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Solution Explanation

We are given a deterministic two‐player zero‐sum turn‐based game where:

One player is MAX (us).
The other player is MIN (the opponent).

We also have an oracle (O(s)) that, given any state (s) in which MIN is to move, returns the move that MIN would choose if they were playing optimally for themselves (i.e., the minimizing move).

We want to reduce this two‐player game to a single‐player search problem that correctly computes our (MAX’s) optimal policy. The key idea is:

When it is our turn (MAX’s turn) in state (s):
- We branch on each possible action (a) we can take.
- Once we pick an action (a), the state transitions to (s' = \text{Result}(s, a)).
- Now it should be MIN’s turn in (s').
- Instead of branching on all possible moves of MIN, we use the oracle (O(s')) to get the single best (for MIN) move (a_{\text{opp}}).
- We then transition immediately to (s'' = \text{Result}(s', a_{\text{opp}})).
- Hence from MAX’s perspective, the next state after our move (a) is deterministic (because the oracle effectively “collapses” MIN’s possible moves to the single best one for MIN).
Hence, the new “single‐player” transition is:
[
s
\xrightarrow[\text{our action}]{a}
s' = \text{Result}(s,a)
\xrightarrow[\text{oracle for MIN}]{O(s')}
s'' = \text{Result}(s',,O(s')).
]
We treat this combined step as a single “transition” in a single‐player game: from (s) to (s'') via our choice (a).
We can then do a standard single‐agent search (e.g., depth‐first with memoization, or any other exhaustive search technique) over this newly defined (deterministic) state space. At each node (s) (where “it is our turn”), we branch only over our possible actions, and the opponent’s reply is fixed by the oracle.
Correctness follows from the fact that in the original two‐player game, MIN would indeed choose that same move if playing optimally. Thus analyzing only that single MIN move in each state is equivalent to standard minimax, but with the branching factor for MIN collapsed to 1 (the known optimal MIN move).

1. Pseudocode

Below is a high‐level pseudocode for the Oracle‐Advised Single‐Player Search that yields the same result as a full minimax search but uses the oracle to avoid branching over MIN’s actions.

FUNCTION OracleAdvisedSearch(state s):
    # If s is a terminal state, return its utility for MAX.
    if IsTerminal(s):
        return Utility(s)

    best_value ← -∞

    # For each action we (MAX) can take in state s:
    for each action a in Actions(s):
        s' ← Result(s, a)              # State after our action a
        a_opp ← O(s')                  # Oracle gives MIN's best response
        s'' ← Result(s', a_opp)        # State after MIN's best response

        # Recursively evaluate the resulting position
        value_a ← OracleAdvisedSearch(s'')

        # Update best value
        if value_a > best_value:
            best_value ← value_a

    return best_value

IsTerminal(s) checks if (s) is a game‐ending state.
Utility(s) returns the terminal payoff (for MAX) if (s) is terminal.
Actions(s) returns the actions available to MAX in state (s).
Result(s, a) returns the next state if action (a) is played in state (s).
O(s') returns the opponent’s (MIN’s) optimal move in state (s'), per the oracle.

In many implementations, one might also keep track of the best action (not just the best value) so we can return the policy. But above we show only the value computation for simplicity.

2. Proof of Correctness (Two‐Column Proof)

Use an inductive argument on the depth of the game tree to show that OracleAdvisedSearch returns the same value that would be computed by the standard Minimax value for MAX.

Statement of What We Prove

For every state (s) in which it is MAX’s turn, the function OracleAdvisedSearch(s) returns the same value as MinimaxValue(s) (the minimax value under optimal play for both sides).

Below is a typical two‐column proof outline.

Left (Statements)	Right (Reasons)
1. We define `MinimaxValue(s)` in the usual way:	Definition of minimax.
( \displaystyle \text{MinimaxValue}(s) = \max_{a ,\in, \text{Actions}(s)} \min_{b ,\in, \text{Actions}(\text{Result}(s,a))} \text{MinimaxValue}(\text{Result}(\text{Result}(s,a), b)) ) if (s) is non‐terminal on MAX’s turn, and similarly for MIN’s turn.	By the standard two‐player game definition, if it’s MIN’s turn, we take a (\min).
2. We define `OracleAdvisedSearch(s)` as in our pseudocode: from (s), we try each action (a), then apply the oracle move for MIN, leading to a single next state, and recurse.	Definition of the OracleAdvisedSearch.
3. Base Case: If (s) is terminal, `OracleAdvisedSearch(s) = Utility(s)` and `MinimaxValue(s) = Utility(s)`. They match immediately.	Directly from the pseudocode and definition of minimax, both agree on terminal nodes.
4. Inductive Step: Assume for all states (t) at depth (d) or less (from the bottom), `OracleAdvisedSearch(t) = MinimaxValue(t)`. We must show it holds at depth (d+1).	Principle of strong (or simple) induction on the game‐tree depth.
5. Let (s) be a state at depth (d+1) where it is MAX’s turn. By definition,	We are focusing on states where MAX moves.
( \text{MinimaxValue}(s) = \max_{a \in \text{Actions}(s)} \min_{b \in \text{Actions}(\text{Result}(s,a))} \text{MinimaxValue}(\text{Result}(\text{Result}(s,a), b)) ). )	Standard minimax rule.
6. By assumption of the oracle, for any state (s') (with MIN to move) we have ( O(s') = b^* ) where ( b^* ) satisfies:	The oracle returns MIN’s best move.
( b^* \in \arg\min_{b \in \text{Actions}(s')} \text{MinimaxValue}(\text{Result}(s', b)). )	Definition of “oracle is always correct.”
7. Hence, in the single‐player transformation, from state (s), if MAX picks (a), we move to (s' = \text{Result}(s,a)), then the oracle’s move is ( b^* = O(s') ). So the resulting state is ( s'' = \text{Result}(s', b^*) ).	This matches the pseudocode transitions.
8. Because (b^*) is MIN’s optimal move from (s'), we have:	By definition of (b^*).
( \min_{b \in \text{Actions}(s')} \text{MinimaxValue}(\text{Result}(s', b)) = \text{MinimaxValue}(\text{Result}(s', b^*)). )	(b^*) attains the min.
9. Therefore,	Combining steps 5 and 8.
( \text{MinimaxValue}(s) = \max_{a \in \text{Actions}(s)} \text{MinimaxValue}(s''), \text{ where } s'' = \text{Result}(\text{Result}(s,a), O(\text{Result}(s,a))). )	Rewriting the minimax expression to factor in the oracle.
10. By the inductive hypothesis, for each “child” (s'') at depth (d) (one level closer to terminal), `OracleAdvisedSearch(s'') = MinimaxValue(s'')`.	We apply the induction assumption to states closer to the bottom.
11. Thus, in computing `OracleAdvisedSearch(s)`, we take the same maximum over exactly the same values (\text{MinimaxValue}(s'')).	From the pseudocode’s recursion.
12. Hence `OracleAdvisedSearch(s) = MinimaxValue(s)`.	They coincide at depth (d+1).
13. By mathematical induction on the depth, these two functions coincide for all states (s).	This completes the proof.

Conclusion: For every state (s), OracleAdvisedSearch(s) returns the same value as the standard minimax search would. Therefore, using the oracle to collapse MIN’s branching to a single move is a correct way to turn a two‐player game (under optimal play by MIN) into a single‐player search problem from MAX’s perspective in a deterministic, zero‐sum setting.Solution Explanation