I mostly restrict attention to the system of gradient dynamics²

where S is a positive-diagonal adjustment matrix, and. A solution to (1) has the

form, where is the trajectory of j. I consider a restricted version of this trajectory map, where initial values are symmetric, i.e.. Then, by symmetry, is the same for all players and solves

I say that an interior symmetric equilibrium is symmetrically stable if the dynamics induced by (2) converge to whenever is close to.³ Hence:

Definition 1 (Symmetric stability) The symmetric equilibrium is symmetrically stable if all eigenvalues of, the Jacobian corresponding to (2), have negative real parts.

is symmetrically unstable if at least one eigenvalue of has positive real part. Stability of (1) implies symmetric stability, but not vice-versa (see Figure 1). Let, and note that

, is a -vector field with Jacobian. A symmetric game is

(symmetrically) regular if i) has only regular zeroes⁴ and ii) points inwards at the boundary of S. Any future reference to a regular game means “symmetrically” regular, which is a weaker condition than general regularity of a symmetric game (see [2] ). The first theorem below reveals the general connection between symmetric stability and the existence of a single symmetric equilibrium, depending on the dimensionality of the strategy space. Its proof exploits an essential relation between, and.

Lemma 1 For:

Proof: The first equality is immediate. Next, decompose. By the Implicit Function Theorem (IFT) which, together with the decomposition of gives the second equality.

Let, , where is the a-th projection of.

Theorem 1 (i) If then is a symmetrically stable equilibrium if or equivalently if

.

(ii) For a symmetric equilibrium is symmetrically stable if has only positive principal minors.

(iii) If and a regular games has multiple symmetric equilibria, then there is at least one symmetrically unstable equilibrium.

Proof: (i) Follows from lemma 1. (ii) Let be the eigenvalues of, and be the diagon-

al entries of S. Lemma 1 and the condition in (ii) imply and

. Hence have negative real parts. (iii) Follows from applying the Index theorem to the symmetric gradient field.⁵ The index of a zero of is if

and if. With multiple symmetric equilibria a symmetric equilibrium

with index exists, and. By lemma 1, hence

, which implies existence of at least one eigenvalue with negative real part.

It follows from (iii) that if each verifies symmetric stability then exactly one symmetric equilibrium exists.⁶ In the one-dimensional case an even stronger relation between symmetric stability and the number of symmetric equilibria applies:⁷

Corollary 1 Let in a regular game. There exists an odd number of symmetrically stable equilibria. Moreover, a symmetric equilibrium is globally symmetrically stable if is the only symmetric equilibrium.

Proof: Given regularity, a zero of is symmetrically stable if. Hence symmetrically sta-

ble equilibria have index and the first claim follows from the Index theorem. Necessity in the second claim is trivial. For sufficiency, note that uniqueness of implies (the corresponding index is). From (3) deduce that. Hence crosses the 45˚-line from above. Thus whenever

. Let. Because it follows from and strong quasicon-cavity that

. Hence whenever (Figure 2). Similarly, whenever.

It may be noted from the above proof (or Figure 2) that the second claim of corollary 1 remains valid if there is a single symmetric equilibrium which belongs to the boundary ().

Best-reply dynamics. Another standard dynamics in the literature are dynamics defined directly over the best-reply functions.⁸ These dynamics are of the form

and the symmetric restriction analogously to (2) yields

A symmetric equilibrium is symmetrically stable with respect to (4) if the Jacobian,

has only eigenvalues with negative real parts. It follows that corollary 1 and theorem 1 (i) and (iii) apply, without modification, to the dynamics (4). The latter follows from (3) and the proof of theorem 1, and the former can be deduced directly from (4) together with Figure 2.⁹

Relation to comparative statics. Typically, the IFT is the main formal tool to (locally) sign the comparative- static effects.¹⁰ Stability conditions allow to robustly sign comparative-static effects [3] and additionally assure local convergence after a small shock, which many deem a natural requirement of a comparative-static predict- tion. A symmetrically unstable equilibrium is not re-established after a symmetric shock. Moreover, symmetri-

cally unstable equilibria may “pervert” the comparative-statics. To illustrate consider a regular game with three symmetric equilibria (see Figure 3) where c is an exogenous common parameter. A and B are symmetrically stable (index), but C is symmetrically unstable (index). Consider a symmetric parameter shift and assume that. As is suggested by the figure (formally

apply the IFT) points A and B both increase to and. As both and are symmetrically stable, the symmetric dynamics (2) converge from A to or from B to, consistent with the suggested shift of. For the symmetrically unstable point C we see that (a consequence of the negative index), contradict-

ing the direction suggested by. As C lies in the basin of attraction of the dynamics do

not move down to but monotonically up to (which is also inconsistent with “small” changes). Hence the comparative-statics suggested by the IFT and the dynamics disagree at the unstable equilibria, and the IFT- prediction could never be supported as a stable equilibrium.

To illustrate symmetric stability in an example consider a payoff of the form

The interpretation is that N contestants choose their strategies, the pairs, to obtain a prize worth, where the value of a prize is endogenously determined. A specific context is provided by [10] , where firms com-

pete in salience and prices for attention-constrained consumers.¹¹ Assume that and (everybody has a chance to seize a prize) for, and (a prize is worthwhile seizing) for. An interior symmetric equilibrium solves

with associated Jacobian

It easily follows from theorem 1 (ii) that is symmetrically stable if both. By contrast, with dynamics (1) we would need to evaluate the eigenvalues of a matrix. The symmetric stability condition states that second-order direct effects of each own strategy (which must be negative by strong

quasiconcavity) are not reversed by the second-order effects of, a property which is typically satisfied in standard functional examples (see [7] ). Moreover, it follows from theorem 1 (iii) that if any verifies this condition and the game is symmetrically regular, a single symmetric and symmetrically stable equilibrium exists.