Consider a town consisting solely of one straight main street along which all the stores are located. Let us depict this situation as follows: The town starts at A and ends at B. People are distributed equally along the main street so that there are as many people between 0 and ¼ as there are between ¾ and 1, or, for that matter, on any two segments of the same length. Two gas stations that are identical in all respects (including price and level of service) want to locate along the main street. Assume that the inhabitants of the town will patronize the gas station that is closest to them. Where, along the main street, will the gas stations position themselves? That is, what are their positions at the Nash equilibrium for this game?
Please explain
We are given that the town starts from point a and ends at point B. Hence if first gas station is located near point A and other gas station is located near point B then the gas station located near point A would make more profit if he decides to shift a bit towards point B. Now if both of them continue this way, they would reach a place where both would be at the same position. This location would be where both will operate.
But now we need to see the location of this position where both are operating. If for instance, they both meet at a point which is 1/4th and 3/4th from A and B respectively. This would mean that both would be getting customers from A , but the B one's would find it far and hence they would try to move towards B to gain the customers from that side of city.
This would continue till the point where both would be located at close proximity to each other in the middle of the town. This would mean maximum profits and Nash equilibrium because now given what the others are doing, the one would have the best profits and would not want to move from that point.
Hence the Nash equilibrium would be at the middle of the city from A to B for both the gas stations.
(You can comment for doubts)
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