Harold Hotelling, the economist who inaugurated the concept of spatial competition almost a century ago, did not actually use the example of two boys arriving to sell ice cream on a long well populated beach on a hot day (as Tim Harford sheepishly acknowledged in this paper a few years ago). But posterity has agreed that the original article would have been much more vivid if he had.

The two boys would naturally gravitate together.  If they were any distance between them, each would gain some of the other’s customers by moving closer. Yet all existing customers would still find their original choice the more convenient.  The result is inefficient – the average walk for an ice cream would be shorter if the carts were more evenly spaced – but the result is inevitable.  And we observe this clustering phenomenon in the similarity of the product offerings of Hertz and Avis, or Unilever and Procter and Gamble. Famously, Australia’s – now wisely abandoned – ‘two airline policy’ led the two ostensibly competing companies to select more or less identical schedules.

       Even in his initial 1929 article, Hotelling noted that his analysis might be relevant to politics as well as economics, a thesis developed  later by the political scientist Anthony Downs  In a two party system, the policy positions would be close together, for the same reason  as the ice cream sellers.   From time to time, one party might move away from the centre ground, but electoral arithmetic would drive it back to the centre. 

       Significantly, a three party version of the game turns out to have no equilibrium. A stable outcome requires that one of the players  is driven out.  As the number of participants  increases, however, the solution approaches one in which ice cream carts, or political parties, are evenly spaced along the beach. Where entry is cheap – as with proportional representation – voters have a wide variety of positions to choose. But where entry is costly – as in a first past the post system – two parties with similar policies will be the norm.

    But the rise of democracy in the selection of party leaders has introduced a complication.  The selection of US presidential candidates was once largely the prerogative of party bosses, but now is made through an exhaustive primary process  Conservative party leaders were once chosen by men in suits in smoke-filled rooms, while Labour left the decision to members of parliament; but both have now given the party at large the final say.

The consequence is that to be in charge of the ice cream cart, you have to be the most popular boy among your own customers, rather than the most popular boy on the beach.  And the ill served customers at the far ends of the beach are likely be most voluble in expressing their views. Those nearer the middle, basking in the sun with an ice cream readily to hand, have little incentive to become involved.

       So the characteristics needed to win the leadership of your party from are very different those needed to win the leadership of your country.  Tea Party candidates are selected who lose races that mainstream Republicans might have won:  and parties revert –  only after long years in opposition – to leaders like Blair and Cameron who are more popular with voters than with members of their own party.

          Hotelling’s model illuminates evident paradoxes of modern politics. Two middle of the road parties with immaterial differences is not a negation of democracy, but an accurate expression of popular will. That outcome leads to complacency and apathy, but given the prevalence of  such apathy, greater opportunity for engagement may lead to less democracy not more, because those who do engage are necessarily unrepresentative. In politics as in  economics, choice works best for you when you do not have to exercise it.