This problem feels like a Dynamic Programming (DP) problem. Except for the need to buy the consoles, this is exactly a knapsack problem where the game prices GPj are the sizes of the items and production values PVj are the values of the items. The budget V is the size of the knapsack. If we ignored the need to buy the consoles, we could define the subproblems by letting~ prod[j][c]~ be the greatest production we can get by using the first ~j~ games and by spending at most ~c \le V~ dollars. The recurrence is then:

prod[j][c] = max(prod[j-1][c], PV_j + prod[j-1][c - GP_j])

because with each new game i, we can either not buy it or we can buy it at a cost of GPi and get a value of PVi.

Next, note that once we've committed to buy a console ~i~, the problem of choosing which games to buy for console ~i~ is a straightforward knapsack problem which we can solve with the recurrence given above. That is, for a given console ~i~, we can calculate ~prod[j][c]~ for console ~i~'s games and that will tell us which of console ~i~'s games to buy for each budget.

The last step is that we need to choose which consoles to buy. Think of this as a knapsack problem on consoles, except that each console doesn't have a single cost and a single value: we can spend more or less on each console depending on which games we buy and we get correspondingly more or less production. Each ~prod[j][c]~ for console ~i~ represents a different "console ~i ~package" that we could purchase. At each iteration ~i~, we could choose to buy console ~i~ and any of these packages for console~ i~, or we could skip console i entirely. This gives us a knapsack dynamic program on consoles where the items we can put in the knapsack at the ~ith~ step are the outputs to the ~prod[j][c]~ dynamic program for the ~ith~ console. We solve this dynamic program and this gives us the optimal value over all consoles.