Subgames and Credible Threats (with perfect information) Econ 171.

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Presentation transcript:

Subgames and Credible Threats (with perfect information) Econ 171

Alice and Bob Bob Go to AGo to B Go to A Alice Go to B Go to A Go to B

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 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 A2,3 0,00,1 Movie B3,21,13,21,0 Alice Bob How many Nash equilibria are there for this game? A)1 B)2 C)3 D)4

The Entry Game Challenger Stay out 0101 Challenge Incumbent Give in Fight 1010

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) -1,-1 0,0 0,1 0,0 Challenge Do not Challenge Challenger Incumbent Give in 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 Bob Go to AGo to B Go to A Alice Go to B Go to A Go to B

Two subgames Bob went ABob went B Alice Go to AGo to B Go to A Go to B

Alice and Bob (backward induction) Bob Go to AGo to B Go to A Alice Go to B Go to A Go to B

Alice and Bob Subgame perfect N.E. Bob Go to AGo to B Go to A Alice Go to B Go to A Go to B

Strategic Form 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 A2,3 0,00,1 Movie B3,21,13,21,0 Alice Bob

A Kidnapping Game Kidnapper Don’t Kidnap 3535 Kidnap Relative Pay ransom Kidnapper Don’t pay Kidnapper 4343 KillRelease KillRelease 1414

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 3535 Kidnap Relative Pay ransom Kidnapper Don’t pay Kidnapper 5353 KillRelease KillRelease 1414

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

Two-Stage Prisoners’ Dilemma Player 1 CooperateDefect Player 2 Cooperate Defect Player 1 C C C C C C D DD D C C CD Pl. 2 Pl 2 20 D D C D C D C D D D

Two-Stage Prisoners’ Dilemma Working back Player 1 CooperateDefect Player 2 Cooperate Defect Player 1 C C C C C C D DD D C C CD Pl. 2 Pl 2 20 D D C D C D C D D D

Two-Stage Prisoners’ Dilemma Working back further Player 1 CooperateDefect Player 2 Cooperate Defect Player 1 C C C C C C D DD D C C CD Pl. 2 Pl 2 20 D D C D C D C D D D

Two-Stage Prisoners’ Dilemma Working back further Player 1 CooperateDefect Player 2 Cooperate Defect Player 1 C C C C C C D DD D C C CD Pl. 2 Pl 2 20 D D C D C D C D D D

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