Efficient extensions of the Myerson value based on endogenous claims from players

We achieve two efficient extensions of the Myerson value by introducing the Shapley value of the original game as endogenous claims of players. Capturing the feature of insufficiency in the surplus to be divided, we identify a graph-induced bankruptcy problem. Two classical bankruptcy rules, namely the constrained equal awards rule and the constrained equal losses rule, are employed to efficiently extend the Myerson value, which correspondingly brings about the efficient constrained equal awards Myerson value and the efficient constrained equal losses Myerson value. The two efficient graph game values are characterized by axiomatic approaches.