Math 486 Lesson 3 Homework - Due Monday June 4 at 11:59 pm 1 TWO stores on Main Street (Gintis/Schecter) In the Lesson 3 videos we described a...

hi, here are three questions I need help on.

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Math 486 Lesson 3 Homework - Due Monday June 4 at 11:59 pm 1 TWO stores on Main Street (Gintis/Schecter) In the Lesson 3 videos we described a discrete location game where firms could choose locationsfrom the set {1,2, . . ., 9}. Here we‘ll consider a game with continuous strategy spaces. Suppose a town has one street—Main Street—and the residents of the town are uniformly distributedalong this street between one end and the other. No companies are considering opening stores on Main Street. Each store must choose a locationalong the street between one end {at 0) and the other end (at 1). Each store will attract the fractionof the population that is closer to it than to the other store. If both stores locate at the same point(which we allow), each attracts half the population. The payoff to each company is the fraction ofthe population that it attracts. Assume that the companies must choose their locations simultaneously. Denote the first company‘sstrategy by a real number a: 6 [0,1] and the second company‘s strategy by y 6 [0,1]. Let u1(I,y)and 21.205, 3;) denote the payoffs to each company. Note the following payoffs: - If a: = y, then in = 1.1.2 = NIH tall- 0 If a: < y, then 11.1(3, y) = (a: +y) and 11.2(3, y) = 1 — %(3: +9).0 If y < x, then u1(:c,y) = 1 — as: + y) and v.2(I, y) = as: + y) I Show that there is exactly one Nash equilibrium: .1: = = %. A suggestion: Start with the case :1: < y. Can company 1 improve its own payofl by changing :1: inthis case (with 3; remaining the same)? 2 Three stores on Main Street (Gintis/Schecter) Use the same set-up as the previous problem, but with three stores instead of two. Each storewill attract the fraction of the population that is closer to them than the other store. If all threeoccupy the same location each will attract 1/3 of the population. Show that this game has no Nashequilibrium (in pure strategies).