# What is a sufficient second order condition for an equality constrained nash equilibrium point?

Given functions $$f_1,f_2:\mathbb{R}^n\rightarrow \mathbb{R}$$ and constraints $$h_1,h_2:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$$ consider the (constrained) Nash equilibrium problem of finding $$x^*=(x_1^*,x_2^*)\in\mathbb{R}^{2n}$$ such that

$$x_1^* \in argmin \{ f_1(x,x_2^*)\quad st \quad h_1(x)=0\},$$

$$x_2^* \in argmin \{ f_2(x_1^*,x)\quad st \quad h_2(x)=0\}.$$

What is the sufficient second order condition in order for $$x^*$$ to be an equilibrium point in this case?

If it was an optimization problem with an equality constraint then it would suffice that $$\nabla^2L(x^*,\lambda^*)>0$$, where $$L(x,\lambda)$$ is the Lagrangian function associated with the problem.

When it comes to Nash equilibriums I'm aware that the necessary 1st order condition is $$\nabla_{x_i} L_i(x^*,\lambda_i)=0\in \mathbb{R}^n$$, for each of the two associated Lagrangians of the problems ($$i=1,2$$), but everywhere I look I don't find the 2nd condition, the texts all mention the first order instead and whenever the search returns something on second order and I open the text it's about optimization instead of equilibrium on the context...

So what would be the 2nd order sufficient condition? Is it really just $$\nabla^2_{x_ix_i}L_i(x^*,\lambda_i^*)>0$$, completely analogous to an optimization problem? I feel like there's something missing here... For instance if you do a Newton method for the optimality conditions with $$f_i$$ quadratic and organize in a linear system then the matrix of the system becomes of size $$2(n+m)\times2(n+m)$$ so I feel like it couldn't be simply a condition on two smaller square matrices and that's that, no?

• Or it could be (I don't know) that thee is either no known 2nd order condition, or at least no known computable (evaluable) 2nd order condition (for instance, as is the case with Semoidefinite Programming problems for which the only known 2nd order conditions seem to be unevaluable, even though there are readily available 1st order conditions). Commented Sep 4 at 23:43