Abstract
In this paper, we extend the equal division and the equal surplus division values for transferable utility cooperative games to the more general setup of transferable utility cooperative games with a priori unions. In the case of the equal surplus division value we propose three possible extensions. We provide axiomatic characterizations of the new values. Furthermore, we apply the proposed modifications to a particular motivating example and compare the numerical results with those obtained with the original values.
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Notes
In this example the quota units are the square meters of the apartments. For the approach we adopt to be meaningful, the quota unit numbers must be integers.
Notice that we have included the efficiency in the definition of value. We could have considered it as one more property and then it would have appeared explicitly in the characterizations; nothing relevant would have changed in that case.
A value g for TU-games with a priori unions satisfies the quotient game property if, for all \(( N,v,P)\in {\mathcal {G}}^U\) with \(P=\{P_1,\ldots ,P_m\}\) and for its quotient game (M, v/P), it holds that \(\sum _{i\in P_k}g_{i}\left( N,v,P\right) =g_{k}\left( M,v/P,P^{m}\right) \) for all \(P_k\in P\), where \(P^m=\left\{ \left\{ 1\right\} ,\left\{ 2\right\} ,\ldots ,\left\{ m\right\} \right\} \).
\(v_{P_{k}}\) denotes the characteristic function of the TU-game \((P_{k},v_{P_{k}})\), where \(v_{P_{k}}(S)=v(S)\) for all \(S\subseteq P_{k}\).
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This work has been supported by the ERDF, the MINECO/AEI grants MTM2017-87197-C3-1-P, MTM2017-87197-C3-3-P, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2017/38 and Centro Singular de Investigación de Galicia ED431G/01). The authors would like to thank two anonymous referees for their helpful suggestions to improve this article.
Appendix
Appendix
(a) Independence of the properties of Theorem 3.2:
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\(\varphi _{i}=\) \(\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum _{l\in M}v(P_{l})}{mp_{k}}\) satisfies ADD, SWU and SAU, but not NPP.
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\(\varphi _{i}=\) \(\frac{v(N)}{n}\) satisfies ADD, SWU and NPP, but not SAU.
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\(\varphi _{i}=\frac{2v(N)}{mp_{k}}\) if \(\displaystyle i=\min _{j\in P_{k}}j\) or \(\varphi _{i}=\frac{(p_{k}-2)v(N)}{mp_{k}(p_{k}-1)}\) if \(i\in P_{k}\) and \(i\ne \min _{j\in P_{k}}j\), satisfies ADD, SAU and NPP, but not SWU.
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\(\varphi _{i}=\frac{2v(N)}{mp_{k}|Z_{k}|}\) if \(\displaystyle i\in Z_{k}=\{j\in P_{k}/v(j)=\min _{z\in P_{k}}v(z)\}\), \(\varphi _{i}=\frac{ (p_{k}-2)v(N)}{mp_{k}(p_{k}-|Z_{k}|)}\) if \(i\in P_{k}\backslash Z_{k}\), satisfies SWU, SAU and NPP, but not ADD.
(b) Independence of the properties of Theorem 4.4:
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\(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum _{l\in M}v(P_{l})}{n}\) satisfies ADD, SWU, DUNPP, but not SAU.
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\(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{2(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}}\) if \(\displaystyle i=\min _{j\in P_{k}}j\) or \(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum _{l\in M}v(P_{l}))}{ mp_{k}(p_{k}-1)}\) if \(i\in P_{k}\) and \(i\ne \min _{j\in P_{k}}j\), satisfies ADD, SAU and DUNPP, but not SWU.
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\(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{2(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}|Z_{k}|}\) if \(\displaystyle i\in Z_{k}=\{j\in P_{k}/v(j)=\min _{z\in P_{k}}v(z)\}\), \(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+ \frac{(p_{k}-2)(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}(p_{k}-|Z_{k}|)}\) if \( i\in P_{k}\backslash Z_{k}\), satisfies SWU, SAU and DUNPP, but not ADD.
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\(\varphi _{i}=v(i)+\frac{v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+\frac{ v(N)-\sum _{l\in M}v(P_{l})}{mp_{k}}\) satisfies ADD, SWU and SAU, but not DUNPP.
(c) Independence of the properties of Theorem 4.5:
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\(\varphi _{i}=v(i)+\frac{v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+\frac{ v(N)-\sum _{l\in M}v(P_{l})}{n}\) satisfies ADD, SWU and DUPP, but not SAU.
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\(\varphi _{i}=v(i)+\frac{v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+\frac{ 2(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}}\) if \(\displaystyle i=\min _{j\in P_{k}}j\) or \(\varphi _{i}=v(i)+\frac{v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+ \frac{(p_{k}-2)(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}(p_{k}-1)}\) if \(i\in P_{k}\) and \(i\ne \min _{j\in P_{k}}j\), satisfies ADD, SAU and DUPP, but not SWU.
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\(\varphi _{i}=v(i)+\frac{v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+\frac{ 2(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}|Z_{k}|}\) if \(\displaystyle i\in Z_{k}=\{j\in P_{k}/v(j)=\min _{z\in P_{k}}v(z)\}\), \(\varphi _{i}=v(i)+\frac{ v(P_{k})-\sum _{j\in P_{k}}v(j)}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum _{l\in M}v(P_{l}))}{mp_{k}(p_{k}-|Z_{k}|)}\) if \(i\in P_{k}\backslash Z_{k}\), satisfies SWU, SAU and DUNPP, but not ADD.
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\(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum _{l\in M}v(P_{l})}{mp_{k}}\) satisfies ADD, SWU and SAU, but not DUPP.
(d) Independence of the properties of Theorem 4.6:
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\(\varphi _{i}=v(i)+\frac{v(N)-\sum _{j\in N}v(j)}{n}\) satisfies ADD, SWU and DPP, but not WSAU.
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\(\varphi _{i}=v(i)+\frac{2(v(N)-\sum _{j\in N}v(j))}{mp_{k}}\) if \( \displaystyle i=\min _{j\in P_{k}}j\) or \(\varphi _{i}=v(i)+\frac{ (p_{k}-2)(v(N)-\sum _{j\in N}v(j))}{mp_{k}(p_{k}-1)}\) if \(i\in P_{k}\) and \( i\ne \min _{j\in P_{k}}j\), satisfies ADD, WSAU and DPP, but not SWU.
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\(\varphi _{i}=v(i)+\frac{2(v(N)-\sum _{j\in N}v(j))}{mp_{k}|Z_{k}|}\) if \( \displaystyle i\in Z_{k}=\{j\in P_{k}/v(j)=\min _{z\in P_{k}}v(z)\}\), \( \varphi _{i}=v(i)+\frac{(p_{k}-2)(v(N)-\sum _{j\in N}v(j))}{ mp_{k}(p_{k}-|Z_{k}|)}\) if \(i\in P_{k}\backslash Z_{k}\), satisfies SWU, WSAU and DPP, but not ADD.
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\(\varphi _{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum _{l\in M}v(P_{l})}{mp_{k}}\) satisfies ADD, SWU, WSAU but not DPP.
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Alonso-Meijide, J.M., Costa, J., García-Jurado, I. et al. On egalitarian values for cooperative games with a priori unions. TOP 28, 672–688 (2020). https://doi.org/10.1007/s11750-020-00553-2
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DOI: https://doi.org/10.1007/s11750-020-00553-2