Week6 Game Theory

Simultanous Games (Normal Form)

Dominant Strategy- optimal to choose that strategy no matter what opponent does.
Player A has a dominant stategy of up. No matter what PB does- A will always go up.

Nash Equilibrium- what's the best stategy an opponent can do, fixing an opponent on a stategy.
Player A fixed at up- B goes left
Player A fixed at down- B goes right
Player B fixed at left- A goes up.
Player B fixed at right- B goes up.
Nash Equilibrium is up and left (10,20). -best response analysis.

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Simultanoues - Infinitely Repeated Games

PV

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Trigger Strategies

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Nash Equilibrium- 0,0.- shows thats if they collude and agree to both go high they can both achieve and higher payoff.
Grim Trigger- when firms collude and agree that if one cheats they will be punished forever after.

A and B agree to both charge high.
A cheats and earns 50. B will charge low every period after. A gets 0 every period after cheating.
If A didnt cheat, it would get 10 in every period.
PVcoop>PVcheat- no incentive to cheat

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Simultanous - Finitely Repeated Games

Firms know when the game is going to end.

  • Firms know the game will be repeated twice.
  • Players cannot be punished in the 3rd period(there isnt one)
  • So firms will go low in the 2nd period.
  • Players cannot be punished in the 2nd period(they're already going low)

Collusion cant work when players know the game is going to end.

Firms charge the one-shot Nash Equlibrium

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Simultanous - Finitely Repeated Games

Firms do not know when the game is going to end

  • The probability that the game will end after a given play is
  • So the probability that the game doesnt end is

If A cheats it earns 50. If it doesnt cheat it gets 10.
Probability that the game wont end is
A earns 10 again if the game doesnt end
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No incentive to cheat if

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        -Game will only be played once. Choose One-Shot Nash Equilibrium.

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Sequentially Played Games (Extensive Form)

Player 1 moves first then player 2 decides.
Role-back equilibrium/Sub-game perfect Nash Equilibrium.
       -start from the end of the game
       -work backwards eliminating actions.

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Example
-at 2.2- P2 goes S- eliminate R
-at 2.1- P2 goes P- eliminate Q
-at 1.1- P1 chooses between A=10 and B=6- eliminate B.
-Sub-game perfect Nash equilibrium= AP= (10,15)

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