more predictive power of the game, although the selection of one PSNE from
the pair still remains indeterminate, as is standard in symmetric settings.8
Theorem 3.2 Assume that U and L are twice continuously differentiable and that
the following holds:
U11(r (z) ,z) - U12(r (z) ,z) ≥ 0 (3)
L11(r (z) ,z) - L12(r (z) ,z) ≥ 0 (4)
then there is exactly one pair of PSNEs.
The next result is devoted to comparing the two asymmetric PSNEs from
any given pair from the point of view of the players’ welfare. Given any pair
of asymmetric PSNEs, it is often of interest to determine circumstances where
a given equilibrium secures better payoffs for a player. In other words, under
what conditions would each player prefer the PSNE where he is the high or the
low-activity player? To this end, we need to impose a condition of monotonicity
on the payoff function of the player in question along his opponent’s best reply
as stated in the following result, which lays out conditions for player 1 (say) to
prefer the PSNE where he is the low-activity player.9
Theorem 3.3 Let x* > y*, so that (x*,y*) and (y*,x*) are equilibria in ∆u
and ∆L, respectively. If A1 - A3 hold and moreover U(r1(y),y) and L(r1 (y), y)
are increasing in y ∈ [0, c] then F(x*,y*) ≤ F(y*,x*').
There is a dual statement giving conditions under which each player would
prefer the high-activity equilibrium, given any pair of PSNEs. Being obvious
from Theorem 3.3, it is omitted for the sake of brevity.
our systematic and robust jump across the diagonal, our findings are actually consistent with
the odd number result in a generic sense.
8As mentioned in the Section 1, assumptions A1 - A3 imply that our game admits a
symmetric mixed-strategy Nash equilibrium. However, in the actual realization of such an
equilibrium, the two players will be heterogeneous with high, if not full, probability.
9 This monotonicity assumption is clearly more general than assuming that each player’s
payoff is increasing in the rival’s action. For an example illustrating this point, see von Stengel
(2003).
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