  This paper focuses on the online saddle point problem, which involves a
sequence of two-player time-varying convex-concave games. Considering the
nonstationarity of the environment, we adopt the duality gap and the dynamic
Nash equilibrium regret as performance metrics for algorithm design. We present
three variants of the proximal point method: the Online Proximal Point
Method~(OPPM), the Optimistic OPPM~(OptOPPM), and the OptOPPM with multiple
predictors. Each algorithm guarantees upper bounds for both the duality gap and
dynamic Nash equilibrium regret, achieving near-optimality when measured
against the duality gap. Specifically, in certain benign environments, such as
sequences of stationary payoff functions, these algorithms maintain a nearly
constant metric bound. Experimental results further validate the effectiveness
of these algorithms. Lastly, this paper discusses potential reliability
concerns associated with using dynamic Nash equilibrium regret as a performance
metric.
