Problem 1. Consider the payoff matrix shown at right. Suppose agent 1 must move first, and agent 2 can see B agent 1's move before agent 2 chooses a move. Draw the game tree, find all pure-strategy Nash equilibria, C and tell which of them are subgame-perfect.
Problem 2. Below is a 3-player extensive-form game. At each terminal node, the numbers are the utility values for players 1, 2, and 3. At each nonterminal node, the number tells which player will move.
(a) Find the subgame-perfect equilibrium. At every nonterminal node, do two things: circle the move that the player will make, and write the expected utility values for all three players.
(b) Suppose that players 2 and 3 play a minimax strategy profile against player 1, and player 1 plays his/her best response to their strategy profile. At every nonterminal node, do two things: circle the move that the player will make, and write the expected utility values for all three players.
Problem 3. Above is a game tree for a perfect-information zero-sum game. Run the alpha-beta algorithm (by hand) on this game tree. Next to each node, write all of the node's intermediate and final values for α, β, and v.
