Received 29 March 2016; accepted 6 June 2016; published 9 June 2016
In this note I develop the idea of symmetric stability of symmetric equilibria in symmetric N-player games. With symmetric equilibria it is reasonable to consider dynamics where the set of trajectories is restricted by symmetric initial conditions. This is particularly relevant when studying the comparative-statics of symmetric equilibria to a common shock, such as changing the prize in a contest or a tax parameter in the Cournot model, since this has symmetric effects on symmetric players both in terms of the initial displacement and the subsequent adjustment process. Symmetric stability conditions can be expressed in terms of a best-reply function obtained by fixing the strategies of all other players to the same action. Given a k-dimensional strategy space, this reduces the dimensionality of the stability problem from Nk to k, while retaining all relevant information about symmetric equilibria and their symmetric stability. By means of this reduced form I prove that the existence of a single symmetric equilibrium is the same formal property as global symmetric stability in regular one-dimensional games, independent of the number of players. Further, stability under symmetric adjustments implies the existence of only one symmetric equilibrium for any finite-dimensional strategy space, and symmetric stability provides a meaningful restriction for the possible comparative-static patterns of symmetric equilibria. All results are independent of the possible existence of asymmetric equilibria, and the practical usefulness of symmetric stability is briefly illustrated by means of a two-dimensional N-player contest.
2. Symmetric Games
I consider games of players. is a strategy of player g, where
with, , and interior. All players have identical strategy space. The payoff of g
is represented by a function, which is strongly quasiconcave1 in. In a symmetric game all players have identical payoff function in the sense that
for any permutation of. With the above assumptions, player g’s best-reply function is continuous, and differentiable at interior points. Given the focus on symmetric equilibria one can restrict attention to a reduced-form problem by picking an indicative player (), and requiring all
opponents to play the same strategies, i.e.,. Let with correspond-
ing best-reply and derivative. Any symmetric equilibrium is identifiable by its first projection, is a symmetric equilibrium if, a symmetric equilibrium always exists and the set of symmetric equilibria is compact  .
2.1. Symmetric Stability
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  ). 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:
Figure 1. Stable (left) and only symmetrically stable (right).
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  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-
Figure 2. Corollary 1.
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.
2.2. Application: Two-Dimensional Contest with Endogenous Price
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  , 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
Figure 3. Symmetric stability and comparative statics.
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  ). 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.
I thank Diethard Klatte and participants at seminars at University of Zurich, Harvard University and at the UECE Lisbon Game Theory meeting for valuable comments, and Ines Brunner for ongoing support.
1Meaning:, (see  ).
2This is a standard dynamics, see e.g.  -  .
3Equivalently, is symmetrically stable if the dynamics (1) converge to for any symmetric starting point close enough to.
5See  .
7The claims in corollary 1 are generally restricted to.
8See e.g.   -  .
9It can be noted occasionally from the figure that strong quasiconcavity is generally indispensable for corollary 1 to apply with the gradient dynamics, but not with (4).
10A well-known exception is supermodular games (see  ).
11Payoff (5) is compatible with multiple prizes.