Symmetric stability in symmetric games

The idea of symmetric stability of symmetric equilibria is introduced, which is relevant, e.g., for the comparative-statics of symmetric equilibria given symmetric shocks. I show that symmetric stability can be expressed in a two-player reduced-form version of the game, derive an elementary relation between symmetric stability and the existence of exactly one symmetric equilibrium and apply symmetric stability to a two-dimensional N -player contest.


Introduction
In this note I develop the idea of symmetric stability of symmetric equilibria in symmetric games.With symmetric equilibria it is reasonable to consider dynamics where the set of trajectories is restricted by symmetric initial conditions.This is most natural when studying the comparative-statics of symmetric equilibria given a common shock, such as changing the prize in a contest or a tax parameter in the Cournot model, 1 since this has symmetric effects on symmetric players both in terms of the initial displacement and the adjustment process.
I show that these 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 N k to k, while retaining all relevant information about symmetric equilibria and their symmetric stability.I then 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, and stability under symmetric adjustments implies the existence of only one symmetric equilibrium for any finite-dimensional strategy space.Further, I discuss the link between symmetric stability and the comparative-statics.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.

Symmeric stability
I mostly restrict attention to the system of gradient dynamics where S is a k × k positive-diagonal adjustment matrix, and ∇ j Π j (x) = ∂Π j (x) ∂x j .A solution to (1) has the form x(t) = (x j (t)) 1≤j≤N , where x j (t) = (x j1 (t), ..., x jk (t)) is the trajectory of j.I consider a restricted version of this trajectory map, where initial values x(0) are symmetric, i.e. x 1 (0) = ... = x N (0).Then, by symmetry, x j (t) is the same for all players and solves ẋ1 = S∇ Π(x 1 ), where I say that an interior symmetric equilibrium x * is symmetrically stable if the dynamics induced by (2) converge to x * whenever x(0) is close to x * .5 Hence: Definition 1 (Symmetric stability) The symmetric equilibrium x * ∈ Int(S) is symmetrically stable if all eigenvalues of Ĵ(x * 1 ), the Jacobian corresponding to (2), have negative real parts.
x * is symmetrically unstable if at least one eigenvalue of Ĵ(x * 1 ) has positive real part.Stability of (1) implies symmetric stability, but not vice-versa (figure 1).
, , Let Cr s = {x 1 ∈ S : ∇ Π (x 1 ) = 0}, and note that ∇ Π (x 1 ) : vector field with k×k Jacobian J(x 1 ).A symmetric game is (symmetrically) regular if i) ∇ Π has only regular zeroes6 and ii) ∇ Π points inwards at the boundary of S. Any reference to a regular game in this article refers to "symmetrically regular", which is a weaker condition than general regularity of a symmetric game (see Hefti (2014)).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 Ĵ, J and φ.
Lemma 1 For x 1 ∈ Int(S): By the Implicit Function Theorem (IFT) gether with the decomposition of J gives the second equality.
(iii) If k ≥ 1 and a regular games has multiple symmetric equilibria, then there is at least one symmetrically unstable equilibrium.
It follows from (iii) that if each x 1 ∈ Cr s verifies symmetric stability then exactly one symmetric equilibrium exists. 8In the one-dimensional case an even stronger relation between symmetric stability and the number of symmetric equilibria applies: 9 Corollary 1 Let k = 1 in a regular game.There exists an odd number of symmetrically stable equilibria.Moreover, a symmetric equilibrium is globally symmetrically stable iff x * is the only symmetric equilibrium.
8 If all eigenvalues of Ĵ(x 1 ) have negative real parts, then Det(− Ĵ(x 1 )) > 0 and by (3) also Det(− J(x 1 )) > 0. Thus every x 1 ∈ Cr s has index +1, which by the index theorem implies existence of a unique symmetric equilibrium.This type of relation between the index of certain vector fields and stability conditions is known in other settings (see Hefti (2016) and the references therein).
9 The claims in corollary 1 are generally restricted to k = 1.
Best-reply dynamics Another standard dynamics in the literature are dynamics defined directly over the best-reply functions.10These 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, J(x 1 ) = −S(I − φ(x 1 )) 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 2.11 Relation to comparative statics Typically, the IFT is the main formal tool to (locally) derive comparative-static effects.12Stability conditions allow to robustly sign comparativestatic effects (Dixit (1986)) and additionally assure local convergence after a small shock, which many deem a natural requirement of a comparative-static prediction.A symmetrically unstable equilibrium is not re-established after a symmetric shock.Moreover, symmetrically unstable equilibria may "pervert" the comparative-statics.To illustrate consider a regular game with three symmetric equilibria x A (c), x B (c), x C (c) (see figure 3) where c is an exogenous common parameter.A and B are symmetrically stable (index +1), but C is symmetrically unstable (index −1).Consider a symmetric parameter shift c → c and assume that φ(x, c ) > φ(x, c).
As is suggested by the figure (formally we would apply the IFT) points A and B both increase to A and B .As both A and B are symmetrically stable, the symmetric dynamics (2) converge from A to A or from B to B , consistent with the suggested shift of φ.For the symmetrically unstable point C we see that C < C (a consequence of the negative index), contradicting the direction suggested by φ(x, c ) > φ(x, c).As C lies in the basin of attraction of B the dynamics do not move down to C but monotonically up to B (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 C → C could never be supported as a stable equilibrium. x

Application: Two-dimensional contest with endogenous price
To illustrate symmetric stability, and its usefulness, consider a payoff of the form The interpretation is that N contestants choose their strategies, the pairs (f j , p j ), to obtain a prize worth V (•), where the value of a prize is endogenously determined.A specific context is provided by Hefti (2015), where firms compete in salience and prices for attention-constrained consumers.13Assume that C (f ) > 0, C (f ) ≤ 0 and π(f, f ) > 0 (everybody has a chance to seize a prize) for f > 0, and V (p, p) > 0 (a prize is worthwhile seizing) for p > 0. An interior symmetric equilibrium (p, f ) > 0 solves It easily follows from theorem 1 (ii) that (p, f ) is symmetrically stable if both Πpp , Πff < 0. By contrast, with dynamics (1) we would need to evaluate the eigenvalues of a N k × N k matrix.
The symmetric stability condition states that second-order direct effects of each own strategy p, f (which must be negative by strong quasiconcavity) are not reversed by the second-order effects of (p, f ), a property which is typically satisfied in standard functional examples (see Hefti (2015)).Moreover, it follows from theorem 1 (iii) that if any (p, f ) ∈ Cr s verifies this condition and the game is symmetrically regular, a single symmetric and symmetrically stable equilibrium exists.