metadata: authors: 0: author: Basu,Kaushik
count: World
docna: 0: docna: The globe and the circle : geometry and
            economic geography as tribute to Thales and Nash
docty: Policy Research Working Paper
subtopic: Social Analysis,Quality of Life & Leisure,National Governance,Government Policies,Youth and Governance
abstracts: cdata!: The geometry of the line and the circle
            has been used in economics for a long time to understand
            location choice and political positioning in democratic
            polity. This paper draws on some elementary ideas from
            geometry and game theory to extend some of the analysis to
            beyond lines and circles, to the globe or the sphere. In the
            spirit of recreational geometry, the paper is focused on
            demonstrating some abstract results concerning Nash
            equilibria on the globe. It is hoped that this will help us
            understand better the robustness of median voter theorems
            and why stores position themselves the way they do.
repnme: repnme: The globe and the circle : geometry and
            economic geography as tribute to Thales and Nash
display_title: The globe and the circle : geometry
            and economic geography as tribute to Thales and Nash
listing_relative_url: /research/2016/09/26782414/globe-circle-geometry-economic-geography-tribute-thales-nash
topicv3: Political Economy of Reform m1326263 836
publishtoextweb_dt: 2016-09-12T00:00:00Z
no_of_pages: 12
author_keys: 0: author_key: 86589
model_extractions: WPS7819 Policy Research Working Paper 7819 The Globe and the Circle Geometry and Economic Geography as Tribute to Thales and Nash _Kaushik Basu_ Development Economics Vice Presidency Office of the Chief Economist September 2016
Policy Research Working Paper 7819

# **Abstract**

The geometry of the line and the circle has been used in economics for a long time to understand location choice and political positioning in democratic polity. This paper draws on some elementary ideas from geometry and game theory to extend some of the analysis to beyond lines and circles, to the globe or the sphere. In the spirit of recreational geometry, the paper is focused on demonstrating some abstract results concerning Nash equilibria on the globe. It is hoped that this will help us understand better the robustness of median voter theorems and why stores position themselves the way they do.

This paper is a product of the Office of the Chief Economist, Development Economics Vice Presidency. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at kbasu@worldbank.org or kb40@cornell.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **The Globe and the Circle**

**Geometry and Economic Geography as Tribute to Thales and Nash**

# **Kaushik Basu**

**The World Bank and Cornell University**

**Key words** : sphere, circle, store location, electoral democracy, Nash equilibrium
# **1. The Problem and the Backdrop**

This paper, located between school geometry, political economy, and game theory, is constructed around a simple problem. Consider a globe, which, somewhat like the Earth, is a sphere. People live all over the globe, uniformly distributed, that is, the number of people living within any two square kilometer spaces is the same. Suppose on this globe there is a rotary or a circle. There are n stores that have to be located on this rotary, and all customers prefer to go to the store nearest to their location (traveling along the globe) 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. 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.

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). Recently, in Basu and Mitra (2016), we tried to provide a characterization of Nash equilibria on a circle. 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, and then to establish some simple properties of Nash equilibria. Given that location problems and the analysis of electoral voting have some common mathematical foundations, 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. 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 and, by analogy, the stretch of voters who may be left without any candidate offering a platform close to their ideal. But the purpose of the paper is not so much the result as an exercise in recreational geometry, in the spirit of Basu (2016), which also hones one’s skill at this kind of analysis, which the reader can then use depending on the problem that he or she has to solve.

# **2. Thales and Nash**

The paper is meant to be a tribute to two remarkable thinkers, Thales of Miletus, who was the progenitor of the geometry of circles, and John Nash of Princeton, the eponymous creator of the concept of equilibrium, which is central to game theory. While we know a lot about Nash, our knowledge of Thales is full of gaps. He was born in Miletus a good half‐century before Pythagoras, but the exact year is not known, probably between 625 and 620 BC. He is considered one of the seven sages of Greece, but some claim he was not Greek, but Phoenician. Thales adopted a child but never got married. Asked in his youth, he allegedly said he was too young to marry; questioned many years later, he said he was too old to marry. There is no record of what answer he gave in the interim. What we do know about him with certainty is that he was a staggering intellectual, with interest in and contribution to philosophy, engineering, astronomy, and, most importantly, geometry. As Bertrand Russell observed in the opening chapter of his classic, History of Western Philosophy, “Western philosophy began with Thales” (Russell, 1946, p.25). He may well be the first person who developed the concept of deductive proofs. And he was the master of the circle. While the properties of the circle were known from the time of the invention of the wheel and certainly to the Egyptians, the first theorems concerning the circle were the discovery of Thales. This paper 2
is a tribute to Thales, since so much of it is about the geometry of circles. In the next section it will be shown how the fact that people live all over the globe can be transformed to a mathematically equivalent problem in which people live all over a circle; and then, using this I will demonstrate some results pertaining to the globe.

The other key figure for this paper is John Nash (1928‐2015). Born on June 13, in Bluefield, West Virginia, Nash made important scientific contributions to numerous fields, including differential geometry, singularity theory, and most importantly game theory and the creation of a concept of equilibrium that is central to contemporary economic analysis. His achievements are all the more remarkable because schizophrenia claimed him by the time he was 30 years old. So all his staggering achievements had to be squeezed into six or seven crowded years. In keeping with this remarkable brevity was his PhD thesis in mathematics from Princeton University—just 28 pages long. A personal reason for me to want to give a tribute to Nash is an encounter with him in 2003. This was at a conference on game theory in Mumbai, partly to honor John Nash, who had by then won the Nobel Prize (1994), and his schizophrenia was in some remission, though he was still clearly distracted and unmindful. As I began presenting my paper, I was touched to see Nash walk in and sit down in the front row. I was nervous at the thought of speaking of Nash equilibria in Nash’s presence. But I need not have been, for he fell asleep within the first five minutes and did not wake up till people clapped at the lecture’s end.

# **3. Spheres and Circles**

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.

To establish this, I first need a sphere. Fortunately, we have a readymade one. The World Bank Group logo turns out to be just what we need for this exercise. The logo is reproduced in Figure1. Suppose people live all over this sphere and the rotary on which all stores are to be located happens to be the circle going through A, J, and B.

What I will first establish is an easy way to locate how a person living anywhere on the sphere will choose which store to go to (recall people go to the nearest store). To do this, label the center of the rotary (on the sphere) as N. This is as shown in Figure 1. I shall refer to this as the ‘relative North Pole,’ that is, relative to the circle. Mark the opposite point (not shown in the figure) as S, the relative South Pole. Next draw the latitude and longitude lines on the sphere with reference to these two poles, N and S. These will be called the ‘relative’ latitude and longitude lines. They are the lines drawn relative to the starting circle. If the starting circle was elsewhere, the relative latitude and longitude lines would, typically, be elsewhere. Luckily, the World Bank logo comes with these latitude and longitude lines marked. These lines are on display in Figure 1.

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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. Fortunately, location analysis in two dimensions 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.<sup>1</sup> Now, using this equivalence between the sphere and the circle, I shall prove two results, one new and one that appears in Basu and Mitra (2016) but is done somewhat differently here.

# **4. Lemma and Theorem**

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. 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. Let us now consider the case where 1 and 2 are in different places, as in Figure 2. 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. Since we reached this conclusion without knowing exactly where 1 and 2 are, half the customers go to 1 and half to 2 no matter where they are. Thus all store placements are Nash equilibria when n=2. From here on we assume n > 2.

> 1 See, for instance, D’Aspremont, Gabszewicz, and Thisse (1979); Salop (1979); Basu (1993); Osborne (1995); Pal (1998).

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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. 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.

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