How to define characteristic function of the coalitional game based on the situation?

Question

The Joy has three sons and there is a testament for their sons that thery should receive 300, 200 and 100 units respectively, after his death. This implies that. give a total amount of $\alpha$ units left after the joy death, the three sons can only claim 300,200,100 respectively, out of the $\alpha$ units. If joy dies, the amount of money left is not enough for this distribution, it is recommended the following:

1. If $\alpha = 100$ is available after the man dies, then each son gets 100/3

2. If $\alpha = 200$ is available after the man dies, son 1 gets 50, and the other two get 75 each.

3. If $\alpha = 300$ is available after the man dies, son 1 gets 50, son 2 gets 100 and son 3 gets 150

As my view, the question is fighting for the heritage. So, we based on it to build the characteristic function. Thus, we can divided three cases for different sons, for example,

• son 1 gets 100/3 when α = 100 that can decribed as son1 {100 units} = 100/3
• son 1 gets 50 when α = 200 that can decribed as son1 {200 units} = 50

• son 1 gets 50 when α = 200 that can decribed as son1 {300 units} = 50

• son2 .....

• son3 .....

I'm not sure that's right to define the characteristic function of the coalitional game. In the subsequent, we based on characteristic function to build the Shapley value to allocate the testament.

Could someone help me?

Answer 1

Your starting point is a general bankruptcy problem that is defined by an ordered pair $(E,\mathbf{d})$, where $E \in \mathbb{R}$ and $\mathbf{d}:=(d_{1},\ldots,d_{n}) \in \mathbb{R}^{n}$ s.t. $d_{i} \ge 0$ for all $1 \le i \le n$ and $0 \le E \le \sum_{i=1}^{n}\,d_{i}$ is given. From this generalized bankruptcy problem a bankruptcy game $(N,v_{E;\mathbf{d}})$ in characteristic function form can be derived while defining

$v_{E;\mathbf{d}}(S):= \max\big(0,E-\sum_{j \in N\backslash S}\,d_{j}\big)$

for all $S \subseteq N$. In case that $E_{1}=100$, and $\mathbf{d}=(100,200,300)$, then for the associated bankruptcy game we get $v_{E_{1},\mathbf{d}}(N)=100$, otherwise zero, and the Shapley value distributes $100/3$ for each player. For $E_{2}=200$, we get $shv(v_{E_{2},\mathbf{d}})=(100,250,250)/3$, and finally for $E_{3}$, we obtain $shv(v_{E_{3},\mathbf{d}})=(50,100,150)$.

These values are different from the allocations assigned by the generalized contested garment principle under the bankruptcy problem, which coincide with the nucleoli of the corresponding bankruptcy games. I left this as an exercise up to you.