Question

Consider the following stage game

Strategy	L	R
T	(3,2)	(1,5)
B	(2,1)	(1,1)

It is repeated over two periods. The payoffs of the players over the two periods is the sum of the payoffs over the two periods. Is there a subgame perfect equilibrium of the repeated game in which (T,L) is played in period 1? If so, describe the equilibrium. If not, explain why not.

Answer 1

Given,

  

		PLAYER N
	Strategy	L	R
PLAYER M	T	(3,2)	(1,5)
	B	(2,1)	(1,1)

Player M’s payoff is shown first and Player N's payoff is shown second.

For Example - If (Player M) plays T and (Player N) plays L then Player M’s payoff is 3 and Player N’s payoff is 2.

And if Player M plays B and Player N plays L then Player M’s payoff is 2 and Player N’s payoff is 1.

A play of the game is a pair such as (T,L) where the 1st element is the strategy chosen by Player M and the 2nd is the strategy chosen by Player N.

What plays are we likely to see for this game? Is (T,L) a likely play?

- If Player M plays T then Player N’s best reply is L since this improves M’s payoff from 3 to 2.

So (T,L) is a likely play.

Is (B,R) a likely play?

- If Player N plays R then Player M’s best reply is B. If Player M plays B then N’s best reply is R. So (D,R) is NOT a likely play.

Is (B,L) a likely play?

- If M plays B then N’s best reply is R. So (B,L) is a likely play.

Is (T,R) a likely play?

- If Player M plays T then Player N’s best reply is R. If Player N plays R then Player M’s best reply is T. So (U,L) is NOT a likely play.

A play of the game where each strategy is a best reply to the other is a Nash equilibrium. Our example has two Nash equilibria are (T,L) and (B,L).

We found that (T,L) and (B,L) are both Nash equilibria for the game. But which will we see? Notice that (T,L) is preferred to (B,L) by both Players. Must we then see (T,L) only?

To see if Pareto-preferred outcomes must be what we see in the play of a game, consider a famous second example of a two-player game called the Prisoner’s Dilemma.

		CLYDE
		S	C
BONIEE	S	(-5,-5)	(-30,-1)
	C	(-1,-30)	(-10,10)

What plays are we likely to see for this game?

If Bonnie plays Silence then Clyde’s best reply is Confess. If Bonnie plays Confess then Clyde’s best reply is Confess.

So no matter what Bonnie plays, Clyde’s best reply is always Confess.Confess is a dominant strategy for Clyde.

Similarly, No matter what Clyde plays, Bonnie’s best reply is always Confess.Confess is a dominant strategy for Bonnie also.

So the only Nash equilibrium for this game is (C,C), even though (S,S) gives both Bonnie and Clyde better payoffs.The only Nash equilibrium is inefficient.

As we saw In both, the players chose their strategies simultaneously. Such games are simultaneous play games.But there are games in which one player plays before another player. Such games are sequential play games.The player who plays first is the leader.The player who plays second is the follower

Sometimes a game has more than one Nash equilibrium and it is hard to say which is more likely to occur. When such a game is sequential it is sometimes possible to argue that one of the Nash equilibria is more likely to occur than the other.

(T,L) and (B,L) are both Nash equilibria when this game is played simultaneously and we have no way of deciding which equilibrium is more likely to occuR.