1 − ρ
3
1 − (1 − π
3
)
k
3
+1
ρ
1
1 − (1 − π
1
)
k
1
(u(|z
j
− x
1
|) + π
1
ω
1
) +
1 − ρ
3
1 − (1 − π
3
)
k
3
+1
1 − ρ
1
1 − (1 − π
1
)
k
1
u(|z
j
− x
2
|) >
δ ρ
3
1 − (1 − π
3
)
k
3
(u(|z
j
− x
3
|) + π
3
ω
3
) +
1 − ρ
3
1 − (1 − π
3
)
k
3
ρ
1
1 − (1 − π
1
)
k
1
(u(|z
j
− x
1
|) + π
1
ω
1
) +
1 − ρ
3
1 − (1 − π
3
)
k
3
1 − ρ
1
1 − (1 − π
1
)
k
1
u(|z
j
− x
2
|) .
(29)
Simplifying yields:
1 − ρ
1
(1 − (1 − π
1
)
k
1
) u(|z
j
− x
2
|) <
[u(|z
j
− x
3
|) + π
3
ω
3
] − ρ
1
(1 − (1 − π
1
)
k
1
) [u(|z
j
− x
1
|) + π
1
ω
1
]
(30)
Because c
13
≤ e
k
3
23
, we can substitute in the utilities for an effective policy 1 instead of an
effective policy 3 into (30), yielding:
1 − ρ
1
(1 − (1 − π
1
)
k
1
) u(|z
j
− x
2
|) < 1 − ρ
1
(1 − (1 − π
1
)
k
1
) [u(|z
j
− x
1
|) + π
1
ω
1
]
(31)
By the fact that e
k
3
23
< c
12
, u(|z
j
− x
1
|) + π
1
ω
1
> u(|z
j
− x
2
|). Therefore, (31) holds for
all k
3
.
Proof of Proposition 2. The second-period strategies are trivial to derive based on the
ideal point location z
j
, the c
ik
(h) cutpoints, and the information revealed from the first-
period experiments. Given the symmetric nature of this case, it is sufficient to establish the
results for the first-period choice between policies 2 and 3.
Given Lemma 1 and c
12
≤ e
k
3
23
, the second-period strategy on both sides of e
k
3
23
is to
select policy 3 iff ω
1
3
= ω
3
and otherwise to adopt policy 2. Thus, policy 3 will be chosen
in the first period if (22) holds. The cutpoint e
k
3
23
is thus defined by (17). This is a unique
solution, in that the left-hand side of (17) is decreasing in e
k
3
23
while the right-hand side is
constant in e
k
3
23
. S
j
experiments with policy 3 if and only if z
j
> e
k
3
23
. Invoking symmetry and
defining the experimentation cutpoint e
k
1
12
analogously to the derivation of (17) completes
the experimental strategies for Proposition 2.
Proof of Proposition 3. The second-period strategies are trivial to derive based on the
ideal point location z
j
, the c
ik
(h) cutpoints, and the information revealed from the first-
period experiments. Given the symmetric nature of this case, it is sufficient to establish the
results for the first-period choice between policies 2 and 3.
Given Lemma 1 and c
13
< e
k
3
23
< c
12
, the second-period strategy on both sides of e
k
3
23
is to
select policy 3 iff ω
1
3
= ω
3
, otherwise to select policy 1 iff ω
1
1
= ω
1
, and otherwise to adopt
29
Convention