_{1}

The idea of symmetric stability of symmetric equilibria is introduced which is relevant, e.g., for the comparative-statics of symmetric equilibria with symmetric shocks. I show that symmetric stability can be expressed in a two-player reduced-form version of the N -player 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.

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.

I consider games of

with

is represented by a ^{1} in

for any permutation

opponents to play the same strategies, i.e.

ing best-reply

I mostly restrict attention to the system of gradient dynamics^{2}

where S is a

form

I say that an interior symmetric equilibrium ^{3} Hence:

Definition 1 (Symmetric stability) The symmetric equilibrium

(symmetrically) regular if i) ^{4} and ii)

Lemma 1 For

Proof: The first equality is immediate. Next, decompose

Let

Theorem 1 (i) If

(ii) For

(iii) If

Proof: (i) Follows from lemma 1. (ii) Let

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

^{5} The index of a zero of

with

It follows from (iii) that if each ^{6} In the one-dimensional case an even stronger relation between symmetric stability and the number of symmetric equilibria applies:^{7}

Corollary 1 Let

Proof: Given regularity, a zero of

ble equilibria have index

It may be noted from the above proof (or

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

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 ^{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 [

cally unstable equilibria may “pervert” the comparative-statics. To illustrate consider a regular game with three symmetric equilibria

apply the IFT) points A and B both increase to

ing the direction suggested by

not move down to

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

The interpretation is that N contestants choose their strategies, the pairs

pete in salience and prices for attention-constrained consumers.^{11} Assume that

with associated Jacobian

It easily follows from theorem 1 (ii) that

quasiconcavity) are not reversed by the second-order effects of

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.

Andreas Hefti, (2016) Symmetric Stability in Symmetric Games. Theoretical Economics Letters,06,488-493. doi: 10.4236/tel.2016.63056