(A30)

(A31)

(A32)

(A33)

With two possible values for θ, there are now sixteen possible sets of {θ, s 1 ,
s 2 , s 3 ′} realizations. Only {s 1 , s 2 , s 3 ′} are observed by the player, resulting in
eight sets of possible signal realizations. When calculating the covariances
of price changes, only half of these realizations can result in unique prod-
ucts of price changes, so we define

Aij≡∆Pi∆PjH,U,G=∆Pi∆PjL,D,B; (A34)

Bij≡∆Pi∆PjH,U,B=∆Pi∆PjL,D,G; (A35)

Cij≡∆Pi∆PjH,D,G=∆Pi∆PjL,U,B; (A36)

Dij≡∆Pi∆PjH,D,B=∆Pi∆PjL,U,G. (A37)

Each of these four possible products must then be weighted by their proba-
bility of occurrence to calculate the expected value of the products of the
price changes (the expected value of each price is zero). The weights for the
Aijcomponent of covariance are:

Pr(H, U, Gθ=+1)+Pr(H, U, Gθ=−1)=pqr/2

+(1−p)(1−q)(1−r)/2,

(A38)

Pr(L, D, Bθ=−1)+Pr(L, D, Bθ=−1)=pqr/2

+(1−p)(1−q)(1−r)/2.

(A39)

Proceeding in this manner, the covariances are:

E[∆Pi∆Pj]=[pqr+(1−p)(1−q)(1−r)]Aij+[pq(1−r)

+(1−p)(1−q)r]Bij

+[p(1−q)r+(1−p)q(1−r)]Cij+[p(1−q)(1−r)

+(1−p)qr]Dij. (A40)

(Earlier calculations of E[∆P 1 ∆P 2 ] had A 12 =B 12 =Xand C 12 =D 12 =Y,
with the rand 1−rfactors from s 3 ′summing to one.) To simplify the algebra,

P

pq r pqr

′ == =Hs Ds′B pq r pqr

=

−−−−

(^323) −−+−
11 1
11 1
s 1 ,,
()()()
()()()
.
P
pqr pqr
′ == =Hs Ds G′ pqr pqr

−−− −
(^323) −+− −
111
111
s 1 ,,
()()()
()()()
;
P
pq r p qr
Hs Us B pq r p qr
CC
CC
′ ===′ =
−−− −
(^323) −+− −
111
111
s 1 ,,
()( )( )
()( )( )
;
P
pqr p q r
Hs Us G pqr p q r
CC
CC
′ ===′ =
−− − −
(^323) +− − −
111
111
s 1 ,,
()()()
()()()
;
INVESTOR PSYCHOLOGY 495