Horizontal Differentiation Spatial Competition Horizontal Differentiation Non Price Competition

1895 – 1973

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: 27-49 (1975)

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)

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 ???

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).

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

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

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

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

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

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

Horizontal Differentiation Varieties of Mustards

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

(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)

(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

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

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)

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.

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.

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

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

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 ??

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

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

k = 3 ¼ ½ ¼ Ice Cream Sellers on a Beach We seek an equilibrium for k = 3, three sellers. 2 ¼ ½ ¼ 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)

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 ½ 2a ¼ a 4a = 1 ¼ ¼ ¼ ½ Each seller earns ¼

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.

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

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:

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

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

Ice Cream Sellers on a Beach k = 6 There is a continuum of equilibria, ranging x x x x x x from: ⅛ ¼ ¼ ¼ 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

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

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.

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.

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

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: 0.5(a +b + c) > b + c a +b 0.5(a +b + c) b +c x 2b 2a x x 2c x

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

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