We investigate first-order notions of correlated equilibria; distributions of actions for smooth, potentially non-concave games such that players do not incur any regret against small modifications to their strategies along a set of continuous vector fields. We define two such notions, based on local deviations and on stationarity of the distribution, and identify the notion of coarseness as the setting where the associated vector fields are in fact gradient fields. For coarse equilibria, we prove that online (projected) gradient decent has a universal approximation property for both variants of equilibrium. In the non-coarse setting, we instead reduce the problem of finding an equilibrium to fixed-point computation via the usual framework of $\Phi$-regret minimisation, and identify tractable instances. Finally, we study the primal-dual framework to our notion of first-order equilibria. For coarse equilibria defined by a family of functions, we find that a dual bound on the worst-case expectation of a performance metric takes the form of a generalised Lyapunov function for the dynamics of the game. Specifically, usual primal-dual price of anarchy analysis for coarse correlated equilibria as well as the smoothness framework of Roughgarden are both equivalent to a problem of general Lyapunov function estimation. For non-coarse equilibria, we instead observe a vector field fit problem for the gradient dynamics of the game. These follow from containment results in normal form games; the usual notion of a (coarse) correlated equilibria is equivalent to our first-order local notions of (coarse) correlated equilibria with respect to an appropriately chosen set of vector fields.

翻译：我们研究相关均衡的一阶概念：针对光滑且可能非凹的博弈，定义一种行动分布，使得参与者不会因沿一组连续向量场对其策略进行微小修改而产生任何遗憾。我们基于局部偏差和分布的平稳性定义了两种此类概念，并将粗糙性概念确定为相关向量场实际上为梯度场的设定。对于粗均衡，我们证明了在线（投影）梯度下降对两种均衡变体均具有通用逼近性质。在非粗糙设定中，我们通过$\Phi$-遗憾最小化的常规框架，将寻找均衡的问题简化为不动点计算，并识别出可处理的实例。最后，我们研究一阶均衡概念的对偶框架。对于由函数族定义的粗均衡，我们发现性能指标最坏情况期望的对偶界具有博弈动力学广义李雅普诺夫函数的形式。具体而言，粗相关均衡的常规对偶无政府价格分析以及Roughgarden的光滑性框架，均等价于广义李雅普诺夫函数的估计问题。对于非粗均衡，我们则观察到博弈梯度动力学的向量场拟合问题。这些结论源于标准形式博弈中的包含关系：常规的（粗）相关均衡概念等价于我们针对适当选择的向量场集所定义的一阶局部（粗）相关均衡概念。