You can take any two player zero-sum game and change its rules, so that they become:
- Start from the game state being evaluated.
- Play for up to N turns.
- If no winner is found after N turns, the winner is the one with the highest heuristic score (calculated in the same way as your minimax algorithm).
This is still a valid game, based on the original. A minimax using the same non-terminal heuristic will solve that derived/limited game optimally.
The solution may be unrelated to solutions to the original game. As a worst case, choose as your heuristic a random oracle (that assigns a random number to each state, in practice this could be done using a cryptographic hashing function). In the extreme case of N=1, then a random oracle heuristic will perform - on the original game - much the same as a random player, until either player is 1 turn away from winning. As N increases to include more possible game ends, the effect of the poor heuristic is reduced, as forced wins/losses are discovered (and usually those parts of the search tree removed because one of the players would reject them).
If you have a perfect heuristic, then the game is solved, and searching with N=1 against that heuristic will result in optimal play for the original game. Here's an example for tic-tac-toe where the choice for search depth 1 is encoded into the grid
After you take the move identified by the heuristic, then the horizon N moves forward. Using the "similar game" idea above, the derived game has changed. It is feasible that a different choice would have been better according to the heuristic evaluated on N+1 turns, or that the search will now find a forced win, tie or loss which was previously beyond its horizon.
Do considerations in the quote (do even better against a suboptimal opponent) still hold?
Yes, but only in terms of potentially being able to score more highly on the heuristic by turn N. The search will identify a next move where the worst case scenario (assuming the other player chooses a response which minimises your possible score by turn N) is a certain heuristic score on turn N. If you are not in that worst case due to the other player's choice, it may be possible to do better and reach a higher score by turn N.
However, because the horizon will move forward following a turn, you now have further information from searching deeper into the game, and that could mean that the other player's move was in fact worse for you. You can still guarantee a heuristic score by turn N at least as good as the one found before, but now you can see turn up to N+2, and a new search may find important differences.
