Answer to Question #192491 in Microeconomics for Narendra

Question

Consider the two period Repeated Prisoner's Dilemma Game where each player is interested in the SUM of the payoffs she gets in each period. Players see the outcome after the play in each period. (The period payoffs are 10,5,1,0.)

(i) Write out this game in its strategic form.

(ii) Find all Nash equilibria and all Subgame Perfect Nash Equilibria.

Answer 1

(i) Strategies in the two-period game:

Player 1:

1. Confess Confess

2. Confess Not Confess

3. Not Confess Confess

4. Not Confess Not Confess

Player 2:

1. Confess Confess

2. Confess Not Confess

3. Not Confess Confess 4.

Not Confess Not Confess

In a prisoner's dilemma game, we know that the two-player have to Confess as the dominant strategy, that is no matter what the other player does, the best response of them either the player is to Confess.

Solving Backwards:

At T-2, the player is bound to return to playing the Nash Equilibrium consisting of the Dominant strategy for the two players. Thus (Confess, Confess) is played in the second period.

At T = 1,

the player would want to play (Not Confess, Not Confess), However, while the other player plays Not Confess the player 1 will have a tendency to deviate from playing its dominant strategy that also pays him better than the (Not Confess, Not Confess) outcome. Given the symmetry of the game, both players deviate and end up playing the (Confess, Confess}

Thus the only equilibrium is: (ConfessConfess, ConfessConfess)

ii)

In a three-period game there will be (n)^3 where n is the number of possible actions that the player can take:

Here n is 2: Confess(C) and Not Confess(NC). So total strategies of player 1: = (2)^3 = 8

1. CCC

2. CCNC

3. CNCC

4. CNCNC

5. NCCC

6. NCCNC

7. NCNCC

8. NCNCNC

Likewise for Player2: 8 strategies as above.

Solving Backwards:

Likewise for Player2: 8 strategies as above.

At T = N,

The players return to playing the Nash Equilibrium or {C, C).

At T = n-1

The players may want to coordinate but because of the dominant strategy of each player to playing Confess(c), each player ends up playing the Nash Equilibrium in the n-1 period as well.

Likewise for T = n-2, T = n-3... T=1. The players end up playing the Nash equilibrium: (Confess, Confess).

Thus the SPNE of the N-period game is (Confess.... Confess...)