Question

4. Consider two hotels considering locating in either Potsdam or Canton. Payoffs from their location decisions are as below:

		Firm B
		Potsdam	Canton
Firm A	Potsdam	18,12	40,30
	Canton	30,40	20,16

a. Suppose that the hotels make their location decisions independently and simultaneously. List all pure strategy Nash Equilibria, and, if there is one, the mixed strategy equilibrium.

b. Suppose the location decisions are sequential in nature and that each firm’s choice is observable to the other. Draw the extensive form game in the case that Firm A chooses first. What is the subgame perfect equilibrium? Will it be different if Firm B chooses first?

Answer 1

a).

Consider the given problem here there are two hotels “A” and “B” both of them having two strategies “P=Potsdam” and “C=Canton”. Now, if “A” chooses “P”, => the optimum choice of “B” is given by “C”, because “30 > 12”. Similarly, if “B” chooses “C”, => the optimum choice of “A” is given by “P”, because “40 > 20”. So, here “P, C” is the pure strategy NE.

Now, if “A” chooses “C”, => the optimum choice of “B” is given by “P”, because 40 > 16”. Similarly, if “B” chooses “P”, => the optimum choice of “A” is given by “C”, because “30 > 18”. So, here “C, P” is the another pure strategy NE.

Now, let’s assume the mixed strategy of both player is given by, “p, 1-p” and “q, 1-q”, => the expected payoff of “A” of choosing “P” and “C” are given by.

=> E(Pa) = 18*q + 40*(1-q), E(Pa) = 40 - 22*q, and => E(Ca) = 30*q + 20*(1-q), E(Ca) = 20 + 10*q. So, at the optimum “E(Pa” must be equal to “E(Ca)”.

=> E(Pa) = E(Ca), => 40 - 22*q = 20 + 10*q, => 20 = 32*q, => q = 20/32 = 5/8, => q = 5/8, 1-q=3/8, be the mixed strategy of “B”.

Similarly, the expected payoff’s of player B are given by.

=> E(Pb) = 40 - 28*p, and => E(Cb) = 16 + 14*p. So, at the optimum “E(Pb)” must be equal to “E(Cb)”.

=> E(Pb) = E(Cb), => p = 12/21, 1-p =9/21, be the mixed strategy of “A”.

b).

Consider the following fig, where “A” be the 1^st mover.

So, here if “A” will choose “P”, => the “B” will choose “C”, => “A” will get “40” as a payoff. Similarly, if “A” will choose “C”, => the “B” will choose “P”, => “A” will get “30” as a payoff. So, here “A” will choose “P” and “B” will choose “C” as a payoff, => the SPNE is given by “P, C”.

Consider the following fig, where “B” be the 1^st mover.

So, here if “B” will choose “P”, => the “B” will choose “C”, => “B” will get “40” as a payoff. Similarly, if “B” will choose “C”, => the “A” will choose “P”, => “B” will get “30” as a payoff. So, here “B” will choose “P” and “A” will choose “C” as a payoff, => the SPNE is given by “P, C”. So, there are not any difference I these two cases.