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The concept of sub-game perfect Nash equilibrium (SPNE) is a refinement of the Nash equilibrium applicable in dynamic games, particularly in games of perfect information where players make decisions at different stages. In game theory, a Nash equilibrium refers to a strategy profile in which no player can unilaterally deviate and improve their payoff, assuming all other players stick to their strategies. However, Nash equilibrium does not always guarantee credible strategies in every part of a dynamic game. To address this issue, the concept of SPNE ensures that strategies constitute a Nash equilibrium in every sub-game of the original game. A *sub-game* is any part of a game that can be considered a game in itself, starting from any decision node to the terminal nodes, maintaining the original rules of the game. A *sub-game perfect Nash equilibrium* requires that players' strategies form a Nash equilibrium not only in the whole game but in every sub-game as well. This means that players’ strategies must be optimal at every point of the game, not just at the initial stage. SPNE rules out non-credible threats or promises that may be part of some Nash equilibria but are not rational to carry out when the time comes. The most common method to derive the SPNE is *backward induction*, especially in games with perfect information and finite stages. Backward induction starts from the end of the game and determines optimal strategies at each decision point, moving backward to the beginning of the game. In doing ____ ______ _________ _________ __________ _________ __________ ______.
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