Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes action $a_i \in A_i$, where $A_i $ is the action set of player $i$. Let $\sigma^*$ be any notion of correlated equilibrium (CE) that is computable and unique. For example, the social optimal correlated equilibrium or max-entropy correlated equilibrium, both can be solved efficiently using linear programming. Thus, $\sigma^*$ is a probability measure on the joint action space $\prod_{i=1}^N A_i$. Then the expected payoff of player $i$ is $$ V_i( u_1, \ldots, u_N) = \mathbb{E}_{(a_1, \ldots, a_N) \sim \sigma^*} \bigl [ u_i(a_1, \ldots a_n) \bigr ] \\ = \sum_{(a_1, \ldots, a_N)\in \prod_{i=1}^N A_i} u_i(a_1, \ldots a_n) \cdot \sigma^* (a_1, \ldots a_n). $$ Note that the value of game at a social optimal CE or a max-entropy CE is unique. I was wondering whether the values of the game $( V_1, \ldots, V_N)\in \mathbb{R}^N$ is Lipschitz with respect to the utility functions. That is, suppose we have two sets of utility functions $\{ u_i \}_{i=1}^N $ and $\{\tilde u_i \}_{i=1}^N $ and we solve for the same kind of CE on both games. Is it possible to show that $$ \max_{i\in \{1, \ldots, N \} } \bigl | V_i ( u_1, \ldots, u_N) - V_i(\tilde u_1, \ldots, \tilde u_N) \bigr | \leq C \cdot \max_{j\in \{1,\ldots, N\} } \| u_j - \tilde u_j \|_{\infty} $$ for some constant $C$?

P.S.: For zero-sum games, it seems that we can show this with $C = 1$.