Abstract
This paper axiomatically studies the equal split-off set (cf. Branzei et al. (Banach Center Publ 71:39–46, 2006)) as a solution for cooperative games with transferable utility which extends the well-known Dutta and Ray (Econometrica 57:615–635, 1989) solution for convex games. By deriving several characterizations, we explore consistency of the equal split-off set on the domains of exact partition games and arbitrary games.
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Notes
Llerena and Mauri (2017) showed that such allocation Lorenz dominates each other core allocation, so at most one such allocation exists.
A solution f is a selector of \(\sigma\) if \(f(N,v)\in \sigma (N,v)\) for all \((N,v)\in \Gamma\).
This fact was pointed out by an anonymous referee.
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Acknowledgements
Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. Two anonymous referees are gratefully acknowledged for helpful comments and suggestions.
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Appendix
Appendix
Theorem 2
The Dutta and Ray (1989) solution is the unique solution on \(\Gamma _{exp}\) satisfying nonemptiness, feasible richness, equal division stability, and rich-restricted max-consistency.
Proof
The Dutta and Ray (1989) solution coincides with the equal split-off set on \(\Gamma _{exp}\). Clearly, the equal split-off set satisfies nonemptiness and feasible richness. By Lemmas 1 and 2, the equal split-off set satisfies equal division stability on \(\Gamma _{exp}\). Llerena and Mauri (2017) showed that the equal split-off set satisfies rich-restricted max-consistency on \(\Gamma _{exp}\).
Let \(\sigma\) be a solution on \(\Gamma _{exp}\) satisfying nonemptiness, feasible richness, equal division stability, and rich-restricted max-consistency. We show by induction on the number of players that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\). By nonemptiness and equal division stability, \(\sigma (N,v)=\{v(N)\}\) for all \((N,v)\in \Gamma _{exp}\) with \(|N|=1\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{exp}\) with \(|N|=k+1\). By nonemptiness, there is \(x\in \sigma (N,v)\). By Lemma 3, \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\) and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\).
Denote \(U^v=\bigcup \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Since (N, v) is an exact partition game, Lemmas 1 and 2 imply that \(U^v\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Suppose that \(R_1^x\ne U^v\). By rich-restricted max-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{max}^x)\). By equal division stability, there is \(i\in U^v{\setminus } R_1^x\) such that
This is a contradiction, so \(R_1^x=U^v\). If \(R_1^x=U^v=N\), then \(\sigma (N,v)=\{(\frac{v(N)}{|N|},\ldots ,\frac{v(N)}{|N|})\}\). Suppose that \(R_1^x=U^v\ne N\). By rich-restricted max-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{max}^x)\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{max}^x)\) consists of one uniquely defined allocation since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\) consists of one uniquely defined allocation. \(\square\)
Theorem 3
The Dutta and Ray (1989) solution is the unique solution on \(\Gamma _{exp}\) satisfying nonemptiness, equal division stability, and rich-restricted marginal-consistency.
Proof
The Dutta and Ray (1989) solution coincides with the equal split-off set on \(\Gamma _{exp}\). Clearly, the equal split-off set satisfies nonemptiness and rich-restricted marginal-consistency. By Lemmas 1 and 2, the equal split-off set satisfies equal division stability on \(\Gamma _{exp}\).
Let \(\sigma\) be a solution on \(\Gamma _{exp}\) satisfying nonemptiness, equal division stability, and rich-restricted marginal-consistency. We show by induction on the number of players that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\). By nonemptiness and equal division stability, \(\sigma (N,v)=\{v(N)\}\) for all \((N,v)\in \Gamma _{exp}\) with \(|N|=1\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{exp}\) with \(|N|=k+1\). By nonemptiness, there is \(x\in \sigma (N,v)\). By equal division stability and rich-restricted marginal-consistency,
This means that \(\sum _{i\in N}x_i=v(N)\), \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\), and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\).
Denote \(U^v=\bigcup \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Since (N, v) is an exact partition game, Lemmas 1 and 2 imply that \(U^v\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Suppose that \(R_1^x\ne U^v\). By rich-restricted marginal-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{marg}^x)\). By equal division stability, there is \(i\in U^v{\setminus } R_1^x\) such that
This is a contradiction, so \(R_1^x=U^v\). If \(R_1^x=U^v=N\), then \(\sigma (N,v)=\{(\frac{v(N)}{|N|},\ldots ,\frac{v(N)}{|N|})\}\). Suppose that \(R_1^x=U^v\ne N\). By rich-restricted marginal-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{marg}^x)\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{marg}^x)\) consists of one uniquely defined allocation since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\) consists of one uniquely defined allocation. \(\square\)
Theorem 4
The Dutta and Ray (1989) solution is the unique solution on \(\Gamma _{exp}\) satisfying nonemptiness, feasible richness, equal division stability, and rich-restricted complement-consistency.
Proof
The Dutta and Ray (1989) solution coincides with the equal split-off set on \(\Gamma _{exp}\). Clearly, the equal split-off set satisfies nonemptiness and feasible richness. By Lemmas 1 and 2, the equal split-off set satisfies equal division stability on \(\Gamma _{exp}\). By Lemma 4, the equal split-off set satisfies rich-restricted complement-consistency since it satisfies efficiency and rich-restricted marginal-consistency.
Let \(\sigma\) be a solution on \(\Gamma _{exp}\) satisfying nonemptiness, feasible richness, equal division stability, and rich-restricted complement-consistency. We show by induction on the number of players that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\). By nonemptiness and equal division stability, \(\sigma (N,v)=\{v(N)\}\) for all \((N,v)\in \Gamma _{exp}\) with \(|N|=1\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\) consists of one uniquely defined allocation for all \((N,v)\in \Gamma _{exp}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{exp}\) with \(|N|=k+1\). By nonemptiness, there is \(x\in \sigma (N,v)\). By Lemma 3, \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\) and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\).
Denote \(U^v=\bigcup \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Since (N, v) is an exact partition game, Lemmas 1 and 2 imply that \(U^v\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). Suppose that \(R_1^x\ne U^v\). By rich-restricted complement-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{comp}^x)\). By equal division stability, there is \(i\in U^v{\setminus } R_1^x\) such that
This is a contradiction, so \(R_1^x=U^v\). If \(R_1^x=U^v=N\), then \(\sigma (N,v)=\{(\frac{v(N)}{|N|},\ldots ,\frac{v(N)}{|N|})\}\). Suppose that \(R_1^x=U^v\ne N\). By rich-restricted complement-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{comp}^x)\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{comp}^x)\) consists of one uniquely defined allocation since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\) consists of one uniquely defined allocation. \(\square\)
Theorem 6
The equal split-off set is the unique maximal solution on \(\Gamma _{all}\) satisfying feasible richness, weak equal division stability, and rich-restricted self-consistency.
Proof
Clearly, the equal split-off set satisfies feasible richness and weak equal division stability. By Lemma 6, the equal split-off set satisfies rich-restricted self-consistency on \(\Gamma _{all}\).
Let \(\sigma\) be a solution on \(\Gamma _{all}\) satisfying feasible richness, weak equal division stability, and rich-restricted self-consistency. We show by induction on the number of players that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\). For all \((N,v)\in \Gamma _{all}\) with \(|N|=1\), \(\sigma (N,v)=\emptyset\) or \(\sigma (N,v)=\{v(N)\}\) by weak equal division stability, so \(\sigma (N,v)\subseteq ESOS(N,v)=\{v(N)\}\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{all}\) with \(|N|=k+1\). If \(\sigma (N,v)=\emptyset\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(\sigma (N,v)\ne \emptyset\) and let \(x\in \sigma (N,v)\). By Lemma 5, \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\) and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). If \(R_1^x=N\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(R_1^x\ne N\). By rich-restricted self-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{self}^f)\) for a selector f of \(\sigma\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{self}^f)\subseteq ESOS(N{\setminus } R_1^x,v_{self}^f)\) since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\subseteq ESOS(N,v)\). \(\square\)
Theorem 7
The equal split-off set is the unique maximal solution on \(\Gamma _{all}\) satisfying feasible richness, weak equal division stability, and rich-restricted complement-consistency.
Proof
Clearly, the equal split-off set satisfies feasible richness and weak equal division stability. By Lemma 4, the equal split-off set satisfies rich-restricted complement-consistency since it satisfies efficiency and rich-restricted marginal-consistency.
Let \(\sigma\) be a solution on \(\Gamma _{all}\) satisfying feasible richness, weak equal division stability, and rich-restricted complement-consistency. We show by induction on the number of players that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\). For all \((N,v)\in \Gamma _{all}\) with \(|N|=1\), \(\sigma (N,v)=\emptyset\) or \(\sigma (N,v)=\{v(N)\}\) by weak equal division stability, so \(\sigma (N,v)\subseteq ESOS(N,v)=\{v(N)\}\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{all}\) with \(|N|=k+1\). If \(\sigma (N,v)=\emptyset\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(\sigma (N,v)\ne \emptyset\) and let \(x\in \sigma (N,v)\). By Lemma 5, \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\) and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). If \(R_1^x=N\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(R_1^x\ne N\). By rich-restricted complement-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{comp}^x)\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{comp}^x)\subseteq ESOS(N{\setminus } R_1^x,v_{comp}^x)\) since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\subseteq ESOS(N,v)\). \(\square\)
Theorem 8
The equal split-off set is the unique maximal solution on \(\Gamma _{all}\) satisfying weak equal division stability and rich-restricted marginal-consistency.
Proof
Clearly, the equal split-off set satisfies weak equal division stability and rich-restricted marginal-consistency.
Let \(\sigma\) be a solution on \(\Gamma _{all}\) satisfying weak equal division stability and rich-restricted marginal-consistency. We show by induction on the number of players that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\). For all \((N,v)\in \Gamma _{all}\) with \(|N|=1\), \(\sigma (N,v)=\emptyset\) or \(\sigma (N,v)=\{v(N)\}\) by weak equal division stability, so \(\sigma (N,v)\subseteq ESOS(N,v)=\{v(N)\}\). Let \(k\in {\mathbb {N}}\) and assume that \(\sigma (N,v)\subseteq ESOS(N,v)\) for all \((N,v)\in \Gamma _{all}\) with \(|N|\le k\). Let \((N,v)\in \Gamma _{all}\) with \(|N|=k+1\). If \(\sigma (N,v)=\emptyset\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(\sigma (N,v)\ne \emptyset\) and let \(x\in \sigma (N,v)\). By weak equal division stability and rich-restricted marginal-consistency,
This means that \(\sum _{i\in N}x_i=v(N)\), \(a_1^x=\max _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\), and \(R_1^x\in \mathop {\mathrm{argmax}}\limits \nolimits _{S\in 2^N{\setminus }\{\emptyset \}}\frac{v(S)}{|S|}\). If \(R_1^x=N\), then \(\sigma (N,v)\subseteq ESOS(N,v)\). Suppose that \(R_1^x\ne N\). By rich-restricted marginal-consistency, \(x_{N{\setminus } R_1^x}\in \sigma (N{\setminus } R_1^x,v_{marg}^x)\), where, by the induction hypothesis, \(\sigma (N{\setminus } R_1^x,v_{marg}^x)\subseteq ESOS(N{\setminus } R_1^x,v_{marg}^x)\) since \(|N{\setminus } R_1^x|\le k\). Hence, \(\sigma (N,v)\subseteq ESOS(N,v)\). \(\square\)
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Dietzenbacher, B., Yanovskaya, E. Consistency of the equal split-off set. Int J Game Theory 50, 1–22 (2021). https://doi.org/10.1007/s00182-020-00713-5
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DOI: https://doi.org/10.1007/s00182-020-00713-5