# Minimum coloring problems with weakly perfect graphs

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## 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.

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## Notes

1. All concepts discussed in the introduction are formally defined in Section 2.

2. The minimum coloring game of Example 2 is not representable as a job scheduling game. To see why, consider the context where each agent’s job must be executed within a given time interval (without any interruption) and two agents are in conflict whenever their jobs’ time intervals overlap. Assume without loss of generality that agent 1’s job has the earliest start time. Then assuming that our conflict graph contains the pairs $$\{1,2\}, \{2,3\}, \{3,4\}$$ but not $$\{1,3\}$$ and $$\{2,4\}$$, it is easy to see that player 4’s time interval cannot overlap with that of player 1. Thus the graphs $${\tilde{G}}=\{\{1,2\}; \{2,3\}; \{3,4\}; \{1,4\}\}$$ and G (from Example 2) are impossible to obtain in the job scheduling model of Bahel and Trudeau (2019). One can construct other examples illustrating this point. The same reasoning applies to altitude problems.

3. For job scheduling games, the method is known as the peak-demand rule. Bahel and Trudeau (2019) introduced another rule, the peak-interval rule, which considers the length in time for the conflicts induced by maximal cliques. There is no equivalent in the minimum coloring game.

4. The authors thank Hans Peters for suggesting the study of this rule.

## References

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## Acknowledgements

We thank two anonymous referees for comments that have improved the article. Christian Trudeau acknowledges financial support by the Social Sciences and Humanities Research Council of Canada (grant number 435-2019-0141).

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Correspondence to Christian Trudeau.

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## 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 ij 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(NGC) 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.

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Bahel, E., Trudeau, C. Minimum coloring problems with weakly perfect graphs. Rev Econ Design 26, 211–231 (2022). https://doi.org/10.1007/s10058-021-00265-4

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• DOI: https://doi.org/10.1007/s10058-021-00265-4