Subgames and Credible Threats with perfect information Econ

Subgames and Credible Threats (with perfect information) Econ 171

Alice and Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2

Strategies • For Bob – Go to A – Go to B • For Alice – Go to A if Bob goes A and go to A if Bob goes B – Go to A if Bob goes A and go to B if Bob goes B – Go to B if Bob goes A and go to A if Bob goes B – Go to B if Bob goes A and go B if Bob goes B • A strategy specifies what you will do at EVERY Information set at which it is your turn.

Strategic Form Alice Bob Go where Bob went. Go to A no matter what Bob did. Go to B no Go where Bob matter what did not go. Bob did. Movie A 2, 3 0, 0 0, 1 Movie B 3, 2 1, 1 3, 2 1, 0 How many Nash equilibria are there for this game? A) 1 B) 2 C) 3 D) 4

The Entry Game Challenger Challenge Stay out Incumbent Give in 1 0 Fight -1 -1 0 1

Are both Nash equilibria Plausible? • What supports the N. E. in the lower left? • Does the incumbent have a credible threat? • What would happen in the game starting from the information set where Challenger has challenged?

Entry Game (Strategic Form) Challenger Challenge Give in Do not Challenge 0, 1 0, 0 -1, -1 0, 0 Incumbent Fight How many Nash equilibria are there?

Subgames • A game of perfect information induces one or more “subgames. ” These are the games that constitute the rest of play from any of the game’s information sets. • A subgame perfect Nash equilibrium is a Nash equilibrium in every induced subgame of the original game.

Backwards induction in games of Perfect Information • Work back from terminal nodes. • Go to final ``decision node’’. Assign action to the player that maximizes his payoff. (Consider the case of no ties here. ) • Reduce game by trimming tree at this node and making terminal payoffs at this node, the payoffs when the player whose turn it was takes best action. • Keep working backwards.

Alice and Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2

Two subgames Bob went A Bob went B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2

Alice and Bob (backward induction) Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2

Alice and Bob Subgame perfect N. E. Bob Go to A Go to B Alice Go to A 2 3 Alice Go to B 0 0 Go to A 1 1 Go to B 3 2

Strategic Form Alice Bob Go where Bob went. Go to A no matter what Bob did. Go to B no Go where Bob matter what did not go. Bob did. Movie A 2, 3 0, 0 0, 1 Movie B 3, 2 1, 1 3, 2 1, 0

A Kidnapping Game Kidnapper Don’t Kidnap Relative Don’t pay Pay ransom Kidnapper Kill 5 1 3 5 Kidnapper Release 4 3 Kill 2 2 Release 1 4

In the subgame perfect Nash equilibrium A) The victim is kidnapped, no ransom is paid and the victim is killed. B) The victim is kidnapped, ransom is paid and the victim is released. C) The victim is not kidnapped.

Another Kidnapping Game Kidnapper Don’t Kidnap Relative Don’t pay Pay ransom Kidnapper Kill 4 1 3 5 Kidnapper Release 5 3 Kill 2 2 Release 1 4

In the subgame perfect Nash equilibrium A) The victim is kidnapped, no ransom is paid and the victim is killed. B) The victim is kidnapped, ransom is paid and the victim is released. C) The victim is not kidnapped.

Does this game have any Nash equilibria that are not subgame perfect? A) Yes, there is at least one such Nash equilibrium in which the victim is not kidnapped. B) No, every Nash equilibrium of this game is subgame perfect.

In the subgame perfect Nash equilibrium A) The victim is kidnapped, no ransom is paid and the victim is killed. B) The victim is kidnapped, ransom is paid and the victim is released. C) The victim is not kidnapped.

Twice Repeated Prisoners’ Dilemma Two players play two rounds of Prisoners’ dilemma. Before second round, each knows what other did on the first round. Payoff is the sum of earnings on the two rounds.

Single round payoffs Player 2 Cooperate P L A Cooperate y E R 1 Defect 10, 10 0, 11 11, 0 1, 1

Two-Stage Prisoners’ Dilemma Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C 20 20 Playe Pl. 2 r 1 D 10 21 D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D Defect C D C Player 1 D C Pl 2 D C D Pl 2 D C 21 11 10 0 11 1 21 11 22 12 10 11 21 22 11 12 10 11 0 1 D C 11 2 12 11 12 1 D 2 2

Two-Stage Prisoners’ Dilemma Working back Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C 20 20 Playe Pl. 2 r 1 D 10 21 D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D Defect C D C Player 1 D C Pl 2 D C D Pl 2 D C 21 11 10 0 11 1 21 11 22 12 10 11 21 22 11 12 10 11 0 1 D C 11 2 12 11 12 1 D 2 2

Two-Stage Prisoners’ Dilemma Working back further Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C 20 20 Playe Pl. 2 r 1 D 10 21 D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D Defect C D C Player 1 D C Pl 2 D C D Pl 2 D C 21 11 10 0 11 1 21 11 22 12 10 11 21 22 11 12 10 11 0 1 D C 11 2 12 11 12 1 D 2 2

Two-Stage Prisoners’ Dilemma Working back further Player 1 Cooperate Defect Player 2 Cooperate Player 1 C C 20 20 Playe Pl. 2 r 1 D 10 21 D C Cooperate Defect Player 1 D C Player 1 C Pl 2 D C D Defect C D C Player 1 D C Pl 2 D C D Pl 2 D C 21 11 10 0 11 1 21 11 22 12 10 11 21 22 11 12 10 11 0 1 D C 11 2 12 11 12 1 D 2 2

Longer Game • What is the subgame perfect outcome if Prisoners’ dilemma is repeated 100 times? How would you play in such a game?