Firm a0 Advertiso (A) Do Not Advertise (DNA $3.150 S4,200 Advertise (A) 100 Firm
ID: 2443765 • Letter: F
Question
Firm a0 Advertiso (A) Do Not Advertise (DNA $3.150 S4,200 Advertise (A) 100 Firm () $2.100 $3,650 Do Not Advertise (DNA) 3,650 Figure 1: Consider the game where two firms, X and Y, are producing under a "duopoly" market (a two-firm oligopoly market). Each firm has two options (strategies), either to spend money on an advertisement (Strategy A), or not (Strategy DNA). The expected payoffs in term of dollar profits are shown in Figure 1. a. Does firm X have a dominant strategy in this game? b. Ifexists, what is this dominant strategy for firm Y in this game? If there is no dominant strategy for firm Y then just type "N/A" c. Is "A-A" a Nash equilibrium for this game? d. Is "DNA-DNA" a Nash equilibrium for this game?Explanation / Answer
The game is a typical Prisoner’s Dilemma game, where producing the Cournot equilibrium output, i.e., choosing "Do Not Advertise", is the dominant strategy for each firm.
If Firm X chooses "A" , the best response for Firm Y is to choose "DNA" because 4200 > 3150 . If Firm X chooses "DNA" , the best response for Firm Y is to choose "DNA" because 3650>2100.
Thus, DNA is the dominant strategy for Firm Y. Since the game is symmetric, we can argue that DNA is also the dominant strategy for Firm X.
This game has a unique Nash equilibrium where each firm plays its dominant strategy. That is, (DNA,DNA) is the unique Nash equilibrium of this game. Note that this is a pure strategy Nash equilibrium.
In this game there is no mixed strategy Nash equilibrium because each firm has a dominant strategy which is to produce the Cournot equilibrium output. Given that firm X is choosing DNA with a 100 percent probability, the best response of firm Y is to choose DNA with a 100 percent probability rather than to choose a randomized strategy over "A" and "DNA" , vice versa.
(a) Yes
(b) Firm Y has also dominant strategy of choosing DNA(explained above).
(c)
Nash equilibrium-a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged.
So A-A is not a nash equilibrium
(d) yes,This game has a unique Nash equilibrium where each firm plays its dominant strategy. That is, (DNA,DNA) is the unique Nash equilibrium of this game. Note that this is a pure strategy Nash equilibrium.
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