Question

The following payoff matrix represents a single-period, simultaneous move game to be played by two firms. Show all the best responses for each player by placing checkmarks next to each payoff that reflects a best response choice. Does Firm A have a dominant strategy and if so, what is it? Does Firm B have a dominant strategy and if so, what is it? Does a Nash Equilibrium (or multiple NE) exist for this game and if so, what is it (or are they)?

Firm A	Firm B
		Advertise Heavily	Advertise Lightly	Do not Advertise
	Advertise Heavily	75, 125	85, 150	200, 100
	Advertise Lightly	100, 175	75, 200	300, 150
	Do not Advertise	150, 300	25, 400	200, 375

Answer 1

Dominant strategies are considered as better than other strategies, no matter what other players might do as these strategies gives the player higher payoff irrespective of what the other player chooses.

Firm A does not have any dominant strategy as no single strategy always gives firm A always a better payoff irrespective of what firm B chooses

Firm B does not have any dominant strategy which is to advertise lightly. This is because irrespective of what Firn A chooses the payoff of firm B is maximized when Firm B chooses to advertise lighlty it can be seen from bullet points 4, 5, 6

The following stars show the best responses.

advertises heavily=AH

do not advertise=NA

Advertise lightly=AL

Checking for nash equilibrium

1. If firm B advertises heavily(AH) then the best response of Firm A is to not advertise(NA) as 150(NA)>100(AL)>75(AH) i.e. payoff from NA > payoff from AL and also from AH. Best response of firm A is NA if firm 1 advertises heavily.
2. If firm B chooses AL then the best response of firm A is AH as payoff of 85>75>25 ( i.e. payoff from advestising heavily is greater than that from AL as well NA.
3. IF firm B chooses NA then the best response of firm 1 is to choose to AL as the payoff from NA is greater than that from AH and AN 300>200=200
4. IF firm A chooses to AH the best response of Firm B is to AL as 150>125>100 i.e. payoff from playing AL > payoff from advertising heavily>payoff from not advertising
5. If firm A chooses to AL, best response of firm B is to AL, as 200>175>150 (i.e. payoff from AL>Payoff from AH>payoff from not advertisng
6. If firm A chooses to not advertise then, the best response of firm B is to choose AL as payoffs from AL>payoff from NA> payoff from AH (400>375>300)

Note that 4 and 2 are mutual best responses and therefore, there exists a single nash equilibrium (Advertise heavily,Advertise lightly).

If firm B chooses AL then the best response of firm A is AH and if firm A chooses to AH the best response of Firm B is to AL. Therefore, it is a mutual best response.