On the importance of reduced games in axiomatizing core extensions

We propose new axiomatizations of the core and three related solution concepts that also provide predictions for (classes of) games in which the core itself is empty. Our results showcase the importance of the reduced game formulation and identify the corresponding converse consistency property as the differentiating characteristic between the core and its various extensions. Existing axiomatizations of the core and similar concepts include the required form of feasibility in the generic definition of a solution concept and/or are restricted to the domain of games for which existence is guaranteed. We dispense of both practices, thus opening up the possibility of comparing, via basic axioms, solution concepts that have different feasibility constraints and domains.

1. The aspiration set is not, strictly speaking, a core extension because it may not coincide with the core when the core is non-empty. However, it is a non-empty superset of the core.

2. Aslan et al. (2019) also stress the importance of considering feasibility requirements, and they use these to define a new concept of duality.

3. Among these, Hwang and Sudhölter (2001) is the only one that axiomatizes the core on the general domain of TU-games, but they maintain feasibility as an implicit axiom. In their definition, any solution concept has to select outcomes that are feasible for the grand coalition of the game. Peleg (1986) also proposes an alternative axiomatization that does not refer explicitly to the domain of balanced games, but his axiom of unanimity states that the solution should coincide with the core for two-person games.

4. For the aspiration set, individual rationality is implied by the other axioms.

5. Other core-like solution concepts such as the \(\varepsilon\)-core, the least core or the D-core (Llerena and Rafels (2007)) maintain the same feasibility assumption of the core vectors but weaken coalitional rationality.

6. Production plans are also known in the literature as balancing weights and each production plan \(\lambda\) gives rise to a balanced family of coalitions \(\{S\in {\mathcal {N}} \mid \lambda _S>0\}\).

7. We follow the literature by using the notation \({\bar{v}}(N)\).

8. Games with non-empty cores are known as balanced games.

9. In Appendix D we discuss a few axiomatizations on various subdomains, and the way our axioms need to be changed to accommodate for subdomains is made explicit there.

10. The 0–1-normalization of the game is obtained by multiplying the game with the scalar \(\frac{1}{v(\{1,2\})-v(\{1\})-v(\{2\})}\) and subtracting the additive game associated with the vector \(\left( \frac{v(\{1\})}{v(\{1,2\})-v(\{1\})-v(\{2\})},\frac{v(\{2\})}{v(\{1,2\})-v(\{1\})-v(\{2\})}\right)\). This translates the worths of the singleton coalitions to 0 and normalizes the worth of the grand coalition to 1.

11. Their Corollary 3.8 uses IR, while their Theorem 4.2 shows that IR can be replaced by a boundedness axiom which is stronger than our BOUND. They also use the stronger versions of RGP and CRGP.

12. Feasibility is not explicitly stated as an axiom in their formulation, but rather embedded directly into the definition of a generic solution concept.

13. Note that this proof shows that the core and the c-core satisfy a stronger version of \(RGP^{p}\), because the cardinality of the subset S is not restricted.

14. Note that this proof shows that the core, the c-core, and the aspiration core satisfy a stronger version of \(RGP^{b}\), because the cardinality of the subset S is not restricted.

15. Of course, all proofs are available from the authors upon request.

16. Note that Example 3.12 does not show that the core does not satisfy \(CRGP^a\) on \(\Gamma _C\).

Authors and Affiliations

1. School of Business, University of Washington, Bothell, 18115 Campus Way NE, Bothell, WA, 98011, USA

   Camelia Bejan & Juan Camilo Gómez

2. Department of Economics, 435 PLC, 1285 University of Oregon, Eugene, OR, 97403-1285, USA

   Anne van den Nouweland

Corresponding author

Correspondence to Camelia Bejan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

A Proofs of Sect. 2 Lemmas

Lemma 2.2

 2.2 If \((N,v) \in \Gamma\) and \(|N|\le 2\), then \(C_c(N,v)=AC(N,v)=Asp(N,v)\).

Proof

Let \((N,v) \in \Gamma\) and \(|N|\le 2\). If \(N=\{i\}\), then \(C(N,v)=C_c(N, v)=AC(N,v)=Asp(N,v)=\{v(N)\}\). So, suppose \(N=\{i,j\}\). If \(v(N)\le v(\{i\})+v(\{j\})\), then \(v_{\Pi }(N)={\bar{v}}(N) = v(\{i\})+v(\{j\})\) and together with coalitional rationality we obtain \(C_c(N, v)=AC(N,v)=Asp(N,v)=\{(v(\{i\}),v(\{j\}))\}\). If \(v(N) > v(\{i\})+v(\{j\})\), then \(v_{\Pi }(N)={\bar{v}}(N) = v(N)\) and together with coalitional rationality we obtain \(C(N,v)=C_c(N, v)=AC(N,v)=Asp(N,v)=\{(x_i,x_j) \mid x_i\ge v(\{i\}),\ x_j\ge v(\{j\}),\ x_i+x_j=v(N)\}\). \(\square\)

Lemma 2.3

2.3 If \((N,v) \in \Gamma\), \(x \in Asp(N,v)\) and \(x_i = v\big (\{i\}\big )\) for every \(i \in N\), then \(Asp(N,v) = AC(N,v)=C_c(N,v)=\{x\}\).

Proof

Let \((N,v) \in \Gamma\) and \(x = \big ( v\big (\{i\}\big )\big )_{i\in N} \in Asp(N,v)\). Let \(y \in {\mathbb {R}}^N\) be a coalitionally rational payoff vector such that \(y_i \ne v\big (\{i\}\big )\) for some \(i\in N\). Then for every coalition S that contains i

so that \(y\not \in X_{\Delta }(N,v)\) and therefore \(y\not \in Asp (N,v)\). This shows that \(Asp(N,v)=\{x\}\).

Using that the aspiration core is non-empty and \(AC(N, v)= Asp(N,v) \cap X_{\Lambda }(N,v)\subseteq Asp(N,v)\), we obtain that \(AC(N,v)=\{x\}\) as well. Since \(\pi :=\left\{ \{i\}\mid i\in N\right\}\) is a partition of N and \(x(S)=v(S)\) for all \(S\in \pi\), we know that \(x\in X_{\Pi }(N,v)\) and therefore \(C_c(N,v)=Asp(N,v) \cap X_{\Pi }(N,v)=\{x\}\).\(\square\)

B Proofs of Sect. 3 Propositions

Proposition

3.1 The aspiration set, the core, the c-core, and the aspiration core satisfy COV.

Proof

Let \((N,v) \in \Gamma\), \(\alpha > 0\), and \(\beta \in {\mathbb {R}}^N\), and define the game (N, w) by \(w(S) = \alpha v(S) + \beta (S)\) for every \(S \in {\mathcal {N}}\).

It is clear that coalitional rationality, which is a requirement for all four solution concepts, is preserved through multiplication by \(\alpha >0\) and translation by \(\beta\). Since \(w(N)=\alpha v(N)+\beta (N)\) by definition, the core thus satisfies COV.

To establish that Asp satisfies COV, it remains to prove individual feasibility. Let \(x \in {\mathbb {R}}^N\) and \(i\in N\). Let \(S\in {\mathcal {N}}\). Because \(\alpha >0\), we have that \(x(S)\le v(S)\) if and only if \(\alpha x(S) + \beta (S)\le \alpha v(S) + \beta (S)=w(S)\). Thus, there exists a coalition \(S\in {\mathcal {N}}\) such that \(S\ni i\) and \(x(S)\le v(S)\) if and only if there exists an \(S\in {\mathcal {N}}\) such that \(S\ni i\) and \(\alpha x(S) + \beta (S)\le w(S)\). Thus, x satisfies individual feasibility in (N, v) if and only if \(\alpha x(S) + \beta (S)\) satisfies individual feasibility in (N, w).

To establish that AC satisfies COV, it remains to prove that \(x\in X_{\Lambda }(N,v)\) if and only if \(\alpha x + \beta \in X_{\Lambda }(N,w)\). We do this by showing that for every production plan \(\lambda \in \Lambda (N)\) it holds that \(w(\lambda )=\alpha v(\lambda )+\beta (N)\), so that \({{\bar{w}}}(N) = \max _{\lambda \in \Lambda (N)} w(\lambda )=\max _{\lambda \in \Lambda (N)} \big ( \alpha v(\lambda )+\beta (N)\big ) = \alpha {{\bar{v}}}(N) +\beta (N)\). The equality follows through simple algebraic manipulations after observing that \(\beta (N)=\sum _{S \in {\mathcal {N}}} \lambda _S \beta (S)\).

To establish that the c-core satisfies COV, it remains to prove that \(x\in X_{\Pi }(N,v)\) if and only if \(\alpha x + \beta \in X_{\Pi }(N,w)\). This follows from our results for production plans because every partition is a production plan. \(\square\)

Proposition

3.3 The core, the c-core, the aspiration core, and the aspiration set satisfy CRGP.

Proof

Let \((N,v)\in \Gamma\) with \(|N|\ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S\in \gamma (S, v_{x,S})\) for every \(S\subset N\), with \(|S|\le 2\), where \(\gamma\) is the core, the c-core, the aspiration core, or the aspiration set. We first prove x is coalitionally rational. Let \(S\subset N\), \(i\in S\), and \(j \in N {\setminus } S\). Since \(x^{\{i,j\}}\in \gamma \left( \{i,j\}, v_{x, \{i,j\}}\right)\) and \(\gamma\) satisfies coalitional rationality, \(x_i \ge v_{x, \{i,j\}}(\{i\}) = \max \{v(\{i\} \cup Q) - x(Q) \mid Q \subseteq N {\setminus } \{i,j\} \}\). Taking \(Q:=S{\setminus } \{i\}\), we have \(Q\subseteq N{\setminus } \{i,j\}\) and thus, by the definition of the reduced game, \(x_i\ge v(S)-x(S{\setminus } \{i\})\), which implies that \(x(S)\ge v(S)\).

Let \(i\in N\). It follows from \(x_i \in \gamma (\{i\}, v_{x,\{i\}})=\{v_{x,\{i\}}(\{i\})\}\) that \(x_i = v(N) - x(N {\setminus } \{i\})\). Thus, \(x(N)=v(N)\). Together with coalitional rationality this implies \(x \in C(N,v)\). CRGP of all four solution concepts follows because \(C(N,v)\subseteq C_c(N,v)\subseteq AC(N,v)\subseteq Asp(N,v)\). \(\square\)

Proposition

3.4 The core and the c-core satisfy \(RGP^{p}\).

Proof

Let \((N,v) \in \Gamma\), \(S \subset N\) and \(x\in C_c(N,v)\).^{Footnote 13} Then \(x(N)=v_{\Pi }(N)\) and thus \(x(S)=v_{\Pi }(N)-x(N{\setminus } S)=v^p_{x, S}(S)\). To prove that the c-core satisfies \(RGP^{p}\) it remains to show that \(x(T)\ge v^p_{x, S}(T)\) for every \(T\subset S\). The inequality \(x(T)\ge v^p_{x, S}(T)\) is equivalent to \(x(T)\ge v(T\cup Q)-x(Q)\), or \(x(T\cup Q)\ge v(T\cup Q)\) for every \(Q\subseteq N{\setminus } S\). The last inequality is true because x is coalitionally rational in (N, v). Hence, \(x^S\in C(S, v^p_{x, S})=C_c(S, v^p_{x, S})\), which proves that the c-core satisfies \(RGP^{p}\).

Let now \(x\in C(N,v)\). Then \(C(N,v)\not =\emptyset\) and thus \(C_c(N,v)=C(N,v)\) and \(v_{\Pi }(N)=v(N)\). Thus, \(x\in C_c(N,v)\) and, as shown above, \(x^S\in C(S, v^p_{x, S})\) for all \(S \subset N\). This proves that the core satisfies \(RGP^{p}\) as well.\(\square\)

Proposition

3.6 The c-core, the aspiration core, and the aspiration set satisfy \(CRGP^{p}\).

Proof

Let \((N,v)\in \Gamma\) with \(|N|\ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S \in C_c(S,v^p_{x,S})\) for every \(S\subset N\) with \(|S| \le 2\), and fix \(i\in N\). Since \(x_i\in C_c\big (\{i\}, v^p_{x, \{i\}}\big )\), it follows that \(x_i=v^p_{x, \{i\}}(\{i\})=v_{\Pi }(N)-x(N{\setminus } \{i\})\), and therefore \(x(N)=v_{\Pi }(N)\).

We prove next that x satisfies coalitional rationality. Let \(S\subset N\) and choose \(i\in S\) and \(j\notin S\). Since \(x^{\{i,j\}}\in C_c\left( \{i,j\}, v^p_{x, \{i,j\}}\right)\), \(x_i\ge v^p_{x, \{i,j\}}(\{i\})\). Taking \(Q:=S{\setminus } \{i\}\), we have \(Q\subseteq N{\setminus } \{i,j\}\) and thus, by the definition of the reduced game, \(x_i\ge v(S)-x(S{\setminus } \{i\})\), which implies that \(x(S)\ge v(S)\). Therefore, the c-core satisfies \(CRGP^{p}\).

To prove that the aspiration core (respectively the aspiration set) satisfies \(CRGP^{p}\) as well, let \(x \in {\mathbb {R}}^N\) such that \(x^S \in AC(S, v^p_{x,S})\) (respectively \(x^S \in Asp(S, v^p_{x,S})\)) for every \(S \subset N\) with \(|S|\le 2\). Then, for every \(S \subset N\) with \(|S|\le 2\), because \(C_c(S, v^p_{x,S})=AC(S, v^p_{x,S})=Asp(S, v^p_{x,S})\) (Lemma 2.2), we know that \(x^S \in C_c(S, v^p_{x,S})\), and thus \(x \in C_c(N,v)\) because the c-core satisfies \(CRGP^{p}\). Therefore, \(C_c(N,v)\ne \emptyset\) and \(x\in C_c(N,v)=AC(N,v)\subseteq Asp(N,v)\). This proves that the aspiration core and the aspiration set satisfy \(CRGP^{p}\). \(\square\)

Proposition

3.8. The core, the c-core, and the aspiration core satisfy \(RGP^b\).

Proof

Let \((N,v)\in \Gamma ,\ S\subset N\), and \(x\in AC(N,v)\).^{Footnote 14} Then \(x(N)={{\bar{v}}}(N)\) and thus \(x(S)={{\bar{v}}}(N)-x(N{\setminus } S)=v^b_{x,S}(S)\). To prove that the aspiration core satisfies \(RGP^{b}\) it remains to show that \(x(T)\ge v^b_{x, S}(T)\) for every \(T\subset S\). The inequality \(x(T)\ge v^b_{x, S}(T)\) is equivalent to \(x(T)\ge v(T\cup Q)-x(Q)\), or \(x(T\cup Q)\ge v(T\cup Q)\) for every \(Q\subseteq N{\setminus } S\). The last inequality is true because x is coalitionally rational in (N, v). Hence, \(x^S\in C(S, v^b_{x, S})=AC(S, v^b_{x, S})\), which proves that the aspiration core satisfies \(RGP^{b}\).

To prove that the c-core satisfies \(RGP^{b}\) as well, let \(x\in C_c(N,v)\). Then \(C_c(N,v)\not =\emptyset\) and thus \(AC(N,v)=C_c(N,v)\) and \({\bar{v}}(N)=v_{\Pi }(N)\). Thus, \(x\in AC(N,v)\) and, as shown above, \(x^S\in C(S, v^b_{x, S})\) for all \(S \subset N\). One implication of this is that, for each \(S \subset N\), \(C(S, v^b_{x, S})\not =\emptyset\), so that \(C(S, v^b_{x, S})=C_c(S, v^b_{x, S})=AC(S, v^b_{x, S})\). Thus, \(x^S\in C_c(S, v^b_{x, S})\) for all \(S \subset N\). This proves that the c-core satisfies \(RGP^{b}\).

The proof that the core satisfies \(RGP^{b}\) is completely analogous to that for the c-core in the previous paragraph. \(\square\)

Proposition

3.10 The aspiration core and the aspiration set satisfy \(CRGP^b\).

Proof

Let \((N,v)\in \Gamma\) with \(|N|\ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S \in AC(S,v^b_{x,S})\) for every \(S\subset N\) with \(|S| \le 2\), and fix \(i\in N\). Since \(x_i\in AC\big (\{i\}, v^b_{x, \{i\}}\big )\), it follows that \(x_i=v^b_{x, \{i\}}(\{i\})={{\bar{v}}}(N)-x(N{\setminus } \{i\})\), and therefore \(x(N)={{\bar{v}}}(N)\).

We prove next that x satisfies coalitional rationality. Let \(S\subset N, S\ne N\) and choose \(i\in S, j\notin S\), arbitrary. Since \(x^{\{i,j\}}\in AC\left( \{i,j\}, v^b_{x, \{i,j\}}\right)\), \(x_i\ge v^b_{x, \{i,j\}}(\{i\})\). Taking \(Q:=S{\setminus } \{i\}\), we have \(Q\subseteq N{\setminus } \{i,j\}\) and thus, by the definition of the reduced game, \(x_i\ge v(S)-x(S{\setminus } \{i\})\), which implies that \(x(S)\ge v(S)\). Therefore, the aspiration core satisfies \(CRGP^{b}\).

To prove that the aspiration set satisfies \(CRGP^{b}\) as well, let \(x \in {\mathbb {R}}^N\) such that \(x^S \in Asp(S, v^b_{x,S})\) for every \(S \subset N\) with \(|S|\le 2\). Then, for every \(S \subset N\) with \(|S|\le 2\), because \(AC(S, v^b_{x,S})=Asp(S, v^b_{x,S})\) (Lemma 2.2), we know that \(x^S \in AC(S, v^b_{x,S})\), and thus \(x \in AC(N,v)\subseteq Asp(N,v)\) because the aspiration core satisfies \(CRGP^{b}\). This proves that the aspiration set satisfies \(CRGP^{b}\). \(\square\)

Proposition

3.13 The set of coalitionally rational allocations satisfies \(RGP^{a}\), \(RGP^{b}\), \(RGP^{p}\) and RGP.

Proof

Denote the set of coalitionally rational allocations by CR(N, v). Let \((N,v) \in \Gamma\), \(S \subset N\) and \(x\in CR(N,v)\). We want to prove that \(x \in CR(S, v^\gamma _{x, S})\), where \(v^\gamma _{x, S}\) represents any one of the four types of reduced games. Let \(T \subseteq S\). As x is coalitionally rational, \(x(T\cup Q)\ge v(T\cup Q)\) or, equivalently, \(x(T)\ge v(T\cup Q)-x(Q)\) for every \(Q\subseteq N{\setminus } S\). Therefore \(x(T)\ge \max \{ v(T\cup Q)-x(Q) \mid Q\subseteq N{\setminus } S \}\). We conclude then that \(x(T)\ge v^\gamma _{x, S}(T)\) if \(T\ne S\) and that \(x(S)\ge v^{a}_{x, S}(S)\).

It remains to show that x(S) is greater than or equal to \(v^{b}_{x, S}(S)\), \(v^{p}_{x, S}(S)\), and \(v_{x, S}(S)\). Because \(x \in CR(N,v)\), by the Bondareva-Shapley Theorem we know that \(x(N) \ge {{\bar{v}}}(N)\). Hence

as desired.\(\square\)

Proposition

3.14 The set of coalitionally rational allocations satisfies \(CRGP^{a}\), \(CRGP^{b}\), \(CRGP^{p}\) and CRGP.

Proof

Let \((N,v)\in \Gamma\) with \(|N|\ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S \in CR(S,v^\gamma _{x,S})\) for every \(S\subset N\) with \(|S| = 2\), where \(v^\gamma _{x, S}\) represents any one of the four types of reduced games.

We first show that \(x(N) \ge v(N)\). Take any \(S \subset N\) such that \(|S|=2\). Since \(x^S \in CR\big (S, v^\gamma _{x, S}\big )\), it follows that \(x(S) \ge v^\gamma _{x, S}(S)\). If \(v^\gamma _{x, S} = v^{a}_{x, S}\), then \(x(S) \ge \max \{v(S \cup Q) - x(Q) \mid Q \subseteq N {\setminus } S \} \ge v(N) - x(N{\setminus } S)\), so \(x(N) \ge v(N)\). If \(v^\gamma _{x, S} = v^{b}_{x, S}\), then \(x(S) \ge {{\bar{v}}}(N) - x(N{\setminus } S)\), so \(x(N) \ge {{\bar{v}}}(N) \ge v(N)\). Similarly, if \(v^\gamma _{x, S} = v^{p}_{x, S}\), then \(x(S) \ge v_{\Pi }(N) - x(N{\setminus } S)\), so \(x(N) \ge v_{\Pi }(N) \ge v(N)\). Finally, If \(v^\gamma _{x, S} = v_{x, S}\), then \(x(S) \ge v(N) - x(N{\setminus } S)\), so \(x(N) \ge v(N)\).

Because \(v^\gamma _{x, \{i,j\}}(\{i\}) = v_{x, \{i,j\}}(\{i\})\) (for \(i,j\in N\)) for each of the four types of reduced games, the first paragraph of the proof of Proposition 3.3 demonstrates that \(x(S) \ge v(S)\) for every \(S \in {{\mathcal {N}}},\ S\ne N\). \(\square\)

C Selected proofs of Sect. 5 Properties

Solution \(\sigma _5\) satisfies \(RGP^p\) and \(CRGP^p\).

Proof

Note first that if \((N,v)\in \Gamma\) and \(x\in {\mathbb {R}}^N\) are such that \(x(S)+x(T)\ge v(S)+v(T)\) for every \(S, T \subset N\) such that \(S{\setminus } T\ne \emptyset , T{\setminus } S\ne \emptyset\), then \(x(N)\ge v_\Pi (N)\). Therefore, if \(x(S)+x(T)\ge v(S)+v(T)\) for every \(S, T \subset N\) with \(S{\setminus } T\ne \emptyset , T{\setminus } S\ne \emptyset\) and \(x(N)=v(N)\), then \(v(N)=v_\Pi (N)\).

Let now \((N,v)\in \Gamma , x\in \sigma _5(N,v)\), and \(S\subset N\). Since \(x(N)=v_\Pi (N)\), we have \(x(S)=v^p_{x,S}(S)\) for every \(S\subset N\). If \(|S|\ge 2\) then, using the reasoning above, it is enough to show that \(x(R)+x(T)\ge v^p_{x,S}(R)+v^p_{x,S}(T)\) for every \(R, T \subset S\) such that \(R{\setminus } T\ne \emptyset , T{\setminus } R\ne \emptyset\) to conclude that \(x^S\in \sigma _5(S, v^p_{x,S})\). Indeed, this an immediate consequence of the definition of \(v^p_{x,S}\) and x satisfying the same inequality in (N, v). If \(S=\{i\}\), then \(x_i=v_\Pi (N)-x(N{\setminus } \{i\})=v^p_{x,S}(S)\) and thus \(x^S\in \sigma _5(S, v^p_{x,S})\).

To prove that \(\sigma _5\) satisfies \(CRGP^p\), let \((N,v)\in \Gamma\) with \(|N|\ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S\in \sigma _5(S, v^p_{x,S})\) for every \(S\subset N,\ |S|\le 2\). Since \(x^S\in \sigma _5(S, v^p_{x,S})\) for \(|S|=1\), we obtain \(x(N)=v_\Pi (N)\). This implies that \(x(S)=v^p_{x,S}(S)\) for every \(S=\{i,j\}\subset N\), and therefore, since \(x^S\in \sigma _5(S, v^p_{x,S})\), \(v^p_{x,S}(S)=x(S)\ge v^p_{x,S}(\{i\})+v^p_{x,S}(\{j\})\) must hold. Hence,

for every \(Q,R\subseteq N{\setminus } \{i,j.\}\). Together with \(x(N)=v_\Pi (N)\), this implies that \(x\in \sigma _5(N,v)\). \(\square\)

Solution \(\sigma _6\) satisfies \(RGP^b\) and \(CRGP^b\).

Proof

The proof that \(\sigma _6\) satisfies \(CRGP^b\) follows the same line of argument as the one used to show that \(\sigma _5\) satisfies \(CRGP^p\).

To prove that \(\sigma _6\) satisfies \(RGP^b\), let \((N,v)\in \Gamma , x\in \sigma _6(N,v)\), and \(S\subset N\), \(|S|\le 2\). Since \(x(N)={{\bar{v}}}(N)\), we have \(x(S)=v^b_{x,S}(S)\) for every \(S\subset N\), which is enough to conclude that \(x^S\in \sigma _6(S,v^b_{x,S})\) for \(S\subset N\) with \(|S|=1\).

Let now \(S = \{i,j\}\), \(i\ne j\). It remains to show that \(x(S)=\overline{v^b_{x,S}}(S)\). Since the only (minimal) balanced families of S are \(\{\{i\},\{j\}\}\) and \(\{S\}\), and \(x(S)=v^b_{x,S}(S)\), it is enough to prove that \(x(S) \ge v^b_{x,S}(\{i\})+v^b_{x,S}(\{j\})\). Let \(Q_i\subseteq N{\setminus } S\) such that \(v^b_{x,S}(\{i\})=v(\{i\}\cup Q_i)-x(Q_i)\) and let \(Q_j\subseteq N{\setminus } S\) such that \(v^b_{x,S}(\{j\})=v(\{j\}\cup Q_j)-x(Q_j)\). Because \(x\in \sigma _6(N,v)\), \(x(\{i\}\cup Q_i)+x(\{j\}\cup Q_j)\ge v(\{i\}\cup Q_i)+v(\{j\}\cup Q_j) = v^b_{x,S}(\{i\})+x(Q_i)+v^b_{x,S}(\{j\})+x(Q_j)\) and thus \(x(\{i\})+x(\{j\})\ge v^b_{x,S}(\{i\})+v^b_{x,S}(\{j\})\). \(\square\)

Solution \(\sigma _7\) satisfies CONS and C-CONS.

Proof

We prove that \(\sigma _7\) satisfies the corresponding consistency and converse consistency axioms.

Let \((N,v) \in \Gamma\), \(x\in \sigma _7(N,v,\gamma (N,v))\), and \(S \subset N\). Since \(\gamma (N,v)\) satisfies the corresponding axiom \(RGP^\gamma\), \(x^S \in \gamma (S, v^\gamma _{x,S})\), where the \(\gamma\) refers to the appropriate solution concept and corresponding reduced game for any of the four CONS and C-CONS properties. As proved in Peleg (1986), \(s_{ij}(v,x,N)=s_{ij}(v^\gamma _{x,S},x^S,S)\). Moreover, if \(T\subseteq S\) is such that \(x^S(T)=v^\gamma _{x,s}(T)\), then there exists \(R\supseteq T\) such that \(x(R)=v(R)\). This proves that \(x^S\in \sigma _7(S, v^\gamma _{x,S},\gamma (S, v^\gamma _{x,S}))\).

To prove that \(\sigma _7(N,v,\gamma (N,v))\) satisfies the corresponding \(CRGP^\gamma\), let \((N,v)\in \Gamma\), \(|N| \ge 3\) and \(x\in {\mathbb {R}}^N\) such that \(x^S\in \sigma _7(S, v^\gamma _{x,S}, \gamma (S, v^\gamma _{x,S}))\) for every \(S\subset N\) such that \(|S|\le 2\). Since \(\gamma\) satisfies \(CRGP^\gamma\), \(x\in \gamma (N,v)\). Let now \(T\subseteq N\) such that \(x(T)=v(T)\) and \(S'=\{i,j\} \subseteq T\) with \(i\ne j\). Notice that \(x(N)=v(N)\) if \(\gamma\) is the core, \(x(N)=v_\Pi (N)\) if \(\gamma\) is the c-core and \(x(N)={{\bar{v}}}(N)\) if \(\gamma\) is the aspiration core. Also, if \(\gamma\) is the aspiration set then \(x(S') = v(T) - x(T {\setminus } S') \le v^\gamma _{x,S'}(S')\). We conclude that, no matter which of the four \(\gamma\) is used, \(x(S')=v^\gamma _{x,S'}(S')\).

Thus

and hence \(x\in \sigma _7(N,v,\gamma (N,v))\).\(\square\)

D Restricted domains

This appendix contains adaptations of our results to specific subclasses of games, such as the family of balanced games and that of totally balanced games. To provide axiomatizations on subclasses of games we need to adapt the axioms to incorporate specific domains of games \(\Gamma _0 \subseteq \Gamma\). This can be done as follows.

Axioms NNE, WNE, BOUND, IR, and WAM are trivially adapted to a subdomain \(\Gamma _0\subseteq \Gamma\) by, in their definitions in Sect. 3, replacing “solution \(\sigma\)” with “solution \(\sigma\) on \(\Gamma _0\),” and replacing “\((N,v)\in \Gamma\)” with “\((N,v)\in \Gamma _0\).” For COV we need to also add the condition \((N, \alpha v + \beta )\in \Gamma _0\):

Covariance under strategic equivalence (COV): A solution \(\sigma\) on \(\Gamma _0 \subseteq \Gamma\) satisfies COV if for every \((N,v) \in \Gamma _0\), \(\alpha > 0\), and \(\beta \in {\mathbb {R}}^N\), it holds that \((N,\alpha v + \beta ) \in \Gamma _0\) and \(\sigma (N,\alpha v + \beta ) = \alpha \sigma (N,v) + \{ \beta \}\).

Adaptation of consistency and converse consistency properties requires the inclusion of reduced games in the subdomains of games under consideration. We illustrate the adaptations of RGP and CRGP below. The adaptations of \(RGP^b\), \(RGP^p\), and \(RGP^a\), as well as of \(CRGP^b\), \(CRGP^p\), and \(CRGP^a\) are analogous.

Reduced game property (RGP): A solution \(\sigma\) on \(\Gamma _0 \subseteq \Gamma\) satisfies RGP if for every \((N,v) \in \Gamma _0\) and every \(x \in {\mathbb {R}}^N\),

Converse reduced game property (CRGP): A solution \(\sigma\) on \(\Gamma _0 \subseteq \Gamma\) satisfies CRGP if for every \((N,v) \in \Gamma _0\) with \(|N|\ge 3\) and every \(x \in {\mathbb {R}}^N\),

In this appendix, we consider the domain of balanced games \(\Gamma _C:= \{(N,v) \in \Gamma \mid C(N,v) \ne \emptyset \},\) the domain of partition-balanced games \(\Gamma _{C_c}:= \{(N,v) \in \Gamma \mid C_c(N,v) \ne \emptyset \}\), and the domain of totally balanced games \(\Gamma ^t_C := \{ (N,v) \in \Gamma _C \mid \forall S \in {\mathcal {N}}, (S,v^S) \in \Gamma _C\},\) where \(v^S\) denotes the restriction of the function v to the set \({\mathcal {S}} = \{T \subseteq S \mid T \ne \emptyset \}\).

Remark D.1

The game \((N,v)\in \Gamma\) with \(|N|=2\), \(v(N)=-1\), and \(v(S)=0\) for every \(S\subset N\), is not balanced. Therefore, on the domain of balanced games, axioms NNE and WNE are equivalent.

Remark D.2

Let \((N,v) \in \Gamma _C\). Then \(v(N) = v_{\Pi }(N) = {{\bar{v}}}(N)\). Therefore, for all \(S \subset N\) such that \(S \ne \emptyset\) and all \(x \in {\mathbb {R}}^N\), the reduced games \((S,v_{x,S})\), the partition-reduced games \((S,v^p_{x,S})\), and the balanced-reduced games \((S,v^b_{x,S})\) coincide. It follows that, on the domain of balanced games, axioms RGP, \(RGP^p\) and \(RGP^b\) are equivalent. Similarly, axioms CRGP, \(CRGP^p\), and \(CRGP^b\) are equivalent on the domain of balanced games.

We omit the proofs of the following three theorems because these are very similar to the proofs that we provided for the corresponding results on the entire domain \(\Gamma\). The minor adaptations needed use the facts that the domains in the three theorems all include all the null games, as well as all 2-player games (N, w) with \(w(N) = 0\) and \(w(S) \le 0\) for all \(S\in {{\mathcal {N}}}\).^{Footnote 15}

Theorem D.3

The core is the only solution concept on the domain \(\Gamma _C\) of balanced games that satisfies NNE, BOUND, COV, IR, WAM, RGP and CRGP.

Theorem D.4

The core is the only solution concept on the domain of totally balanced games, \(\Gamma ^t_C\), that satisfies NNE, BOUND, COV, IR, WAM, RGP and CRGP.

The previous two theorems demonstrate that the axioms that characterize the aspiration core on the entire domain of TU-games, also characterize the core on the domain of balanced games, as well as on the domain of totally balanced games, because RGP and CRGP are equivalent to \(RGP^b\) and \(CRGP^b\) on the domains \(\Gamma _C\) and \(\Gamma ^t_C\).

Theorem D.5

The c-core is the only solution concept on the domain of partition-balanced games, \(\Gamma _{C_c}\), that satisfies WNE, BOUND, COV, IR, WAM, \(RGP^p\) and \(CRGP^p\).

The previous theorem demonstrates that the axioms that characterize the aspiration core on the entire domain of TU-games, also characterize the c-core on the domain of partition-balanced games, because \(RGP^p\) and \(CRGP^p\) are equivalent to \(RGP^b\) and \(CRGP^b\) on the domain \(\Gamma _{C_c}\).

The axioms that characterize the aspiration set on the entire domain \(\Gamma\) do not characterize the core on the domain of balanced games because, as we show in the example below, the core does not satisfy \(CRGP^a\) on \(\Gamma _C\).^{Footnote 16}

Example D.6

The core does not satisfy \(CRGP^{a}\) on \(\Gamma _C\). Consider the game \((N,v)\in \Gamma\) defined by \(N=\{1,2,3,4\}\), \(v(S)=9\) if \(|S|\ge 3\) and \(1 \in S\), and \(v(S)=0\) for all other \(S\in {{\mathcal {N}}}\). \((N,v) \in \Gamma _C\) because \((9,0,0,0) \in C(N,v)\). Consider the payoff vector \(x:=(3,3,3,3) \notin C(N,v)\) and the coalitions \(T \subset N\) of size 2. If \(T = \{1,i\}\) for some \(i \in \{2,3,4\}\), then \(v^a_{x,T}(\{1\}) = 3\), \(v^a_{x,T}(\{i\}) = 0\) and \(v^a_{x,T}(T) = 6\), so that \(x^T = (3,3) \in C(T,v^a_{x,T})\). Similarly, if \(T = \{i,j\} \subset \{2,3,4\}\) with \(i \ne j\), then \(v^a_{x,T}(\{i\})=v^a_{x,T}(\{j\}) = 3\) and \(v^a_{x,T}(T) = 6\), so again \(x^T = (3,3) \in C(T,v^a_{x,T})\).

The next proposition shows that the core satisfies a weaker version of the aspiration reduced game property, which is defined below.

Weak converse aspiration-reduced game property (\(wCRGP^a\)): A solution \(\sigma\) satisfies \(wCRGP^a\) on \(\Gamma _0\subseteq \Gamma\) if for every \((N,v) \in \Gamma _0\) with \(|N|\ge 3\) and every \(x \in {\mathbb {R}}^N\),

Proposition D.7

The core satisfies \(wCRGP^a\) on \(\Gamma _C\).

Proof

Let \((N,v) \in \Gamma _C\) with \(|N|\ge 3\) and \(x \in {\mathbb {R}}^N\) such that \((S,v^a_{x,S})\in \Gamma _C\) and \(x^S \in C(S,v^a_{x,S})\) for all \(S\subset N\) with \(S\ne \emptyset\). We first prove that x is coalitionally rational in (N, v). Let \(i\in N\) and consider the aspiration reduced game \((\{i\},v^a_{x,\{i\}})\). Because \(x_i\in C(\{i\},v^a_{x,\{i\}})\), we have \(x_i=v^a_{x,\{i\}}=\max \left\{ v(\{i\}\cup T)-x(T)\mid T\subseteq N{\setminus }\{i\}\right\}\). Thus, \(x(\{i\}\cup T)\ge v(\{i\}\cup T)\) for every \(T\subseteq N{\setminus }\{i\}\). This establishes that x is coalitionally rational in (N, v).

We prove next that \(x(N)=v(N)\). For every \(i\in N\), consider the aspiration reduced game \((N{\setminus } \{i\},v^a_{x,N{\setminus } \{i\}})\). Because \(x^{N{\setminus } \{i\}}\in C(N{\setminus } \{i\},v^a_{x,N{\setminus } \{i\}})\), we have \(x(N{\setminus } \{i\})=v^a_{x,N{\setminus } \{i\}}(N{\setminus } \{i\})=\max \left\{ v(N{\setminus } \{i\}\cup \{i\})-x_i, v(N{\setminus } \{i\})\right\} =\max \left\{ v(N)-x_i, v(N{\setminus } \{i\})\right\}\).

If there exists an \(i\in N\) such that \(v(N)-x(i)\ge v(N{\setminus } \{i\})\), then \(x(N{\setminus } \{i\})=v^a_{x,N{\setminus } \{i\}}(N{\setminus } \{i\})=v(N)-x(i)\) and thus \(x(N)=v(N)\).

If \(v(N)-x(i) < v(N{\setminus } \{i\})\) for all \(i\in N\), then \(x(N{\setminus } \{i\})=v^a_{x,N{\setminus } \{i\}}(N{\setminus } \{i\})=v(N{\setminus } \{i\})\) for each \(i\in N\). Note that the family \(\mathcal {B}:=\left\{ N{\setminus }\{i\}\right\} _{i\in N}\) of coalitions is balanced with balancing weights \(\left\{ \lambda _S=\frac{1}{n-1}\right\} _{S\in {\mathcal {B}}}\). We derive

where the last inequality follows from the Bondareva-Shapley theorem applied to the balanced game (N, v). Because \(x(N)\ge v(N)\) by coalitional rationality of x, we obtain \(x(N)=v(N)\). \(\square\)

An alternative characterization of the core on the domain of balanced games can be obtained using the standard version of the aspiration-reduced game property (defined below) and the weak converse aspiration-reduced game property.

Standard aspiration-reduced game property (\(sRGP^a\)): A solution \(\sigma\) on \(\Gamma _0 \subseteq \Gamma\) satisfies \(sRGP^a\) if for every \((N,v) \in \Gamma _0\) and every \(x \in {\mathbb {R}}^N\),

Theorem D.8

The core is the only solution concept on \(\Gamma _C\) that satisfies NNE, BOUND, COV, WAM, \(sRGP^a\) and \(wCRGP^a\).

Proof

We already established in Theorem D.3 that the core satisfies NNE, BOUND, COV, and WAM on \(\Gamma _C\). Bejan and Gómez (2012) show that the core satisfies \(sRGP^{a}\) on \(\Gamma _C\), and Proposition D.7 demonstrates that the core satisfies \(wCRGP^a\) on \(\Gamma _C\). Thus, the core satisfies all the desired axioms on \(\Gamma _C\).

Let now \(\sigma\) be a solution concept that satisfies NNE, BOUND, COV, WAM, \(sRGP^a\) and \(wCRGP^a\) on \(\Gamma _C\). The proof of Proposition 4.5 demonstrates that \(\sigma (N,v) \subseteq Asp(N,v)\) for every \((N,v) \in \Gamma _C\). The proofs of Lemmas 4.2 and 4.6 (part 4) can be adapted to demonstrate that \(\sigma (N,v) = Asp(N,v)\) for every \((N,v)\in \Gamma _C\) with \(|N|\le 2\). Thus, using Lemma 2.2, \(\sigma (N,v) = C(N,v)\) for every \((N,v)\in \Gamma _C\) with \(|N|\le 2\).

Having demonstrated that \(\sigma (N,v) = C(N,v)\) for every \((N,v)\in \Gamma _C\) with \(|N|\le 2\), we now use induction over |N| to prove that \(\sigma (N,v) = C(N,v)\) for every \((N,v)\in \Gamma _C\) with \(|N|\ge 3\). So, let \(n\ge 2\) and suppose that \(\sigma (N,v) = C(N,v)\) for every \((N,v)\in \Gamma _C\) with \(|N|\le n\). Let \((N,v) \in \Gamma _C\) with \(|N|=n+1\).

We prove first that \(\sigma (N,v)\subseteq C(N,v)\). Let \(x\in \sigma (N,v)\). Since \(\sigma\) satisfies \(sRGP^a\) on \(\Gamma _C\) we have that \((S, v^a_{x,S})\in \Gamma _C\) and \(x^S\in \sigma (S, v^a_{x,S})\), for every \(S\in {\mathcal {N}}, S\ne N\). By the induction hypothesis, \(\sigma (S, v^a_{x,S})=C(S, v^a_{x,S})\) thus \(x^S\in C(S, v^a_{x,S})\) for every \(S\in {\mathcal {N}}, S\ne N\). Since the core satisfies \(wCRGP^a\), this implies \(x\in C(N,v)\) and therefore \(\sigma (N,v)\subseteq C(N,v)\).

For the reverse inclusion, let \(x\in C(N,v)\). Since the core satisfies \(sRGP^a\), \((S, v^a_{x,S})\in \Gamma _C\) and \(x^S\in C(S, v^a_{x,S})\), for every \(S\in {\mathcal {N}}, S\ne N\). The induction hypothesis implies that \(x^S\in \sigma (S, v^a_{x,S})\), and \(wCRGP^a\) of \(\sigma\) implies that \(x\in \sigma (N,v)\). \(\square\)

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