# Game Theory - Centipede Game

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.

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.

• 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. Nov 22, 2012 at 0:29