Prospect theory or skill signaling?

Rick Harbaugh, Indiana University

Princeton UniversityApril 2008

1 / 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

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])

12 / 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

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

14 / 43

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�

15 / 43

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

16 / 43

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

17 / 43

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

18 / 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

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]

21 / 43

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

36 / 43

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

37 / 43

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]

39 / 43

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

40 / 43

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

41 / 43

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

42 / 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

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