Subgames and Credible Threats

Nuclear threat USSR Invade Don’t Invade Hungary US 1 Give in Bomb USSR 1 Give in Bomb USSR 5 -10 -5

Nuclear threat (strategic form) Soviet Union Invade Don’t Invade -5,-10 1,0 Give in if USSR Invades Bomb if USSR Invades 0, 5 1,0 United States How many pure strategy Nash equilibria are there? A) 1 B) 2 C) 3 D) 4

Are all Nash Equilibria Plausible? What supports the no-invasion equilibrium? Is the threat to bomb Russia credible? What would happen in the game starting from the information set where Russia has invaded Hungary? What if the U.S. had installed a Doomsday machine, a la Dr. Strangelove?

Similar structure, but less terrifying: The entry game Challenger Challenge Stay out Incumbent 1 Give in Fight 1 -1 -1

Alice and Bob Revisited: (Bob moves first) Go to A Go to B Alice Alice Go to B Go to A Go to A Go to B 2 3 1 3 2

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

Now for some theory… Reinhard Selten John Harsanyi John Nash Thomas Schelling

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

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

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

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

How many subgame perfect N.E. does this game have? There is only one and in that equilibrium they both go to movie A. There is only one and in that equilbrium they both go to movie B. There are two. In one they go to movie A and in the other tney go to movie B. There is only one and in that equilibrium Bob goes to B and Alice goes to A.

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.

A Kidnapping Game Kidnapper Kidnap Don’t Kidnap Relative 3 5 Pay ransom Kidnapper Don’t pay Kidnapper Kill Release Kill Release 4 3 22 5 1 1 4

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

Another Kidnapping Game Kidnapper Kidnap Don’t Kidnap Relative 3 5 Pay ransom Kidnapper Don’t pay Kidnapper Kill Release Kill Release 5 3 22 4 1 1 4

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

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

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 10, 10 0, 11 11, 0 1, 1 Player 2 Cooperate Defect

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

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

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

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

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