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I am working on a game theory question. The question is as follows:

Characterize the unique subgame-perfect equilibrium in the following game. Can you find any Nash equilibrium which is not subgame perfect? Explain.

gametheoryquestiondiagram

First, I will explain how I think this game is to be interpreted. There are $N$ players in this game. Each player is presented with a choice: continue right ($R$) or defect ($D$). A player $p$, where $1 \leq p \leq (N-1)$, can either defect, receive $\frac{1}{p}$, and award $\frac{1}{p}$ to every other player, or continue to play by choosing $R$. Player $N$ can defect and receive $\frac{1}{N}$ (along with all other players), or choose $R$, receive $2$, and award all other players with $2$.

I will address the first portion of the question. I believe I approach this part by solving for the outcome using backwards induction. So, player $N$ decides between $R$ or $D$. He sees that the greater payoff is choosing $R$, as $2 \geq 1/N$. Player $N-1$ can now either choose $R$ or $D$. But, he knows player $N$ will choose $R$, and so he chooses $R$, which will result in a payoff for him of $2$, and $2 \geq \frac{1}{N-1}$. This continues until the first player.

I think the subgame perfect equilibrium is a move that every player will make given the game. So the subgame perfect equilibrium is to choose $R$. Does what I have make sense?

Now, I don't have the answer for the second part of the question. Is there any Nash equilibrium that is not subgame-perfect? I don't think there is any other way to play the game if the player's are rational. Does this mean every Nash equilibrium is subgame-perfect?

Thanks for any help.

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up vote 1 down vote accepted

You are right about the unique subgame-perfect equilibrium of the game. But there are other Nash equilibria. It is easily checked that a profile in which both 1 and 2 play down is a Nash equilibrium, no matter what the other players do.

Btw: The original centipede game, introduced by Robert Rosenthal in Games of Perfect Information, Predatory Pricing, and the Chain Store, is a two-player game and has a very different structure.

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Thank you for your reply. How is it checked that a profile in which both 1 and 2 play down is a Nash equilibrium? It seems this would only be the choice if they thought the next player would choose down. –  Jerzy_Gregor Nov 22 '12 at 0:14
    
If player 1 plays down, the outcome is uneffected by what other players do. Everything they do is a best response. So we only have to check that player 1 plays a best response in such a profile. Now, if player 2 plays down, the payoff of player 1 would be reduced from to $1$ to $1/2$ when playing $R$. So the unique best response of player 1 to such a profile is to play down. –  Michael Greinecker Nov 22 '12 at 0:29
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