Game Theory with Applications to Finance and Marketing

Let the set of correlated equilibria be A. Then,
A = {(a, b, c, d) : a + b + c + d = 1, a, b, c, d ≥ 0, a, d ≥ b, c.}.
Note that all NE’s are contained in A, 4 and if (a, b, c, d) is a totally
mixed strategy NE, then it must satisfy
a
b = c d , a
c = b d .
16. Example 6. In a duopolistic industry two risk neutral firms (i.e.
expected profits maximizers) that produce respectively products A and
B are faced with three segments of consumers:
Segment Population Valuation for A Valuation for B
L A α V 0
L B β 0 V
S 1 − α − β v v
where 0 < β ≤ α < α + β < 1, and 0 ≤ v < V . These segments are
loyals to the two firms and the switchers who regard the two products
as perfect substitutes. We have assumed that a loyal is willing to pay
more than the switcher to obtain the product.
For simplicity the two firms have no production costs, and they compete
in price in a simultaneous game. We shall demonstrate the equilibrium
dealing behavior of the two firms.
17. First, we look for a pure strategy NE. Suppose (p A , p B ) is an equilibrium.
There are 3 possibilities: (i) p A , p B > v; (ii) p A , p B ≤ v;
and (iii) max(p A , p B ) > v ≥ min(p A , p B ). For case (i), we must have
p A = p B = V , and for this to be an NE, we must require
βV ≥ (1 − α)v, αV ≥ (1 − β)v. (1)
4 There are 3 NE’s for this game, and they are respectively
(a, b, c, d) = (1, 0, 0, 0), (a, b, c, d) = (0, 0, 0, 1), (a, b, c, d) = ( 1 4 , 1 4 , 1 4 , 1 4 ).
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