There's no stable equilibrium. The ones on the edges are constantly incentivized to move towards the center, but at some point the guy in the middle is better off moving anywhere else. Kind of like the "divide the dollar" game.

I doubt that position is important if the 3 are directly adjacent one another, or at least not as important as say the appearance of your store front or your prices or your quality or your reputation or your display of well known logos ....

If I walk all the way down the beach 3m further to get to a nicer looking icecream shop is nothing. Indeed I think I'd look at the price of the icecreams first, then the ones people are leaving with and make a cost analysis, then probably choose the one selling pistachio flavoured icecream.

3 miles or 3 meter?

There should probably have been some punctuation: if I walk down the [whole length of the] beach, 3m is not much further.

Metres, not "mi" miles.

Thanks. I am not too familiar with the non-SI systems.

Actually, the ones on the edges should spread out a bit, they will reduce the competition for those coming from the ends of the beach who will now have to walk further to get to another shack, and they still split the difference between themselves and the next shack. How MUCH they should spread out and what more competition will do is unstable.

Sure there is equilibrium. Differentiation on price points, quality, service and length of lines will balance things out.

That said, the difference between a third ice cream cart and a 30th ice cream cart in the area is great. But then, the market should correct itself so that the optimal number of ice cream sellers are available.

I've been thinking about that. I don't know what the eventual distribution would look like, there doesn't appear to be a stable solution. But at the beginning, he should locate at a point one-third of the way from the center to an end - at that point 41% of the beach is closer to him than to either of the original competitors.

The three get together, but the middle one get no clients.

And the middle one moves to the end of the line, grabbing half the clients. Now, the problem is the same but someone else is in the middle, who then must relocate himself, ad infinitum.

Two of them get together and manage to keep the likely thirds marginal?

The one in the middle forecloses on his house.