My analysis long ago was that for this case, the equilibrium mixed strategy with three players was a random choice with uniform probability everywhere within th

My analysis long ago was that for this case, the equilibrium mixed strategy with three players was a random choice with uniform probability everywhere within the interval from 1/4 to 3/4 and no probability in the segments at each end. This means that for the people at the ends of the beach, the competitive solution with 3 vendors is not as good as the centrally planned solution with 2 vendors, but for those in the middle of the beach it is very likely to be much better. This lines up with the conclusion for the 2 vendor case that Hotelling found: competition with few vendors for somewhat differentiated products will favor consumers with average preferences.