The core -symmetric players

Question

We have $n$-persons ($n\ge 3$) cooperative game. And we know that player $1$ and $2$ are symmetric. So for each element $(x_1,x_2,...,x_n)$ from the core we have $x_1=x_2$ ? Is that true ? Never seen example when this is not true, but how do it come from ?

Answer 1

Consider the following three person TU-game in the usual lexi-order of coalitions

v=[0,0,0,8,5,5,14]

Notice, that in this game players $1$ and $2$ are substitutes, i.e., the transposition (1,2) is a symmetry in the game. Computing the core-vertices of the game, we get

      0   8   6
      0   9   5
      9   5   0
      5   9   0
      8   0   6
      9   0   5 

From these vertices, we observe that it does not hold for all core elements $x_{1}=x_{2}$.