The other player becomes current. Zero-sum turns two players’ totals into one signed advantage.

Instead of tracking two scoreboards, track “how far ahead can the player to move finish?” After taking a value, the opponent owns the smaller game’s advantage, so subtract it.

What the function promises: Return the first player’s maximum final score minus the second’s. Empty input returns zero.

def score_advantage(values):
    n = len(values)
    if not n: return 0
    dp = list(values)
    for length in range(2, n+1):
        for left in range(n-length+1):
            right = left+length-1
            dp[left] = max(values[left]-dp[left+1], values[right]-dp[left])
    return dp[0]

assert score_advantage([1,5,2]) == -2
assert score_advantage([1,5,233,7]) == 222

Work and storage: O(n²) time and O(n) space.

When to recognize it: Zero-sum turns two players’ totals into one signed advantage.

Check before applying it: This recurrence assumes perfect play, alternating turns, and endpoint-only moves.

One difference may not summarize both players’ objectives; keep the relevant payoff pair or use game-specific equilibrium reasoning.