Core:
Given a game v ∈ G N , the core of v is the set of all imputations x
in I(c) such that x (S) ≥ v (s) for all non-empty coalitions S ⊂ N.
The core of a game v ∈ G N is denoted by C (v ).
Example 1
coalitions v()
∅ 0
{1} v ({1}) = 0
{2} v ({2}) = 0
{3} v ({3}) = 0
{1, 2} v ({1, 2}) = 1
{1, 3} v ({1, 3}) = 1
{2, 3} v ({2, 3}) = 1
{1, 2, 3} v ({1, 2, 3}) = 3
We want to find the core allocation of the above coalition game.
We have v ({1}) = 0 ≤ xA1 , v ({2}) = 0 ≤ xA2 , v ({3}) = 0 ≤ xA3 .
v ({1, 2}) = 1 ≤ xA1 + xA2 , v ({1, 3}) = 1 ≤ xA1 + xA3 , v ({2, 3}) =
1 ≤ xA2 + xA3 .
v ({1, 2, 3}) = 3 ≤ xA1 + xA2 + xA3 .
Substituting v ({2, 3}) = 1 ≤ xA2 + xA3 in
v ({1, 2, 3}) = 3 = xA1 + xA2 + xA3 . We have 2 ≥ xA1 , similarly we
get 2 ≥ xA2 and 2 ≥ xA3 .
Therefore, core allocations are 2 ≥ xA1 ≥ 0, 2 ≥ xA2 ≥ 0,
2 ≥ xA3 ≥ 0 and 3 = xA1 + xA2 + xA3 .
It is shown in figure 1 and 2.
Example 2
coalitions v()
∅ 0
{1} v ({1}) = 1
{2} v ({2}) = 1
{3} v ({3}) = 1
{1, 2} v ({1, 2}) = 2
{1, 3} v ({1, 3}) = 2
{2, 3} v ({2, 3}) = 2
{1, 2, 3} v ({1, 2, 3}) = 3
We want to find the core allocation of the above coalition game.
We have v ({1}) = 1 ≤ xA1 , v ({2}) = 1 ≤ xA2 , v ({3}) = 1 ≤ xA3 .
v ({1, 2}) = 2 ≤ xA1 + xA2 , v ({1, 3}) = 2 ≤ xA1 + xA3 , v ({2, 3}) =
2 ≤ xA2 + xA3 .
v ({1, 2, 3}) = 3 ≤ xA1 + xA2 + xA3 .
Substituting v ({2, 3}) = 2 ≤ xA2 + xA3 in
v ({1, 2, 3}) = 3 = xA1 + xA2 + xA3 . We have 1 ≥ xA1 , similarly we
get 1 ≥ xA2 and 1 ≥ xA3 .
And we have
v ({1}) = 1 ≤ xA1 , v ({2}) = 1 ≤ xA2 , v ({3}) = 1 ≤ xA3 .
Therefore, core allocation is (xA1 , xA2 , xA3 ) = (1, 1, 1).
See figure 3 and 4.
Example 3
coalitions v()
∅ 0
{1} v ({1}) = 1
{2} v ({2}) = 1
{3} v ({3}) = 1
{1, 2} v ({1, 2}) = 2
{1, 3} v ({1, 3}) = 2
{2, 3} v ({2, 3}) = 2
{1, 2, 3} v ({1, 2, 3}) = 4
We want to find the core allocation of the above coalition game.
We have v ({1}) = 1 ≤ xA1 , v ({2}) = 1 ≤ xA2 , v ({3}) = 1 ≤ xA3 .
v ({1, 2}) = 2 ≤ xA1 + xA2 , v ({1, 3}) = 2 ≤ xA1 + xA3 , v ({2, 3}) =
2 ≤ xA2 + xA3 .
v ({1, 2, 3}) = 4 ≤ xA1 + xA2 + xA3 .
Substituting v ({2, 3}) = 2 ≤ xA2 + xA3 in
v ({1, 2, 3}) = 4 = xA1 + xA2 + xA3 . We have 2 ≥ xA1 , similarly we
get 2 ≥ xA2 and 2 ≥ xA3 .
And we have
v ({1}) = 1 ≤ xA1 , v ({2}) = 1 ≤ xA2 , v ({3}) = 1 ≤ xA3 .
Therefore, core allocation are
2 ≥ xA1 ≥ 1, 2 ≥ xA2 ≥ 1, 2 ≥ xA3 ≥ 1 and 4 = xA1 + xA2 + xA3
It is shown in figure 5 and 6.
Example 4
Suppose one person A owns an old car which values nothing to
him. There are two potential buyers, Buyer B values it at 1000 and
buyer C values it at 1050. The trade between these people can be
analysed based on coalition formation.
Coalitions Value or worth of coalitions
{A} 0
{B} 0
{C } 0
{A, B} 1000
{A, C } 1050
{B, C } 0
{A, B, C } 1050
{∅} 0
What is the core allocation?
We have xA ≥ 0, xB ≥ 0, xC ≥ 0,
xA + xB ≥ 1000, xA + xC ≥ 1050, xB + xC ≥ 0
xA + xB + xC ≥ 1050.
Substituting xA + xC ≥ 1050 in xA + xB + xC = 1050, we get
xB ≤ 0. We have xB ≥ 0, thus xB = 0.
Substituting xA + xB ≥ 1000 in xA + xB + xC = 1050, we get
xC ≤ 50. We have xC ≥ 0, thus 50 ≥ xC ≥ 0.
From this we get the core allocation as
C (v ) = {(xA , xB , xc ) = (1050 − d, 0, d)|0 ≤ d ≤ 50}.
