Saturday 27 May 2023

A lame lemma from Kaushik Basu

What would be the distribution of shops selling a particular commodity which everybody needs on a globe where population is

uniformly distributed, that is, the number of people living within any two square kilometer spaces is the same?

The answer depends on the cost of travel and information acquisition. If both are zero, Location doesn't matter. If information is slightly costly and entry is sequential, all shops will locate in the same place. Why risk setting up in the boondocks if everybody thinks that all the shops are in the shopping metropole? After all travel isn't costly. If we assume that travel cost is a function of distance to the shop then what matters is whether travel cost is 'material' relative to expenditure. Suppose I am buying an aircraft carrier. The cost of visiting shipyards is not 'material' given the budget. Ceteris paribus, if entry is sequential and there is some information cost or other external economy, we will still see concentration. This is like 'Hotelling's law' which gets reinforced because of external economies of scope and scale.

Basu, cretin that he is, takes a different view-

Suppose on this globe there is a rotary or a circle.

The shortest path between two points on a sphere- called the geodesic- is always a segment of a 'great circle' (i.e. one which would cut the sphere in two equal parts)

There are n stores that have to be located on this rotary,

is it a 'great circle' ? Basu does not say

and all customers prefer to go to the store nearest to their location (traveling along the globe)

in which case, if the rotary is a 'great circle', choice of store is uniquely determined. The relevant geodesic is orthogonal to the rotary. There is always one and only one 'shortest path' save at the two relevant poles. But it is unlikely that any person happens to be constantly hovering exactly over one or other pole

and, when indifferent between k stores in terms of distance, each person chooses to buy from a store chosen by applying equal probability to all k stores.

This is nonsense. Basu is forgetting that there is a unique great circle between any two points on a globe. That segment which is is shortest is the geodesic. A few years after publishing this nonsense, Basu claimed that there would be no concentration because it is impossible for two stores to be at the same location. He should have used a notion of 'materiality'. Then, he would have seen that, for all practical purposes, a number of stores can be said to share a location.

The aim of each store is to maximize the number of customers, that is, the people who shop from the store. This paper is an analysis of where the stores will locate themselves. After all stores have chosen a location, I shall refer to that as a ‘placement’ of stores. A placement of stores is an equilibrium if, given the location of the other stores, no single store in that placement can do better by changing its location unilaterally.

But if changing your location can't worsen the outcome for you, why bother with location or 'placement'? The thing doesn't matter in the slightest. It isn't a Nash or other type of equilibrium any more than you deciding when to scratch your arse.

There is literature investigating similar questions for cases where people live on a line or a plane. In some ways, it dates back to the early contributions to economic geography, such as von Thunen (1826).

which does not deal with a homogeneous terrain.

Recently, in Basu and Mitra (2016), we tried to provide a characterization of Nash equilibria on a circle.

Anything at all can be a Nash equilibria if there is reason to fear that if you change your behavior somebody or everybody will fuck you up. The fact is that this is a 'optimal transport problem' (Monge-Kantorovich Transportation problem) and that will the basis of Muth Rational Expectations.

Suppose the free market doesn't get to the cost minimizing solution, then, absent non-market constraints, there will be a process of merger and acquisition and 'internalization of externalities' till that convergence occurs- if the game is worth the candle.

The main aim of the present paper is to extend the analysis to the globe and develop a methodology for translating the globe to the circle,

In which case, you need to talk about geodesics.

and then to establish some simple properties of Nash equilibria.

Absent market failure, the system will solve the Transportation problem- if it is worth doing. In this case, if travel costs are 'material', then you'd have distribution hubs and home delivery.

Given that location problems and the analysis of electoral voting have some common mathematical foundations,

But Chichilnisky & Heal showed that there will be no coordination device- market based, voting based, administrative or otherwise- if there is no preference or endowment diversity. Basu assumed a homogenous population distribution and, implicitly, made geodesic distance the only item of diversity.  There would be literally nothing for a population of this sort to vote about.

it is hoped that this exercise will be of interest to those analyzing voting patterns and electoral politics, even though the problem is presented in this paper as an abstract exercise in geometry.

This is ignorant shit.

The result established in the paper tells us about the largest stretch of consumers who may be left unattended by a nearby store in equilibrium

Either this doesn't matter to anybody, or else population won't remain homogeneously distributed.

and, by analogy, the stretch of voters who may be left without any candidate offering a platform close to their ideal.

This is stupid shit. It would be like a very tall or short person complaining that the average or the median does not pander to their notion of what is a proper and respectable height. It is a separate point that a minority might be slaughtered by the majority. But in that case, there is a discoordination game- some people should be selling up and running away. Politics, in such cases, is a waste of time. A Bengali Hindu should understand that.

The location problem described in the opening section has people staying all over a globe or a sphere and buying from the nearest store located on a rotary or a circle. The first result I want to establish is an equivalence. If the people, instead of being uniformly distributed on the sphere, were instead uniformly distributed on the rotary or circle where the stores are located, with the rest of the sphere uninhabited, the mathematical problem would be exactly the same. Indeed, there is a general principle regarding how any distribution (not necessarily uniform) on a sphere can be converted to an equivalent distribution on a circle and then for the analysis to be done on a circle.

There is a unique geodesic (save at the poles) to the 'rotary'. This is the one whose great circle is exactly orthogonal to the rotary. One can then see which shop is closest.

How stupid are Basu's students if they don't know this? The following scarcely deserves the grand name of

general equivalence result. Start with any distribution of population on the sphere. It does not have to be uniform. It is easy to convert this to an equivalent distribution on the circle on which the stores are to be located so that a Nash equilibrium based on the distribution on the circle will be identical to the Nash equilibrium using the original distribution on the sphere. For this, the rule is the following. Move all the people on each relative longitude line to the 5 point of intersection of this line with the original circle. In Figure 1, this means moving all those on the circle through NJP to be relocated at J. If this is done for all points, the new distribution of population on the circle AJIB is, for purposes of Nash equilibrium analysis, identical to when people were living all over the sphere. In other words, we can, from now on, convert all location problems where people live all over the globe to a two‐dimensional analysis.

This is unnecessary. The 'rotary is one dimensional and every inhabitant (save at the relevant poles) has a unique mapping on to it. But firms don't have to locate on the same great circle. The actual solution to the transportation problem involves some sort of 'tessellation' or tiling pattern.  (e.g the multiplicatively weighted Voronoi diagram which is also called circular Dirichlet tessellation). But this is useful stuff- Bezos probably hires guys with PhDs in such things so as to make yet more money- not the useless shite that is Basu's stock in trade. Come to think of it, the billionaire Engineering & OR savant Purnendu Chatterjee must be about Basu's age.

Fortunately, location analysis in two dimensions

or three since about the time of Gauss or, at least, Lobachevski

has a long history, from Hotelling (1929) to recent times, such as Gulati and Ray (2015), Basu and Mitra (2016), and many contributions in between.1 Now, using this equivalence between the sphere and the circle,

The thing could be a straight line if the cut is made at the right place (i.e. equidistant between two store locations). But this is just the unique distance function.

I shall prove two results, one new and one that appears in Basu and Mitra (2016) but is done somewhat differently here.

They are both nonsense.

Given the result in the previous section, we can now pretend that all people live uniformly distributed along the circle, where the stores have to choose their location. We shall, for simplicity, assume that the length of the circle, that is, of the circumference, is 1 and the total population living on the circle is 1. So the number of people or customers living on an arc of length x is x. People buy from the nearest store. Each store’s aim is to maximize the number of customers, i.e., people who buy from the store. A ‘Nash equilibrium’ is a choice of location by all firms such that no firm can do better by making a unilateral move to another location on the circle.

Why the fuck would all stores locate on the same great circle? Either they all locate at the same spot or else there is convergence to the Muth Rational solution to the Transportation problem.

Given a placement of stores on the circle and given any point, we refer to the maximum arc of the circle around that point with no stores (except possibly some stores at that point itself) as the ‘customer neighborhood’ of that point. A store is described as ‘isolated’ if there is no other store at the same point. It is useful for the uninitiated to hone intuition by checking that all locations are Nash equilibria when n=2. Suppose there are two stores, 1 and 2, located on the circle. If they are at the same point, everybody is indifferent between the two stores and so each store will get an expected half the customers.

But, by relocating to the 'antipode', that expectation- in this model- turns into a certainty (assuming travel cost is 'material') Even minimal risk-aversion does the rest.

Let us now consider the case where 1 and 2 are in different places, as in Figure 2.

Then each expects half the total custom no matter where they are located. All that matters in this model is the geodesic distance which is unique and gives exactly half the population one and only one 'nearest' store.

Let A and B be the midpoints on the two arcs between 1 and 2. Clearly, half the people living on the stretch between 1 and 2 on the eastern arc (the halfway point being marked by A) will go to 1 and half to 2. And half the people on the western arc will go to 1 and half to 2. Hence, half the entire population will go to 1 and half to 2.

This analysis is otiose. We arrive at the conclusion on the basis of the principle of maximal uncertainty or what obtains when we have no relevant information.

Basu now gives us a foolish lemma

Lemma. If there is a placement of stores such that 3 or more stores are located at one point, then that placement cannot be a Nash equilibrium.

All stores can be located at one point.  What happens when a third store enters a market where the existing two firms are separated? The answer is that expected market share is one third provided the new entrant is equi-distant between the two existing firms & on a great circle orthogonal to the center of the geodesic between then . The fourth should also have this property and so on an so forth. There is a problem where entry is 'cascading' rather than sequential but the steady state involves tessellation.

Basu is writing nonsense because he didn't take the trouble to define geodesics and made the absurd stipulation that firms can only locate on the same great circle. Basu's proof relies on this arbitrary and foolish assumption. Furthermore, he is assuming sequential entry- which is silly. If you can enter a market so can anyone like you.  If entry is turbulent, there are certain conditions under which the right tessellation will be achieved quite quickly. If this is not the case there is a Muth Rational correlated equilibrium which can be implemented in different ways. It is foolish to think that Nash is the right solution concept here. The fact is, if there is net entry or exit then there is no steady state- least of all a Nash equilibrium. What Basu has written is puerile nonsense. Why has the World Bank published it?

The paper is meant to demonstrate the elegance and aesthetics of geometry on a sphere,

Calculating the shortest route 'as the crow flies' between two places on Earth is very very fucking useful. Elegance and aesthetics certainly characterize Algebraic geometry. Most kids find tessellation beautiful. Teachers often show slides of the Alhambra or the paintings of Escher to motivate their students.

which can be used to understand the economics of location or economic geography.

Basu has understood nothing. He would later claim to have disproved Hotelling's law because two firms can't locate at the same point. He doesn't get that a few hundred yards distance is not 'material'.

The one theorem that was proved, though new, is sufficiently easy not to constitute a challenge. This is one area where one does not need pre‐established results. I wanted to demonstrate that with one’s school geometry honed, one can be ready to take on problems as and when they come along.

Only if you make crazy assumptions- e.g all shops must be on exactly the same great circle.

What was important was the equivalence claims in section 3, which shows that a large class of location problems on a sphere can be converted to a two‐dimensional exercise on a circle.

This is not true. There is a 'representation' of anything in the mathematical universe as anything else. But such conversions are only useful if they represent 'natural transformations' or solve a specific optimization problem.

Honing the technique of analysis used here is important because it constitutes the abstract backdrop of understanding location economics and the political economy of electoral democracy.

But Basu, in his long career, has learned nothing, understood nothing, illumined nothing. India , being an electoral democracy, has got the message. Mathsy Bengali economists know neither Math nor Econ. Moreover, even if the Dynasty puts in a Punjabi Economist as a figurehead PM, the markets won't be fooled by the hiring of Cornel Professors as 'Chief Economics Advisors'.