Explain the concept of backward induction in sequential games.

Backward induction is a method used to solve sequential games by working backwards from the end.

In sequential games, players take turns to make decisions, and the outcome of each decision affects the subsequent decisions and payoffs. Backward induction starts at the end of the game and works backwards to determine the optimal strategy for each player at each decision point.

To use backward induction, we first identify the final outcome of the game and the payoffs associated with it. Then, we consider the second-to-last decision point and determine the optimal strategy for each player given the payoffs from the final outcome. We continue this process, working backwards through each decision point until we reach the beginning of the game.

For example, consider the following game:

Player 1 chooses A or B
Player 2 chooses X or Y

If Player 1 chooses A, Player 2 can choose X or Y. If Player 1 chooses B, Player 2 can choose X or Y. The payoffs are as follows:

(A, X) = (5, 3)
(A, Y) = (1, 1)
(B, X) = (0, 0)
(B, Y) = (4, 2)

To use backward induction, we start at the end of the game and consider the payoffs for each possible outcome. The highest payoff for Player 2 is (5, 3) if Player 1 chooses A and Player 2 chooses X. Therefore, we can eliminate the possibility of Player 1 choosing B. Now, we consider the optimal strategy for Player 2 given that Player 1 will choose A. The highest payoff for Player 2 is (5, 3) if they choose X. Therefore, the optimal strategy for Player 2 is to choose X if Player 1 chooses A. Thus, the optimal strategy for Player 1 is to choose A. The final outcome is (A, X) with payoffs (5, 3).