Theorem 4.2 Assume that B10 and B2 - B5 hold then there exist at least one
pair of asymmetric PSNEs and no symmetric one.
The proofs of these results follow the same reasoning as the proofs of the
precedent ones, however, existence of PSNE is now guaranteed through the
application of Brouwer’s Fixed Point Theorem (which is suitable given that
payoff functions are quasiconcave).
Uniqueness of a pair of equilibria can be shown if reaction functions are
contractions by using Banach’s Fixed Point Theorem as in the previous section.
A comparison of equilibria when payoff functions are partially supermodular
(or quasiconcave) can be done in the same spirit of Theorem 3.3. Instead of
assuming A1 - A4 we assume B1 - B5 (B10 - B5). In the proof, the second
inequality follows now from the fact that x* ≥ d and Assumption B3.11
Adopting the results of Echenique (2004), wherein the order on the action
spaces is not exogenously given would enlarge the scope of supermodular games.
In particular, since our game here has at least two PSNEs, one can always find
a partial order such that it becomes a supermodular game (Echenique, 2004,
Theorem 5).
4.2 Applications: Quality Investment
We illustrate the results of this section with two papers dealing with quality
investment problems. The first paper we analyze is Aoki and Prusa (1996). In
this paper, two identical firms produce products differentiated by quality, in a
two-stage setting. In the first stage firms 1 and 2 decide the level of quality
investment x ∈ [0, c] and y ∈ [0, c] respectively, and in the second stage they
simultaneously announce prices.12
Consumers are diversified in their willingness to pay for quality. Production
cost is assumed to be 0 and firm 1 (firm 2) incurs a cost of quality investment
11It is easy to see that Assumption B3 means that for z>d, U (r1 (z) ,z) <L(r1 (z) ,z) .
12Aoki and Prusa (1996) consider unlimited quality investments. We impose the upper limit
c, arbitrarily big, such that the strategy spaces are compact.
23
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