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Consistency of the equal split-off set

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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

  1. Klijn et al. (2000) called this the bounded maximum payoff property and Llerena and Mauri (2017) called this property rich player feasibility.

  2. Llerena and Mauri (2017) showed that such allocation Lorenz dominates each other core allocation, so at most one such allocation exists.

  3. A solution f is a selector of \(\sigma\) if \(f(N,v)\in \sigma (N,v)\) for all \((N,v)\in \Gamma\).

  4. 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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Correspondence to Bas Dietzenbacher.

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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 (Nv) 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

$$\begin{aligned}x_i\ge \frac{v_{max}^x(U^v{\setminus } R_1^x)}{|U^v{\setminus } R_1^x|}\ge\, \frac{v(U^v)-\sum _{i\in R_1^x}x_i}{|U^v{\setminus } R_1^x|}=\frac{|U^v|a_1^x-|R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=\,\frac{|U^v{\setminus } R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=a_1^x.\end{aligned}$$

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,

$$\begin{aligned} v(N)&\ge \sum _{i\in N}x_i=\sum _{l =1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l -1}^x}x_i=\sum _{l=1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l -1}^x}a_l ^x\\&\ge \sum _{l =1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l-1}^x}\max _{S\in 2^{N{\setminus } R_{l -1}^x}{\setminus }\{\emptyset \}}\frac{v(S\cup R_{l-1}^x)-v(R_{l-1}^x)}{|S|}\\&\ge \sum _{l=1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l-1}^x}\frac{v(R_l^x)-v(R_{l-1}^x)}{|R_l^x{\setminus } R_{l-1}^x|}\\&=\sum _{l=1}^{|N|}\left( v(R_l^x)-v(R_{l-1}^x)\right) =v(R_{|N|}^x)-v(R_0^x)=v(N)-v(\emptyset )=v(N). \end{aligned}$$

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 (Nv) 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

$$\begin{aligned} x_i\ge \frac{v_{marg}^x(U^v{\setminus } R_1^x)}{|U^v{\setminus } R_1^x|}=\frac{v(U^v)-v(R_1^x)}{|U^v{\setminus } R_1^x|}=\frac{|U^v|a_1^x-|R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=\frac{|U^v{\setminus } R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=a_1^x. \end{aligned}$$

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 (Nv) 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

$$\begin{aligned} x_i\ge \frac{v_{comp}^x(U^v{\setminus } R_1^x)}{|U^v{\setminus } R_1^x|}=\frac{v(U^v)-\sum _{i\in R_1^x}x_i}{|U^v{\setminus } R_1^x|}=\frac{|U^v|a_1^x-|R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=\frac{|U^v{\setminus } R_1^x|a_1^x}{|U^v{\setminus } R_1^x|}=a_1^x. \end{aligned}$$

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,

$$\begin{aligned} v(N)&\ge \sum _{i\in N}x_i=\sum _{l=1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l -1}^x}x_i=\sum _{l =1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l -1}^x}a_l ^x\\&\ge \sum _{l =1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l-1}^x}\max _{S\in 2^{N{\setminus } R_{l-1}^x}{\setminus }\{\emptyset \}}\frac{v(S\cup R_{l -1}^x)-v(R_{l -1}^x)}{|S|}\\&\ge \sum _{l =1}^{|N|}\sum _{i\in R_l ^x{\setminus } R_{l -1}^x}\frac{v(R_l ^x)-v(R_{l -1}^x)}{|R_l ^x{\setminus } R_{l -1}^x|}\\&=\sum _{l=1}^{|N|}\left( v(R_l^x)-v(R_{l-1}^x)\right)=v(R_{|N|}^x)-v(R_0^x)=v(N)-v(\emptyset )=v(N). \end{aligned}$$

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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