Ice Cream Vendors on the Beach

Another example of selfish behavior is the ice cream vendors on a beach. Suppose we have two ice cream vendors at a beach. Both offer the same flavors, and customers choose the closest vendor. The beach can be seen as a one-dimensional line, and the vendors have to choose a spot where they open their stall.

On the first day, both decide to open their stall at opposite sides of the beach as shown in the following figure. We assume that the number of customers doesn’t depend on their distance. But customers choose the vendor who is closest to them.

Both ice cream vendors have access to half of the guests at the beach. Guests left from the middle go to vendor A and guests right from the middle go to vendor B.

On the second day, B decides to change his spot by moving a bit more towards the middle. Now, he has access to more customers. He increased his market share. All customers right of B will visit him. And customers between A and B will go to the vendor who is closest.

Vendor A follows the next day to win the lost customers back. Again, both vendors share the customers equally.

For the customers, this is the most beneficial spot. The total distance to the vendors is minimized.

This process repeats until all vendors reach the middle of the beach.

Both vendors still share their customers equally, like at the beginning. But for the customers, this is again a situation with long distances to the vendors. This situation is also called Hotelling’s law. It describes why products can be very similar, although producers face strong competition. It can also be found in politics and is known there as the median voter theorem.