Socially optimal solution vs Nash equilibrium

Why are gas stations always built right next to other gas stations?

Why can I drive for a mile without finding a coffee shop and then stumble across three on the same corner?

Why do grocery stores, auto repair shops, and restaurants always seem to exist in groups instead of being spread evenly throughout the community?

While there are several factors that might go into deciding where to place your business, clusters of similar companies can be explained by a very simple story called Hotelling’s Model of Spatial Competition.

Imagine that you sell ice cream at the bench.
Your bench is one mile long and you have no competition.
Where would you place your cart in order to sell the most product?

In the middle.

The one-half-mile walk may too be far for some people at end of the beach, but your cart seaves as many as possible.

One day you show up at work just as your cousin teddy is arriving at the beach with his own ice cream cart.
In fact, He's selling exactly the same type of ice cream as you are.

You agree that you'll split the beach in harf.
In order to ensure that customer don't have to walk too far you set up your cart quarter mile south of the beach center, right in the middle of terriitory.

Teddy sets up a quarter mile north of the center, in the middle of Teddy territory.
With this agreement, everyone south of you buys ice cream from you.
Everyone north of Teddy buys from him, and the 50% of beachgores in between walk to the closet cart.

No one walks more than a quarter of a mile, and both venders sell to harf of the beachgores.
Game theorists consider this a socially optimal solution.
It minimizes the maximum number of steps any visior must take in order to reach an ice cream cart.

The next day when you arrive at work, Teddy sets up his cart in the middle of the beach.
You return to your location a quarter mile south of center and get the 25% of quarters to the south of you.
Teddy still gets all the customers north inTeddy territory, but now you split the 25% of people in the between the two carts.

Day three of the ice cream wars, you get to the beach early, and set up in the center oh Teddy territory, assuming you'llserve the 75% of beachgores to your south, leaving your cousin to sell to the 25% of customers to the north.

When Teddy arrives, he sets up just south of you stealing all the southrly customers, and leaving you with a small group of people to the north.
Not tobe outdone, you move 10 paces south of Teddy to regain your customers.

When you take a brake , Teddy shuffles 10 paces south of you, and agaiin steals back all the customers to the far end of the beach.
Throughout the course of the day, both of you continue to periodcally move south towards the bulk of ice cream buyers, until both of you eventually end up at the center of the beach, back to back, each serving 50% of the ice cream-hungry beachgores.

At this point, you and your competitive cousin have reached what game theorists call a Nash Equilibrium -the point where neither of you can improve your position bu deviating from your current strategy.

Your original strategy, where you were each a quarter mile from the middle of the beach,  didn't last, becauze it wasn't a Nash Equilibrium.
Either of you could move yourcart towards the other to sell ice cream.

With both of you now in the center of the beach, you can't reposition your cart closer to your furthest customers, without making your current customers worse off.
However you no longer have a social optimal solution, since customers at either of the beach have to walk further than necessary to get a sweet treat.

Think about all the fast food chains, clothing boutiques, or mubile phone kiosks at the mall.
Customers may be better served by distributing services throughout a community, but this leaves businesses vulnerable to aggresive competition.

In tne real world, customers come from more than one direction, and businesses are free to compete with marketing strategies, by differentiaiting their product line and with price cuts, but the heart of their strategy,
company like to keep their competition as close as possible.