Lecture 7: Cooperative (collusive) oligopoly

Players (firms) can coordinate their strategies.

Group of players (firms) that coordinate their strategies is called a coalition. The coalition of all players is called a grand coalition.

The description of all coalitions in a game is called coalition structure. One firm is also a coalition.

We assume that the coalition structure is free and disjunct.

The higher the number of players, the higher the number of coalitions.

Function v(K) that assigns to coalition K its payoff is called characteristic function. Compare: characteristic function deals with coalitions, payoff function with individual players.

Cooperation has to offer some benefit for all members of coalition in comparison to competition:

v(1,2) ³ v(1) + v(2)

The difference v(1,2) - v(1) + v(2)

is called superadditive effect.

The vector of final divisions of payoffs (a1,…,aN) is called imputation.

The set of all possible imputations is the core.

Assumptions:

a1 + a2 = v(1,2)

a1 ³ v(1)

a2 ³ v(2)

There is an infinite number of imputations that satisfy these conditions.

One good solution:

a1 + a2 = v(1,2)

a1 = v(1) + 0,5[v(1,2) - v(1) + v(2)]

a2 = v(2) + 0,5[v(1,2) - v(1) + v(2)]

Cournot cooperative duopoly

strategic variable is quantity.

x1 – output 1, cost function c1(x1) = 150 + 12x1,

x2 – output 2, cost function c2(x2) = x22

Price function: p = f(x1 , x2) = 100 – (x1 + x2).

Profit function:

M(x1, x2) = f(x1 , x2) (x1 + x2) – c1(x1) – c2(x2)

M = a1 + a2, a1 ³ M1, a2 ³ M2.

M = [100 – x1 – x2)](x1 + x2) – (150+12x1) – x22

x1° = 38, x2° = 6, M° = 1822, p° = 100 – 44 = 56.

Non-cooperative solution: M1=1146, M2= 512.

Distribution of profit

a1 = 1146 + 0,5(1822 – 1658) = 1228,

a2 = 512 + 0,5(1822 – 1658) = 594.

Comparison of models / Output / Price
Non-cooperative solution / 52 / 48
Cooperative solution / 44 / 56
Perfect competition / 88 / 12


An example:

N = 3

v(1,2,3) = 9,125

v(1,2) = 10,125

v(2,3) = 2,25

v(1,3) = 4

v(1) = 5

v(2) = 3,25

v(3) = -1

ü  principle of collective rationality – find the maximum total profit

ü  principle of group stability


The Shapley Value

Proposed by L. S. Shapley in 1953.

A measure of the negotiating power of players. It is a descriptive concept, it does not determine an optimal coalition structure.

The value is interpreted as a mean contribution of the player to all coalitions in which he may be a member.