González-Díaz and Sánchez-Rodríguez introduced in 2007 the so-called Core-Center, a new and natural solution concept in Game Theory. It is basically defined as follows:
Core-Center: Let $(N,v)$ be a game with non-empty Core. The Core-Center of $(N,v)$ is defined as $\mathcal{K}(N,v)=\mathbb{E}[U(C)]$.
Where $\mathbb{E}[U(C)]$ is the result of endowing the Core with a uniform distribution and taking the expectation. In different words, $\mathbb{E}[U(C)]$ is the expected value of the core if all points in it are equally valuable.
Consider the following game $(N,v)$ with $N=3$, $v(1,2)=0.75$, $v(1,3)=0.50$, $v(2,3)=0.25$, $v(N)=1$ and $0$ else. If I did it correctly, the Core-Center for this game should be:
$\mathcal{K}(N,v)=\left(\frac{13}{24},\frac{7}{24},\frac{4}{24}\right)$.
I have two questions for this wonderful community:
- Could anyone prove or disprove my calculations for this example?
- Could anyone provide another $3$ or $4$ Player example and its Core-Center?
Please, show with detail how you made your computations!
Thank you all very much for your time.
EDIT: Just in case someone is interested, it turns out that the vector $\mathcal{K}(N,v)$ that I computed in my original question is not the Core Centre but the Alexia Value (Tijs, Borm, Lohmann and Quant, 2011), another Core-selector within the class of vectorial solution concepts in cooperative Game Theory.