What is the concept of backward induction in Game Theory and how is it applied in solving extensive form games?

Backward induction is a concept in game theory that involves reasoning backward from the end of a game to determine the optimal strategy for each player. It is commonly used to solve extensive form games, which are games that are represented by a game tree.

In extensive form games, players make sequential moves, and each player's decision at each node of the game tree depends on the decisions made by the previous players. Backward induction starts at the final node of the game tree and works its way backward, considering the optimal strategies for each player at each node.

To apply backward induction, we begin by analyzing the final nodes of the game tree. At these nodes, we determine the payoffs for each player and assign them accordingly. Then, we move one step backward to the previous nodes and consider the optimal strategies for the players at these nodes.

At each node, we consider the payoffs and the strategies available to the players. We assume that each player is rational and aims to maximize their own payoff. Therefore, we eliminate any strategies that are dominated, meaning that there is always another strategy that yields a higher payoff regardless of the opponent's strategy.

By iteratively eliminating dominated strategies and considering the optimal strategies for each player at each node, we eventually reach the initial node of the game tree. At this point, we have determined the optimal strategy for each player throughout the game.

Backward induction is a powerful tool in solving extensive form games as it allows us to determine the subgame perfect Nash equilibrium, which is a strategy profile where no player can unilaterally deviate and improve their payoff. This concept ensures that each player is playing their best response to the strategies chosen by the other players, resulting in a stable outcome.

In summary, backward induction is a concept in game theory that involves reasoning backward from the end of a game to determine the optimal strategy for each player. It is applied in solving extensive form games by iteratively eliminating dominated strategies and considering the optimal strategies at each node of the game tree, ultimately leading to the determination of the subgame perfect Nash equilibrium.