Question

Assume the following game is played one time only. Based on the information in the payoff matrix below, PNC Bank and Citizens Bank are considering an implicit collusive agreement on interest rates. Payoffs to the two firms are represented in terms of profits in thousands of dollars.

		Citizens Bank
		Collude: Raise Rates	Defect: Keep Rates Where They Are
PNC	Collude: Raise Rates	(900, 600)	(700, 800)
	Defect: Keep Rates Where They Are	(1100, 300)	(800, 400)

Does PNC have a dominant strategy? What is it?

Does Citizens have a dominant strategy? What is it?

Does the result of your answer change if the game is played an infinite number of times? Why or why not? Properly use game theoretic terminology in your answer.

Answer 1

Solution:-

A dominant strategy is one which is optimal for a player no matter what the other player is playing.

1). In the above game, to find the dominant strategy of PNCs, we observe the best response when Citizens Bank plays either of the two strategies. When Citizens Bank plays Collude, PNC maximizes its payoff by playing strategy Defect (1100>900) if Citizens Bank chooses to play Defect, then PNC maximizes its payoff by choosing to Defect (800>700). Thus, we conclude that PNC chooses to defect no matter what Citizens Bank plays; hence Defect is the dominant strategy of PNC in this game.

2). To find the dominant strategy of Citizens Bank we have to check its best response when PNC plays either of the two strategies.  When PNC plays Collude, Citizens Bank maximizes its payoff by playing strategy Defect (800>600) if PNC plays Defect, then Citizens Bank maximizes its payoff by choosing to Defect (400>300).Hence Defect is the dominant strategy of Citizens Bank in this game.

A defect is the dominant strategy of both the players in this game. Thus, (Defect, Defect) is the Nash equilibrium in this game.

3). The result may change when the game is played an infinite number of times. When a game is played an infinite number of times then one of the players may choose “Collude”, it may cost him in the current game, but believing that if he chooses “Collude,” the other player will be more likely to also choose “Collude” in future plays of the game. If both PNC and Citizens bank chooses to collude, then they can increase their pay-off to (900,600) which is higher than (800,400).