Nash implementation in production economies with unequal skills: a characterization

The present study examines production economies with unequal labor skills, where the planner is ignorant of the set of feasible allocations in advance of production. In particular, we characterize Nash implementation by canonical mechanisms by means of Maskin monotonicity and a new axiom, non-manipulability of unused skills (NUS), where the latter represents a weak independence property with respect to changes in skills. Following these characterizations, we show that some Maskin monotonic social choice correspondences are not implementable if information about individual skills is absent.

1. Sustainable management requires knowledge on the size of the stocks, their age composition, their distribution, and the environment in which they live. Every year, data from Norwegian scientific surveys and from fishers are compared with data from other countries (Norwegian marine scientists cooperate closely with researchers from other countries, especially Russia) and assessed by ICES, the International Council for the Exploration of the Sea.

2. This practice is in accordance with international agreements including the 1982 UN Law of the Sea Convention, the 1995 UN Fish Stocks Agreement, and the 1995 FAO Code of Conduct for Responsible Fisheries.

3. Recently, an ecosystem approach is increasingly being applied to Norwegian fisheries management. This takes into account not only how harvesting affects fish stocks, but also how fisheries affect the marine environment for living marine resources in general.

4. Given that the total allowable catch in the Barents Sea is allocated through negotiations under international agreements, the country’s quotas are distributed among different groups of fishers and then subdivided and allocated among fishing boats in each group. With this in mind, the resource management mechanism in Norway is expected to implement fishing allocations by monitoring and punishing each fishing boat’s overexploitation of resources.

5. In addition to the above-mentioned studies, for instance, Suh (1995), Yoshihara (1999), Kaplan and Wettstein (2000), and Tian (2009) all proposed simple or natural mechanisms to implement particular SCCs, whereas Shin and Suh (1997) and Yoshihara (2000) characterized SCCs implementable by simple or natural mechanisms.

6. The symbol \(\mathbb {R}_{+}\) denotes the set of non-negative real numbers.

7. The symbol \(\mathbb {R}_{++}\) denotes the set of positive real numbers.

8. For any two sets X and Y, \(X\subseteq Y\) whenever any \(x\in X\) also belongs to Y, and \(X=Y\) if and only if \(X\subseteq Y\) and \(Y\subseteq X\).

9. It might be more natural to define labor skill and labor intensity in a discriminative way: for example, if \({\overline{s}}_{i}\in {\mathcal {S}}\) is i ’s labor skill, then i’s labor intensity is a variable \(s_{i}\), where \( 0<s_{i}\le {\overline{s}}_{i}\). In such a formulation, we may view the amount of \(s_{i}\) as being determined endogenously by agent i. In spite of this more natural view, we assume in the following discussion that labor intensity is a constant value, \(s_{i}={\overline{s}}_{i}\), for the sake of analytical simplicity. The main theorems in the following discussion remain valid with a few changes to the settings of the economic environments, even if labor intensity was assumed to be varied.

10. See Hong (1995), Suh (1995), Shin and Suh (1997), Tian (1999, 2000), and Yoshihara (2000).

11. Hong (1995) also discusses implementation problems for the case of withholding as a simple extension of the case of not withholding.

12. Even if f, \({\varvec{x}}\), and w are observed, the exact location of the true skill profile \({\varvec{s}}\) cannot be known before or after the production process. For a detailed discussion on this point, see Yamada and Yoshihara (2008).

13. It is also possible in the same class of production economies with unknown skills to provide a full characterization of Nash implementation without any restriction on available mechanisms. In this case, the necessary and sufficient condition for Nash implementation is a variation of Condition M (originally introduced in Moore and Repullo (1990)), which has a highly complicated form. For this issue, see Yoshihara and Yamada (2017).

14. The forthrightness condition was first introduced by Saijo et al. (1996) for implementation problems in pure exchange economies. Lombardi and Yoshihara (2013) then formulated this condition for implementation in abstract social choice problems.

15. For a detailed explanation about information smuggling and about how forthrightness can exclude this problem, see Lombardi and Yoshihara (2013).

16. Moreover, in the abstract social choice environments, the monotone transformation takes place within the set of feasible alternatives.

17. By presuming exactly the same type of skill changes as that of NUS, the independence of unused skills (IUS) axiom introduced by Yamada and Yoshihara (2007) straightforwardly requires that the \(\varphi \)-optimal allocation at the economy \({\varvec{e}}\) should also be \(\varphi \)-optimal at the new economy \({\varvec{e}}^{\prime }\). NUS is much weaker than IUS.

18. Van Parijs (1995) formulated the EOB principle as undominated diversity (Van Parijs 1995), which is stronger than SIU.

19. Note \(\partial X\) denotes the upper boundary of the set \(X\subseteq \mathbb {R }_{+}^{2}\).

20. From now on, we simply write a value of \(g^{*}\) as \(g^{*}\left( {\varvec{a}}\right) \) instead of \(g^{*}\left( {\varvec{a}};f\left( \sum s_{k}x_{k}\right) \right) \), without loss of generality. Moreover, let \( g_{j2}^{*}\left( {\varvec{a}}\right) \) be the second component of \( g_{j}^{*}\left( {\varvec{a}}\right) \), which is the share of the output produced to agent j specified by the mechanism under \({\varvec{a}} \).

Authors and Affiliations

1. Department of Economics, University of Massachusetts Amherst, Amherst, MA, 01002, USA

   Naoki Yoshihara

2. School of Management, Kochi University of Technology, Eikokuji, Kochi City, Kochi, 780-8515, Japan

   Naoki Yoshihara

3. Institute of Economic Research, Hitotsubashi University, Kunitachi, Tokyo, 186-8603, Japan

   Naoki Yoshihara

4. School of Society and Collaboration, Sapporo University, 7-3-1 Nishioka 3-jo, Toyohira-ku, Sapporo, Hokkaido, 062-8520, Japan

   Akira Yamada

Corresponding author

Correspondence to Naoki Yoshihara.

Publisher's Note

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

We wish to give our special thanks to the editor in charge and two reviewers of this journal, whose comments and suggestions substantially improved the paper. We are also grateful to Arunava Sen, Guoqiang Tian, Michele Lombardi, Hans Peters, Yoshi Saijo, William Thomson, and Takuma Wakayama for their helpful discussions and to all the participants of the International Meeting of the Society for Social Choice and Welfare in Moscow in July 2010, the Annual Meeting of the Japanese Economic Association in June 2010, and the workshops at Maastricht University, Ecole Polytechnique, and Keio University.

Appendix

As a preliminary step, we construct two auxiliary functions, which are used in the proofs of Theorem 2. Given \({\varvec{x}}\in \left[ 0,{\bar{x}} \right] ^{n}\) and \(i\in N\), let

Let the notation \(\sigma _{i}\in {\mathcal {S}}\) represent the announced skill of agent i, which is not necessarily identical to a truthful skill \( s_{i}\in {\mathcal {S}}\) of agent i, and \(\varvec{\sigma }=\left( \sigma _{i}\right) _{i\in N}\in {\mathcal {S}}^{n}\) represent a profile of the announced skills of all agents. Then,

• Let \(d^{{\varvec{y}}}\) be such that for each \(w\in \mathbb {R} _{+}\), each \(\left( \varvec{\sigma }\mathbf {,}{\varvec{x}} \mathbf {,}{\varvec{y}}\right) \in {\mathcal {S}}^{n}\times \left[ 0, {\bar{x}}\right] ^{n}\times \mathbb {R}_{+}^{n}\), and each \(i\in N\),\(d_{i}^{{\varvec{y}}}\left( \varvec{\sigma }\mathbf {,} {\varvec{x}}\mathbf {,}{\varvec{y}},w\right) =\left\{ \begin{array}{ll} w &{} \text {if }x_{i}=\pi \left( {\varvec{x}}_{-i}\right)<{\bar{x}} \text { or }x_{i}<\pi \left( {\varvec{x}}_{-i}\right) ={\bar{x}}\text {, and }y_{i}=0\text {,} \\ 0 &{} \text { otherwise.} \end{array} \right. \)

• Let \(d^{\varvec{\sigma }}\) be such that for each \(w\in \mathbb {R} _{+}\), each \(\left( \varvec{\sigma }\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \in {\mathcal {S}}^{n}\times \left[ 0, {\bar{x}}\right] ^{n}\times \mathbb {R}_{+}^{n}\), and each \(i\in N\),\(d_{i}^{\varvec{\sigma }}\left( \varvec{\sigma }\mathbf {,} {\varvec{x}}\mathbf {,}{\varvec{y}},w\right) =\left\{ \begin{array}{ll} w &{} \begin{array}{l} \text {if }x_{i}=0\text {, }\sigma _{i}>\sigma _{j}\text { for each }j\ne i\text {, } \\ \text {and }y_{i}>\max \left\{ f\left( \sum _{j\ne i}\sigma _{j}{\overline{x}} \right) ,w\right\} \end{array} \\ 0 &{} \text {otherwise.} \end{array} \right. \)

The function \(d^{{\varvec{y}}}\) assigns all of the output produced to only one agent who provides the maximal positive amount, but less than \({\bar{x}}\), of labor time and reports zero demand for output. The function \(d^{\varvec{\sigma }}\) assigns all of the output produced to only one agent who reports the highest skill and does not work.

Let us assume that a game form in \(\Gamma \) requires agents to announce their own skills. Moreover, assume that this game form specifies the share of output by using the function \(d^{\varvec{\sigma }}\) whenever the expected output \(f\left( \sum \sigma _{k}x_{k}\right) \) derived from the deta \(\left( \varvec{\sigma }\mathbf {,}{\varvec{x}}\right) \) differs from the realized output \(w\left( {\varvec{s}},{\varvec{x}}\right) \). Then, it has an interesting property as the following Lemma A1 shows.

Lemma A1

Let Assumption 1hold. Let \( \gamma =\left( A,g\right) \in \Gamma \) be a game form, where every agent i is requested to announce his/her own skill, \(\sigma _{i} \), as a part of a message, and the outcome function g specifies the share of the output according to the function \(d^{ \varvec{\sigma }}\) whenever \(f\left( \sum \sigma _{i}x_{i}\right) \ne w\left( {\varvec{s}},{\varvec{x}}\right) \) . Given \(\left( {\varvec{u}}\mathbf {,}{\varvec{s}}\right) \in {\mathcal {E}}\), let \( {\varvec{a}}=\left( {\varvec{m}}\mathbf {,}\widehat{{\varvec{x}}} \right) \in \times _{i\in N}A_{i}\)bea Nash equilibrium of \(\left( \gamma ,{\varvec{u}}\mathbf {,}{\varvec{s}}\right) \) such that \(f\left( \sum \sigma _{k}{\widehat{x}} _{k}\right) =w\left( {\varvec{s}},\widehat{{\varvec{x}}}\right) \) . Then,for each \(i\in N\) with \({\widehat{x}}_{i}>0\), \( \sigma _{i}=s_{i}\).

This proof is presented exactly as Lemma 1 in Yamada and Yoshihara (2007). Note that the mechanism, \(\gamma ^{*}\), constructed in the proof of Theorem 2 meets the premise of Lemma A1.

Proof of Theorem 2

As a preliminary step, given \({\varvec{s}}\in {\mathcal {S}}^{n}\), \( {\varvec{z}}=\left( {\varvec{x}},{\varvec{y}}\right) \in Z^{n}\), and \(p\in \Delta \), let \(B\left( p,s_{i},z_{i}\right) \equiv \left\{ z_{i}^{\prime }\in Z\mid p_{y}y_{i}^{\prime }-p_{x}s_{i}x_{i}^{\prime }\le p_{y}y_{i}-p_{x}s_{i}x_{i}\right\} \). Moreover, define \(B\left( \Delta ^{P}\left( {{\varvec{e}}}, {{\varvec{z}}}\right) ,s_{i},z_{i}\right) \equiv \cup _{p\in \Delta ^{P}\left( {\varvec{e}},{\varvec{z}}\right) }B\left( p,s_{i},z_{i}\right) \).

Let \(p_{\alpha ^{{}}}\left( x_{i};{\varvec{x}}_{-i},{ {\varvec{s}}}\right) \equiv \lim _{x_{i}^{\prime }\rightarrow x_{i}} \frac{f\left( \sum _{j\ne i}s_{j}x_{j}+s_{i}x_{i}^{\prime }\right) -f\left( \sum _{j\ne i}s_{j}x_{j}+s_{i}x_{i}\right) }{s_{i}x_{i}^{\prime }-s_{i}x_{i}} s_{i}\), where \(\alpha =`+\)’ if \(x_{i}^{\prime }>x_{i}\); and \(\alpha =\)‘−’ if \(x_{i}^{\prime }<x_{i}\). Given \(\left( {\varvec{u}},\varvec{ \sigma }\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \in {\mathcal {U}}^{n}\times {\mathcal {S}}^{n}\times \left[ 0,{\bar{x}}\right] ^{n}\times \mathbb {R}_{+}^{n}\), let

Let us see how the set \(N\left( {\varvec{u}},\varvec{\sigma } \mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \) can function in the mechanism. When \(i\in N\left( {\varvec{u}},\varvec{\sigma }\mathbf {,}{\varvec{x}} \mathbf {,}{\varvec{y}}\right) \), there are two cases: one is \(x_{i}=0\), and the other is \( x_{i}>0\).

Let \(x_{i}=0\) for \(i\in N\left( {\varvec{u}},\varvec{\sigma } \mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \). Then, \(\sigma _{i}\) may be a false announcement whenever there exists \(\left( \sigma _{i}^{\circ },x_{i}^{\circ },y_{i}^{\circ }\right) \), rather than just \(\left( x_{i}^{\circ },y_{i}^{\circ }\right) \), such that \(\left( \left( x_{i}^{\circ },{\varvec{x}}_{-i}\right) ,\left( y_{i}^{\circ },{\varvec{y}}_{-i}\right) \right) \in \varphi \left( {\varvec{u}},\left( \sigma _{i}^{\circ }, \varvec{\sigma }_{-i}\right) \right) \) holds. If such a profile exists, then this agent could be a potential deviator in her announcements of not only her consumption vector but also her skill. Therefore, the mechanism would assign a ‘punishment outcome’ to this agent by taking \(\left( \sigma _{i}^{\circ },x_{i}^{\circ },y_{i}^{\circ }\right) \) as a potential true message. However, such a potential true message would not necessarily be uniquely specified, in that there may be multiple potential true messages. In such a case, we specify how to select one from the set of multiple potential true messages in the following way. For \(i\in N\left( {\varvec{u}},\varvec{\sigma } \mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \) with \(x_{i}=0\), let \(\left( \sigma _{i}^{\mu ^{{}}},x_{i}^{\mu ^{{}}},y_{i}^{\mu ^{{}}}\right) \) be selected by:

Here, if \(\sigma _{i}^{\mu ^{{}}}x_{i}^{\mu ^{{}}}>0\), then \( y_{i}^{\mu ^{{}}}- p_{\alpha ^{{}}}\left( x_{i}^{\mu ^{{}}}; {\varvec{x}}_{-i}, \left( \sigma _{i}^{\mu ^{{}}},\varvec{\sigma }_{-i}\right) \right) x_{i}^{\mu ^{{}}}=y_{i}^{\mu ^{{}}}- p_{-}\left( x_{i}^{\mu ^{{}}}; {\varvec{x}}_{-i},\left( \sigma _{i}^{\mu ^{{}}},\varvec{\sigma } _{-i}\right) \right) x_{i}^{\mu ^{{}}}\), while if \(\sigma _{i}^{\mu ^{{}}}x_{i}^{\mu ^{{}}}=0\), then \(y_{i}^{\mu ^{{}}}-p_{\alpha ^{{}}}\left( x_{i}^{\mu ^{{}}};{\varvec{x}}_{-i},\left( \sigma _{i}^{\mu ^{{}}}, \varvec{\sigma }_{-i}\right) \right) x_{i}^{\mu ^{{}}}=y_{i}^{\mu ^{{}}}-p_{+}\left( x_{i}^{\mu ^{{}}};{\varvec{x}}_{-i},\left( \sigma _{i}^{\mu ^{{}}},\varvec{\sigma }_{-i}\right) \right) x_{i}^{\mu ^{{}}}=y_{i}^{\mu ^{{}}}\).

Next, consider \(x_{i}>0\) for \(i\in N\left( {\varvec{u}},\varvec{ \sigma }\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \). In this case, \(\sigma _{i}\) may be the ture information as discussed below. Then, as there exists \(\left( x_{i}^{\circ },y_{i}^{\circ }\right) \) such that \(\left( \left( x_{i}^{\circ },{\varvec{x}}_{-i}\right) ,\left( y_{i}^{\circ },{\varvec{y}}_{-i}\right) \right) \in \varphi \left( {\varvec{u}},\varvec{\sigma }\right) \) by definition of \(N\left( {\varvec{u}} ,\varvec{\sigma }\mathbf {,}{\varvec{x}}\mathbf {,} {\varvec{y}}\right) \), such \(\left( x_{i}^{\circ },y_{i}^{\circ }\right) \) is a potential true message for agent i. Then, the mechanism would assign a ‘punishment outcome’ to this agent by taking \(\left( x_{i}^{\circ },y_{i}^{\circ }\right) \) as a potential true message. Note that, again there may be multiple potential true messages. However, in this case, any selection from the set of potential true messages leaves the agent indifferent, as \(u_{i}\left( x_{i}^{\circ },y_{i}^{\circ }\right) =u_{i}\left( x_{i}^{\prime \circ },y_{i}^{\prime \circ }\right) \) holds for any different potential true messages \(\left( x_{i}^{\circ },y_{i}^{\circ }\right) \) and \(\left( x_{i}^{\prime \circ },y_{i}^{\prime \circ }\right) \), given that \(\varphi \left( {\varvec{u}},\varvec{\sigma }\right) \subseteq P\left( {\varvec{u}},\varvec{\sigma }\right) \). Therefore, in this case, the specification of a ‘punishment outcome’ is not difficult, as discussed below.

Now, we are ready to discuss the construction of a mechanism. Denote the upper boundary of \(L\left( z_{i},u_{i}\right) \) by \(\partial L\left( z_{i},u_{i}\right) \equiv \left\{ z_{i}^{\prime }\in L\left( z_{i},u_{i}\right) \mid u_{i}\left( z_{i}^{\prime }\right) =u_{i}\left( z_{i}\right) \right\} \). For the sake of simplifying notations, given \( \left( {\varvec{u}},\varvec{\sigma }\right) \in {\mathcal {E}}\) and \(\left( z_{j}^{\circ },\left( {\varvec{x}}_{-j}, {\varvec{y}}_{-j}\right) \right) \in \varphi \left( {\varvec{u}}, \varvec{\sigma }\right) \), we will sometimes use \(L\left( z_{j}^{\circ },u_{j};\varvec{\sigma }\right) \) instead of the precise statement \(L\left( \left( z_{j}^{\circ },\left( {\varvec{x}}_{-j}, {\varvec{y}}_{-j}\right) \right) ,u_{i};\varvec{\sigma }\right) \) in the following discussion. We define a canonical mechanism \( \gamma ^{*}=\left( A^{*},g^{*}\right) \in \Gamma _{C}^{*}\) with \(A_{i}^{*}\equiv \)\(M_{i}\times \left[ 0,{\bar{x}}\right] \), where \( M_{i}\equiv {\mathcal {U}}^{n}\times {\mathcal {S}}\times Z\) with generic element \( \left( {\varvec{u}}^{i},\sigma _{i},z_{i}\right) \), for each \(i\in N\) , as follows:

For each \({\varvec{a}}=\left( \left( {\varvec{u}} ^{i}\right) _{i\in N},\varvec{\sigma }\mathbf {,}{\varvec{z}}, \widehat{{\varvec{x}}}\right) \equiv \left( {\varvec{u}}^{i},\sigma _{i},z_{i}, {\widehat{x}}_{i}\right) _{i\in N}\in \times _{i\in N}\left( M_{i}\times \left[ 0,{\bar{x}}\right] \right) \) and for each output \(w=w\left( {\varvec{s}}, \widehat{{\varvec{x}}}\right) \in \mathbb {R}_{+}\),

Rule 1: if \(f\left( \sum \sigma _{k}{\widehat{x}} _{k}\right) =w\), and

1–1: there exists \({\varvec{u}}\in {\mathcal {U}} ^{n} \) such that \({\varvec{u}}^{i}={\varvec{u}}\) for each \( i\in N\), and \(\sum y_{k}\le w\) and

1–1-a): if \(\left( \widehat{{\varvec{x}} }\mathbf {,}{\varvec{y}}\right) \in \varphi \left( {\varvec{u}},\varvec{\sigma }\right) \), then \(g^{*}\left( {\varvec{a}} ;w\right) =\left( \widehat{{\varvec{x}}},{\varvec{y}}\right) \),

1–1-b): if \(\left( \widehat{{\varvec{x}} }\mathbf {,}{\varvec{y}}\right) \notin \varphi \left( {\varvec{u}},\varvec{\sigma }\right) \), then \(g^{*}\left( {\varvec{a}} ;w\right) =\left( \widehat{{\varvec{x}}},{\mathbf {0}}\right) \),

1–2: there exists \(j\in N\) such that \({\varvec{u}} ^{i}={\varvec{u}}\ne {\varvec{u}}^{j}\) for each \(i\ne j\), \( \left( \widehat{{\varvec{x}}}\mathbf {,}{\varvec{y}}\right) \notin \varphi \left( {\varvec{u}}^{j},\varvec{\sigma }\right) \), and

1–2-a): if \(j\in N\left( {\varvec{u}},\varvec{ \sigma }\mathbf {,}\widehat{{\varvec{x}}}\mathbf {,} {\varvec{y}}\right) \), then \(g_{i}^{*}\left( {\varvec{a}};w\right) =\left( {\widehat{x}}_{i},0\right) \) for each \(i\ne j\), and \(g_{j}^{*}\left( {\varvec{a}};w\right) =\left\{ \begin{array}{ll} \left( {\widehat{x}}_{j},\min \left\{ y_{j}^{\prime \prime },w\right\} \right) &{} \text {if }y_{j}>f\left( \sum \sigma _{k}{\bar{x}}\right) \\ \left( {\widehat{x}}_{j},0\right) &{} \text {otherwise,} \end{array} \right. \)

where \(y_{j}^{\prime \prime }\) is given by^{Footnote 19}

\({\left( {\widehat{x}}_{j},y_{j}^{\prime \prime }\right) \in \left\{ \begin{array}{ll} \partial \left[ L\left( z_{j}^{\circ },u_{j};\varvec{\sigma }\right) \cup B\left( \Delta ^{P}\left( \left( {\varvec{u}},\varvec{ \sigma }\right) ,\left( z_{j}^{\circ },\left( \widehat{{\varvec{x}}} _{-j},{\varvec{y}}_{-j}\right) \right) \right) ,\sigma _{j},z_{j}^{\circ }\right) \right] &{} \text {if }{\widehat{x}}_{j}>0 \\ \left\{ \left( 0,y_{j}^{\mu ^{{}}}-p_{\alpha ^{{}}}\left( x_{j}^{\mu ^{{}}}; \widehat{{\varvec{x}}}_{-j},\left( \sigma _{j}^{\mu ^{{}}},\varvec{\sigma }_{-j}\right) \right) x_{j}^{\mu ^{{}}}\right) \right\} &{} \text {otherwise,} \end{array} \right. }\)

for \(z_{j}^{\circ }=(x_{j}^{\circ },y_{j}^{\circ })\) with \(\left( \left( x_{j}^{\circ },\widehat{{\varvec{x}}}_{-j}\right) ,\left( y_{j}^{\circ },{\varvec{y}}_{-j}\right) \right) \in \varphi \left( {\varvec{u}},\varvec{\sigma }\right) \),

1–2-b): if \(j\notin N\left( {\varvec{u}}, \varvec{\sigma }\mathbf {,}\widehat{{\varvec{x}}}\mathbf {,} {\varvec{y}}\right) \) and there exists \((x_{j}^{\circ },y_{j}^{\circ })\in Z\) such that

\(\left( \left( x_{j}^{\circ },\widehat{{\varvec{x}}} _{-j}\right) ,\left( y_{j}^{\circ },{\varvec{y}}_{-j}\right) \right) \in P\left( {\varvec{u}}^{j},\varvec{ \sigma }\right) \), then \(g_{i}^{*}\left( {\varvec{a}};w\right) =\left( {\widehat{x}}_{i},0\right) \) for each \(i\ne j\), and

\(g_{j}^{*}\left( {\varvec{a}};w\right) =\left\{ \begin{array}{ll} \left( {\widehat{x}}_{j},w\right) &{} \text {if }{\widehat{x}}_{j}>0\text { and } y_{j}>w \\ \left( {\widehat{x}}_{j},0\right) &{} \text {otherwise,} \end{array} \right. \)

1–3: in any other case, \(g^{*}\left( {\varvec{a}} ;w\right) =\left( {\varvec{x}},d^{{\varvec{y}}}\left( \varvec{ \sigma }\mathbf {,}\widehat{{\varvec{x}}}\mathbf {,}{\varvec{y}},w\right) \right) \),

Rule 2: if \(f\left( \sum \sigma _{k}{\widehat{x}} _{k}\right) \ne w\), then \(g^{*}\left( {\varvec{a}};w\right) =\left( {\varvec{x}},d^{\varvec{\sigma }}\left( \varvec{\sigma }\mathbf {,} \widehat{{\varvec{x}}}\mathbf {,}{\varvec{y}},w\right) \right) \).

Lemma A2

Let Assumption 1hold and \( n\ge 3\). Then,\(\gamma ^{*}\) implements any interior and efficient SCC \(\varphi \) satisfyingM and NUS in Nash equilibria.

Proof

Let \(\varphi \) be an interior and efficient SCC satisfying M and NUS. Let \({\varvec{e}}=\left( {\varvec{u}}, {\varvec{s}}\right) \in {\mathcal {E}}\).

(1) First, we show that \(\varphi \left( {\varvec{e}} \right) \subseteq NA\left( \gamma ^{*},{\varvec{e}}\right) \). Let \({\varvec{z}}=\left( {\varvec{x}}\mathbf {,} {\varvec{y}}\right) \in \)\(\varphi \left( {\varvec{e}}\right) \). Let \( {\varvec{a}}=\left( \left( {\varvec{u}}^{i}\right) _{i\in N}, {\varvec{s}}\mathbf {,}{\varvec{z}}\mathbf {,}{\varvec{x}}\right) \in \times _{i\in N}\left( M_{i}\times \left[ 0,{\bar{x}}\right] \right) \) be such that \({\varvec{u}}^{i}={\varvec{u}}\) for each \(i\in N\). Then, \(g^{*}\left( {\varvec{a}}\right) =\left( {\varvec{x}},{\varvec{y}}\right) \) from Rule 1–1.^{Footnote 20} Suppose \(j\in N\) deviates to \(a_{j}^{\prime }=\left( {\varvec{u}} ^{j\prime },s_{j}^{\prime },z_{j}^{\prime },x_{j}^{\prime }\right) \in M_{i}\times \left[ 0,{\bar{x}}\right] \). From Assumption 1 and the continuity of the utility functions, if \(g_{j2}^{*}\left( a_{j}^{\prime }, {\varvec{a}}_{-j}\right) =0\), it implies the worst outcome for j.

If \(a_{j}^{\prime }\) induces Rule 2, then \(x_{j}^{\prime }>0\) and \( g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) =0\). If \( a_{j}^{\prime }\) induces Rule 1-3, then either \(\left( \left( x_{j}^{\prime },{\varvec{x}}_{-j}\right) ,\left( y_{j}^{\prime }, {\varvec{y}}_{-j}\right) \right) \in \varphi \left( {\varvec{u}}^{j},\left( s_{j}^{\prime },{\varvec{s}}_{-j}\right) \right) \) or \(\sum _{i\ne j}y_{i}+y_{j}^{\prime }>f\left( \sum _{i\ne j}s_{i}x_{i}+s_{j}x_{j}^{\prime }\right) \). In either case, \(y_{j}^{\prime }>0\) holds, and so \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}} _{-j}\right) =0\). If \(a_{j}^{\prime }\) induces Rule 1–2-b, then \( x_{j}=0=x_{j}^{\prime }\) and \(s_{j}^{\prime }\ne s_{j}\). Thus, \( g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) =0\).

If \(a_{j}^{\prime }\) induces Rule 1–2-a, then either \(x_{j}^{\prime }>0\) or \( x_{j}^{\prime }=0\). In the former case, where \(s_{j}^{\prime }=s_{j}\) holds, Rule 1–2-a implies that

since any price vector \(p\in \Delta ^{P}\left( {\varvec{e}},{\varvec{z}} \right) \) implies \(B\left( p,s_{j},z_{j}\right) \subseteq L\left( z_{j},u_{j}\right) \). When \(x_{j}^{\prime }=0\), there exists \(\left( \sigma _{j}^{\mu ^{{}}},x_{j}^{\mu ^{{}}},y_{j}^{\mu ^{{}}}\right) \) such that \( \left( \left( x_{j}^{\mu ^{{}}},{\varvec{x}}_{-j}\right) ,\left( y_{j}^{\mu ^{{}}},{\varvec{y}}_{-j}\right) \right) \in \varphi \left( {\varvec{u}},\left( \sigma _{j}^{\mu ^{{}}},{\varvec{s}}_{-j}\right) \right) \) and \(y_{j}^{\mu ^{{}}}-p_{-}\left( x_{j}^{\mu ^{{}}};{\varvec{x}} _{-j},\left( \sigma _{j}^{\mu ^{{}}},{\varvec{s}}_{-j}\right) \right) x_{j}^{\mu ^{{}}}\le y_{j}-p_{-}\left( x_{j}; {\varvec{x}}_{-j},{\varvec{s}}\right) x_{j}\) or \(y_{j}^{\mu ^{{}}}\le y_{j}-p_{-}\left( x_{j}; {\varvec{x}}_{-j},{\varvec{s}}\right) x_{j}\) hold. Let \(p=\left( p_{x},p_{y}\right) \) be the efficiency price which supports \({\varvec{z}}\) as a \(\varphi \)-optimal allocation at \({\varvec{e}}\). Then, \(y_{j}-\frac{p_{x}}{ p_{y}}s_{j}x_{j}\ge y_{j}-p_{-}\left( x_{j};{\varvec{x}}_{-j}, {\varvec{s}}\right) x_{j}\), so that \(g_{j2}^{*}\left( a_{j}^{\prime }, {\varvec{a}}_{-j}\right) =y_{j}^{\mu ^{{}}}-p_{\alpha ^{{}}}\left( x_{j}^{\mu ^{{}}};{\varvec{x}}_{-j},\left( \sigma _{j}^{\mu ^{{}}}, {\varvec{s}}_{-j}\right) \right) x_{j}^{\mu ^{{}}}\le y_{j}-\frac{p_{x}}{p_{y}} s_{j}x_{j}\). This implies \(g_{j}^{*}\left( a_{j}^{\prime },{\varvec{a}} _{-j}\right) \in L\left( z_{j},u_{j}\right) \). Finally, if \(a_{j}^{\prime }\) induces Rule 1–1, then \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}} _{-j}\right) =y_{j}^{\prime }\le f(\sum _{i\ne j}s_{i}x_{i}+s_{j}x_{j}^{\prime })-\sum \nolimits _{i\ne j} y_{i}\). Thus, since \({\varvec{z}}\in \)\(P\left( {\varvec{e}}\right) \), \( u_{j}\left( x_{j}^{\prime },y_{j}^{\prime }\right) \le u_{j}\left( z_{j}\right) \). In summary, j has no incentive to switch to \(a_{j}^{\prime }\).

(2) Second, show \(NA\left( \gamma ^{*},{\varvec{e}}\right) \subseteq \varphi \left( {\varvec{e}}\right) \). Since \( g^{*}\) is constant with \({\varvec{x}}\) in \({\varvec{z}}\) , let \({\varvec{a}}=\left( \left( {{\varvec{v}}}^{i}\right) _{i\in N},\varvec{\sigma },{\varvec{z}},{\varvec{x}} \right) \in NE\left( \gamma ^{*},{\varvec{e}}\right) \) without loss of generality.

Suppose that \({\varvec{a}}\) induces Rule 2. Then, either \(N^{0}\left( {\varvec{x}}\right) \equiv \left\{ i\in N\mid x_{i}=0\right\} =\varnothing \) or \(N^{0}\left( {\varvec{x}}\right) \ne \varnothing \) . If \(N^{0}\left( {\varvec{x}}\right) =\varnothing \), then for each \( i\in N\), \(g_{i2}^{*}\left( {\varvec{a}}\right) =0\). Then, if \( \sum _{i\ne k}\sigma _{i}x_{i}=\sum _{i\ne k}s_{i}x_{i}\) holds for each \( k\in N\), then \(\left( n-1\right) \cdot \left( \sum \sigma _{i}x_{i}\right) =\left( n-1\right) \cdot \left( \sum s_{i}x_{i}\right) \), which contradicts Rule 2. Thus, for some \(j\in N\), \(\sum _{i\ne j}\sigma _{i}x_{i}\ne \sum _{i\ne j}s_{i}x_{i}\). If j switches to \(a_{j}^{\prime }=\left( {{\varvec{v}}}^{j\prime },\sigma _{j}^{\prime },z_{j}^{\prime },x_{j}^{\prime }\right) \) with \(\sigma _{j}^{\prime }>\max \left\{ \sigma _{i}\mid i\ne j\right\} \), \(y_{j}^{\prime }>\max \left\{ f\left( \sum _{k\ne j}\sigma _{k}{\overline{x}}\right) ,w\right\} \), and \( x_{j}^{\prime }=0\), then \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}} _{-j}\right) >0\) under Rule 2.

Let \(N^{0}\left( {\varvec{x}}\right) \ne \varnothing \) with \( \#N^{0}\left( {\varvec{x}}\right) \ge 2\). Then, for each \(j\in N^{0}\left( {\varvec{x}}\right) \), if j’s deviating strategy \( a_{j}^{\prime }\) is such that for each \(i\ne j\), \(\sigma _{j}^{\prime }>\sigma _{i}\), \(y_{j}^{\prime }>\max \left\{ f\left( \sum _{k\ne j}\sigma _{k}{\overline{x}}\right) ,w\right\} \), and \(x_{j}^{\prime }=0\), then \( g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) =f\left( \sum s_{k}x_{k}\right) \) under Rule 2.

Let \(\#N^{0}\left( {\varvec{x}}\right) =1\) and \(\#N\backslash N^{0}\left( {\varvec{x}}\right) \ge 2\). Then, there exists \(j\in N\backslash N^{0}\left( {\varvec{x}}\right) \) such that \(\sum _{i\in N\backslash \left( N^{0}\left( {\varvec{x}}\right) \cup \left\{ j\right\} \right) }\sigma _{i}x_{i}\ne \sum _{i\in N\backslash \left( N^{0}\left( {\varvec{x}}\right) \cup \left\{ j\right\} \right) }s_{i}x_{i}\). Thus, j can switch to \(a_{j}^{\prime }\) such that \( g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) >0\) under Rule 2. This can be shown in a similar way to the case of \(N^{0}\left( {\varvec{x}}\right) =\varnothing \).

Suppose that \({\varvec{a}}\) induces Rules 1–2 or 1–3. Then, there exists \( j\in N\) such that \(g_{j2}^{*}\left( {\varvec{a}}\right) =0\). From Lemma A1, \(\sigma _{j}=s_{j}\) or \(x_{j}=0\). Suppose \({\varvec{a}}\) induces Rule 1-2. Then, \(g_{j2}^{*}\left( {\varvec{a}}\right) =0\) implies that \(y_{j}\le f\left( \sum \sigma _{k}{\overline{x}}\right) \). Then, j can either deviate to Rule 1-3 with \(\sigma _{j}^{\prime }=s_{j}\), \( x_{j}^{\prime }=\pi \left( {\varvec{x}}_{-j}\right) <{\overline{x}}\), and \(y_{j}^{\prime }=0\) or get \( g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) >0\) under Rule 1–2 by \(y_{j}^{\prime }>f\left( \sum _{i\ne j}\sigma _{i}{\bar{x}}+\sigma _{j}^{\prime }{\bar{x}}\right) \). Suppose \({\varvec{a}}\) induces Rule 1–3. Then, there exists \(a_{j}^{\prime }\) such that \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) >0\) under Rule 1–3. In summary, \( {\varvec{a}}\) induces neither Rule 1–2 nor Rule 1–3.

Suppose that \({\varvec{a}}\) induces Rule 1–1-b. Then, \(g^{*}\left( {\varvec{a}}\right) =\left( {\varvec{x}},{\mathbf {0}}\right) \). Then, some \(j\in N\) can deviate to induce Rule 1–2, so that \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) >0\), which is a contradiction.

Thus, \({\varvec{a}}\) induces Rule 1–1-a, and \(g^{*}\left( \varvec{a }\right) =\left( {\varvec{x}},{\varvec{y}}\right) \). From the definition of Rule 1–1-a, \(\left( {\varvec{x}},{\varvec{y}} \right) \in \varphi \left( {\varvec{u}}^{\prime },\varvec{\sigma }\right) \) where \({\varvec{u}}^{\prime }={\varvec{v}}^{i}\) for all \(i\in N\). Since \({\varvec{a}}\in NE\left( \gamma ^{*}, {\varvec{e}}\right) \), \(\sigma _{i}=s_{i}\) holds for any \(i\in N\) with \(x_{i}>0\) according to Lemma A1. Assume, without loss of generality, that there exists at most one unique individual j such that \(x_{j}=0\). Let us consider the following two cases:

Case 1: Let \(\left( {\varvec{x}}, {\varvec{y}}\right) \in P\left( {\varvec{u}}^{\prime },{\varvec{s}} \right) \). Then, we can show that \(\left( {\varvec{x}}, {\varvec{y}}\right) \in \varphi \left( {\varvec{u}}^{\prime },{\varvec{s}}\right) \). Suppose that \(\left( {\varvec{x}}, {\varvec{y}}\right) \notin \varphi \left( {\varvec{u}} ^{\prime },{\varvec{s}}\right) \). Then, for the individual \(j\in N\) with \(x_{j}=0\), \(\sigma _{j}\ne s_{j}\). Then, if this j takes the strategy \(a_{j}^{\prime }=\left( {\varvec{u}}^{j},s_{j},z_{j}^{ \prime },x_{j}^{\prime }\right) \) with \({\varvec{u}}^{j}\ne {\varvec{u}}^{\prime }\), \(x_{j}^{\prime }>0\), and \(y_{j}^{\prime }>f\left( \sum s_{k}{\overline{x}}\right) \), then Rule 1–2-b can be applied. This is because NUS implies \(j\notin N\left( {\varvec{u}} ^{\prime },{\varvec{s}}\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \) by \(\left( {\varvec{x}}, {\varvec{y}}\right) \in P\left( {\varvec{u}}^{\prime },{\varvec{s}} \right) \backslash \varphi \left( {\varvec{u}}^{\prime }, {\varvec{s}}\right) \) and \(x_{j}=0\). Then, if \(x_{j}^{\prime }>0\) is sufficiently small, \(u_{j}\left( g_{j}^{*}\left( a_{j}^{\prime }, {\varvec{a}}_{-j}\right) \right) =u_{j}\left( x_{j}^{\prime },f\left( \sum _{i\ne j}s_{i}x_{i}+s_{j}x_{j}^{\prime }\right) \right) >u_{j}\left( 0,y_{j}\right) =u_{j}\left( g_{j}^{*}\left( {\varvec{a}}\right) \right) \) holds by the fact that \(\left( {\varvec{x}}, {\varvec{y}}\right) \) is an interior allocation, since \(\left( {\varvec{x}}, {\varvec{y}}\right) \in \varphi \left( {\varvec{u}}^{\prime },\varvec{\sigma }\right) \). This implies \(\left( {\varvec{x}}, {\varvec{y}}\right) \notin NA\left( \gamma ^{*},\left( {\varvec{u}},{\varvec{s}}\right) \right) \), which is a contradiction. Thus, \( \left( {\varvec{x}},{\varvec{y}}\right) \in \varphi \left( {\varvec{u}}^{\prime }, {\varvec{s}}\right) \). Note that \(\left( {\varvec{x}}, {\varvec{y}}\right) \in NA\left( \gamma ^{*},\left( {\varvec{u}},{\varvec{s}}\right) \right) \) implies that \(\partial L\left( \left( x_{i},y_{i}\right) ,u_{i}^{\prime };{\varvec{s}}\right) \subseteq L\left( \left( x_{i},y_{i}\right) ,u_{i}\right) \cap Z_{i}\left( {\varvec{s}}, {\varvec{z}}_{-i}\right) \) for each \(i\in N\) by Rule 1–2-a. Thus, \(\left( {\varvec{x}},{\varvec{y}}\right) \in \varphi \left( {\varvec{u}},{\varvec{s}}\right) \) by M.

Case 2: Let \(\left( {\varvec{x}}, {\varvec{y}}\right) \notin P\left( {\varvec{u}}^{\prime }, {\varvec{s}}\right) \). Since \(\left( {\varvec{x}},{\varvec{y}} \right) \in P\left( {\varvec{u}}^{\prime },\varvec{\sigma }\right) \) , \(\sigma _{j}<s_{j}\) holds for the agent \(j\in N\) with \(x_{j}=0\). In this case, either \(j\in N\left( {\varvec{u}}^{\prime }, {\varvec{s}}\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \) or not.

First, suppose that \(j\notin N\left( {\varvec{u}}^{\prime }, {\varvec{s}}\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \). Take \({\varvec{u}} ^{j}=\left( u_{j}^{\prime \prime },{\varvec{u}}_{-j}^{\prime }\right) \) such that

Then, \(\left( {\varvec{x}},{\varvec{y}}\right) \in P\left( {\varvec{u}}^{j},{\varvec{s}}\right) \) holds. Then, by \(a_{j}^{\prime }=\left( {\varvec{u}}^{j},s_{j},z_{j}^{*},x_{j}^{*}\right) \) with \(x_{j}^{*}>0\) and \(y_{j}^{*}>f\left( \sum s_{k}{\overline{x}}\right) \), j can induce Rule 1–2-b, and get \(g_{j2}^{*}\left( a_{j}^{\prime },{\varvec{a}} _{-j}\right) =f\left( \sum _{i\ne j}s_{i}x_{i}+s_{j}x_{j}^{*}\right) \). Thus, if \(x_{j}^{*}>0\) is sufficiently small, \(\left( {\varvec{x}},{\varvec{y}}\right) \notin NA\left( \gamma ^{**},\left( {\varvec{u}},{\varvec{s}}\right) \right) \), which is a contradiction. Thus, \(j\notin N\left( {\varvec{u}}^{\prime }, {\varvec{s}}\mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \) does not hold.

Second, let \(j\in N\left( {\varvec{u}}^{\prime },{\varvec{s}} \mathbf {,}{\varvec{x}}\mathbf {,}{\varvec{y}}\right) \). Since \(\left( {\varvec{x}},{\varvec{y}}\right) \notin P\left( {\varvec{u}}^{\prime }, {\varvec{s}}\right) \), there exists \(z_{j}^{\prime }\equiv \left( x_{j}^{\prime },y_{j}^{\prime }\right) \gg {\mathbf {0}}\) such that \(\left( z_{j}^{\prime },\left( {\varvec{x}}_{-j},{\varvec{y}}_{-j}\right) \right) \in \varphi \left( {\varvec{u}}^{\prime }, {\varvec{s}}\right) \). In this case, \(y_{j}^{\prime }\ge y_{j}+\frac{p_{x}}{p_{y}} s_{j}x_{j}^{\prime }\) holds for \(p\equiv \left( p_{x},p_{y}\right) \in \Delta ^{P}\left( \left( {\varvec{u}}^{\prime },{\varvec{s}}\right) ,\left( z_{j}^{\prime },\left( {\varvec{x}}_{-j}, {\varvec{y}}_{-j}\right) \right) \right) \), since \(z_{j},z_{j}^{\prime }\in \left\{ \left( x,y\right) \in Z\mid y=f(\sum _{i\ne j}s_{i}x_{i}+s_{j}x)-\sum _{i\ne j}y_{i}\right\} \) and f is concave. Moreover, since \(z_{j}^{\prime }\notin L\left( \left( x_{j},y_{j}\right) ,u_{j}^{\prime }\right) \) and \( z_{j}^{\prime }\in B\left( p,s_{j},z_{j}^{\prime }\right) \), \(B\left( p,s_{j},z_{j}^{\prime }\right) \nsubseteq L\left( \left( x_{j},y_{j}\right) ,u_{j}^{\prime };{\varvec{s}}\right) \) holds. Then, since \({\varvec{a}} \in NE\left( \gamma ^{*},{\varvec{e}}\right) \), (i) \(\partial \left[ L\left( \left( x_{j},y_{j}\right) ,u_{j}^{\prime };{\varvec{s}}\right) \cup \right. \left. B\left( p,s_{j},z_{j}^{\prime }\right) \right] \subseteq L\left( \left( x_{j},y_{j}\right) ,u_{j}\right) \) and (ii) \(y_{j}^{\prime }=y_{j}+\frac{p_{x}}{p_{y}}s_{j}x_{j}^{\prime }\) hold. Indeed, if (i) does not hold, then j can induce Rule 1–2-a by suitably choosing \(a_{j}^{\prime }=\left( {\varvec{u}}^{j},s_{j},z_{j}^{*},x_{j}^{*}\right) \) with \({\varvec{u}}^{j}\ne {\varvec{u}}^{\prime }\), \(x_{j}^{*}>0\), and \(y_{j}^{*}>f\left( \sum s_{k} {\overline{x}}\right) \), so that \(g_{j2}^{*}\left( a_{j}^{\prime }, {\varvec{a}}_{-j}\right) \ge y_{j}+\frac{p_{x}}{p_{y}}s_{j}x_{j}^{*}\) and \(g_{j}^{*}\left( a_{j}^{\prime },{\varvec{a}}_{-j}\right) \notin L\left( \left( x_{j},y_{j}\right) ,u_{j}\right) \), which is a contradiction from \({\varvec{a}}\in NE\left( \gamma ^{**},{\varvec{e}}\right) \). The same argument also follows if (ii) does not hold.

Consider \({\varvec{u}}^{\prime \prime }\equiv \left( u_{j}, {\varvec{u}}_{-j}^{\prime }\right) \). Note that by M, \( \left( {\varvec{x}},{\varvec{y}}\right) \in \varphi \left( {\varvec{u}}^{\prime \prime },\varvec{\sigma }\right) \). Then, by Rule 1–1-a, for \({\varvec{a}}^{\prime \prime }=\left( \left( { {\varvec{v}}}^{\prime \prime i}\right) _{i\in N},\varvec{\sigma } ,{\varvec{z}},{\varvec{x}}\right) \) with \( {\varvec{v}}\)\(^{\prime \prime i}={\varvec{u}}^{\prime \prime }\) (\(\forall i\in N\)), \(g^{*}\left( {\varvec{a}}^{\prime \prime }\right) =\left( {\varvec{x}},{\varvec{y}}\right) \) and \( {\varvec{a}}^{\prime \prime }\in NE\left( \gamma ^{*}, {\varvec{e}}\right) \) hold. Suppose \(\left( {\varvec{x}},{\varvec{y}} \right) \notin \varphi \left( {\varvec{u}}^{\prime \prime }, {\varvec{s}}\right) \). Then, since \(\left( {\varvec{x}}, {\varvec{y}}\right) \in P\left( {\varvec{u}}^{\prime \prime },{\varvec{s}}\right) \), j can induce Rule 1–2-b by the same reasoning as in Case 1, so that \(\left( {\varvec{x}},{\varvec{y}}\right) \notin NA\left( \gamma ^{*}, {\varvec{e}}\right) \), which is a contradiction. Thus, \(\left( {\varvec{x}}, {\varvec{y}}\right) \in \varphi \left( {\varvec{u}}^{\prime \prime },{\varvec{s}}\right) \). Then, \({\varvec{a}} ^{\prime \prime }\in NE\left( \gamma ^{*},{\varvec{e}}\right) \) and \(\left( {\varvec{x}},{\varvec{y}}\right) \in NA\left( \gamma ^{*},{\varvec{e}}\right) \) imply that \(\partial L\left( \left( x_{i},y_{i}\right) ,u_{i}^{\prime };{\varvec{s}}\right) \subseteq L\left( \left( x_{i},y_{i}\right) ,u_{i}\right) \) holds for each \(i\in N\backslash \left\{ j\right\} \) by Rule 1–2-a. Thus, \(\left( {\varvec{x}}, {\varvec{y}}\right) \in \varphi \left( {\varvec{e}}\right) \) by M. \(\square \)

Proof of Theorem 2. From Lemma A2, we obtain the desired result. \(\square \)

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