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

The present paper constitutes an attempt to contribute to this rich debate
along standard lines of argument in applied game theory and industrial orga-
nization. Consider a two-player symmetric normal-form game characterized by
two key properties: (a) actions form strategic substitutes, and (b) each player’s
payoff, though continuous, admits a key nonconcavity along the diagonal in ac-
tion space, which results in a jump of the reaction correspondence across the 45^o
line. Such a game always admits pure-strategy Nash equilibrium points due sim-
ply to the property of strategic substitutes. Furthermore, due to property (b),
no such equilibrium could ever be symmetric. At any of the possibly multiple
equilibria, which obviously occur in pairs due to the symmetry of the game, oth-
erwise identical agents will necessarily take different equilibrium actions. While
this description exactly fits the main result of the paper, we consider two other
related classes of games that always possess asymmetric, but never symmetric,
pure-strategy equilibria although they are, strictly speaking, not of strategic
substitutes. This suggests that the latter property is not as critical as the diag-
onal nonconcavity property in generating exclusively asymmetric outcomes.

Since payoffs are continuous in actions in all three classes of games under con-
sideration, these games will typically admit a symmetric mixed-strategy Nash
equilibrium (Dasgupta and Maskin, 1986). As this would be the only focal
equilibrium in the sense of Schelling (1960), it may reasonably be advanced as a
plausible outcome of such a game. Nevertheless, in the actual realization of the
equilibrium randomizations, the players will still end up playing different actions
with high, if not full, probability. Hence, given a focus on explaining observed
heterogeneity, this approach need not rule out mixed strategies a priori.

Towards the goal of generating endogenous heterogeneity, this approach is
obviously closest in spirit to Matsuyama’s symmetry-breaking explanation. By
allowing for suitable discontinuities in the players’ reaction curves, it dispenses
with the need to interconnect two separate games in the somewhat complex
(and subtle) manner proposed by Matsuyama. More importantly, it also pro-
vides a framework that is independent of the controversial argument of outright
Schendel and Teece (1991), Roller and Sinclair-Desgagne (1996).

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