Question

1. Consider the following normal form game: 112 L CR T 10 102 12 0 13 M 12 25 5 0 0 B|13 010 011 a) (Level A) First suppose this game is played only once. What are the pure strategy Nash equilibria? (b) (Level B) Now suppose this game is played twice. Players observe the actions chosen in the first period prior to the second period. Each player's total payoff is the sum of his/her payoff in the two periods. Consider the following strategy profile Player 1: plays T in period 1 M in period 2 if the action profile played in period 1 is (T, L); and B otherwise Player 2: plays .L in period 1 C in period 2 if the action profile played in period 1 is (T, L); and R otherwise Argue that the above strategy profile constitutes a subgame perfect equilib- rium of the two-period game (c) (Level C) Recall from Lecture 8 that, in a finitely repeated Prisoner's Dilemma, the only subgame perfect equilibrium involves playing the stage game Nash equilibrium at every history. In the above subgame perfect however, the play Date: 24th Jan 2017 in period 1 is not a Nash equilibrium of the stage game. What is the difference between the above game and Prisoner's Dilemma?

Answer 1

Answer :

a).

Consider the given problem here there are two players “P1” and “P2” having three possible strategies. Now, if “P1” will choose “T” then the optimum choice for “P2” is “R” and if “P2” choose “R” then the optimum choice for “P1” is “B”, => “T, R” is not the optimum choice or the solution of the game.

Similarly, if “P1” will choose “M” then the optimum choice for “P2” is “C” and if “P2” choose “C” then the optimum choice for “P1” is “M”, => “M, C” is the NE of the game. Similarly, “B, R” is also another NE of this game. So, there are two pure strategy NE of this game, these are “(M, C)=(5, 5)” and “(B, R)=(1, 1)”.

b).

Now, let’s assume that the game is played twice, => in the 2^nd or last round the outcome will be either “M, C” or “B, R”. So, given the strategy profile given in the above game the payoff matrix will be given by.

112 L 10+5, 10+5 2+1,12+1 0+1, 13+1 15, 15 3, 13 1,14 12+1,2+1 5+1,5+1 0+1, 0+1 -13,36,6 1,1 14,11,12,2 So here there are three possible equilbrium of this game.

So, given the strategy profile if “P1” choose “T” then then the optimum choice is given by “L” and if “P2” chooses “L” then the optimum choice is given by “T”, => “T, L” is the optimum solution of the game, => “T, L” is the SPNE of this game. Similarly there are other two SPNE these are “M, C” and “B, R”.