Question

3. There are two firms, 1 and 2, who simultaneously announce prices from {1, 2, 3, 4} and 12 buyers, each of whom is willing to pay at most $4 for one item and nothing for any additional items. Each unit costs a firm $1 to make. Three of the buyers are loyal to firm 1, that is they will buy firm l's product if its price is less than or equal to 4. Likewise 3 buyers are loyal to firm 2. The other six will buy from whichever firm charges the lowest price. Find all Nash equilibria in the game.

Answer 1

firm 2 Amy We can write this game in mabix form 1 2 3 4 110,00, 310, 60, 91 23,0 6,6 9,6 9,9 Firm 1 4 9,0 9,9 1,18 (15,18 when bath changes $2, then and each will get o as pay off as cast of producten of 1 unit is = @ $1 whenever firm charges 1 it cwill get o pay off when, (firm 1=2, firm 2 = 1) then the & T = (2-1)3 = 3 when, (f1 = 3, f2= 1), ^, = (3-1)3 = 6 - (F124, F221), a, = (4-1)3 = 9 when both charges 2 then prafit of Tata (2-1) 6 = 6 when both changes (3-1) 6 = = 12 ni = 72 = 12 when both charges 4 hahz a/4-1) 6 =18 wher, f1= 3, f2= 2 then n = (3-1)3 = 6 1 T2 = (2-1) 9 = 9 when, fi=4 i D = (4-1) 3 = 9 1 1 2 = (2-1) 9 = 9 ehen, f1=2, f2=3 T = (2-1) 9 = 9 1 2 = (3-1)3 = 6 = 4,82= 2 2 = (2-1) 9 = 9

wher, flau, f223 T = (4-1)3 =9 | T2 = (3-1) 9 2 18 No firm cuill change & 1 as it is strictly dominated shakegy. and $2 is weakly dominated strategy by 83. & ($3,43) is an nash equilibrium. as 1279, o and is and no one has incentina to deriate from this situaren 144, 44) is a nash equilibrium. as 1879 and o and no one has incentive to deviate from this position,