Sellers on a street stand back to back

In 1929 the economist Harold Hotelling asked why so many things are so alike — why competing shops cluster on the same street, why rival cider makers sell nearly the same cider, why the two American parties of his day offered platforms nobody could tell apart. His model was a single street with customers spread evenly along it, each buying from whichever seller is nearest. A seller can stand anywhere, and the custom a seller wins is the stretch of street closer to it than to anyone else. The question is where the sellers stand when none of them can win more by moving.

That is a question about equilibrium in the sense the road that makes everyone later was careful to distinguish from the sense of a good outcome: a state nobody can improve alone, not a state anybody would choose. The street makes the distinction vivid, because the equilibrium can be found exactly and compared with the arrangement that would serve customers best. For two sellers the comparison is stark. For three there is no equilibrium to compare. And once prices enter the model, the clustering Hotelling explained turns into its opposite.

Where sellers stand when nobody wants to move #

On a street of 25 evenly spaced stands, the equilibria can be found by brute force: try every arrangement of the sellers, and for each, check whether any seller can gain by moving to any other stand. Sellers at the same stand split its custom equally. The figure shows what survives.

Two sellers have exactly one equilibrium: both at the middle stand, back to back, each with half the street. Three sellers have none at all. Four have their equilibrium in pairs, two at a quarter and two at three quarters, each seller with a quarter of the street; on this grid there are three such arrangements, differing only by a single stand, which is the grid’s coarseness rather than a new equilibrium. Five sellers stand in pairs at a sixth and five sixths with one alone in the middle — and the lone seller’s share is a third of the street, twice its neighbours’. Six sellers already have 25 equilibria on this street, one of them drawn, and on a continuous street they have infinitely many. These are the results B. Curtis Eaton and Richard Lipsey derived in 1975, and the search recovers every one.

Why the outer sellers come in pairs #

The pattern has a logic that can be followed without any search. Look at the leftmost seller. Everything between the start of the street and that seller is its custom, and so is half the gap to its neighbour. If it stands strictly left of its neighbour, stepping right gains it half the step from the gap and loses nothing behind it, since the customers behind still have nobody nearer. So in equilibrium the leftmost seller stands right against its neighbour, and by the same argument so does the rightmost. With two sellers that means both together, and the only place where neither can gain by hopping to the other side is the middle.

A seller strictly between two others serves half of each gap beside it. It could instead move into any other gap between two sellers, and wherever it stands in that gap it wins exactly half of it. So it is content only if no gap anywhere on the street is more than twice the custom it already has. With three sellers the two outer ones must each be paired with somebody, and there is only the middle seller to pair with; it cannot stand against both at once, and so there is no equilibrium at all. With four sellers there are two pairs. With five the fifth sits alone between the pairs: each pair’s sellers have a sixth of the street apiece, the two gaps beside the lone seller are a third each, and moving into either would win a sixth — exactly what a paired seller already has, so the balance is exact and the lone seller’s share comes out to a third. From six on the inner sellers have slack: gaps can vary within the factor-of-two condition, and the equilibria form a continuum.

The same argument explains why the equilibria waste so much. Every condition in it concerns the seller’s share, which depends on the gaps to its neighbours, and nothing in it concerns how far any customer walks. An arrangement can therefore meet every seller’s conditions while serving customers badly, and the pairs at the ends of the row are the clearest case: two sellers at one spot serve exactly the custom one seller there would serve, while a whole stretch of street has nobody near it.

Two sellers walk to the middle #

The equilibrium for two sellers is also where they end up if they simply take turns choosing their best stand.

The first move is a leap: the seller at 0.1 jumps across the street to stand just inside its rival, on the rival’s longer side, which wins it everything from that point back to the start of the street. The rival, now squeezed against the far end, replies by stepping just inside on the other side, and so it goes, each move one stand nearer the middle, until after fifteen turns both stand at 0.5. The turn-taking is the simplest kind of best reply to the past: each seller answers the rival’s current position as if it were permanent, and here, unlike in Shapley’s game in that essay, the process settles. At no point did either seller consider the customers. Every move was made to take a little more of the street from the rival, and the place where that stops is the middle.

For the customers the middle is as bad as standing at the two ends of the street would be. Spread out, at a quarter and three quarters, the two sellers would still split the street evenly, and nobody would walk more than a quarter of its length; back to back in the middle, the customers at the ends walk half its length. Neither seller has any reason to stay at the quarter mark, because a single step towards the other wins more custom.

Three sellers never settle #

With three sellers there is no arrangement to walk to, and the turns go on for ever.

The outer sellers close in on the middle one immediately, for the same reason two sellers do. Then whichever seller is in the middle has only the narrow stretch between its two neighbours, and it does better by leaping just outside one of them — after which a different seller is in the middle, and it leaps in turn. The three wobble round the centre of the street within a stand or two of one another, and the wobble never stops. Trying every arrangement of three sellers on 101 stands confirms what the dynamics suggest: none of the 176,851 arrangements is one from which nobody wants to move.

This is the street’s version of a game with no pure equilibrium, the case a third of random games have no pure equilibrium counted among games picked at random. Here the absence is not a matter of chance but of geometry, and it is robust: it does not depend on the grid, the starting positions or the order of moves. What the three sellers can have is a mixed equilibrium, the kind a mixture that is a population read as the share of a large population playing each strategy, in which each chooses a stand at random from a distribution; Shaked found one in 1982, with each seller spreading its position uniformly over the middle half of the street. Watching three real sellers would show something like the wobble, not like the distribution.

Bunching doubles the walk #

The cost of the equilibrium to customers can be measured exactly, by the average distance from a customer to the nearest seller.

For two sellers the equilibrium makes customers walk a quarter of the street on average, against an eighth if the sellers stood at the quarter marks. For four sellers in pairs the average walk is an eighth, against a sixteenth if they were spread evenly — the four sellers in two pairs serve the customers exactly as well as two sellers spread evenly would. Five sellers in the equilibrium arrangement make customers walk 0.083 against 0.050. In each case the equilibrium is close to twice as wasteful as necessary.

The arrangement that minimises the walk has a name elsewhere in this collection. The stretch of street a seller serves is its Voronoi cell, the region nearer to it than to any other seller, the one-dimensional case of nearest neighbour divides the plane. The walk is smallest when every seller stands at the centre of its own cell, which is what every site in the middle of its own cell reached by Lloyd’s iteration — move each site to the centre of its cell, recompute the cells, repeat. Competition does the opposite. A seller’s custom depends only on where its cell’s edges are, and the edges are halfway to the neighbours; so a seller gains by moving towards a neighbour and away from the centre of its own cell, and the equilibrium puts sellers at the edges of the cells that the planner would have them centred in.

Two parties meet at the median voter #

Hotelling’s last example was political, and in 1957 Anthony Downs made it the basis of a theory of two-party elections. Replace the street by a line of political positions from left to right, the customers by voters who each vote for the party nearest their own view, and the sellers by two parties choosing platforms. Voters are not spread evenly in reality, so let them be skewed: many towards the left, a long tail towards the right.

The parties converge as the sellers did, step by step, and they meet at the median voter, at 0.264, the position with exactly half the voters on each side. They do not meet at the mean, 0.286, nor at the most common view, 0.200. The median is the only position from which a party cannot be outflanked: a rival standing anywhere else takes less than half the vote, because more than half the voters are on the median’s side of the rival. This is the same median that a lie that pays met in Duncan Black’s theorem, where voters’ preferences rising and falling along one dimension guarantee a candidate who beats every other in a two-way vote, and it is the same reason, approached from the side of the candidates instead of the voters.

The prediction has well-known limits. Real electorates are not one-dimensional, turnout depends on how distinct the platforms are, and a third party can enter — after which the model has the three-seller instability, with no platform safe for anyone. That instability among candidates is the counterpart of the instability among voters that how often the majority goes in a circle measured: once a choice has more than one dimension, or more than two contenders, majority stops having a resting point. Downs knew all of this. What the model contributes is a reason for convergence that needs no assumption about anyone’s motives except wanting votes.

With prices, the sellers move apart #

Hotelling believed the clustering was general, and the phrase that became attached to it is the principle of minimum differentiation. His own analysis included prices: the sellers choose where to stand and then what to charge, and customers pay the price plus a cost of travelling. Fifty years later Claude d’Aspremont, Jean Gabszewicz and Jacques-François Thisse found that the analysis was wrong. With a travel cost proportional to the distance, as Hotelling assumed, the price competition has no equilibrium at all when the sellers stand close together, because each can undercut the other and capture the whole street. They repaired the model by making travel cost grow with the square of the distance, which makes the price competition well behaved everywhere, and then the conclusion reversed. The failure with linear costs is easy to see. When two sellers stand close together, a customer near one of them pays almost the same to walk to the other, so a small price cut by either captures nearly the whole street at once. Each seller’s best price then jumps between undercutting the rival and retreating to its own loyal customers, and no pair of prices is a best reply to each other. A cost that grows with the square of the distance makes the customers near a seller much more loyal than those far away, so demand changes smoothly with price and the jumps disappear.

Both curves fall all the way. A seller moving towards its rival brings more customers within easy reach, but the two sellers become closer substitutes, the price war intensifies, and the prices fall faster than the custom rises. At the two ends of the street each seller charges the full cost of crossing the street and earns half of that; standing together in the middle, they would compete their prices down to nothing. So with prices in the model the equilibrium is maximal differentiation: the sellers stand as far apart as the street allows.

Which of the two models describes a given market depends on whether prices are fixed. Where prices are fixed — by regulation, by custom, or because there are none, as with political parties — the first model applies and sellers cluster. Where sellers compete freely on price, the second applies and they differentiate. Hotelling’s street holds both conclusions, depending on one assumption, and his own reading of it took the wrong one.

What the pictures cannot show #

The equilibria on the street are found exactly, but on a grid, and a grid can both add and remove equilibria. The extra arrangements for four and five sellers differ from the true ones by a single stand and are artefacts of the spacing; the absence of any equilibrium for three sellers, found on 101 stands, agrees with the continuous theorem. Nothing in the figures shows that the six-seller equilibria form a continuum on a continuous street; the 25 on the grid are a sample of them.

The dynamics are one particular rule — take turns, move to the best stand, break ties towards the nearest — and other rules converge differently or not at all. The price game is computed for one shape of travel cost, the square of the distance; with the linear cost Hotelling used it has no price equilibrium for sellers close together, and that failure is described here rather than drawn. And none of the figures shows mixed equilibria, which exist for three sellers and are what game theory would predict for them.

Still open: many sellers in two dimensions #

On a line the equilibria are understood for every number of sellers. On a plane — a town rather than a street, with customers spread over a square or a disc — they are not. For two sellers on a disc the equilibrium is both at the centre, by the same argument as on the street. For more, pure equilibria often fail to exist at all, the cells are polygons whose shapes depend on every seller’s position, and which arrangements are equilibria for a given number of sellers and a given shape of town has been settled only in special cases.

For n sellers in a square, which arrangements are pure equilibria, and for which n does one exist? The Voronoi geometry that made the street tractable is still there; what changes is that a seller’s cell can gain area by moving in any of infinitely many directions, and no ordering of the sellers from left to right exists to organise the argument. The street answered Hotelling’s question about clustering in both directions, depending on prices. The town has not yet answered it in either.