1. Horizontal Differentiation
Spatial Competition
Horizontal Differentiation
Non Price Competition

2. 1895 – 1973

3. Hotelling, H.,
Stability in competition. Economic Journal 39:41-57 (1929)
Eaton, B.C. and R. Lipsey,
The Principle of Minimum Differentiation Reconsidered: Some new Development in the Theory of Spatial Competition,
Review of Economic Studies 42: (1975)

4. Ice Cream Sellers on a Beach
Competition purely in Location – non-price competition
Consumers are uniformly distributed on an interval (the beach).
Each wants to buy one unit of a homogeneous good (ice cream).
Ice cream sellers are located at various points on the beach .
All sell at the same price, and choose their location to max profit
(his market share).
A consumer pays linear transportation costs for walking to a seller.
Each consumer chooses a seller to minimize his costs
(i.e. he chooses the nearest seller)

5. Ice Cream Sellers on a Beach
This defines a non-cooperative game between the sellers
Sellers choose location (simultaneously).
The payoff of each seller is his market share
(the customers nearest to him).
What are the equilibria of this game ???

6. Ice Cream Sellers on a Beach
This defines a non-cooperative game between the sellers
Sellers choose location (simultaneously).
The payoff of each seller is his market share
(the customers nearest to him).

7. Ice Cream Sellers on a Beach
Interpretations of the `Beach’
Location.
Quality Space.
Ideology space (voters). 

8. Ice Cream Sellers on a Beach
Interpretations of the `Beach’
Location.
Quality Space.
Ideology Space (voters).  
NOT a Single Peaked Preference
Single Peaked Preferences
Preferences
Differentiated Goods

9. Ice Cream Sellers on a Beach
Interpretations of the `Beach’
Location.
Quality Space.
Ideology Space (voters).  
Voter’s Preferences L C R
Ideology or Policy Space

10. Ice Cream Sellers on a Beach
Interpretations of the `Beach’
Location.
Quality Space.
Ideology Space (voters). 
The individual votes for the nearest party
(when the preferences are symmetric) x x R B L C R
Ideology or Policy Space

11. Ice Cream Sellers on a Beach
Interpretations of the `Beach’
Location.
Quality Space.
Ideology Space (voters).
The individual
does not necessarily
vote for the nearest party
when the preferences are
not symmetric  x x x R B L C R
Ideology or Policy Space

12. Ice Cream Sellers on a Beach
Horizontal Differentiation
Ice Cream Sellers on a Beach
A major characteristic of
Single Peaked Preferences of consumers distributed on a line
is the high degree of `disagreement’ between them
For any two products (A,B) between the two customers
A 1 B
B 2 A x x x x 2 A 1 B

13. Horizontal Differentiation
Varieties of Mustards

15. Ice Cream Sellers on a Beach
Assume, first, that there are 2 sellers on the beach. x x |
mid point
(between the two sellers)

16. (between the two sellers)
Ice Cream Sellers on a Beach
2 sellers on the beach. x x |
All these customers go
to this seller
mid point
(between the two sellers)

17. (between the two sellers)
Ice Cream Sellers on a Beach
2 sellers on the beach.
But this seller could increase his market share by moving towards the other firm
New Market Share x x | |
All these customers go
to this firm
mid point
(between the two sellers)
new

18. Ice Cream Sellers on a Beach
2 sellers on the beach.
Hence, in equilibrium the two sellers should choose the same spot – and each will serve half the population x x

19. Ice Cream Sellers on a Beach
2 sellers on the beach.
> ½ x x
But, if one side has more than ½ of the population then a seller gains by
moving to that side (sufficiently close to the other seller)

20. Ice Cream Sellers on a Beach
2 sellers on the beach. x x
Hence, the unique equilibrium is for the two sellers to locate at the mid-
point of the interval.

21. x x x x x x Ice Cream Sellers on a Beach Some General Observations
Suppose there are many sellers on the beach. x x x x x x
In equilibrium, there cannot be single sellers at the edges.
For they can increase their share by moving inwards.

22. x x x x x k 2b a Ice Cream Sellers on a Beach
Some General Observations
So, in equilibrium, there must be two or more sellers in the extreme positions (the last locations of the beach) k x x x 2b a x x
Each of the sellers in the cluster earns
In equilibrium: x

23. x x x x x k 2a 2b a Ice Cream Sellers on a Beach
Some General Observations
So, in equilibrium, there must be two or more sellers in the extreme positions. k x x x 2a 2b a x x
Each of the sellers in the cluster earns
In equilibrium:
The payoff of each seller of this pair is a

24. x x x x x x a 2a b 2b 2a a Ice Cream Sellers on a Beach
Some General Observations
This also holds for the other extreme position a x x 2a b 2b 2a a x x x x
But a and b should be equal. WHY ??

25. x x x x x x k 2b 2a Ice Cream Sellers on a Beach
Some General Observations
Similarly, any other cluster of sellers cannot have more than 2 sellers. k x x x 2b 2a x x x
IF k = 2 then a = b

26. Ice Cream Sellers on a Beach
k = 3
We seek an equilibrium for k = 3, three sellers.
In a pure strategy equilibrium all three should be in a cluster
(else there will be a single firm in an extreme position)
Each serving a 1/3 of the total population. x x x
But this is not an equilibrium.
There is at least one side containing more than a 1/3 of the population

27. k = 3 ¼ ½ ¼ Ice Cream Sellers on a Beach
We seek an equilibrium for k = 3, three sellers. 2
There exists a symmetric mixed strategy equilibrium, in which each
seller chooses locations in the centre half with equal probabilities.
(A uniform distribution over the marked interval)

28. Ice Cream Sellers on a Beach
k = 4
We seek an equilibrium for k = 4, four sellers.
There cannot be single cluster. (?)
The only candidate for a pure strategy equilibrium is:
Prove that it is an equilibrium
By checking all possible deviations x x x x a 2a a
4a = 1
Each seller earns ¼

29. Ice Cream Sellers on a Beach
k = 5
We seek an equilibrium for k = 5, five sellers.
There cannot be a single cluster.
The only candidate for a pure strategy equilibrium is:
Prove that it is an equilibrium x x x x x a 2a 2a a
6a = 1 ⅙ ⅓ ⅓ ⅙ ⅙ ⅓ ⅓ ⅙
This is a non-symmetric equilibrium:
The central seller earns ⅓
The others earn ⅙ each.

30. Ice Cream Sellers on a Beach
k = 6
We seek an equilibrium for k = 6, six sellers.
There are two candidates for equilibrium.
The first candidate: x x x x x x ⅙ a 2a ⅓ ⅓ 2a ⅙ a
6a = 1 ⅙ ⅓ ⅓ ⅙

31. Ice Cream Sellers on a Beach
k = 6
The second candidate for equilibrium:
a + b a x x x x x x a 2a 2b 2a a
6a + 2b = 1
Payoffs:

32. Ice Cream Sellers on a Beach
k = 6
The second candidate for equilibrium:
a + b a x x x x x x a 2a 2b 2a a
a  ⅙
6a  1
6a + 2b = 1
b  ⅛
8b  1
a  b

33. Ice Cream Sellers on a Beach
k = 6
The second candidate for equilibrium:
Prove !!! x x x x x x a 2a 2b 2a a
6a + 2b = 1
For 0  b  ⅛,
choose a = ⅙ (1-2b),
the pair a,b is defines an equilibrium.
b  a
a  ⅙
b  ⅛

34. Ice Cream Sellers on a Beach
k = 6
There is a continuum of equilibria, ranging x x x x x x
from: ⅛ ⅛
b = ⅛
and all
the
intermediate
ones: x x x x x x x x x x x x
to: ⅙ ⅓ ⅓ ⅙
b = 0

35. Ice Cream Sellers on a Beach
Find all equilibria for any n sellers

36. Entry Given fixed costs of entry f. How many firms will enter?
As long as there is a firm with market share of  4f,
another firm could enter and earn at least f.
Hence there are at least 1/4f firms.
As f →0, the number of firms approaches 
and the market share of each tend to 0.

37. A Biological Example of Choice of Location
Hamilton, W.D.: Geometry for the selfish herd.
J. Theoret. Biol 31 (2), 295–311 (1971)
A finite number, n, of prey individuals are located on a circle with circumference 1.
A predator chooses a point randomly and catches a prey that is nearest to him.
If there is a number of prey individuals nearest to him, he catches one with equal probability.
Prey individuals choose a location to minimize their probability of being caught.

38. a +b b +c 2b 2a 2c Consider two single neighbors,
their expected probabilities of being caught are:
Assume this is an equilibrium,
then neither can gain by joining the other.
a +b
b +c 2a 2b 2c x

39. 0.5(a +b + c) > a + b a +b 0.5(a +b + c) b +c 2b 2a 2c
Consider two single neighbors,
their expected probabilities of being caught are:
Assume this is an equilibrium,
then neither can gain by joining the other.
0.5(a +b + c) > a + b
and similarly: (a +b + c) > b + c
a +b
0.5(a +b + c)
b +c x 2b 2a x x 2c x

40. Hence, in equilibrium there cannot be two neighboring singles.
Consider two single neighbors,
their expected probabilities of being caught are:
Assume this is an equilibrium,
then neither can gain by joining the other.
0.5(a +b + c) > a + b
c > a + b
a > b + c
0.5(a +b + c) > b + c
c > 2b +c
Impossible !!
Hence, in equilibrium there cannot be two neighboring singles. x 2b 2a x x 2c x

41. and hence to formation of a single herd
Similarly, if in equilibrium there are two neighboring clusters
m, n > 1 then no one can gain by joining the other.
Show that this leads to the complete desertion of one cluster
and hence to formation of a single herd
Impossible !!! n x m x x x x x x 2b 2a x x 2c x