00Thaler_FM i-xxvi.qxd

For part 2, note that 
*=k 1 s 1 +k 2 s 2 , where

(A8)

(A9)

This implies that the distribution of 
* conditional on θ+
is nor-

mal with mean

(A10)

and variance

(A11)

The complement of the standardized cumulative normal distribution

function of a normal random variable with nonzero mean and vari-

ance is increasing in its mean. Since E[
*θ+
] is proportional to θ+
,

the probability conditional on P 1 that 
* exceeds a given threshold

value (indicating occurrence of the positive event) is increasing in θ+
.

The reverse holds for a negative event, proving part (2).

Appendix B: Discrete Model of Outcome-dependent

Overconfidence

At time 0, θhas a value of +1 or −1 and an expected value of zero. At time
1, the player receives a signal s 1 , and, at time 2, a signal s 2. s 1 may be either
Hor Lwhile s 2 may be either Uor D. After each signal, the player updates
his prior expected value of θ.

Pr(s 1 =Hθ=+1)=p=Pr(s 1 =Lθ=−1), (A12)

Pr(s 2 =Uθ=+1)=q=Pr(s 2 =Dθ=−1). (A13)

The probabilities that θ=+1, given s 1 and s 2 are

(A14)

Pr

Pr Pr

Pr

()

()()

()

/

/( )/

.

θ

θθ

=+ = =

==+ =+

=

=

+−

=

1

11

2

21 2

1

1

1





sH

sH

sH

p

pp

p

[( ) ]

().

kk 122 k 1 22

22

++

+

+

σσ

σσ

θ θ

θ

()

()

kk 122 k 1 2

22

++

+

+

σσ

σσ

θ θ

θ

k

Cp p

2

22

=− (^22) ++ 2 22
σσ
σσ σ σσ
θ
θθ


()
.
k
p
Cp p
1
22 2
= 22 2 22

• ++
  σσ σ
  σσ σ σσ
  θ
  θθ
  
  ()
  ()
  ,
  492 DANIEL, HIRSHLEIFER, SUBRAHMANYAM