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.

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. $\square$