Harold Hoteling analyzed a model of spatial competition; i.e. the location of different sellers in a market respect to one another. Anthony Downs saw that this model could explain some aspects of political competition of candidates with respect to ideological position.
Hoteling investigated how sellers would choose locations along a a linear market, say ice cream stands along a beach. He assumed that the product was uniform so customers would buy from the nearest seller. Suppose as is shown in the diagram below that there are only two sellers and they are located at the quarter points of the beach. The two sellers would then each capture one half of the beach ice cream market.
If the customers were not evenly distributed along the beach the tendency of the ice cream sellers under competition would be to locate at the median point, the point where half of the customers are to the west of that point and half are to the east of it.
Hoteling also considered what the effect of having more than two sellers would be on the location of the sellers.
Anthony Downs noted that Hoteling's model could explain political competition. Instead of a beach the linear market is the political spectrum. The assumption is that each voter will vote for the candidate or party that is closest to his or her political position. Thus when a candidate takes a position to the right of the other candidate he or she will get the votes of all to the right of that position. As with the ice cream sellers on the beach the political candidates will choose a political position that is virtually the same as their opponent's. Furthermore the candidates will be driven to select the political position of the median voter.