* Assistant Professor of Operations Management at INSEAD ...

the two points. By symmetry, everything we have said about firm 1 is true
for firm 2 as well.
We have thus verified the conditions for the generalized existence theorem
(Proposition 7.4. Harker and Pang 1990) and can conclude that an
equilibrium to the game with two customer groups exists.
Furthermore, Proposition 7.3. in Harker and Pang (1990) describes the
equivalent variational inequality problem associated with our equilibrium
problem. Strict concavity of U E (U1 , . , U N)T guarantees that the function
F in the variational inequality is strictly monotone. Thus, Proposition 3.2.
in Harker and Pang (1990) tells us that there is at most one solution to the
variational inequality (and therefore at most one equilibrium). We have now
established existence of a unique equilibrium, which must be symmetric by
the symmetry of the firms. o
A.2 Proof of Proposition 2
First, I show that the full information equilibrium is obtained if all customers
tell the truth. This follows from
E [qn(,-,01 + 4 _ +
An An mn 7 m„ z
= 1147:
= pn,
— An P", (An)
where the second equality follows from the expected value of the exponential
distribution, and the third equality holds because A n , B implement p7„, which
is shown in Theorem 3.2 in Mendelson and Wha.ng (1990).
We now show that the proposed contract is incentive-compatible. The con-
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