The problem of $\delta$-meeting on a $3$-dimensional ball

An infinite non-zero-sum two-person game with payoff function
$$ L_{\delta}(x,y)= \begin{cases} 1, &\rho(x,y)\le\delta,\\ 0, &\rho(x,y)>\delta, \end{cases} \qquad(x\in X,y\in Y) $$
(where $X$, $Y$ are $3$-dimensional unit balls) is considered. A solution of this game for $0,475<\delta<1$ is given.