Question

Consider the following advertising game between two restaurants in a small town in Raub.

	Do Not advertise	Advertise
Do Not advertise	3,3	0,4
Advertise	4,0	1,1

a) Find the Nash equilibrium if the restaurants need to make their decision
simultaneously.

b) If the game is played repeatedly for an infinite number of times, calculate the discount
factor that will make cooperation sustainable if the restaurants are using grim trigger
strategy.

Answer 1

a)

	Do Not advertise	Advertise
Do Not advertise	(3,3)	(0,4)#
Advertise	(4,0)*	(1,1)#*

- If Restaurant 1 Chooses to play "Do not advertise", then the best response of Restaurant 2 is to choose "Advertise" (marked by #)

- If Restaurant 1 Chooses to play "advertise", then the best response of Restaurant 2 is to choose "Advertise" (marked by #)

Similarly,

- If Restaurant 2 Chooses to play "Do not advertise", then the best response of Restaurant 1 is to choose "Advertise" (marked by *)

- If Restaurant 2 Chooses to play "advertise", then the best response of Restaurant 1 is to choose "Advertise" (marked by *)

Hence, the Nash Equilibrium of the game is : <Advertise, Advertise>

b)

Both Plays the Grimm Trigger Strategy:

Let be the discount factor

if Both restaurants cooperate, then following will be series of pay-off:

Vc = 3 + 3 + 32 + 33 + ...........

Vc = 3 / (1 - )

Now, if one restaurant deviates then the restaurant will first get 4 and then play <Advertise, Advertise> and then gets 1 going forward, so following is the series of pay-off in the deviation case:

Vd = 4 + + 2 + 3 + ......

Vd = 4 + / (1-)

Both Restaurants will cooperate iff:

Vc Vd

3 / (1 - ) 4 + / (1-)

3 4 - 4 +

3 1

0.33

Discount Factor of 0.33 will make the cooperation sustainable

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