Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

3.2.2 Provision of Information

The second example deals with the provision of information in Bertrand oligopoly,
see Ireland (1993). Two a priori symmetric firms produce a homogeneous prod-
uct and play a two stage game. In the first stage, each firm sets the level of its
product information and in the second stage they compete in prices. Informa-
tion regards only the existence of the product. Consumers may obtain costless
information about prices of products that they know to exist. The number of
consumers is normalized to 1.Thevariablesx and y are the proportions of
consumers who know about product 1 and 2 respectively. Each consumer is not
willing to pay a unit price higher than 1. Firm 1’s sales are given by:

{X if P1 < P2

x - ^xy if P1 = P2                         (¹⁰)

x - xy if p1 >p2

For x = y =1 the Bertrand oligopoly has a pure strategy Nash equilibrium
at p1 = p2 =0. If information is not full (x or y or both are less than 1),
no pure strategy Nash equilibrium exists. There exists a mixed strategy Nash
equilibrium given by the distribution function Gi (pi) that has the following
form respectively for firm 1 and 2 .

f ^1-(1-x)/p if 1 - x ≤ p ≤ 1               f ^1-(1-x)/p if 1 - x ≤ p ≤ 1

Gι(p) = < ^x           G^y-     G2(p) = { ^y            -

[ 0 if 0 ≤ p ≤ 1 - x                     [ 0 if 0 ≤ p ≤ 1 - x

(11)

The overall payoff for (say) firm 1 in the game, upon substituting the second
stage equilibrium payoffs, is given by F(x, y)=Ep (p1Q1) or

U(x, y)=x (1 - y) if x ≥ y

L (x, y)=x (1 - x) , if x ≤ y

It is trivial to show that assumptions A1, A2 and A3 are verified in this
example. There exists an equilibrium and this equilibrium cannot be symmetric
for p ∈ (0, 1).MoreoverU1 (0, 0) > 0 and L1 (1, 1) < 0. So we can conclude
from Theorem 3.1 that no symmetric equilibria exist for p ∈ [0, 1] .

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