On games on the unit square with a “roof” payoff

This paper describes all the optimal strategies of players on the unit square witk the payoff
$$ K(x,y)= \begin{cases} 1+\frac1\delta(x-y)&\text{if}\quad|x-y|\le\delta,\quad x\le y, \\ 1+\frac1\delta(y-x)&\text{if}\quad|x-y|\le\delta,\quad x>y, \\ p&\text{if}\quad|x=y|>\delta, \end{cases} $$
where $\delta>0$.