Question

7. Untied and Air "R" Us are the only two airlines operating flights between Collegeville and Bigtown. That is, they operate in a duopoly. Each airline can charge either a high price or a low price for a ticket. The accompanying matrix shows their payoffs, in profits per seat (in dollars), for any choice that the two airlines can make. Air "R" Us Low price High price $20 profit s0 profit Low price $20 profit $50 profit $50 profit $40 profit High price s0 profit $40 profit Suppose the two airlines play a one-shot game-that is, they interact only once and never again. What will be the Nash (noncooperative) equilibrium in this one- shot game? b. Now suppose the two airlines play this game twice. And suppose each airline can play one of two strategies: it can play either always charge the low price or tit for tat-that is, it starts off charging the high price in the first period, and then in the second period it does whatever the other airline did in the previous period. Write down the payoffs to Untied from the following four possibilities: i. Untied plays always charge the low price when Air "R" Us also plays always charge the low price. ii. Untied plays always charge the low price when Air "R" Us plays tit for tat. iii. Untied plays tit for tat when Air "R" Us plays always charge the low price. a. iv. Untied plays tit for tat when Air "R" Us also plays tit for tat. Untied

Answer 1

a).

Consider the given problem here there are two players “P1=Untied” and P2= Air R Us”. So, of then have two possible strategies “LP=low price” and “HP=high price”. Now, if “P1” charge LP then the optimum price for “P2” is “LP” because the “P2” gets “$20” as profit which is more than “$0”. Similarly, if “P1” charge HP then the optimum price for “P2” is “LP” because the “P2” gets “$50” as profit which is more than “$40”. So, in both the cases “P2” choose “LP”, => “LP” is the dominant strategy for “P2”. Similarly with the same logic we can prove that “LP” is the dominant strategy for “P1”, => the only NE is “LP, LP”, => both the player will end uo by choosing “LP” and will get “$20” as a profit.

b).

i).

Now, let’s assume that the same game will be played twice and “Untied=P1” always charge LP when “P2 = Air R Us” also play LP, => in both the round both the player will choose “LP” and will get “$20” as a profit, => the payoff are given by “20+20=$40”. So, at the end of 2 round both the player will get “$40” as a profit.

ii).

Now, “P1” always choose “LP” but “P2” choose “tit for tat”, => in the 1^st round “P1” choose “LP” and “P2” choose “HP”, => “P1” gets “$50” and “P2” gets “$0” as a profit. Now, in the 2^nd round “P1” choose “LP” and “P2” choose “LP”, => both the players gets “$20” as a profit. So, the payoff under this case is “50+20=$70” for “P1” and “$20” for “P2”.

iii).

Now, “P1” always choose “tit for tat” but “P2” choose “LP”, => in the 1^st round “P1” choose “HP” and “P2” choose “LP”, => “P1” gets “$0” and “P2” gets “$50” as a profit. Now, in the 2^nd round “P1” choose “LP” and “P2” choose “LP”, => both the players gets “$20” as a profit. So, the payoff under this case is “0+20=$20” for “P1” and “$50+$20=$70” for “P2”.

iv).

Now, if both the players choose “tit for tat”, => in the 1^st round both choose “HP”, => both gets “$40” as a profit. Now, in the 2^nd round both further choose “HP”, => both the player gets “$40” as a profit in the 2^nd round. So, the payoff under this case is “40+40=$80”, => both the player gets “$80” as a profit at the end of the 2^nd round.