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Suppose that two players play each other for two periods. In the first period they play the first game below, and in the second period they play the second game below. There is no discounting between periods. Players observe the action their opponent took in the first period before choosing their second period actions. \begin{bmatrix}(2,2)&(-10,x)\\(y,0)&(0,0)\end{bmatrix} \begin{bmatrix}(8,4)&(0,0)\\(0,0)&(4,8)\end{bmatrix}

(a) For x ≤ 2 and y ≤ 6, find a subgame perfect equilibrium in which player 1 receives a payoff of 10. (b) For x = 5 and y = 3 find a subgame perfect equilibrium in which player 2 receives a payoff of 10. (c) For x = y = 4, show that there is no subgame perfect equilibrium in which (U,L) is played in the first period.

I'm totally lost as how to find subgame perfect equilibria. Aren't these games simultaneous games? Why would the players consider the choices made in the first game to play the second game?

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I am not sure that I am able to fully answer your problem, but I can give you a short answer to your questions. "Aren't these games simultaneous games? Why would the players consider the choices made in the first game to play the second game?"

I would refer to collusion and repeated games. Every stage game is simultaneous, but the fact that they are played sequentially makes them a super-game. In super-games the outcome of the first stage game influences the second, because in this way every player can introduce a punishment at the second stage if the other player did not cooperate. So pretty much like in collusion games, what matters are the threats one player can pose to the other through the second stage to obtain collaboration at the first stage.

Having said that, I'd suggest some easy readings: Introduction: Cabral "introduction to industrial organization" chapter on game theory and collusion" Game theory: Chapter 14 and 15 Osborne "Introduction to game theory"

Regards.

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  • $\begingroup$ Exactly! Strategies in the second stage game are functions of the actions in the previous game. OP has to think of strategies more broadly, i.e. less like "Left-Right / Top-Bottom", and more like "if Left in previous period then Top in this period" etc. $\endgroup$ Mar 20 at 12:20

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