Policy Research Working Paper 7819 e Globe and the Circle Geometry and Economic Geography as Tribute to ales and Nash Kaushik Basu Development Economics Vice Presidency Office of the Chief Economist September 2016 WPS7819 Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized
Transcript
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 PresidencyOffice of the Chief EconomistSeptember 2016
WPS7819P
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
edP
ublic
Dis
clos
ure
Aut
horiz
ed
Produced by the Research Support Team
Abstract
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.
Policy Research Working Paper 7819
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 [email protected] or [email protected].
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 geom-etry 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 theo-rems and why stores position themselves the way they do.
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
2
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
3
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.
4
Figure 1
Consider two stores A and B and a person or customer located at some randomly‐chosen point,
say P, on the globe. Now suppose for the person at P, instead of going directly to the store, he chooses
the following route. First, travel along the (relative) longitude line to the rotary, i.e., the circle going
through A, J, and B, and then travel on the rotary to the store. Let me call this kind of travel as the ‘scenic
route’. So if a person at P wants to go to A by the scenic route, she first goes to J and then goes along the
rotary to A.
It is easy to verify that for a person at P, A is closer than B if and only if the scenic route to A is
shorter than the scenic route to B. Here is the hint of how to prove this. Consider a point on the rotary
that is to the east of J such that the distance AJ is the same as the distance between J and this point.
Clearly I is such a point. It is obvious (by visualizing an isosceles triangle on the sphere) that the distance
AP is the same as IP (since the distance AJ equals IJ). Hence, if store A is closer to P than B by the direct
route, it must be closer also by the scenic route.
Now suppose we relocate each person from where she lives to the point of intersection of the
(relative) longitude line on which she lives and the rotary. Given the relative equivalence of the direct and
scenic routes, this will cause no change in the decision of which store the person goes to.
If we do this kind of relocation of people, the entire population of the world will be living on the
rotary, uniformly distributed, with the rest of the world uninhabited. Since this does not change their
choice of store, for a store trying to decide where to locate, this world is identical to the previous one
where people live all over the globe. In other words, we are now in familiar two‐dimensional territory,
where people live uniformly distributed on a circle, and stores face the challenge of where to locate.
Indeed, now we are in a position to state a more 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. 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.1 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).
6
Figure 2
Let me now state a lemma (Basu and Mitra, 2016) and prove it, for completeness.
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.
Proof. Let there be ( n) stores at one point. Suppose is the length of the customer
neighborhood of this point. It follows that each store at that point gets customers equal to a payoff of
. By deviating, one of those stores can earn at least as much as .
But if > 2, then > . Hence, wherever 3 or more stores are located at one point, that
cannot be a Nash equilibrium. □
In stating the next theorem, let me clarify that after the stores have chosen their location, that is,
we have a placement of stores, any stretch of the circle that does not have a store will be referred to as
an ‘empty stretch’.
There can be many ways to characterize the Nash equilibrium placements. Most such exercise can
be complex. One full characterization was provided in Basu and Mitra (2016). What I will provide here is
one simple property of Nash equilibria, which can be proved using no more than school geometry.
Theorem. If a placement of stores is such that there is an empty stretch with length greater than
n/2, then that placement cannot be a Nash equilibrium.
Proof. Suppose the longest empty stretch is between stores 1 and 2 and it is of length x > n/2.
First consider the case where 1 and 2 do not share their location with any other store. It is possible to see
that the average number of consumers going to each of the other stores 3, 4, …, n, is at most
.
7
To see this, suppose the stores neighboring 1 and 2 (call them 3 and n) are very close to 1 and 2.
Then 3, 4, …, n get almost all the customers living on the stretch of length 1 – x, which is the complement
of the empty stretch between 1 and 2. This is illustrated in Figure 3.
It is easy to see, as 3 and n get closer to 1 and 2, respectively, the average market share of firms
3, 4, …, n, approaches
.
Figure 3
If any of these firms deviates to the empty stretch between 1 and 2, it will obviously get x/2
customers. It follows that this placement is not Nash if
>
. (1)
It is easy to verify, if x > , then (1) has to be true. Hence, the placement is not Nash.
Now consider the case where 1 and 2 are not isolated. Assume 3 is located at the same spot as 2.
Let 4 be the next store and let the distance between 3 and 4 be y. Since x was the longest empty stretch,
y ≤ x. Suppose y < x. Then it cannot be Nash, since store 3, that is currently earning
could do better
by moving to the stretch between 1 and 2, which would give it a payoff of x/2. So now suppose y = x.
Assume, first, store 4 is alone at that point, as shown in Figure 4. Then the remaining n‐4 stores,
by the same logic as before, would be earning on average at most
.
8
If one of those firms deviates to the longest empty stretch, it will earn x/2. It follows the current
placement is not Nash if
>
. (2)
Figure 4
It is easy to verify, if x > , (2) must be true. Hence, the placement is not Nash.
What if 4 is not alone where it is but shares the spot with store 5? By the same logic as before, it
can be shown that the placement cannot be Nash if x > . Up to now we have considered cases where
stores are either alone at a spot or in pairs. What happens if more than 2 stores locate at the same point?
By the above lemma, such a placement cannot be Nash. □
What we have thus proved is that when n stores locate themselves on a notary if there is an empty
stretch longer than , the placement cannot be a Nash equilibrium.
5. Conclusion and Apology
The paper is meant to demonstrate the elegance and aesthetics of geometry on a sphere, which
can be used to understand the economics of location or economic geography. 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. What was important was the equivalence
9
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.
10
References
Basu, K. (1993), Lectures in Industrial Organization Theory, Oxford: Blackwell Publishers.
Basu, K (2016), ‘A New and Quite Long Proof of the Pythagoras Theorem By Way of a Proposition on
Isosceles Triangles,’ Journal of College Mathematics, forthcoming.
Basu, K. and Mitra, T. (2016), ‘Nash on a Rotary: Two Theorems with Implications for Electoral
Politics,’ World Bank Policy Research Working Paper, No. 7701.
D’Aspremont, C., Gabszewicz, J. J. and Thisse, J. F. (1979), ‘On Hotelling’s Stability in Competition,’
Econometrica, vol. 47.
Gulati, N. and Ray, T. (2016), ‘Inequality, Neighbourhoods and Welfare of the Poor,’ Journal of
Development Economics, vol. 122.
Hotelling, H. (1929), ‘Stability in Competition,’ Economic Journal, vol. 39.
Osborne, M. J. (1995), ‘Spatial Models of Political Competition under Plurality Rule,’ Canadian Journal of
![· Web viewWhen true, members from the two groups are combined and span the whole circle. size [Reference to Field] Optional. A field that gives a relative strength of connections](https://static.documents.pub/doc/80x56/5b3f5ee17f8b9aff118c0a5c/-web-viewwhen-true-members-from-the-two-groups-are-combined-and-span-the-whole.jpg)
