Question

A question of game theory

2. Imagine that a large number of consumers are uniformly distributed along a boardwalk that is 1 mile long. (Hint: "Uniformly distributed" is a mathematical expression which refers to the uniform distribution of a continuous random variable. In this exercise, it essentially describes a situation such that half of the consumers are located along the first half mile of the boardwalk, and the other half of the consumers are located along the second half of the boardwalk; and . in general, r% of the consumers are located along the first r/100 miles of the board- walk, where r can take any value between 0 and 100.) Ice-cream prices are regulated, so consumers go to the nearest vendor because they dislike walking (assume that at the regulated prices all consumers will purchase an ice cream even if they have to walk a full mile). If more than one vendor is at the same location, they split the business evenly. . Consider a game in which two ice-cream vendors pick their location simultaneously Show that there exists a unique pure-strategy Nash equilibrium and that it involves both vendors locating at the midpoint of the boardwalk.

Answer 1

Consumer have to buy ice cream from nearest shop, suppose there is a only one vender and he want maximum customer so vender will open shop at mid-point of broad walk so shop will be close to maximum number of customer. When vender move from mid-point to left or right closeness to customer will start fall and at the right and left corner vender will get minimum no of consumer.

            When two vender simultaneously choose a location both will choose mid-point because it is given maximum no of customer. And if both of them choose mid-point location then they have to divide market equally. So both will lose half market. So mid-point a Nash equilibrium point where each vender have half market.