Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

Consider a two-player normal form game Γ given by the tuple (X, Y, F, G).
Let X and Y be the action sets of player 1 and 2 respectively, such that X =
Y =[0,c] ⊂ R. The maps F and G : X × Y → R are the payoff functions of
players 1 and 2 respectively and F can be expressed as:

{U(x,y), x ≥ y

,    ,                                                 (1)

L(x, y),x < y

By symmetry of the game Γ, G can be expressed as:

{L(y, x), x ≥ y

,    ,                                                 (2)

U(y, x),x < y

Observe that, somewhat contrary to standard practice, the first argument of
F is the action of player 1 while the first argument of G is the action of player
2. It is useful to define the following sets:

∆U = {(x, y) ∈ R² : x ≥ y} and ∆L = {(x, y) ∈ R² : x ≤ y}.

It will be assumed throughout the paper that U, L, F and G are jointly con-
tinuous functions of the two actions. Define the best response correspondences
(reaction curves) for players 1 and 2 respectively as r1 (y) = arg max{F (x, y):
x ∈ [0, c]} and r2 (x) = arg max{G(y, x):y ∈ [0, c]}.

As usual, a pure strategy Nash equilibrium (or PSNE for short), (x*,y*) ∈
[0, c]² is said to be symmetric if x* = y*, and asymmetric otherwise. It follows
from the symmetry of the game that if (x*,y*) is a PSNE, (y*,x*) is also a
PSNE.

Each of the next three sections investigates a separate class of normal-form
symmetric games that always possess asymmetric Nash equilibria and no sym-
metric Nash equilibria. For each of the three classes, we provide a general result
establishing both the existence and the inexistence conclusions and an illustra-
tion based on previous studies where a special case of the result was derived in
a specific setting.

The definitions and main results from the theory of supermodular games
used in this paper are reviewed in the appendix in a very simple way, which is
sufficient for the purposes of this paper.

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