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Facets of the cone of totally balanced games


The class of totally balanced games is a class of transferable-utility coalitional games providing important models of cooperative behavior used in mathematical economics. They coincide with market games of Shapley and Shubik and every totally balanced game is also representable as the minimum of a finite set of additive games. In this paper we characterize the polyhedral cone of totally balanced games by describing its facets. Our main result is that there is a correspondence between facet-defining inequalities for the cone and the class of special balanced systems of coalitions, the so-called irreducible min-balanced systems. Our method is based on refining the notion of balancedness introduced by Shapley. We also formulate a conjecture about what are the facets of the cone of exact games, which addresses an open problem appearing in the literature.

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Correspondence to Tomáš Kroupa.

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This research has been supported by the Grant GAČR No. 16-12010S. We are grateful to Eric Quaeghebeur for providing us with results of his computations related to his PhD thesis.

A Min-balanced systems for a small number of players

A Min-balanced systems for a small number of players

Here we give a list of all permutational types of non-trivial min-balanced systems for at most four players. We present a type representative, indicate what is the type of the complementary system (Definition 3.4) and whether the type is irreducible, give the standard inequality ascribed to the system (see Sect. 3.3) and say what is the number of systems of this type. In order to shorten the notation we write abc instead of \(\{a,b,c\}\).

1.1 A.1 Two players

The only non-trivial min-balanced system on \(N=\{a,b\}\) is as follows.

  1. 1.

    \(\mathcal{B}=\{a,b\}\) self-complementary, irreducible

    \(m(ab)-m(a)-m(b)+m(\emptyset )\ge 0\)\(1\times \)

1.2 A.2 Three players

The following are all three types of 5 non-trivial min-balanced systems on \(N=\{a,b,c\}\).

  1. 1.

    \(\mathcal{B}=\{ a, b, c\}\)complementary type 3.

    \(m(abc)-m(a)-m(b)-m(c)+2\cdot m(\emptyset )\ge 0\)\(1\times \)

  2. 2.

    \(\mathcal{B}=\{ a, bc\}\) self-complementary, irreducible

    \(m(abc)-m(a)-m(bc)+m(\emptyset )\ge 0\)\(3\times \)

  3. 3.

    \(\mathcal{B}=\{ ab, ac , bc\}\)complementary type 1., irreducible

    \(2\cdot m(abc)-m(ab)-m(ac)-m(bc)+m(\emptyset )\ge 0\) \(1\times \)

Thus, one has two types of 4 irreducible min-balanced systems on \(N=\{a,b,c\}\).

1.3 A.4 Four players

The following are all nine types of 41 non-trivial min-balanced system on \(N=\{a,b,c,d\}\).

  1. 1.

    \(\mathcal{B}=\{ a, b, c, d\}\)complementary type 9.

    \(m(abcd)-m(a)-m(b)-m(c)-m(d)+3\cdot m(\emptyset )\ge 0\)\(1\times \)

  2. 2.

    \(\mathcal{B}=\{a, b, cd\}\)complementary type 6.

    \(m(abcd)-m(a)-m(b)-m(cd)+2\cdot m(\emptyset )\ge 0\)\(6\times \)

  3. 3.

    \(\mathcal{B}=\{ ab, cd\}\)self-complementary, irreducible

    \(m(abcd)-m(ab)-m(cd)+m(\emptyset )\ge 0\)\(3\times \)

  4. 4.

    \(\mathcal{B}=\{ a, bcd\}\)self-complementary, irreducible

    \(m(abcd)-m(a)-m(bcd)+m(\emptyset )\ge 0\)\(4\times \)

  5. 5.

    \(\mathcal{B}=\{a, bc, bd, cd\}\)complementary type 8.

    \(2\cdot m(abcd)-2\cdot m(a)-m(bc)-m(bd)-m(cd)+3\cdot m(\emptyset )\ge 0\)\(4\times \)

  6. 6.

    \(\mathcal{B}=\{ ab, acd, bcd\}\)complementary type 2., irreducible

    \(2\cdot m(abcd)-m(ab)-m(acd)-m(bcd)+m(\emptyset )\ge 0\)\(6\times \)

  7. 7.

    \(\mathcal{B}=\{ a, bd, cd, abc\}\)self-complementary

    \(2\cdot m(abcd)-m(a)-m(bd)-m(cd)-m(abc)+2\cdot m(\emptyset )\ge 0\)\(12\times \)

  8. 8.

    \(\mathcal{B}=\{ ab, ac, ad, bcd\}\)complementary type 5., irreducible

    \(3\cdot m(abcd)-m(ab)-m(ac)-m(ad)-2\cdot m(bcd)+2\cdot m(\emptyset )\ge 0\)\(4\times \)

  9. 9.

    \(\mathcal{B}=\{ abc, abd, acd, bcd\}\)complementary type 1., irreducible

    \(3\cdot m(abcd)-m(abc)-m(abd)-m(acd)-m(bcd)+m(\emptyset )\ge 0\)\(1\times \)

Thus, there are five types of 18 irreducible min-balances systems on \(N=\{a,b,c,d\}\).

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Kroupa, T., Studený, M. Facets of the cone of totally balanced games. Math Meth Oper Res 90, 271–300 (2019).

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  • Coalitional game
  • Totally balanced game
  • Balanced system
  • Polyhedral cone