dynamic games with complete information

dynamic games with complete information

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dynamic games with complete information

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1. dynamic games with complete information

2. SPNEdefinition of subgame perfect Nash equilibrium A subgame perfect Nash equilibrium is a vector of strategies that, when confined to any subgame of the original game, have the players playing a Nash equilibrium within that subgame. In a game of perfect information, the SPNE coincides with the backward induction solution.

3. SPNE: example dynamic game with imperfect information -2,-1 T Coke T Pepsi A -3, 1 E A T 0,-3 Coke O Coke 0, 5 A 1, 2

4. SPNE: example dynamic game with imperfect information Nash equilibria

5. SPNE: example dynamic game with imperfect information q 1-q Post-entry subgame p 1-p

6. SPNE: example dynamic game with imperfect information Nash equilibria of the post-entry subgame: (p=1, q=1) (p=0, q=0) (p=1/3, q=1/2) SPNE: [(Enter, p=0), q=0] [(Out, p=1), q=1] [(Out, p=1/3), q=1/2]

7. SPNE: repeated prisoner’s dilemma -6,-6 c 2 c -3,-9 1 nc c -9,-3 nc -4,-4 nc c c -3,-9 2 c 2 1 0,-12 nc nc c -6,-6 nc c -1,-7 nc 1 c -9,-3 2 c nc -6,-6 c 1 nc c -12,0 nc -7,-1 nc nc c -4,-4 2 c -1,-7 nc c 1 -7,-1 nc -2,-2 nc

8. SPNEinfinitely repeated games • Discount factor δ: • discounting the future (ex: impatience) • uncertainty about the future: positive probability that the game will be repeated in the future • If payoff of the stage game is Π, discounted total payoff is Π+δΠ+ δ2Π+…+ δtΠ+…= Π/(1-δ)

9. SPNEinfinitely repeated games: trigger strategies • the (grim) trigger strategy comprises two parts: a desired behavior on the part of the players and a punishment regime that is triggered whenever either player violates the desired behavior • ex: infinitely repeated prisoner’s dilemma: • «start by playing “don’t confess”; continue playing “don’t confess” if nobody has confessed in the past; if someone has deviated in the past, play “confess” forever»

10. SPNEinfinitely repeated games: trigger strategies • Check for all possible histories: 2 types of subgames • no deviation until then: • Payoff of conforming: -1/(1-δ) • Payoff of deviating: 0+δ.[-3/(1-δ)] • a deviation has occurred in the past: • Payoff of conforming: -3/(1-δ) • Payoff of deviating: -6+δ.[-3/(1-δ)]

11. SPNEinfinitely repeated games: trigger strategies • the (grim) trigger punishment can sustain the “nice” behavior provided the players have a sufficiently high discount factor. • ex: infinitely repeated prisoner’s dilemma: • δ>1/3