Drawing two pieces of curves with Tikz package

Question

Does anyone have a way to write a code in tikz package to obtain the graphics shown in the below image: (Where the two pieces of curves are arbitrary arcs.)

I try to mimic the code from How to shift a curve with TiKZ as follow

\documentclass{article}
\usepackage{tikz,pgfplots}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[>=latex]
\begin{axis}[
    axis x line=bottom,
    axis y line=left,
    xmin=0, xmax=10,
    ymin=0, ymax=10,
    xlabel style={at={(current axis.right of origin)}},
    ylabel style={at={(current axis.above origin)}, rotate=270,},
    xlabel={$\tau_1$},
    ylabel={$\tau_2$},
    ytick=\empty,
    xtick=\empty,
    ]
    \draw (axis cs:0,6.5) to [bend right=30] coordinate[pos=0.2] (l_i) (axis cs:6,0);
    \fill (l_i) circle (1.2pt) node[above right] {$f_1(\tau_2)$};
\draw (axis cs:.0,-2) to [bend right=10] coordinate[pos=0.8] (dl_j) (axis cs:6,-4);
\fill (dl_j) circle (1.2pt) node[above right] {$f_2(\tau_2)$};


\end{axis}
\end{tikzpicture}
\caption{Preferences}
\end{figure}
\end{document} 

Thank you very much in advance!