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A proportional value for cooperative games with a coalition structure

Abstract

We introduce a solution concept for cooperative games with transferable utility and a coalition structure that is proportional for two-player games. Our value is obtained from generalizing a proportional value for cooperative games with transferable utility (Ortmann 2000) in a way that parallels the extension of the Shapley value to the Owen value. We provide two characterizations of our solution concept, one that employs a property that can be seen as the proportional analog to Myerson’s balanced contribution property; and a second one that relies on a consistency property.

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Notes

  1. 1.

    This type of ambient tax is also postulated by Segerson (1988). Note that situations with multiple polluters usually induce further incentive problems, which are not taken into consideration in our paper.

  2. 2.

    The idea of proportional sharing is also central to the CS-value introduced by Alonso-Meijide and Carreras (2011), although their solution concept yields the standard solution for two-player games.

  3. 3.

    The external game is also known as the quotient game (Owen 1977).

  4. 4.

    In this paper we focus on positive games. Thus, the axioms are only states for positive games.

  5. 5.

    The intermediate game property is also known as the Game between Coalitions Property (Winter 1992).

  6. 6.

    Formally, \(\pi \) is also an argument of the payoff function but for notational parsimony we write \(v_{\phi }^{C}\) instead of \(v_{\phi }^{C,\pi }\).

  7. 7.

    Note that the external game \(v^{\pi }\), the internal games \(v_{\psi }^{\pi \left( i\right) }\), and all games \(\left( v,\pi \right) |_{S}, S\subseteq N \) are positive if the original game \(\left( v,\pi \right) \) is positive. Hence, the value introduced in the theorem is well defined.

  8. 8.

    Balanced contributions property: For all \(N\in \mathcal {N}, v\in \mathcal {G}^{N}\), and \(i,j\in N, i\ne j,\) we have \(\phi _{i}\left( v\right) -\phi _{i}\left( v|_{N{\setminus }\left\{ j\right\} }\right) =\phi _{j}\left( v\right) -\phi _{j}\left( v|_{N{\setminus }\left\{ i\right\} }\right) .\)

References

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  15. Vorob’ev, N. N., & Liapounov, A. N. (1998). The proper Shapley value. In L. A. Petrosjan & V. V. Mazalov (Eds.), Game theory and applications IV (pp. 155–159). New York: Nova Science Publishers.

  16. Winter, E. (1992). The consistency and potential for values of games with coalition structure. Games and Economic Behavior, 4, 132–144.

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Acknowledgments

I would like to thank Sylvain Béal, André Casajus, Philippe Solal, Winfried Hochstättler, Harald Wiese, and an anonymous referee for helpful comments on previous versions of this paper. Financial support from the Université Charles-de-Gaulle—Lille 3 is gratefully acknowledged.

Author information

Correspondence to Frank Huettner.

Appendix

Appendix

We prepare the proofs of the following results by a technical lemma, stating that it does not matter whether we consider the restriction of an internal game or the internal game of the corresponding restriction of the original game.

Lemma 5

Let \(\phi \) be a TU-value. For all \(N\in \mathcal {N} , v\in \mathcal {G}^{N},\pi \in \Pi ^{N},\) and \(C^{\prime }\subseteq C\in \pi \), we have \(\big ( v_{\phi }^{C}\big ) |_{C^{\prime }}=\left( v|_{N{\setminus } D}\right) _{\phi }^{C^{\prime }}\) where \(D:=C{\setminus } C^{\prime }\).

Proof

For \(S\subseteq C^{\prime }\) we have \(\left( \pi {\setminus }\left\{ C\right\} \right) \cup \left\{ S\right\} =\left( \pi |_{N{\setminus } D}\setminus \left\{ C^{\prime }\right\} \right) \cup \left\{ S\right\} \). Moreover,

$$\begin{aligned} v^{\pi }{\big |}_{\left( \pi {\setminus }\left\{ C\right\} \right) \cup \left\{ S\right\} }\left( \mu \right) =\left( v{\big |}_{N{\setminus } D}\right) ^{\left( \pi {\big |}_{N{\setminus } D}\setminus \left\{ C^{\prime }\right\} \right) \cup \left\{ S\right\} }\left( \mu \right) \end{aligned}$$

for all \(\mu \subseteq \left( \pi {\setminus }\left\{ C\right\} \right) \cup \left\{ S\right\} \). Thus,

$$\begin{aligned} \phi _{S}\left( v^{\pi }{\big |}_{\left( \pi {\setminus }\left\{ C\right\} \right) \cup \left\{ S\right\} }\right) =\phi _{S}\left( \left( v{\big |}_{N{\setminus } D}\right) ^{\left( \pi {\big |}_{N{\setminus } D}\setminus \left\{ C^{\prime }\right\} \right) \cup \left\{ S\right\} }\right) ^{\pi {\big |}_{\left( N{\setminus } C\right) \cup S}} \end{aligned}$$

for all \(S\subseteq C^{\prime }\), i.e., \(\big ( v_{\phi }^{C}\big ) |_{C^{\prime }}\left( S\right) =\left( v|_{N{\setminus } D}\right) _{\phi }^{C^{\prime }}\left( S\right) \) for all \(S\subseteq C^{\prime }\). \(\square \)

Proof of Lemma 1

Let \(\phi \) satisfy E*, let \(\varphi \) be given by Eq. 7, and let \(v\in \mathcal {G}^{N}\). Consider \(\pi =\left\{ N\right\} \). Then, the internal game equals the original game, \(v_{\phi }^{N}=v\), since

$$\begin{aligned} v_{\phi }^{N}\left( S\right) \overset{\text {(5)}}{=}\phi _{S}\left( \left( v{\big |}_{N{\setminus }\left( C\setminus S\right) }\right) ^{\left( \pi {\setminus }\left\{ C\right\} \right) \cup \left\{ S\right\} }\right) \overset{\mathbf{E*}\text { of }\phi }{=}v\left( S\right) \end{aligned}$$

for all \(S\subseteq N\). This implies \(\phi \big ( v_{\phi }^{N}\big ) =\phi \left( v\right) \) and with Eq. 7 we obtain

$$\begin{aligned} \varphi \left( v,\left\{ N\right\} \right) =\phi \left( v\right) \text { for all }N\in \mathcal {N},\text { and }v\in \mathcal {G}^{N}. \end{aligned}$$
(10)

Now, consider arbitrary \(\pi \in \Pi ^{N}\). For all \(C\in \pi \), we have

$$\begin{aligned} v_{\phi }^{C}\left( C\right) \overset{\text {(7)}}{=}\phi _{C}\left( v^{\pi }\right) \overset{\text {(10)}}{=}\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) . \end{aligned}$$

This yields

$$\begin{aligned} \sum _{i\in C}\varphi _{i}\left( v,\pi \right) \overset{\text {(7)}}{=}\sum _{i\in C}\phi _{i}\left( v_{\phi }^{C}\right) \overset{\mathbf{E*}\text { of }\phi }{=}v_{\phi }^{C}\left( C\right) =\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) , \end{aligned}$$

what establishes IG. Regarding E of \(\varphi \), we have

$$\begin{aligned} \sum _{C\in \pi }\sum _{i\in C}\varphi _{i}\left( v,\pi \right) \overset{\mathbf{IG \text{ o }f }\varphi }{=}\sum _{C\in \pi }\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) \overset{\text {(10)}}{=} \sum _{C\in \pi }\phi _{C}\left( \pi ,v^{\pi }\right) \overset{\mathbf{E*}\text { of }\phi }{=}v\left( N\right) \end{aligned}$$

\(\square \)

Proof of Theorem 2

Existence: By Definition 1, E* of \(\psi \), and Lemma 1, \(\varpi \) fulfills E and IG. Further, for all \(N\in \mathcal {N}, v\in \mathcal {G}_{++}^{N},\pi \in \Pi ^{N},C\in \pi \), and \(i,j\in C\) we have

$$\begin{aligned} \frac{\varpi _{i}\left( v,\pi \right) }{\varpi _{j}\left( v,\pi \right) } \!=\! \frac{\psi _{i}\left( v_{\psi }^{C}\right) }{\psi _{j}\left( v_{\psi } ^{C}\right) } \!=\! \frac{\psi _{i}\left( v_{\psi }^{C}{\big |}_{C{\setminus }\left\{ j\right\} }\right) }{\psi _{j}\left( v_{\psi }^{C}{\big |}_{C{\setminus }\left\{ i\right\} }\right) } \!=\! \frac{\psi _{i}\left( \left( v{\big |}_{C{\setminus }\left\{ j\right\} }\right) _{\psi }^{C{\setminus }\left\{ j\right\} }\right) }{\psi _{j}\left( \left( v{\big |}_{C{\setminus }\left\{ i\right\} }\right) _{\psi }^{C{\setminus }\left\{ i\right\} }\right) } \!=\! \frac{\varpi _{i}\left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }}{\varpi _{j}\left( v,\pi \right) {\big |}_{N{\setminus }\left\{ i\right\} }}, \end{aligned}$$

where the first and the fourth equation follow from the definition of \(\varpi \); the second equation is due to the fact that \(\psi \) fulfills PR ;and the third equation follows from Lemma 5 with \(D=\left\{ j\right\} ,\left\{ i\right\} \), respectively. Therefore, \(\varpi \) fulfills IPR.

Uniqueness: Let \(\varphi \) be a value that obeys E, IG, and IPR. We need to show \(\varphi =\varpi \).

For all \(N\in \mathcal {N}, v\in \mathcal {G}_{++}^{N},\) let us first consider \(\pi =\left\{ N\right\} \). IPR requires \(\varphi \) to preserve ratios among all pairs of players \(i,j\in N\) in \(\left( v,\pi \right) \) and in all subgames \(\left( v,\pi \right) |_{S}\) since \(\pi =\left\{ N\right\} \) implies \(\pi |_{S}=\left\{ S\right\} \); hence, IPR becomes as strong as PR. According to Ortmann (2000 Theorem 2.6), PR and E characterize the proportional TU-value \(\psi \). Thus, we have (*) \(\varphi \left( v,\left\{ N\right\} \right) =\psi \left( v\right) \) for all \(N\in \mathcal {N}, v\in \mathcal {G}_{++}^{N}\).

Now, consider arbitrary \(\pi \in \Pi ^{N}\). Using (*) for external games, we get (**) \(\varphi \left( v^{\pi },\left\{ \pi \right\} \right) =\psi \left( v^{\pi }\right) \) for all \(N\in \mathcal {N},v\in \mathcal {G}_{++}^{N}\), and \(\pi \in \Pi ^{N}\). We now obtain

$$\begin{aligned} \sum _{i\in C}\varphi _{i}\left( v,\pi \right) \overset{\mathbf{IG}}{=}\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) \overset{\text {(**)}}{=}\psi _{C}\left( v^{\pi }\right) \quad \text { for all }\;C\in \pi , \end{aligned}$$
(11)

and it is left to show how \(\psi _{C}\left( v^{\pi }\right) \) is divided within the component if \(\left| C\right| >1\). By IPR,

$$\begin{aligned} \varphi _{j}\left( v,\pi \right) =\frac{\varphi _{i}\left( v,\pi \right) \cdot \varphi _{j}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ i\right\} }\right) }{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) } \end{aligned}$$

for all \(j\in \pi \left( i\right) \). Insertion into Eq. 11 yields

$$\begin{aligned} \psi _{\pi \left( i\right) }\left( v^{\pi }\right) =\varphi _{i}\left( v,\pi \right) +\sum _{j\in \pi \left( i\right) {\setminus }\left\{ i\right\} }\frac{\varphi _{i}\left( v,\pi \right) \cdot \varphi _{j}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ i\right\} }\right) }{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) }. \end{aligned}$$

Solving for \(\varphi _{i}\left( v,\pi \right) \) gives

$$\begin{aligned} \varphi _{i}\left( v,\pi \right) =\frac{1}{\psi _{\pi \left( i\right) }\left( v^{\pi }\right) }\cdot \left( 1+\sum _{j\in \pi \left( i\right) {\setminus } \left\{ i\right\} }\frac{\varphi _{j}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ i\right\} }\right) }{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) }\right) . \end{aligned}$$

With the previous equation, we have obtained a recursive description of \(\varphi \) that determines \(\varphi \) uniquely since \(\varphi _{i}\left( v,\pi \right) =\psi _{\pi \left( i\right) }\left( v^{\pi }\right) \) if \(\left| \pi \left( i\right) \right| =1\) according to Eq. 11. \(\square \)

Proof of Lemma 3

Let \(N\in \mathcal {N}, v\in \mathcal {G}^{N},\pi \in \Pi ^{N},R\subseteq C\in \pi \), and \(\phi \) and \(\varphi \) as stated in the lemma. For \(Q\subseteq R\) set \(U:=R{\setminus } Q\). By E* of \(\phi \) and Lemma 1, \(\varphi \) fulfills E. Thus, Eq. 8 becomes

$$\begin{aligned} \tilde{v}_{\varphi }^{R}\left( Q\right) =\sum _{i\in Q}\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus } U}\right) , \end{aligned}$$
(12)

for all \(Q\subseteq R\). Now, consider the internal game of \(C\), reduced to \(R\subseteq C\). For \(Q\subseteq R\), we have

$$\begin{aligned} \widetilde{\left( v_{\phi }^{C}\right) }_{\phi }^{R}\left( Q\right)&= v_{\phi }^{C}\left( C{\setminus } U\right) -\sum _{j\in C{\setminus } R}\phi _{j}\left( v_{\phi }^{C}{\big |}_{C{\setminus } U}\right) \\&\overset{\mathbf{E*}\text { of }\phi }{=}\sum _{i\in Q}\phi _{i}\left( v_{\phi }^{C}{\big |}_{C{\setminus } U}\right) \overset{\text {Lemma 5}}{=}\sum _{i\in Q}\phi _{i}\left( \left( v{\big |}_{N{\setminus } U}\right) _{\phi }^{C{\setminus } U}\right) \overset{\text {(9),(12)}}{=}\tilde{v}_{\varphi }^{R}\left( Q\right) \end{aligned}$$

Thus, we have \(\varphi _{i}\left( \tilde{v}_{\varphi }^{R},\left\{ R\right\} \right) =\phi _{i}\left( \widetilde{\big ( v_{\phi }^{C}\big ) }_{\phi } ^{R}\right) \) for all \(i\in R\). By C* of \(\phi , \phi _{i}\left( \widetilde{\big ( v_{\phi }^{C}\big ) }_{\phi }^{R}\right) =\phi _{i}\big ( v_{\phi }^{C}\big ) \). Finally, Eq. 9 implies \(\varphi _{i}\left( \tilde{v}_{\varphi }^{R},\left\{ R\right\} \right) =\varphi _{i}\left( v,\pi \right) \). \(\square \)

Proof of Theorem 4

Existence: By Theorem 2, \(\varpi \) fulfills 2P and IG. By Lemma 3 and Ortmann (2000, Theorem 2.11), \(\varpi \) satisfies C.

Uniqueness: Let \(\varphi \) satisfy 2P, C, and IG. By Theorem 2, it suffices to show that \(\varphi \) fulfills E and IPR. Let \(N\in \mathcal {N}, v\in \mathcal {G}_{++}^{N},\) and \(\pi =\left\{ N\right\} \). By 2P, C, and Ortmann (2000, Theorem 2.11), \(\varphi \left( v,\left\{ N\right\} \right) =\psi \left( v\right) \). In the following, consider arbitrary \(\pi \in \Pi ^{N}\). Then, we have \(\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) =\psi _{C}\left( v^{\pi }\right) \). By IG, \(\sum _{i\in C}\varphi _{i}\left( v,\pi \right) =\varphi _{C}\left( v^{\pi },\left\{ \pi \right\} \right) \). This entails

$$\begin{aligned} \sum _{C\in \pi }\sum _{i\in C}\varphi _{i}\left( v,\pi \right) =\sum _{C\in \pi }\psi _{C}\left( v_{C}^{\pi }\right) =v\left( N\right) . \end{aligned}$$

Therefore, \(\varphi \) satisfies E. By E of \(\varphi \), Eq. 8 now becomes

$$\begin{aligned} \tilde{v}_{\varphi }^{R}\left( Q\right) =\sum _{i\in Q}\varphi _{i}\left( \left( v,\pi \right) {\big |}_{\left( N{\setminus } R\right) \cup Q}\right) . \end{aligned}$$
(13)

It remains to show that \(\varphi \) fulfills IPR. Let \(C\in \pi \) and \(i,j\in C\). Consider the reduced game of \(v\) to \(\left\{ i,j\right\} \). By Eq. 13,

$$\begin{aligned} \tilde{v}_{\varphi }^{\left\{ i,j\right\} }\left( \left\{ i\right\} \right)&=\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) ,\\ \tilde{v}_{\varphi }^{\left\{ i,j\right\} }\left( \left\{ i,j\right\} \right)&=\sum _{\ell \in \left\{ i,j\right\} }\varphi _{\ell }\left( v,\pi \right) . \end{aligned}$$

By 2P, player \(i\)’s payoff in the reduced game according to \(\varphi \) must be

$$\begin{aligned} \varphi _{i}\left( \tilde{v}_{\varphi }^{\left\{ i,j\right\} },\left\{ \left\{ i,j\right\} \right\} \right) =\frac{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) }{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) +\varphi _{j}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ i\right\} }\right) }\cdot \sum _{\ell \in \left\{ i,j\right\} }\varphi _{\ell }\left( v,\pi \right) . \end{aligned}$$

For the ratio, we get

$$\begin{aligned} \frac{\varphi _{i}\left( \tilde{v}_{\phi }^{\left\{ i,j\right\} },\left\{ \left\{ i,j\right\} \right\} \right) }{\varphi _{j}\left( \tilde{v}_{\phi }^{\left\{ i,j\right\} },\left\{ \left\{ i,j\right\} \right\} \right) }=\frac{\varphi _{i}\left( \left( v,\pi \right) {\big |}_{N{\setminus }\left\{ j\right\} }\right) }{\varphi _{j}\left( \left( v,\pi \right) {\big |}_{N{\setminus } \left\{ i\right\} }\right) }. \end{aligned}$$

By C, the left-hand side equals \(\varphi _{i}\left( v,\pi \right) /\varphi _{j}\left( v,\pi \right) \). Thus, \(\varphi \) fulfills IPR. \(\square \)

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Huettner, F. A proportional value for cooperative games with a coalition structure. Theory Decis 78, 273–287 (2015). https://doi.org/10.1007/s11238-014-9420-9

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Keywords

  • Shapley value
  • Owen value
  • Proportional value
  • Consistency