In a zero-sum game where two players choose strategies simultaneously, what is the relationship between Nash equilibrium and subgame perfect equilibrium when both players are rational and the game is played in a sequential manner?

In a zero‑sum game, a Nash equilibrium is a pair of strategies that no player can improve by changing strategy alone. When the game is played sequentially, we can split it into subgames, and a subgame‑perfect equilibrium (SPE) requires optimal play in every subgame, removing non‑credible threats. For rational players, backward induction shows that any SPE must also satisfy the Nash condition for the whole game, but a Nash equilibrium might rely on threats that are not credible in subgames. For instance, if the first player chooses A or B and the second responds with X or Y, backward induction forces the second player to pick the best reply, yielding an SPE that is also a Nash equilibrium of the whole game. Thus, every SPE is a Nash equilibrium, but not every Nash equilibrium is subgame‑perfect.