Question

Bar A and Bar B are located across the street from each other, and are both deciding how much to charge for a pint of beer: $10 or $4. There are 100 potential customers. Of these, x are loyal customers of Bar A: they will each buy their pint of beer from Bar A, no matter what prices each bar charges. The remaining 100 - x customers will buy their pint from whichever bar charges the lowest price. If both bars charge the same price, each of these 100 – x customers will choose randomly between the two bars so that, on expectation, half of them will end up at each bar. Payoffs for each bar are equal to expected revenues. (a) Fill in the payoffs in the following table. Bar B $10 $4 $10 Bar A (b) True or false: “Both bars charging $10 is a Nash equilibrium for any 100 > x > 0.” Provide a proof of your answer. (c) Fill in the blank, and show your work: “If x < , both bars charging $4 is a Nash equilibrium.”

Answer 1

2 2 100-1 And Revenue = p X q when, Bar I changes $ 10 and Bar B charges $10 then, total no. of customers for Bar A, q= at 1002 100++ and for Bar B, q = -100- when, Bar A chooses $10, and Bar B charges $4 then, q A = n and q B = (1002) so the matrix becomes :- - Bar B 100+% (100-2)(0) 10x 2) 2 10x, 4/100-x) Bar A $4 400, 0 4647), 4(100->> $ 4 ale can simplify the game as - Bar B $ 10 Bart $10 | 5(10), 5/10), 108, 4/10-x) 44 (400,0 anos), (1-2) b) both will change $10 is a nash equat ip 1007070 I True If Bar A changes $4 I proff proof i Pof Bar A chooses $10, it is better for Bon i o to choose $10 as 4(100 ) 25/100-x) for any (positive) vahu ef a. and, wher, Bar B chooses. 410 Bar A will always choose $10 as for any positive value of a 5/100+) 7400 so ($10, $10) is a nauh equith for no

Any And, ($4, $4) will be noush when , 0 < 2(100-) on (100-a) 20 on 100% and lon L2(100+) an 50 £ 100+ an 40 2100 x 24 a £25 So ff na 25