Weak local Nash equilibrium (Q378476)

Given metric spaces \((X_1,d_1),\dots, (X_n,d_n)\) and functions \(p_1,\dots,p_n:\) \(S_1\times\dots \times S_n\to\mathbb R\), the authors study the existence of a weak local equilibrium for \(p_1,\dots,p_n\), which is an \(\tilde s=(\tilde s_1,\dots,\tilde s_n)\in S_1\times\dots \times S_n\) such that for all \(\varepsilon>0\) there exists a \(\delta>0\) with \(p_i(\tilde s_1,\dots,\tilde s_{i-1},s_i,\tilde s_{i+1},\dots,\tilde s_n)\leq p_i(\tilde s)+\varepsilon d_i(s_i,\tilde s_i)\) whenever \(d_i(s_i,\tilde s_i)<\delta\) and \(1\leq i\leq n\).