\section{Experiments}\label{exexex}

In this section we provide additional numerical experiments to support our theoretical findings. In each experiments we construct periodic games with common equilibrium, and provide numerical results on the mixed strategies and KL-divegence of (Extra-MWU) and (OMWU) on these games. 

\subsection{Experiment 1}

The payoff matrix in this experiment is a 2-periodic game defined by
\begin{align*}
    A_t=
\begin{cases}
\left( (0, 0.25, 0.75), (1.5, 0, 0), (0, 1, 0) \right), & t \textnormal{\ \ is \ odd.} \\ 
\\
\left((0, 0.75, 0.25), (1.5, 0, 0), (0, 0, 1) \right), & t\textnormal{\ \ is \ even.}
\end{cases}
\end{align*}
In Figure (\ref{em1}), we present experimental results for (Extra-MWU). In (a) of Figure (\ref{em1}), we can see the mixed strategy of 2-player converge to the equilibrium point $(1/3 , 1/3, 1/3)$. In (b) of Figure (\ref{em1}), we can see $\KL \left( (\tb{x}_1^*,\tb{x}_2^*), (\tb{x}_1^t,\tb{x}_2^t) \right) \to 0$ as time $\to \infty$. This support the result in Theorem \ref{T2}.

In Figure (\ref{om1}), we present experimental results for (OMWU). In (a) of Figure (\ref{om1}), we can see the mixed strategy of 2-player do not converge to the equilibrium. In (b) of Figure (\ref{om1}), we can see $\KL \left( (\tb{x}_1^*,\tb{x}_2^*), (\tb{x}_1^t,\tb{x}_2^t) \right) \to \infty$ as time $\to \infty$. In (b) of Figure (\ref{om1}), we can see $\KL \left( (\tb{x}_1^*,\tb{x}_2^*), (\tb{x}_1^t,\tb{x}_2^t) \right) \to \infty$ as time $\to \infty$, this implies players' mixed strategies will diverge to the boundary of the simplex, thus the phenomenon here is similar to Theorem \ref{thm: OMWU fails}.

\begin{figure}[h]
    \centering
    \subfigure[Mixed strategy of 2-player]{
        \includegraphics[width=1.5in]{Extra_1.png}
        \label{}
    }
    \subfigure[KL-divergence of 2-player]{
	\includegraphics[width=1.4in]{Extra_KL_1.png}
        \label{}
    }
    \caption{First experimental results for Extra-MWU.}
    \label{em1}
\end{figure}

\begin{figure}[h]
    \centering
    \subfigure[Mixed strategy of 2-player]{
        \includegraphics[width=1.5in]{OMWU_1.png}
        \label{}
    }
    \subfigure[KL-divergence of 2-player]{
	\includegraphics[width=1.4in]{O_KL_1.png}
        \label{}
    }
    \caption{First experimental results for OMWU.}
    \label{om1}
\end{figure}


\subsection{Experiment 2}

The payoff matrix in this experiment is a 4-periodic game defined by
\begin{align*}
    A_t=
\begin{cases}
\left( (0, -1, 1), (1, 0, -1), (-1, 1, 0) \right),\ t \mod 4 = 0 \\ 
\\
\left((0, 1, -1), (-1, 0, 1), (1, -1, 0) \right),\ t \mod 4 = 1 \\
\\
\left((1, -3, 2), (-2, 1, 1), (1, 2, -3) \right),\ t \mod 4 = 2 \\
\\
\left((1, -2, 1), (-2, 1, 1), (1, 1, -2) \right),\ t \mod 4 = 3
\end{cases}
\end{align*}
In Figure (\ref{em2}), we present experimental results for (Extra-MWU). In (a) of Figure (\ref{em2}), we can see the mixed strategy of 2-player converge to the equilibrium point $(0.25 , 0.375, 0.375)$. In (b) of Figure (\ref{em2}), we can see $\KL \left( (\tb{x}_1^*,\tb{x}_2^*), (\tb{x}_1^t,\tb{x}_2^t) \right) \to 0$ as time $\to \infty$. This support the result in Theorem \ref{T2}.

In Figure (\ref{om2}), we present experimental results for (OMWU). (a) of Figure (\ref{om2}) shows mixed strategy do not converge to the equilibrium, and the strategy 3 tends to $0$ as time process. In (b) of Figure (\ref{om2}), we can see $\KL \left( (\tb{x}_1^*,\tb{x}_2^*), (\tb{x}_1^t,\tb{x}_2^t) \right) \to \infty$ as time $\to \infty$. This implies the players mixed strategies tends to the boundary of the simplex constrains. 
 
 


\begin{figure}[h]
    \centering
    \subfigure[Mixed strategy of 2-player]{
        \includegraphics[width=1.5in]{Extra_2.png}
        \label{}
    }
    \subfigure[KL-divergence ]{
	\includegraphics[width=1.4in]{Extra_KL_2.png}
        \label{}
    }
    \caption{Second experimental results for Extra-MWU.}
    \label{em2}
\end{figure}

\begin{figure}[h]
    \centering
    \subfigure[Mixed strategy of 2-player]{
        \includegraphics[width=1.5in]{OMWU_2.png}
        \label{}
    }
    \subfigure[KL-divergence ]{
	\includegraphics[width=1.4in]{O_KL_2.png}
        \label{}
    }
    \caption{Second experimental results for OMWU.}
    \label{om2}
\end{figure}