Question

You are matched with someone and each of you chooses R or B. The payoffs are determined as follows. If you each play your red card, you will each earn $10.If you each play the black card, you will each earn $20. If you play your black card and the other person plays red, then you earn $0 and the other person earns $30. If you play red and the other person plays black, then you earn $30and the other person earns nothing.

(a)Express the game in matrix form. [5pts]

(b)Find all the Nash equilibria (possibly involving mixed strategies). [5pts]

(c)Derive the set of the subgame perfect equilibria when you play this game T times with the same student. [5pts]

(d)For what values of the discount factor does the combination of the follow-ing trigger strategies form a subgame-perfect Nash equilibrium when thetwo students play the game repeatedly infinitely many times? [10pts]

a_it= (B if t= 1 or both players have observed (a_1t0, a_2t0) = (B,B) for all t' ≤ t−1; R otherwise), where a_it is player i’s action at the t ^th stage.

Answer 1

Let the red card played be represented by R and the black card be represented by B. There are two players in the game. The first player is 'You' and the second player is 'Other person '.The game matrix for this game is shown below :-

The image clearly shows the strategies played by the two players. To determine the Nash equilibrium, let us consider that the 'other person ' plays R then the dominant strategy for 'You ' is to play R because it gives a pay off of 10 whereas playing B gives a payoff of 0. Considering the other case, if other person plays B then the dominant strategy for You is play R because it gives a payoff of 30 whereas playing B would give a payoff of 20.

Now, similarly let us consider that You plays R then Other person would play R and if 'You' plays B then also the other person would play R. Therefore, irrespective of each others strategy, the dominant strategy for both players is to play R. Hence, the Nash equilibrium is (R, R) in which both the players get a pay off of (10,10).

Now the subgame perfect nash equilibrium can be determined through the diagram shown below assuming that 'You' moves first.

Through backward induction, it can be said that (R, R) is the sub game perfect nash equilibrium.

If this game would be played T times then the outcome would be same because if a game with unique nash equilibrium is played in a finite number many times then there we will always get the same subgame perfect equilibrium outcome as both players will play their dominant strategy R.

The discount values for which these strategies will converge to sub game perfect nash equilibrium can be determined through this formula-

This sum will be equal to 10

Therefore, the discount factor will be equal to 10/11.