In a sequential game where players can use mixed strategies, what is the significance of subgame perfect equilibrium in determining optimal strategies?

In a sequential game with mixed strategies, a subgame perfect equilibrium (SPE) guarantees that each player’s strategy is optimal not just overall but also at every possible point in the game, after any history of play. This means that after any sequence of moves, the players’ choices form a Nash equilibrium in the remaining subgame. By using backward induction, we can eliminate strategies that would be irrational once a subgame is reached, ensuring that no player has an incentive to deviate at any stage. For example, in a game where a player can choose to cooperate or defect, SPE requires that if a player knows the future payoff, they will defect only if it is better at that specific subgame; otherwise, they cooperate. Thus, SPE helps identify the truly optimal strategies that remain stable throughout the entire game.