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 $C = 1$. 



Perturbation of the value of a general-sum game at a equilibirium
Consider the following 2x2 two player game
\begin{array}{c|c}
1,1 & 0,1 \\
\hline
1,0 & 0,0
\end{array}
In this game, all strategy profiles are Nash equilibria, and consequently every point in the unit square is an equilibrium payoff (and a correlated equilibrium payoff).
Take now the following perturbation of this game (for positive $\epsilon$)
\begin{array}{c|c}
1+\epsilon,1+\epsilon & \epsilon,1 \\
\hline
1,\epsilon & 0,0
\end{array}
In this game each player has a dominant strategy, hence the unique correlated equilibrium payoff is $(1+\epsilon,1+\epsilon)$.
Take the same game with negative $\epsilon$. The unique correlated equilibrium payoff is (0,0).
Does this example answer you question?


Regarding zero-sum games: in a zero-sum game, the unique correlated equilibrium payoff coincides with the value. Since the value is 1-Liphschitz in the maximum norm, then the answer to your question is positive (for zero-sum games).


