Assume that the characteristic functions of different coalitions in a three-person group are given by: V(1,2,3) = $60
V(1,2) = $40, V(2,3) = $30, and V(1,3) = $35.
The values created by the individuals separately are given by:
V(1) = $20, V(2) = $16, V(3) = $10.
Which of the following allocations will lie within the core?

Question

Quizplus · Accepted Answer

To check if an allocation lies within the core, we need to verify if there is no coalition that can gain more by forming and deviating from the current allocation. Let's calculate the total value of the grand coalition and each possible coalition:
- Grand coalition (1,2,3): $60
- Coalition (1,2): $40
- Coalition (2,3): $30
- Coalition (1,3): $35
- Individual 1: $20
- Individual 2: $16
- Individual 3: $10

Now let's verify each option:
A) Payoffs to individuals 1,2, and 3 are $21, $17, and $22 respectively.
The total payoff is $60, which is the value of the grand coalition. However, coalition (1,2) can deviate and form a coalition worth $41, where each member gets $20 and $21, respectively. Therefore, this allocation is not in the core.

B) Payoffs to individuals 1,2, and 3 are $31, $17, and $12 respectively.
The total payoff is $60, which is the value of the grand coalition. However, coalition (2,3) can deviate and form a coalition worth $35, where member 2 gets $16 and member 3 gets $19. Therefore, this allocation is not in the core.

C) Payoffs to individuals 1,2, and 3 are $21, $26, and $13 respectively.
The total payoff is $60, which is the value of the grand coalition. However, coalition (1,3) can deviate and form a coalition worth $45, where member 1 gets $21 and member 3 gets $24. Therefore, this allocation is not in the core.

D) Payoffs to individuals 1,2, and 3 are $25, $18, and $17 respectively.
The total payoff is $60, which is the value of the grand coalition. There is no coalition that can deviate and form a coalition worth more than $60. Therefore, this allocation is in the core.

Assume That the Characteristic Functions of Different Coalitions in a Three-Person