Answer
Both sellers would set up at the middle of the beach. This outcome reminds us of the median voter result.
Work Step by Step
Let's say the two sellers are set up at miles 0.9 and 1 of the beach. The 0.95 part of the beach would have beach goers split between which stand they would go to. Anyone closer to the .949999 point on the beach would go to the stand at mile 0, and anyone closer to the .950001 point on the beach would go to the stand at mile 1. Additionally, the stand at mile 0.9 would get any beach goers between miles 0 and .95. However, the stand at mile 1 would only get the beach goers between miles .95 and 1.
The stand at mile 1 would want to move to mile .9 to gain the beach goers between miles 0 and 0.925 (since there is still the stand at mile .95). The stand at mile .95 would have customers from mile .925 to mile 1, so the stand at mile .95 would move to mile .85.
This shifting of stands continues until there is a stand at mile .5 and mile .55. The stand at mile .5 gets customers from mile 0 to mile .525, and the stand at mile .55 gets customers from mile .525 to mile 1. The stand at mile .55 should move to mile .5 at this point. (Moving to mile .45 would have the same effect as staying at mile .55.)
Moving to mile .5 would equally split the beach goers to both ice cream sellers.
