Consider a two-period repeated game. The stage game is the following Find all pu
ID: 1212446 • Letter: C
Question
Consider a two-period repeated game. The stage game is the following Find all pure-strategy Nash equilibria of the stage gam in other words when the above game is played only once. Now consider the two-period repeated game. Suppose the discount factor delta = 1 for both player 2 players. Find a pair of subgame perfect equilibrium strategies in which equilibrium. Note plays R in the first period. Explain why the pair of strategies for each that the game is not symmetric so you need to separately player.Explanation / Answer
A) The first step in this game is to apply IDSDS. This will indicate that Player 1 will never choose ‘B’ strategy. Hence it is a strictly dominated strategy and can be deleted. Similarly, Player 2 will never choose ‘R’ strategy. Hence it is a strictly dominated strategy and can be deleted
Player 2
Player 1
L
C
T
4, 3
0, 1
M
1, 0
1, 2
The game is now reduced to a 4×4 matrix that can be solved for pure strategy Nash equilibrium. Note that there are two Nash equilibria as determined by the circles through best reply methods. These are the stage equilibria. Hence there are two Nash equilibria, (T, L) and (M, C).
2) With the discounting factor being 1, players are perfectly patient between both time periods. Therefore, the total payoffs for the entire game are simply the sum of the payoffs in each time period.
Now suppose the stage game is played by two players in periods t = 1 and t = 2. In second period, players can remeber the historical Nash equibria and will behave accordingly. If we focus on pure strategies, this repeated game has 16 total possible outcomes, 4 for each in each period. Make only two combinations of period that are Nash equlibrium so that the players might pick one of them in t = 2, as a part of SPNE
If (T, L) was played in t = 1, then the matrix become this t=2
Player 2
Player 1
L
C
T
8, 6
4, 4
M
5, 3
5, 5
If (M, C) was played in t = 1, then the matrix become this t=2
Player 2
Player 1
L
C
T
5, 5
1, 3
M
2, 2
2, 4
Note that the Subgame has the same strategy as the players have in the game itself. In any SPNE, players MUST play a Nash equilibrium in the subgame played in t=2. Here the subgame has the same Nash equilibria as the stage game. Playing a Nash equilibrium of the stage game in period t=1 and then a Nash equilibrium of the stage game in period t=2 constitutes an SPNE.
Thus, having either (T,L) or (M,C) in period t=1 followed by either (T,L) or (M,C) in period t=2 can be supported as the outcome of SPNE
Player 2
Player 1
L
C
T
4, 3
0, 1
M
1, 0
1, 2