## Discussion Papers on Economics,

University of Southern Denmark, Department of Economics

# No 6/2020: Characterization of TU games with stable cores by nested balancedness

*Michel Grabisch* () and *Peter Sudhölter* ()

Additional contact information

Michel Grabisch: Paris School of Economics, Postal: University of Paris I, 106.112, Bd. de l'Hôpital, 75013 Paris, France

Peter Sudhölter: Department of Business and Economics, Postal: University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

**Abstract:** A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T ) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.

**Keywords:**
Domination; stable set; core; TU game

**JEL-codes:**
C71

20 pages, June 15, 2020

**Full text files**

dpbe6_2020.pdf Full text

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