pr[winjg ] > Pr[winjb]
Performance skill
Re�ects private info about skill
So Pr[s jg ] > Pr[s jb]Accepting a gamble can show con�dence in skill
Refusing a gamble can be admission that one expects to lose
Evaluation skill
Re�ects private info about odds of gamble
Signal more informative if decider is skilled
Pr[winjs, g ]� Pr[winjs, b] � Pr[winju, g ]� Pr[winju, b]Accepting or refusing gamble need not re�ect on skill
29 / 43
Now have true signaling game
Look at perfect Bayesian equilibria surviving D1
Unexpected acceptance? D1 predicts was by type g
Unexpected refusal? D1 predicts was by type b
Neither-gamble equilibrium:
πθ = v(Pr[s ])� E [v(Pr[s jg , x ])jθ]Separating equilibrium:
πθ = v(Pr[s jb])� E [v(Pr[s jg , x ])jθ]Both-gamble equilibrium:
πθ = v(Pr[s jb])� E [v(Pr[s jx ])jθ]
30 / 43
Now have true signaling game
Look at perfect Bayesian equilibria surviving D1
Unexpected acceptance? D1 predicts was by type g
Unexpected refusal? D1 predicts was by type b
Neither-gamble equilibrium:
πθ = v(Pr[s ])� E [v(Pr[s jg , x ])jθ]Separating equilibrium:
πθ = v(Pr[s jb])� E [v(Pr[s jg , x ])jθ]Both-gamble equilibrium:
πθ = v(Pr[s jb])� E [v(Pr[s jx ])jθ]
30 / 43
Overcon�dence and Loss Aversion
The more the decider knows about her own skill the better to takea chance than admit lack of skill ... especially if gamble is long shot
Theorem
Suppose v 0 > 0. In any pure strategy equilibrium the risk premiaπθ are negative for a) any given Pr[s jg ]� Pr[s jb] > 0 andsu¢ ciently low Pr[winjθ], or b) any given Pr[winjθ] and su¢ cientlylarge Pr[s jg ]� Pr[s jb].
Loss aversion kicks in if pressure to show o¤ is not too strong
Theorem
Suppose v 00 < 0. In any pure strategy equilibrium the risk premiaπθ are positive for any given Pr[winjθ] and su¢ ciently smallPr[s jg ]� Pr[s jb].
31 / 43
Overcon�dence and Loss Aversion
The more the decider knows about her own skill the better to takea chance than admit lack of skill ... especially if gamble is long shot
Theorem
Suppose v 0 > 0. In any pure strategy equilibrium the risk premiaπθ are negative for a) any given Pr[s jg ]� Pr[s jb] > 0 andsu¢ ciently low Pr[winjθ], or b) any given Pr[winjθ] and su¢ cientlylarge Pr[s jg ]� Pr[s jb].
Loss aversion kicks in if pressure to show o¤ is not too strong
Theorem
Suppose v 00 < 0. In any pure strategy equilibrium the risk premiaπθ are positive for any given Pr[winjθ] and su¢ ciently smallPr[s jg ]� Pr[s jb].
31 / 43
Long shot bias still holds
De�nition
Two gambles F and G are symmetric pair if PrF [win] = PrG [lose],PrF [winjs, θ]� PrF [winju, θ] = PrG [winjs, θ]� PrG [winju, θ],PrF [qjθ] = PrG [qjθ], and PrF [θ] = PrG [θ] = 1/2 for q 2 fu, sg,θ 2 fb, gg.
Theorem
Suppose v 0 > 0, v 00 < 0, and v 000 > 0 and consider a symmetricpair of gambles F and G. (i) In any given pure strategyequilibrium the average risk premium π is lower for F than for G,and (ii) a neither-gamble equilibrium exists for G if it exists for F ,and a both-gamble equilibrium exists for F if it exists for G.
32 / 43
Imputed w(p) for example with informed decision maker
lose = 0, win = 10, v = �1/Pr[s ]Pr[s jg ]� Pr[s jb] = 1/10, Pr[g ] = 1/2
33 / 43
Multiple equilibria captures di¤erential risk taking acrossgroups?
In signaling games with discrete choices multiple equilibria can arise
Easy if allow arbitrary beliefs for play o¤ the equilibrium path
Possible even with strong equilibrium re�nements such as D1
In our context depends on priors, odds, strength of private info, etc.
If expect neither-gamble equilibrium then not gambling is safe
But if expect separating or both-gamble equilibrium then notgambling reveals of lack of con�dence
Empirical literature �nds that di¤erent genders display di¤erentrisk aversion
Could be due to real di¤erences
Or could re�ect di¤erent expectations/equilibria
34 / 43
Multiple equilibria captures di¤erential risk taking acrossgroups?
In signaling games with discrete choices multiple equilibria can arise
Easy if allow arbitrary beliefs for play o¤ the equilibrium path
Possible even with strong equilibrium re�nements such as D1
In our context depends on priors, odds, strength of private info, etc.
If expect neither-gamble equilibrium then not gambling is safe
But if expect separating or both-gamble equilibrium then notgambling reveals of lack of con�dence
Empirical literature �nds that di¤erent genders display di¤erentrisk aversion
Could be due to real di¤erences
Or could re�ect di¤erent expectations/equilibria
34 / 43
Multiple equilibria captures di¤erential risk taking acrossgroups?
In signaling games with discrete choices multiple equilibria can arise
Easy if allow arbitrary beliefs for play o¤ the equilibrium path
Possible even with strong equilibrium re�nements such as D1
In our context depends on priors, odds, strength of private info, etc.
If expect neither-gamble equilibrium then not gambling is safe
But if expect separating or both-gamble equilibrium then notgambling reveals of lack of con�dence
Empirical literature �nds that di¤erent genders display di¤erentrisk aversion
Could be due to real di¤erences
Or could re�ect di¤erent expectations/equilibria
34 / 43
Multiple equilibria captures aspect of �framing�?
Subjects refuse gamble whengambling outcomes are framedas in gains domain
Subjects gamble when gamblingoutcomes are framed as in lossesdomain
Framing of the gamble can givesome indication of receiver�sbeliefs about which equilibriumis being played
So depending on how gamble isframed, subject can be "dared"into taking a gamble
35 / 43
More about �evaluation skill�
Pr[winjg , s ]� Pr[winjb, s ] > Pr[winjg , u]� Pr[winjb, u]
Some people better at evaluating odds of di¤erent gambles
Skilled broker picks right stocks
Skilled handicapper picks right horses
Also standard in career concerns literature
Holmstrom (1982/1999) �distorted investment decisions
Prendergast and Stole (1996) � sunk costs
36 / 43
More about �evaluation skill�
Pr[winjg , s ]� Pr[winjb, s ] > Pr[winjg , u]� Pr[winjb, u]
Some people better at evaluating odds of di¤erent gambles
Skilled broker picks right stocks
Skilled handicapper picks right horses
Also standard in career concerns literature
Holmstrom (1982/1999) �distorted investment decisions
Prendergast and Stole (1996) � sunk costs
36 / 43
More about �evaluation skill�
Pr[winjg , s ]� Pr[winjb, s ] > Pr[winjg , u]� Pr[winjb, u]
Some people better at evaluating odds of di¤erent gambles
Skilled broker picks right stocks
Skilled handicapper picks right horses
Also standard in career concerns literature
Holmstrom (1982/1999) �distorted investment decisions
Prendergast and Stole (1996) � sunk costs
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Evaluation skill generates same type of results
Losing doubly unfortunate since looks like one foolishly took abad gamble
Same tradeo¤ as with performance skill
So lower risk premium for long shot gamble than equivalentnear sure-thing gamble
If pure performance skill then risk premia are always positive
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 long shot gamble like taking a near-sure thing
And refusing a near-sure thing like taking a long shot
Average risk premium for low Pr[win] gambles negative � justas if low Pr[win] gambles overweighted
Average risk premium for high Pr[win] gambles positive � justas if high Pr[win] gambles underweighted
Hard to distinguish from prospect theory�s weighting function
38 / 43
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 ignore skillsignaling
a) Skill gap =2Pr [win]Pr [lose]
b) Skill gap =Pr [win]Pr [lose]
c) Informed decider
d) Outcome of refusedgamble observed
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Relation to Atkinson�s model of achievement motivation
Atkinson�s (1957) achievement motivation model was long astandard psychology model of risk taking
Atkinson (and others earlier) noted di¤erent probabilitygambles convey di¤erent skill information
Argued with reduced form model that people should avoidgambles with equal chance of success or failure since mostrevealing
Experiments found people also favored long shots over nearsure-things, which was considered anomalous
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Implied expected utility in achievement motivation model
Atkinson model can beformalized by initial example
Pr [winjs ]� Pr [winju] =Pr [win]Pr [lose] to generatehis updating
Piecewise linear v with kink atPr [s ] = 1/2 to generate hisutility function
If instead v 000 < 0 thenAtkinson model predicts thedata
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Relevance for prospect theory and career concerns
Can skill signaling provide a foundation for prospect theory?
Experiments done without any explicit skill component sowould seem unlikely
Except most tests are thought experiments without real stakes
So subjects have to imagine what they would do - and in realworld almost all risk has skill component
Can prospect theory be a reduced form model of career concerns?
Lots of evidence that managers do engage in skill signaling
And that their behavior is consistent with prospect theory
So why not just use prospect theory?
Skill signaling is simplest career concern model - can get verydi¤erent behavior as change information and incentives
43 / 43
Relevance for prospect theory and career concerns
Can skill signaling provide a foundation for prospect theory?
Experiments done without any explicit skill component sowould seem unlikely
Except most tests are thought experiments without real stakes
So subjects have to imagine what they would do - and in realworld almost all risk has skill component
Can prospect theory be a reduced form model of career concerns?
Lots of evidence that managers do engage in skill signaling
And that their behavior is consistent with prospect theory
So why not just use prospect theory?
Skill signaling is simplest career concern model - can get verydi¤erent behavior as change information and incentives
43 / 43
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