Boolean Game

Consider $N$ binary variables $x_1, x_2, \ldots, x_N$. For each valid $i$, the $i$-th of these variables can be $x_i = 0$ or $x_i = 1$; therefore, there are $2^N$ possible assignments of values to the variables. For each valid $i$, setting $x_i = 1$ gives you score $g_i$. In addition, there are $M$ special intervals (numbered $1$ through $M$). For each valid $i$, the $i$-th interval is $[u_i, v