d y
Figure 1: Decreasing reaction curves have a jump along the diagonal, and there
is no symmetric equilibrium
reader that the continuity of the reaction curves in each triangle over and below
the diagonal is only there for the sake of a clearer figure. It needs not hold under
assumptions A1- A3). The first result gives us existence of equilibrium via the
strategic substitutes property, and the second precludes symmetric equilibria.
The complete proof can be found in the appendix.
Theorem 3.1 does not rule out the existence of multiple pairs of PSNEs. In-
deed, the two reaction curves may intersect several times above and below the
diagonal. In case of multiple pairs of PSNEs, there will typically be co-existence
of pairs of Cournot-stable and pairs of Cournot-unstable PSNEs. Neverthe-
less, Theorem 3.1 does imply that all of these PSNEs are asymmetric. Hence
symmetry-breaking in this context does not rely on the rejection of Cournot-
unstable symmetric PSNEs. In the same vein, this type of symmetry-breaking
is not at odds with Schelling’s (1960) notion of focalness of PSNEs.
The next result adds further restrictions of a general nature on the payoff
components of our game that lead to a unique pair of PSNEs, which are then
necessarily Cournot-stable.7 In this case, symmetry-breaking is coupled with
7It is worthwhile to point out here that our results indicate an even total numb er of PSNEs,
in apparent conflict with the well-known odd number results. The explanation is that the latter
results are based on degree theory and require continuity of the best-response form. G iven
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