Appendix
1.1 1. Graphical representation of a BCBM with \(\alpha = 1\)
1.2 2. Final payoffs obtainable by players by calling the grand coalition depending on the identity of first proposer
1.3
j
first proposer
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \pi _j&= \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{2\alpha (1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )^2}-\frac{2(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{3-2\alpha }-\frac{(6-8\alpha +3\alpha ^2)v(j,k)}{(3-2\alpha )(2-\alpha )^2}+\frac{(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{(4-3\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{(6-6\alpha +\alpha ^2)v(j,k)}{(3-2\alpha )(2-\alpha )^2}+\frac{(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}.\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _j&= \frac{v(N)}{3-2\alpha }+\frac{(2-6\alpha +3\alpha ^2)v(i,j)}{(3-2\alpha )(2-\alpha )^2}+\frac{1}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) ,\\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}\\&\quad -\frac{(5-9\alpha +7\alpha ^2-2\alpha ^3)}{(3-2\alpha )}\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) , \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{(1-\alpha )(4-3\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )^2}\\&\quad +(1-\alpha )\frac{(4-5\alpha +2\alpha ^2)}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) . v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _j&= \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}-\frac{2(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )}-\frac{(4-3\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}+\frac{(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{v(i,j)}{(3-2\alpha )}+\frac{v(j,k)}{(3-2\alpha )}+\frac{(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}.\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _j&= \frac{v(N)}{3-2\alpha }+\frac{(2-6\alpha +3\alpha ^2)v(i,j)}{(3-2\alpha )(2-\alpha )^2}+\frac{v(j,k)}{(2-\alpha )}-\frac{2(12-25\alpha +16\alpha ^2-8\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^4},\\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{v(j,k)}{(2-\alpha )}+\frac{(12-21\alpha +12\alpha ^2-2\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{(1-\alpha )(4-3\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )^2}+\frac{2\alpha (1-\alpha )(2-3\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4}.\\ \end{aligned}$$
1.4
i
first proposer
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _i&= \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(12-16\alpha +5\alpha ^2)v(j,k)}{(3-2\alpha )(2-\alpha )^3}+\frac{(4-8\alpha +3\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4},\\ \pi _j&= \frac{(1-\alpha )(v(N)+v(i,j))}{3-2\alpha }+\frac{(1-\alpha )}{3-2\alpha }\left( \frac{\alpha v(j,k)}{(2-\alpha )^2}+\frac{(4-\alpha )v(i,k)}{(2-\alpha )^3}\right) , \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{(4-3\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{(12-18\alpha +8\alpha ^2-\alpha ^3)v(j,k)}{(3-2\alpha )(2-\alpha )^3}\\&\quad +\frac{(4-6\alpha +4\alpha ^2-\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^4}.\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _i&= \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(1-\alpha )(7-3\alpha )}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^3}-\frac{v(i,k)}{(2-\alpha )^4}\right) ,\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(1-\alpha )(4-\alpha )}{(3-2\alpha )}\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) , \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(1-\alpha )(1-3\alpha +\alpha ^2)}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^3}-\frac{v(i,k)}{(2-\alpha )^4}\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _i&= \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{2v(j,k)}{3-2\alpha }\\&\quad +\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)+v(j,k)}{3-2\alpha }-\frac{(16-28\alpha +17\alpha ^2-4\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } -\frac{(4-3\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{v(j,k)}{3-2\alpha }-\frac{(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}.\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _i&= \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{v(j,k)}{2-\alpha }+\frac{(10-15\alpha +8\alpha ^2-2\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{v(j,k)}{2-\alpha }-\frac{(12-19\alpha +10\alpha ^2-2\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _k&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{2(1-\alpha )^2v(i,k)}{(3-2\alpha )(2-\alpha )^3}. \end{aligned}$$
k
first proposer
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _k&= \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(12-23\alpha +15\alpha ^2-3\alpha ^3)v(j,k)}{(3-2\alpha )(2-\alpha )^3}\\&\quad +\frac{\alpha (1-3\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4},\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{3-2\alpha }+\frac{\alpha (1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha ^3)}-\frac{(1-\alpha )(4-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^4}, \\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{v(i,j)}{3-2\alpha }-\frac{(6-8\alpha +3\alpha ^2)v(j,k)}{(3-2\alpha )(2-\alpha )^2}+\frac{(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}.\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _k&= \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{(3-2\alpha )(2-\alpha )}+\alpha \left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) ,\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{(1-\alpha )(3-\alpha )}{(3-2\alpha )}\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) , \\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(3-3\alpha +\alpha ^2)}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _k&= \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}\\&\quad -\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)+v(j,k)}{3-2\alpha }-\frac{(16-28\alpha +17\alpha ^2-4\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha } +\frac{v(i,j)}{3-2\alpha } - \frac{(4-3\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}+\frac{(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3}.\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _k&= \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2\alpha (1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^3},\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{v(j,k)}{2-\alpha }-\frac{(12-19\alpha +10\alpha ^2-2\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}, \\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha } +\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{v(j,k)}{2-\alpha } +\frac{(12-21\alpha +12\alpha ^2-2\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^3}. \end{aligned}$$
1.1 3. Final payoffs obtainable by players by calling their most profitable 2-players coalition depending on the identity of first proposer
1.2
j
first proposer (calling \(\left\{ j,k \right\} \))
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(12-23\alpha +15\alpha ^2-3\alpha ^3)v(j,k)}{(3-2\alpha )(2-\alpha )^3}\right. \\&\quad \left. + \frac{(1-3\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4},0\right) .\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\alpha \Bigr (\frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\Bigl ),0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(i,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}\right. \\&\left. \quad -\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(i,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2\alpha (1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )},0\right) .\\ \pi _k^2&> 0 \wedge v(i,k)> (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-9\alpha +4\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}\right. \\&\left. \quad -\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ \pi _k^2&\le 0 \wedge v(i,k) > (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _j&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-9\alpha +4\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2\alpha (1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )},0\right) .\\ \end{aligned}$$
1.3
i
first proposer (calling \(\left\{ i,k \right\} \))
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(j,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(12-23\alpha +15\alpha ^2-3\alpha ^3)v(j,k)}{(3-2\alpha )(2-\alpha )^3}\right. \\&\left. \quad + \frac{(1-3\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4},0\right) .\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(j,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\alpha \Bigr (\frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\Bigl ),0\right) .\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(j,k)> (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-8\alpha +4\alpha ^2) v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(12-23\alpha +15\alpha ^2-3\alpha ^3)v(j,k)}{(3-2\alpha )(2-\alpha )^3}\right. \\&\left. \quad -\frac{\alpha (1-3\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^4},0\right) .\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(j,k)> (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-8\alpha +4\alpha ^2) v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\alpha \Bigr (\frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\Bigl ),0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(j,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )^2}\right. \\&\quad \left. -\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(j,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{\alpha v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2\alpha (1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(j,k)> (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-8\alpha +4\alpha ^2) v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )^2}\right. \\&\left. \quad -\frac{4(1-\alpha )(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(j,k) > (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _i&= v(i,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{(6-8\alpha +4\alpha ^2) v(j,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{3-2\alpha }+\frac{2\alpha (1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) . \end{aligned}$$
1.4
k
first proposer (calling \(\left\{ j,k \right\} \))
$$\begin{aligned} v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0\\ \pi _k&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}\right. \\&\left. \quad -\frac{3(1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{2\alpha (1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )^2} - \frac{2(2-2\alpha +\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) .\\ v(i,k)&\le (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0\\ \pi _k&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{3(1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{(2-6\alpha +3\alpha ^2)v(i,j)}{(3-2\alpha )(2-\alpha )^2}\right. \\&\left. \quad +\frac{1}{3-2\alpha }\Bigr (\frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3} \Bigl ),0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2> 0 \wedge v(i,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _k&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{3(1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}-\frac{2(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ v(i,k)&> (2-\alpha )\frac{v(j,k)}{3-2\alpha } \wedge \pi _k^2 \le 0 \wedge v(i,k) \le (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _k&= \quad v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{3(1-\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{(2-6\alpha +3\alpha ^2)v(i,j)}{(3-2\alpha )(2-\alpha )^2}+\frac{v(j,k)}{2-\alpha }\right. \\&\left. \quad -\frac{2(12-25\alpha +16\alpha ^2-8\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^4},0\right) .\\ \pi _k^2&> 0 \wedge v(i,k)> (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _k&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(6-5\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{2(1-\alpha )v(j,k)}{(3-2\alpha )(2-\alpha )}-\frac{2(8-12\alpha +5\alpha ^2)v(i,k)}{(3-2\alpha )(2-\alpha )^3},0\right) .\\ \pi _k^2&\le 0 \wedge v(i,k) > (2-\alpha )\frac{v(i,j)}{3-2\alpha }\\ \pi _k&= v(j,k) - \alpha \max \left( \frac{v(N)}{3-2\alpha }+\frac{(1-2\alpha )v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(6-5\alpha )v(i,k)}{(3-2\alpha )(2-\alpha )^2},0\right) +\\&\quad -(1-\alpha )\max \left( \frac{v(N)}{3-2\alpha }+\frac{(2-6\alpha +3\alpha ^2)v(i,j)}{(3-2\alpha )(2-\alpha )^2}+\frac{v(j,k)}{2-\alpha }\right. \\&\quad \left. -\frac{2(12-25\alpha +16\alpha ^2-8\alpha ^3)v(i,k)}{(3-2\alpha )(2-\alpha )^4},0\right) . \end{aligned}$$
1.5 4. Example of computation procedure to find the payoffs for \(\alpha \in (0,1)\)
Case: \(v(i,k) \le (2-\alpha )\frac{v(j,k)}{3-2\alpha }\), j selected as proposer choosing \(\left\{ i, j\right\} \). The grand coalition is not present any more.
$$\begin{aligned}&j: v(i,j) - x_i,\\&x_i = \alpha \left( (3-2\alpha )\frac{v(i,k)}{(2-\alpha )^2} - \frac{v(j,k)}{2-\alpha }\right) + (1-\alpha )(v(i,j) - x_j), \end{aligned}$$
\(x_i\) is the minimal offer that i would accept, that is therefore equal to the expected payoff i would get by rejecting j’s offer. Note that \((3-2\alpha )\frac{v(i,k)}{(2-\alpha )^2} - \frac{v(j,k)}{2-\alpha }\) is the payoff that i would obtain if, by rejecting the proposal of j, v(i, j) gets burned, leaving only v(j, k) and v(i, k) as non–zero coalitions. Since v(i, j) gets burned with probability \(\alpha \), this value is multiplied by it. The best option for i, in case v(i, j) does not get burned, is to call back \(\left\{ i,j\right\} \), but then i needs to offer j an acceptable offer \(x_j\). We then need to write down this last and t0 solve the system of two equations in two unknowns to have \(x_i\) and, then, to compute the payoff of j as proposer of \(\left\{ i,j\right\} \).
$$\begin{aligned} {\left\{ \begin{array}{ll} x_i = \alpha \left( (3-2\alpha )\frac{v(i,k)}{(2-\alpha )^2} - \frac{v(j,k)}{2-\alpha }\right) + (1-\alpha )(v(i,j) - x_j),\\ x_j = \alpha \left( \frac{v(j,k)}{2-\alpha } - \frac{v(i,k)}{(2-\alpha )^3}\right) +(1-\alpha )(v(i,j) - x_i). \end{array}\right. } \end{aligned}$$
Similar to before, \(\frac{v(j,k)}{2-\alpha } - \frac{v(i,k)}{(2-\alpha )^3}\) is the expected payoff j can get if, rejecting i’s offer, v(i, j) gets burned. By solving this system we get \(x_i = (1-\alpha )\frac{v(i,j)}{2-\alpha }-(1-\alpha )\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) \), and the payoff obtainable by j as proposer of v(i, j) is therefore equal to \(v(i,j) - x_i = \frac{v(i,j)}{2-\alpha }+(1-\alpha )\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) \).
Let us now consider an example of computation of payoffs when the grand coalition is called.
Case: \(v(i,k) \le (2-\alpha )\frac{v(j,k)}{3-2\alpha }\) and \(\pi _k^2 > 0\). k, as proposer, calls the grand coalition.
We then have \(k: v(N) - x_j - x_i\) and the system of equations to be solved is:
$$\begin{aligned} {\left\{ \begin{array}{ll} x_j = \alpha \left( \frac{v(i,j)}{2-\alpha }+(1-\alpha )\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) \right) + (1-\alpha )(v(N) - x_i - x_k),\\ x_i = \alpha \left( \frac{v(i,j)}{2-\alpha }-\frac{v(j,k)}{(2-\alpha )^2}+\frac{v(i,k)}{(2-\alpha )^3}\right) + (1-\alpha )(v(N) - x_i - x_k),\\ x_k = \alpha \left( \frac{v(i,j)}{2-\alpha }+(1-\alpha )\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^2}\right) \right) + (1-\alpha )(v(N) - x_i - x_k), \end{array}\right. } \end{aligned}$$
Note that, in the brackets following \(\alpha \), we have the expected payoff of each player in case she rejects the offer of the proposer and the grand coalition gets burned. In particular, note that, for \(x_j\), this is exactly the payoff we obtained before since the underlying assumptions are the same. Bw solving this system of three equations in three unknowns, we get the same result of 2 with k first proposer and \(v(i,k) \le (2-\alpha )\frac{v(j,k)}{3-2\alpha }\) and \(\pi _k^2 > 0\), namely:
$$\begin{aligned} \pi _k&= \frac{v(N)}{3-2\alpha }-\frac{2v(i,j)}{(3-2\alpha )(2-\alpha )}+\alpha \left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) ,\\ \pi _j&= \frac{(1-\alpha )v(N)}{3-2\alpha }+\frac{v(i,j)}{(3-2\alpha )(2-\alpha )}+\frac{(1-\alpha )(3-\alpha )}{(3-2\alpha )}\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) , \\ \pi _i&= \frac{(1-\alpha )v(N)}{3-2\alpha } - \frac{v(i,j)}{(3-2\alpha )(2-\alpha )}-\frac{(3-3\alpha +\alpha ^2)}{3-2\alpha }\left( \frac{v(j,k)}{(2-\alpha )^2}-\frac{v(i,k)}{(2-\alpha )^3}\right) . \end{aligned}$$
