Minimum coloring problems with weakly perfect graphs

Abstract

The minimum coloring game allows to model economic situations where costly and potentially conflicting tasks have to be executed. The primitives of the game are a graph (describing conflicts between the respective tasks) and a cost function (giving the cost of completing any number of pairwise conflicting tasks). This setting gives rise to a cooperative game where agents have to split the minimum cost of completing all tasks. Our study of the core allows to find a necessary and sufficient condition guaranteeing its non-vacuity; and we describe a subset of allocations that are always in the core (when it is nonempty). These allocations put weights on the incompatible groups with highest cardinality, called maximal cliques. We then propose two cost sharing rules and their axiomatizations. The first rule assigns shares in proportion to the number of maximal cliques an agent belongs to; and the key property in its axiomatization prevents splitting and merging manipulations. The second rule is an extension of the well-known airport rule; and it requires every agent to pay a minimal fraction of the total cost.

A Appendix: Independence of properties

1.1 A.1 Theorem 3

1.1.1 A.1.1 No Merging No Splitting, Group Similarity but not Equal Cost per Element

Let $z_{i}$ be the number of cliques in ${\mathcal {Q}}$ that i belongs to. Let $y^{z}$ be such that $y_{i}^{z}=\frac{z_{i}}{\sum _{j\in N}z_{j}}C\left( \omega (G)\right) .$ We can verify that $y^{z}$ satisfies No Merging No Splitting and Group Similarity but not Equal Cost per Element.

1.1.2 A.1.2 No Merging No Splitting, Equal Cost per Element but not Group Similarity

Any graph is an intersection graph: Represent G by assigning objects to each i,  such that i, j are both assigned object k if and only if $(i,j)\in G.$ Let $O_{i}$ be the set of objects assigned to i. Arbitrarily number objects. In this context, a splitting manipulation for agent i is such that (a) $N^{\prime }=(N\backslash i)\cup \left\{ i_{1},\dots ,i_{K}\right\}$, with $K>1$, (b) $k\in O_{j}\Leftrightarrow k\in O_{j}^{\prime },$ for all $j\in N\backslash i,$ (c) $\bigcup _{k=1}^{K}O_{i_{k}}^{\prime }=O_{i}$ and (d) $\cap _{k}O_{k}^{i}=\emptyset .$ Put a weight of a(N, G, C) on the first object generating a maximal clique, and a weight of zero on all others. Let $y^{w}$ be the allocation method of the form (2) built from these weights. We can verify that $y^{w}$ satisfies No Merging No Splitting and Equal Cost per Element but not Group Similarity.

1.1.3 A.1.3 Group Similarity, Equal Cost per Element but not No Merging No Splitting

For every $S\in {\mathcal {M}}(G),$ let $\alpha _{S}=\sum _{i\in S}z_{i}.$ Let w be the weights on elements of ${\mathcal {M}}(G)$ such that $w_{S}=\frac{ \alpha _{S}}{\sum _{T\in {\mathcal {M}}(G)}\alpha _{T}}a(N,G,C).$ Let $y^{\alpha }$ be the allocation method of the form (2) built from these weights. We can verify that $y^{w}$ satisfies Group Similarity, Equal Cost per Element but not No Merging No Splitting.

1.2 A.2Theorem 4

1.2.1 A.2.1 Cost Additivity, Independence of Irrelevant Agents, Cost Rank Independence but not Unanimity Lower Bound

We can verify that $y^{MC}$ satisfies Cost Additivity, Independence of Irrelevant Agents, Cost Rank Independence but not Unanimity Lower Bound.

1.2.2 A.2.2 Cost Additivity, Independence of Irrelevant Agents, Unanimity Lower Bound but not Cost Rank Independence

Order agents such that $q_{1}\ge ...\ge q_{n}.$ Let $y^{0}$ be such that $y_{1}^{0}=\frac{C(1)}{\left| N\right| }+C(\omega (G))-C(1)$ and $y_{i}^{0}=\frac{C(1)}{\left| N\right| }$ for all $i>1.$ We can verify that $y^{0}$ satisfies Cost Additivity, Independence of Irrelevant Agents, Unanimity Lower Bound but not Cost Rank Independence.

1.2.3 A.2.3 Cost Additivity, Cost Rank Independence, Unanimity Lower Bound but not Independence of Irrelevant Agents

Let ${\bar{y}}$ be such that ${\bar{y}}_{i}=\frac{C(\omega (G))}{\left| N\right| }$ for all $i\in N.$ We can verify that ${\bar{y}}$ statisfies Cost Additivity, Cost Rank Independence, Unanimity Lower Bound but not Independence of Irrelevant Agents.

1.2.4 A.2.4 Independence of Irrelevant Agents, Cost Rank Independence, Unanimity Lower Bound but not Cost Additivity

Let $y^{p}$ be such that $y_{i}^{p}=\frac{C(1)}{\left| N\right| } +\left( C(\omega (G))-C(1)\right) \frac{q_{i}}{\sum _{j\in N}q_{j}}$ for all $i\in N.$ We can verify that $y^{p}$ satisfies Independence of Irrelevant Agents, Cost Rank Independence, Unanimity Lower Bound but not Cost Additivity.