Consider a sequential game which is known as the Centipede Game. In this game, each of two players chooses between “Left” and “Right” each time he or she gets a turn. The game does not, however, automatically proceed to the next stage unless players choose to go “Right” rather than “Left”.
A: Player 1 begins — and if he plays “Left”, the game ends with payoff of (1,0) (where here, and throughout this exercise, the first payoff refers to player 1 and the second to player 2). If however, he plays “Right”, the game continues and it’s player 2’s turn. If player 2 then plays “Left”, the game once again ends, this time with payoffs (0,2), but if she plays “Right”, the game continues and player 1 gets another turn. Once again, the game ends if player 1 decides to play “Left”— this time with payoffs of (3,1), but if he plays “Right” the game continues and it’s once again player 2’s turn. Now the game ends regardless of whether player 2 plays “Left” or “Right”, but payoffs are (2,4) if she plays “Left” and (3,3) if she plays “Right”.
(a) Draw out the game tree for this game. What is the sub game perfect Nash Equilibrium of this game.
(b) Write down the 4 by 4 pay off matrix for this game. What are the pure strategy Nash Equilibria in this game? Is the sub game perfect Nash Equilibrium you derived in (a) among these?
(c) Why are the other Nash Equilibria in the game not sub game perfect?
(d) Suppose you changed the (2,4) payoff pair to (2,3). Do we now have more than 1 sub game perfect Nash Equilibrium?
(e) How does your answer to (b) change?
(f) Consider again the original game but suppose I came as an outsider and offered to change the payoff pairs in the final stage from(2,4) and (3,3) to (2,2) and (4,4). How much would each of the two players be willing to pay me to change the game in this way (assuming we know that players always play sub game perfect equilibria)?
B: Consider the original centipede game described in part A. Suppose that, prior to the game being played, Nature moves and assigns a type to player 2, with type 1 being assigned with rob ability ρ and type 2with probability (1−ρ). Throughout, type 1 is a rational player who understands sub game perfection.
(a) Suppose type 2 is a super-naive player that simply always goes “Right” whenever given a chance. For what values of ρ will player 1 go “Right” in the first stage?
(b) Suppose instead that type 2 always goes “Right” the first time and “Left” the second time. How does your answer change?
(c) We have not explicitly mentioned this in the chapter — but game theorists often assume that payoffs are given in utility terms, with utility measured by a function u that allows gambles to be represented by an expected utility function. Within the context of this exercise, can you see why?
(d) Suppose the payoffs in the centipede game are in dollar terms, not in utility terms. What do your answers to (a) and (b) assume about the level of risk aversion of player 1?