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STRONG NASH IMPLEMENTATION AND
ECONOMIC APPLICATIONS

Mohamed Bouchta, Ahmed Doghmi, Abderrahmane Ziad

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Mohamed Bouchta, Ahmed Doghmi, Abderrahmane Ziad. STRONG NASH IMPLEMENTATION AND ECONOMIC APPLICATIONS. 2025. hal-05047583

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STRONG NASH IMPLEMENTATION AND
ECONOMIC APPLICATIONS

Mohamed Bouchta

National Institute of Statistics and Applied Economics, Madinat Al Irfane,
Rabat Institutes, 10100 Rabat, Morocco.

Ahmed Doghmi

National Institute of Statistics and Applied Economics, Madinat Al Irfane,
Rabat Institutes, 10100 Rabat, Morocco.

Abderrahmane Ziad

CNRS, CREM-UNICAEN-University of Caen Normandy-NU-Normandy
University, Esplanade de la paix 14000 Caen, France and Laboratory of Applied
Mathematics (LaMA), Ferhat Abbas University of Setif 1, Setif, Algeria.

Date: April 2025

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MISE EN ŒUVRE SELON L'ÉQUILIBRE DE NASH
FORT ET APPLICATIONS ÉCONOMIQUES

Résumé: La mise en œuvre selon l'équilibre de Nash fort représente un défi
complexe, pour lequel aucun théorème équivalent à celui de Maskin (1977/1999)
n'existe, offrant des conditions simples et une grande diversité d'applications.
Dans cet article, nous examinons les origines du théorème de Maskin (1979)
dans ce cadre. Nous supposons qu'il existe au moins une paire d'agents associée à
des alternatives spécifiques, appelées *alternatives exclusivement séparées*. Une
telle paire se caractérise par le fait que chaque agent préfère son alternative
à celle de l'autre, tandis que, pour chaque agent, tous les autres membres de
la société préfèrent toute alternative socialement choisie à leurs alternatives
respectives. Nous démontrons que, lorsqu'il y a au moins trois agents, toutes les
correspondances de choix social vérifiant une version forte de la monotonie de
Maskin et l'unanimité peuvent être mises en œuvre dans des équilibres de Nash
fort. Nous appliquons nos résultats à des problèmes d'allocation équitable finie
sur des marchés d'appariement, à des problèmes d'allocation équitable infinie
avec des préférences uni-modales, à des règles de partage efficaces pour une
technologie partagée et à des jeux de coalition.

*Mots-clés :* Mise en œuvre ; Équilibre de Nash fort ; Problèmes d'allocation équitable.
STRONG NASH IMPLEMENTATION AND
ECONOMIC APPLICATIONS

Mohamed Bouchta<sup>a</sup>, Ahmed Doghmi<sup>a</sup> and Abderrahmane Ziad<sup>b,c</sup>

<sup>a</sup> National Institute of Statistics and Applied Economics, Madinat Al Irfane, Rabat Institutes, 10100 Rabat, Morocco. <sup>b</sup>CNRS, CREM-UNICAEN-University of Caen Normandy-NU-Normandy University, Esplanade de la paix 14000 Caen, France. <sup>c</sup>Laboratory of Applied Mathematics (LaMA), Ferhat Abbas University of Setif 1, Setif, Algeria.

**Abstract.** Strong Nash implementation is a challenging problem for which there is no similar theorem to that of Maskin (1977/1999) for Nash implementation with simple conditions, which involves a wide range of applications. In this paper, we explore the origins of Maskin's theorem (1979) in this context. We assume that there exists at least one pair of agents who are associated with specific alternatives, which we refer to as separately exclusive alternatives. A pair of agents with separately exclusive alternatives is one where each agent prefers their own alternative over the other agent's alternative, and for each agent, all other agents in society prefer any socially chosen alternative over their respective alternatives. We show that when there are at least three agents, all social choice correspondences satisfying a strong version of Maskin monotonicity and unanimity can be implemented in Nash equilibria. We apply our findings to finite fair allocation problems in matching markets, to infinite fair allocation problems with single-peaked preferences, to efficient sharing Rules for a commonly owned technology and coalitional games.<fnref n="1" />

**Keywords:** Implementation; Strong Nash equilibrium; Fair allocation problems.

**JEL classification:** C72; D71

<footnote n="1">¹The first version, April 2025.</footnote>
# 1 Introduction

When there is a social goal to achieve, a planner (who could be a representative of the state, a local collectivity, a region, a manager of a firm, etc.) can use an objective function known as the social choice correspondence (SCC) or rule to choose the desired outcomes. However, the challenge lies in the fact that the planner may not have complete information about the preferences of the agents involved. Implementation theory, which is a branch of game theory, deals with the question of whether a planner can design a set of mechanisms or institutions in such a way that the predicted outcomes of the individual strategic interactions align with the desired outcomes represented by SCCs. To achieve this, certain conditions need to be met regarding the behavior of the individuals involved. Thus, Maskin (1977,1999) was the first who discovered an important connection between the class of implementable SCC in Nash equilibria and a property called *Maskin monotonicity*. He proved that any Nash implementable SCC must satisfy Maskin monotonicity. However, this condition alone can not guarantee the implementability of an SCC. Hence, Maskin provided an additional condition called *no-veto power*<fnref n="2" />. He showed that any SCC satisfying monotonicity and no-veto power can be implemented in Nash equilibria with at least three agents. Nevertheless, no-veto power is not a necessary property for implementation and it is not always checked in many economic and political applications. Thus, many kinds of literature have emerged to refine Maskin’s result, either by offering alternative sufficient or necessary conditions or by providing full characterizations<fnref n="3" />. To show the part of sufficiency in this literature, we distinguish between two cases. The first one is for three or more agents. In this case, all the above studies are based on the canonical mechanism of Maskin (1977,1999) to construct their game forms in which each agent reports a preference profile, an outcome, and a positive integer. These types of mechanisms are known in the literature as integer/modulo game forms where the integers are used for a possible deviation and a tie-breaking in the case of multiple deviations. The considered strategy spaces in these mechanisms are judged to be

<footnote n="2">²The two properties of Maskin monotonicity and no-veto power will be defined in Sub-section 2.2</footnote>

<footnote n="3">³See for example Repullo (1987), McKelvey (1989), Moore and Repullo (1990), Dutta and Sen (1991a), Sjöström (1991), Danilov (1992), Yamato(1992), Ziad (1997,1998), Doghmi and Ziad (2008ab,2015), and Doghmi(2016) among others.</footnote>
large. Therefore, several authors (like Saijo (1988) and McKelvey (1989) among others) tried to reduce these spaces. The second case is for exactly two agents. This case is much harder than the case of three or more agents (see, for example, Moore and Repullo (1990), or Dutta and Sen (1991a)), this is because an additional intersection condition must hold for lower contour sets. This needs an additional component for which the agents are asked to raise or not a "flag". Now, the question is: What happens about these mechanisms and hence these previous results when we extend the solution concept from Nash equilibrium where the deviations are unilateral to the general case of strong Nash equilibrium in which the deviations are coalitional?

Different from the extensive literature on Nash implementation that builds on the celebrated results of Maskin (1977, 1999), there have been few studies focused on strong Nash implementation theory. This theory was explored by Maskin (1979), who showed that the property of monotonicity remains a necessary condition for an SCC to be implementable in strong Nash equilibria. However, alongside the absence of veto power, monotonicity fails to form a sufficient condition for strong implementation<fnref n="4" />. In this context, aside from the full characterization by Dutta and Sen (1991b) and Suh (1996), which was inspired by the works of Dutta and Sen (1991a) and Moore and Repullo (1990), as well as the contributions of Moulin and Peleg (1982), Holzman (1987), and Fristrup and Keiding (2001), there are no comparable sufficient conditions to those found in Nash implementation theory. To our knowledge, two papers have presented sufficient conditions for strong implementation. The first is that of Korpela (2013), who introduced a new property, called the axiom of sufficient reason, which forms together with weak Pareto optimality and a certain Holocaust alternative a new sufficient condition for strong implementation. The second is the result of Savva (2018) that has a relation with the growing literature on implementation problems with partial honesty<fnref n="5" />. Savva (2018) provided a sufficient condition for an SCC to be implementable in strong Nash equilibrium in the presence of partially honest agents. He relaxed the axiom

<footnote n="4">⁴The solution concept of strong Nash equilibrium used by Maskin (1979) is due to Aumann (1959).</footnote>

<footnote n="5">⁵For more details on this literature, see, for example, Dutta and Sen (2012), Doghmi and Ziad (2013), Kartik et al. (2014), Korpela (2014), Saporiti (2014), Ortner (2015), and Lombardi and Yoshihara (2018), among others.</footnote>
of sufficient reason provided by Korpela (2013) to the property of weak Pareto dominance and the Holocaust alternative to the universally worst alternative, and He proved that these properties are sufficient along with weak Pareto optimality. However, the sufficient conditions proposed by Korpela (2013) and Savva (2018) that are based on the Holocaust mechanism do not apply to different domains in which there is no Holocaust alternative and universally worst alternative (UWA). Therefore, the use of these latter requirements is very restrictive. This restriction is pointed out by Savva himself (2018) “...the domain restriction of UWA could be avoided and more clear-cut results could be obtained.”. Hence, although the results of Korpela (2013) and Savva (2018) provided some nice applications, they remained silent for others.

Most mechanisms of these previous works of literature on strong implementation are inspired by the hard case of two agents in Nash implementation. This, because in the case of hybrid announcements for a big coalition, some mechanisms are based on certain Holocaust alternative types like that of Korpela (2013). Hence, following these works, the case of three or more agents turns out to be much harder than that in Nash implementation if we want to implement in strong equilibrium. In this context, Dutta and Sen (1991b) write “...The game form described above is not appropriate when strong Nash equilibrium is the relevant equilibrium concept. This is because any coalition of two or more agents can destroy truth-telling as a strong Nash equilibrium by making an hybrid announcement and precipitating the integer game...”. For the case of two agents, the strong implementation does not much differ from Nash implementation beyond the additional requirement of weak Pareto optimality.

In this paper, we reexamine the origins of the Maskin theorem (1979) and present our findings on implementation problems for multiple agents. We assume that there exists at least one pair of agents associated with specific alternatives, which we refer to as separately exclusive alternatives. A pair of agents with separately exclusive alternatives is one in which each agent prefers their own alternative over the other agent's alternative, and for
each agent, all other agents in society prefer any socially chosen alternative over their respective alternatives. We demonstrate that when there are at least three agents, any weakly Pareto optimal social choice correspondence (SCC) that satisfies a strong variant of Maskin monotonicity-introduced by Yi (2012), which we refer to as *Y*-monotonicity-can be implemented in a strong equilibrium. Based on these findings, we provide interesting applications for strong Nash implementation.

The first area of application is several-to-one matching markets. In this setting, we show that if there is at least one pair of agents with separately exclusive alternatives, and if an improper sub-solution of the group stable matching correspondence satisfies *Y*-monotonicity, it can be implemented in strong equilibria, provided that there are at least three agents in the society. Consequently, we build on the findings of Shin and Suh (1996) and Korpela (2013) regarding strong implementation, extending the results from one-to-one and pure one-to-one matching rules to general several-to-one matching rules. Additionally, we generalize the results of Doghmi and Ziad (2015) on several-to-one matching markets, extending them from Nash implementation to strong implementation.

For the second domain of applications, which is the fair allocation problems with single-peaked preferences, we are exploring whether certain solutions can be strongly implemented. We present examples of solutions that can and cannot be strongly implemented. Our result shows that the individually rational and no-envy solutions, as well as their intersections, cannot be implemented in strong equilibria. However, we have demonstrated that the Pareto selection, core from equal division, and group no-envy solutions can be implemented in strong equilibria. This extends previous findings on Nash implementation to strong implementation in this environment, building on the work of Thomson (2010) and Doghmi and Ziad (2008b).

Additionally, we examine two other domains of application. The first involves commonly owned technology used by a group of agents to produce a consumption good from a (possibly heterogeneous) input. We analyze the implementability of efficient sharing rules for such
commonly owned technology. We show that, in environments where there exists at least one pair of agents with separately exclusive alternatives, and where utility functions are not necessarily monotonic, a general efficient sharing rule satisfies the properties of Y-monotonicity and unanimity, and is therefore implementable in strong equilibria. This extends its implementability from Nash equilibria to strong equilibria, and from strictly monotonic utility functions to non-monotonic ones, thereby generalizing the work of Corchón and Puy (2002). The second domain we explore concerns coalitional games, where we investigate the implementability of the weak core in strong equilibria. This involves extending the results of Shinotsuka and Takamiya (2003), moving from implementability of the weak core in Nash equilibrium for simple games to strong equilibrium in general coalitional games. We also extended the results of Doghmi and Ziad (2008b) regarding the implementation of the weak core in coalitional games, moving from Nash equilibrium to strong equilibrium, and improved the results of Bouchta et al. (2025) by generalizing their findings from simple games to general coalitional games.

The paper is structured as follows: In section 2, we introduce notations and definitions and review the origins of Maskin's results (1979). In Section 3, we describe an environment of agents with separately exclusive alternatives. In Section 4, we present new sufficient condition for strong implementation when there are at least three agents. In Section 5, we give applications in finite allocation problems for several-to-one matching markets. In Section 6, we provide applications in infinite allocation problems with single-peaked preferences. In Sections 7 and 8, we apply our results to coalitional games efficient sharing rules for commonly owned technologies and coalitional games. Finally, in Section 9, we conclude.

# 2 Preliminaries

## 2.1 Notations and Definitions

Assume that we have a planner who faces a certain number of agents $N = \{1, 2, ..., n\}$ and a finite or infinite set of alternatives $A$. Each individual $i$ is characterized by a preference
relation $R_i$ defined over $A$, which is a complete, and transitive relation in some class $\Re_i$ of admissible preference relations. Let $\Re = \Re_1 \times \dots \times \Re_n$ and $\Re \subset \Pi_{i \in N} \Re_A$, where $\Re_A$ is the set of all possible orderings on $A$. An element $R = (R_1, \dots, R_n) \in \Re$ is a preference profile. The relation $R_i$ indicates the individual $i$'s preference. For $a, b \in A$, the notation $aR_i b$ means that the individual $i$ weakly prefers $a$ to $b$. The asymmetrical and symmetrical parts of $R_i$ are noted respectively by $P_i$ and $\sim_i$. Planner has the goal to improve the social welfare, which is represented by a social choice correspondence (or rule). This correspondence, denoted $F$, is a multi-valued mapping from $\Re$ into $2^A \setminus \{\emptyset\}$, that associates with every preference profile $R$ a nonempty subset of $A$. An sub-correspondence of $F$, denoted $f$, if for all $R \in \Re$, $f(R) \subseteq F(R)$. The sub-correspondence $f$ is proper if for all $R \in \Re$, $f(R) \subset F(R)$, and is improper, if there exists $R \in \Re$ such that $f(R) = F(R)$. For all $R_i \in \Re_i$ and all $a \in A$, the lower contour set for agent $i$ at alternative $a$ is noted by: $L_i(a, R) = \{b \in A | aR_ib\}$.

The planner does not know the exact preferences of agents. So that the agents reveal their true preferences, the planner organizes a non-cooperative game, where the strategies of players depend on the preference profiles class $\Re$, the alternatives set $A$, and the nonnegative integers $N$ such that for each agent $i$, the strategy set is $S_i = \Re \times A \times N$ and $S = \Pi_{i \in N} S_i$, to implement a mechanism. Let $G = (S, g)$ be this mechanism where $g$ is a function from $S$ to $A$. The elements of $S$ are denoted by $s = (s_1, s_2, \dots, s_n)$. For any coalition (non-empty subset of agents), $C \subseteq N$, the strategy set $S_C = \Pi_{i \in C} S_i$, $S_{N \setminus C} = \Pi_{i \in N \setminus C} S_i$, and $s = (s_C, s_{-C})$, where $s_C = (s_i)_{i \in C} \in S_C$ and $s_{-C} = (s_i)_{i \in N \setminus C} \in S_{N \setminus C}$.

The solution concept of the organized game is a strong (Nash) equilibrium of the game $(G, R)$. The well-known concept of strong equilibrium used in such games was introduced by Aumann (1959), which is a vector of strategies $s \in S$ such that for any $C \subseteq N$, and for all $\tilde{s}_C \in S_C$, there exists $i \in C$ such that $g(s)R_ig(\tilde{s}_C, s_{-C})$. Given $SE(S, g, R)$, the set of strong equilibria (SE) of the game $(G, R)$, let

$$g(SE(S, g, R)) = \{a \in A \mid \exists s \in SE(S, g, R) \text{ such that } g(s) = a\}.$$

A mechanism $G = (S, g)$ implements an SCC $F$ in strong equilibria if, for all $R \in \Re$,
$F(R) = g(SE(S, g, R))$. We say that an SCC $F$ is implementable in strong equilibria if there is a mechanism that implements it in these equilibria.

## 2.2 Maskin's Results

To present our main results in the next sections, it would be helpful first, before proceeding though, to review some fundamental results of Maskin (1977,1979,1999). This will enable us to outline more clearly the new sufficient conditions for strong implementation that we will derive later.

We now present the first key property, which is currently referred to as Maskin monotonicity. In simpler terms, Maskin monotonicity requires that if an alternative $a$ is chosen under a certain profile of preferences $R$, and the preference for $a$ only strengthens or remains the same relative to other alternatives in a new profile $R'$, then $a$ should still be chosen under $R'$. In formal terms, Maskin monotonicity can be defined as follows:

*Maskin monotonicity:* A SCC $F$ satisfies monotonicity if for all $R, R' \in \Re$, for any $a \in F(R)$, if for any $i \in N$, $L_i(a, R) \subseteq L_i(a, R')$, then $a \in F(R')$.

Maskin (1977,1999) proved that any Nash implementable correspondence must satisfy Maskin monotonicity.

**Theorem 1 (Maskin, 1977, 1999)**

*If an SCC can be implemented for Nash equilibrium, then it is Maskin monotonic.*

Although Maskin monotonicity is a necessary condition that encourages individuals to truthfully reveal their preferences, it alone cannot ensure that an SCC can be implemented in Nash equilibria in a general environment. Therefore, Maskin (1977, 1999) introduced an additional property called no-veto power. This property requires that if an alternative is top-ranked for every possible profile of individual preferences, whenever all individuals except one, this alternative must be chosen by the society. Formally:
**No-veto power:** An SCC $F$ satisfies no-veto power if for $i, R \in \mathbb{R}$, and $a \in A$, if $L_j(a, R) = A$ for all $j \in N \setminus \{i\}$, then $a \in F(R)$.

Maskin (1977, 1999) demonstrated that, if at least three agents participate in the game, any SCC that satisfies Maskin monotonicity and no-veto power can be implemented in Nash equilibria.

**Theorem 2 (Maskin, 1977, 1999)**

Let $n \ge 3$. Any SCC satisfying no-veto power can be implemented for Nash equilibrium iff it is monotonic.

In the case of strong Nash equilibria, Maskin (1979) proved that monotonicity remains a necessary condition for implementation.

**Theorem 3 (Maskin, 1979)**

*If an SCC can be implemented for strong equilibrium, then it is Maskin monotonic.*

However, Maskin (1979) illustrated that the no-veto power property becomes incompatible with the strong implementability of SCCs, at least in an unrestricted domain of individual preferences when there are three or more alternatives. In the next, we present Maskin's impossibility theorem.

**Theorem 4 (Maskin, 1979)**

If $|A| \ge 3$, and if an SCC satisfies no-veto power, it cannot be implemented for strong equilibrium.

As a result of this impossibility in strong equilibrium and in comparison with his finding in Nash implementation, Maskin (1979) states: *"The results of my previous paper were positive. If an SCC satisfies a monotonicity property and a much weaker requirement called no veto power, it can be implemented by Nash equilibrium. The results for strong equilibrium, on the other hand, are on the whole negative. I show (theorem 2) that SCC's satisfying no veto power cannot in general be implemented for strong equilibrium when the number of alternatives is at least three"*. Thus, Maskin (1979) investigated the properties of weak Pareto optimality and
individually rational to reach a positive result for a particular case of strong implementation

*Weak Pareto optimality:* Given $R \in \mathbb{R}$, an alternative $a$ is weakly Pareto optimal if there exists no alternative $b \in A$ such that $bP_ia$ for all $i \in N$. Given any $R \in \mathbb{R}$, the set of all weakly Pareto optimal alternatives is denoted by $P_w(R)$. The weak Pareto rule is an SCC, $F$, such that, for all $R \in \mathbb{R}$, $F(R) = P_w(R)$. A SCC $F$ satisfies weak Pareto optimality if for any $R \in \mathbb{R}$, $F(R) \subseteq P_w(R)$.

Maskin (1979) demonstrated that weak Pareto optimality is a necessary condition for strong implementation through a surjective mechanism<fnref n="6" />.

**Proposition 1** (Maskin, 1979)

*If an SCC $F$ is implementable in strong equilibria by a surjective mechanism, then $F$ is weakly Pareto optimal.*

For sufficiency, Maskin (1979) introduced the additional property of individual rationality.

*Individual rationality:* Let $a_0 \in A$ be identified as the status quo. An alternative $a$ is individually rational if $aR_ia_0$ for all $i \in N$. Given any $R \in \mathbb{R}$, the set of all individually rational alternatives is denoted by $IR(R)$. The individually rational rule is an SCC, $F$, such that, for all $R \in \mathbb{R}$, $F(R) = IR(R)$. A SCC $F$ satisfies individual rationality if for any $R \in \mathbb{R}$, $F(R) \subseteq IR(R)$.

**Theorem 5** (Maskin, 1979)

*If an SCC $F$ satisfies individual rationality and weak Pareto optimality, then $F$ can be implemented in strong equilibrium.*

In this paper, we present a new perspective on strong implementation that may lead to more positive results. Our approach is based on some agents that have special alternatives, which we describe in Section 3. We investigate the conjunction of some properties that form

<footnote n="6">⁶A mechanism $G = (S, g)$ is said to be surjective if for each $b \in A$, there exists a profile $\tilde{s} \in S$ such that $b = g(\tilde{s})$.</footnote>
new sufficient condition. This condition is examined in Section 4 and it characterizes the
family of SCCs that can be implemented in strong equilibria.

# 3 Agents with Separately Exclusive Alternatives

Agents who have a preference profile based on a particular pair of alternatives will be called
*agents with separately exclusive alternatives*. In other words, our domain of admissible
preference profiles includes alternatives that exhibit specific characteristics, which will be
possessed by certain pairs of agents.

**Definition 1 (Agents with separately exclusive alternatives)**

We say that agents *i* and *j* in a society possess separately exclusive alternatives if there exist
two alternatives, $a_{sea}^i$ and $a_{sea}^j$, such that the following conditions hold:

i) For each $R$, $a_{sea}^k P_k a_{sea}^{k'}$ for each $k, k' \in \{i, j\}$ with $k \neq k'$, (Mutual Preferences)

ii) For each $R'$ and for each $a \in F(R')$, $aR'_t a_{sea}^i$ for all $t \in N \setminus \{i\}$, and $aR'_t a_{sea}^j$ for all $t \in N \setminus \{j\}$ (F-optimality).

Roughly speaking, we can interpret this definition as follows: The first part expresses
that each agent prefers their own alternative over the other agent's alternative. Specifically,
agent *i* prefers $a_{sea}^i$ over $a_{sea}^j$, and agent *j* prefers $a_{sea}^j$ over $a_{sea}^i$. The second part suggests
that all other agents in the society prefer any F-optimal alternative over $a_{sea}^i$ and $a_{sea}^j$. These
alternatives, $a_{sea}^i$ and $a_{sea}^j$, are referred to as *separately exclusive alternatives*.

It is important to note that the separately exclusive alternatives, $a_{sea}^i$ and $a_{sea}^j$, can also be
F-optimal alternatives. In this case, although agents *i* and *j* may prefer their own exclusive
alternatives, these alternatives could also be socially chosen, for instance, if they are ranked
highly in terms of fairness or efficiency within the society. Therefore, $a_{sea}^i$ and $a_{sea}^j$ may not
only be preferred by the agents themselves, but they could also be preferred by others if they
satisfy the F-optimality criterion.

Let $A_{sea}^i$ and $A_{sea}^j$ denote the sets of separately exclusive alternatives for agents *i* and *j*,
respectively. Agents with at least one separately exclusive alternative are called *agents with*
separately exclusive alternatives. The pairwise set of all such agents is represented as:

$$N_{\text{sea}} = \{\{i, j\} \subset N \mid A_{\text{sea}}^{i} \neq \emptyset \text{ and } A_{\text{sea}}^{j} \neq \emptyset\}$$

### Example 1: A Simple Case with Three Agents

Consider a set of agents $N = \{1, 2, 3\}$ and a set of alternatives $A = \{a, b, c, d, e, f, g\}$. Suppose we define preference relations $R, R' \in \Re$ for the agents as follows:

<table><thead><tr><th>R:</th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th></tr></thead><tbody><tr><td></td><td>c</td><td>a</td><td>a</td></tr><tr><td></td><td>b</td><td>b</td><td>b</td></tr><tr><td></td><td>a,g</td><td>c,e</td><td>c,f,g</td></tr><tr><td></td><td>d</td><td>f</td><td>d</td></tr><tr><td></td><td>e</td><td>d,g</td><td>e</td></tr><tr><td></td><td>f</td><td></td><td></td></tr></tbody></table>

<table><thead><tr><th>R':</th><th>R'<sub>1</sub></th><th>R'<sub>2</sub></th><th>R'<sub>3</sub></th></tr></thead><tbody><tr><td></td><td>b</td><td>b</td><td>b</td></tr><tr><td></td><td>c,g</td><td>a,c</td><td>a,c</td></tr><tr><td></td><td>a,d</td><td>e</td><td>g</td></tr><tr><td></td><td>e</td><td>f</td><td>d,f,e</td></tr><tr><td></td><td>f</td><td>d,g</td><td></td></tr></tbody></table>

Let $F(R) = \{a, c\}$ and $F(R') = \{b, g\}$. We can easily verify that the quadruple $(d, e, f, g)$ is a collection of separately exclusive alternatives. For the agents 1 and 2, these specific alternatives are respectively: $a_{\text{sea}}^1 = d$, $a_{\text{sea}}^2 = e$.

To determine separately exclusive alternatives, for each agent, we need to check the two conditions for separately exclusive alternatives:

* For Agents 1 and 2:

- Condition (i) (Mutual Preferences):

* In $R$: $dP_1e$ and $eP_2d$, $gP_1f$ and $fP_2g$.

* In $R'$: $dP'_1e$ and $eP'_2d$, $gP'_1f$ and $fP'_2g$.

- Condition (ii) (F-optimality):

* In $R$: $a, c \in F(R)$, and agents 2 and 3 both prefer $a, c$ over $d, g$, so $d, g$ are candidates to be separately exclusive alternatives for Agent 1. Agent 1 and
3 both prefer a, c over e, f, so e, f are candidates to be separately exclusive alternatives for Agent 2.

* In $R'$: $b, g \in F(R')$, and agents 2 and 3 both prefer $b, g$ over $d, g$, so $d, g$ are separately exclusive alternatives for Agent 1. Agent 1 and 3 both prefer $b, g$ over $e, f$, so $e, f$ are separately exclusive alternatives for Agent 2.

Thus, Agents 1 and 2 have $d, e, f, g$ as separately exclusive alternatives:

$$A_{\text{sea}}^{1} = \{d, g\} \text{ and } A_{\text{sea}}^{2} = \{e, f\}$$

* For Agent 3, when paired with either Agent 1 or Agent 2, conditions 1 and 2 of mutual preferences and F-optimality do not hold. Therefore, Agent 3 has no separately exclusive alternatives:

$$A_{\text{sea}}^{3} = \emptyset$$

Therefore, the set of agents with separately exclusive alternatives is:

$$N_{\text{sea}} = \{1, 2\}$$

In the applications that we will explore in Sections 5 (Labor Matching Markets) and 6 (Allocation of Divisible Goods), the existence of such alternatives is evident as we illustrate in Examples 2 and 3 below.

## Example 2: Allocation of Divisible Goods

In the case of a divisible good, the total amount $\Omega \in \mathbb{R}_{++}$ is to be allocated among $n$ agents and the set of feasible allocations $X$ is defined as

$$X = \left\{ x \equiv (x_j)_{j \in N} \in \mathbb{R}_+^n : \sum_{j \in N} x_j = \Omega \right\}$$
The relationship between two feasible allocations, $x$ and $y$, in $X$, denoted by $xR_iy$, implies that $x_iR_iy_i$. Separately exclusive alternatives (or allocations) can be viewed as vectors of shares, where there are at least two agents. For example, one agent receives the entire amount $\Omega$, while the others receive nothing. This holds as long as each agent prefers their own share of the entire amount $\Omega$ over the share where the other agent receives zero, and the other agents prefer their own share in any socially chosen allocation over receiving nothing. Thus, conditions 1 and 2 in Definition 1 of mutual preferences and $F$-optimality hold. Specifically, the separately exclusive allocations for two agents $i$ and $j$ are given by:

$$a_{\text{sea}}^{i} = (0, 0, \dots, \underbrace{\Omega}_{\text{position } i}, \dots, 0)$$

and

$$a_{\text{sea}}^{j} = (0, 0, \dots, \underbrace{\Omega}_{\text{position } j}, \dots, 0).$$

To illustrate this, suppose $N = \{1, 2, 3\}$ and $X = \{x, y, z, w\}$, where each allocation is a vector of shares that each agent receives, and the sum of the shares equals $\Omega$. Specifically, let $x = (x_1, x_2, x_3)$, $y = (\Omega, 0, 0)$, and $z = (0, \Omega, 0)$. For allocations $y$ and $z$, agents 1 and 2 can either receive $\Omega$ or nothing, and $w = (\frac{\Omega}{3}, \frac{\Omega}{3}, \frac{\Omega}{3})$.

Now, consider the solution to the fair allocation problem for divisible goods: the individually rational correspondence from equal division $I_{ed}$, which selects the allocations where each agent weakly prefers their share over the average. For simplicity, let's assume a single preference profile $R$ such that:

$$R: \begin{array}{c} R_1 \\ R_2 \\ R_3 \\ x_1, \Omega \\ 0 \\ \frac{\Omega}{3} \end{array} \begin{array}{c} \Omega \\ x_2, 0 \\ \frac{\Omega}{3} \end{array} \begin{array}{c} x_3, 0 \\ \frac{\Omega}{3} \\ \Omega \end{array}$$

Thus, $I_{ed}(R) = \{x, y\}$.
In the following, we will illustrate that the conditions of mutual preferences and $I_{ed}$-optimality for agents 1 and 2, as defined in Definition 1, are satisfied.

• Mutual preferences:

$$ \Omega P_{10}, \text{ i.e., } y = (\Omega, 0, 0) P_{1z} = (0, \Omega, 0). $$

and

$$ \Omega P_{20}, \text{ i.e., } z = (0, \Omega, 0) P_{2y} = (\Omega, 0, 0). $$

• F-optimality:

– For agent 1:

$$ x_2 R_{20}, \text{ i.e., } x = (x_1, x_2, x_3) R_{2y} = (\Omega, 0, 0). $$

$$ x_3 R_{30}, \text{ i.e., } x = (x_1, x_2, x_3) R_{3y} = (\Omega, 0, 0). $$

$$ 0 R_{20}, \text{ i.e., } y = (\Omega, 0, 0) R_{2y} = (\Omega, 0, 0). $$

$$ 0 R_{30}, \text{ i.e., } y = (\Omega, 0, 0) R_{3y} = (\Omega, 0, 0). $$

– For agent 2:

$$ x_1 R_{10}, \text{ i.e., } x = (x_1, x_2, x_3) R_{1z} = (0, \Omega, 0). $$

$$ x_3 R_{30}, \text{ i.e., } x = (x_1, x_2, x_3) R_{3z} = (0, \Omega, 0). $$

$$ \Omega R_{10}, \text{ i.e., } y = (\Omega, 0, 0) R_{1z} = (0, \Omega, 0). $$

$$ 0 R_{30}, \text{ i.e., } y = (\Omega, 0, 0) R_{3z} = (0, \Omega, 0). $$

Thus, $a_{\text{sea}}^1 = y$ and $a_{\text{sea}}^2 = z$, which means $A_{\text{sea}}^1 = \{y\}$ and $A_{\text{sea}}^2 = \{z\}$. Therefore, $N_{\text{sea}} = \{1, 2\}$.
## Example 3: Several-to-One Matching Markets

Consider a firm-worker matching market, where a set of firms $\mathcal{F}$ must be matched with a set of workers $\mathcal{W}$. Each firm has a preference list for workers, and each worker has a preference list for firms. These labor markets can be structured to include separately exclusive alternatives (matchings), which occur when certain firms and workers have mutual preferences, meaning each agent prefers their own matching over the other agent's matching. Additionally, optimality condition holds when all other agents prefer any socially chosen matching over any matching satisfying mutual preferences condition.

In such markets, the set of all possible matchings is denoted by $\mathcal{M}$, where a matching $\mu \in \mathcal{M}$ is represented as a list of matched pairs from the sets of firms $\mathcal{F}$ and workers $\mathcal{W}$. For each $i \in \mathcal{F} \cup \mathcal{W}$, $\mu(i)$ represents the agent (firm or worker) assigned to $i$ under matching $\mu$.

In our model of several-to-one matching, discussed in Section 5, a matching must satisfy the following conditions:

(i) For all $w \in \mathcal{W}$, $|\mu(w)| \le 1$, and if $\mu(w) \ne \emptyset$, then $\mu(w) \in \mathcal{F}$.

(ii) For all $f \in \mathcal{F}$, $0 \le |\mu(f)| \le q_f$, where $q_f$ is the capacity of firm $f$, and if $\mu(f) \ne \emptyset$, then $\mu(f) \subset \mathcal{W}$.

(iii) For all $(f, w) \in \mathcal{F} \times \mathcal{W}$, $\mu(w) = f$ if and only if $w \in \mu(f)$.

For two matchings, $\mu, \nu \in \mathcal{M}$, the expression $\mu R_i \nu$ implies that $\mu(i) R_i \nu(i)$.

In this context, separately exclusive alternatives (matchings) arise when at least two agents-either two firms, two workers, or a combination of both-are involved. For example:

(1) Each firm is matched to only one worker, while the other firms remain unmatched.

(2) Each worker is matched, while the other workers remain unmatched.

Condition 1 of mutual preferences is satisfied by assuming that each agent prefers their assigned matching to being unmatched. Condition 2 is satisfied by supposing that all other agents prefer any socially chosen matching over any matching satisfying Condition 1.
To illustrate this, consider $\mathcal{F} = \{f_1, f_2, f_3\}$, a set of firms, and $\mathcal{W} = \{w_1, w_2, w_3\}$, a set of workers, and consider the following matchings:

$$ \nu = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{array} \right), \text{ and } \rho = \left( \begin{array}{ccccc} w_1 & w_2 & w_3 & \emptyset & \emptyset \\ \emptyset & \emptyset & \{f_1\} & \{f_2\} & \{f_3\} \end{array} \right) $$

$$ \xi = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{array} \right), \text{ and } \rho = \left( \begin{array}{ccccc} w_1 & w_2 & w_3 & \emptyset & \emptyset \\ \{f_2\} & \emptyset & \emptyset & \{f_1\} & \{f_3\} \end{array} \right) $$

For simplicity, we consider a single preference profile consisting of 6-tuples of preference relations $R = (R_{f_1}, R_{f_2}, R_{f_3}; R_{w_1}, R_{w_2}, R_{w_3}) \in \Re$, given by:

<table><thead><tr><th>R:</th><th>R<sub>f1</sub></th><th>R<sub>f2</sub></th><th>R<sub>f3</sub></th><th>R<sub>w1</sub></th><th>R<sub>w2</sub></th><th>R<sub>w3</sub></th></tr></thead><tbody><tr><td>μ(f<sub>1</sub>)</td><td>μ(f<sub>2</sub>)</td><td>μ(f<sub>3</sub>)</td><td></td><td>μ(w<sub>1</sub>)</td><td>μ(w<sub>2</sub>)</td><td>μ(w<sub>3</sub>)</td></tr><tr><td>o(f<sub>1</sub>), ν(f<sub>1</sub>)</td><td>ρ(f<sub>2</sub>), ξ(f<sub>2</sub>)</td><td>ρ(f<sub>3</sub>), ν(f<sub>3</sub>), ξ(f<sub>3</sub>)</td><td></td><td>ξ(w<sub>1</sub>), ρ(w<sub>1</sub>)</td><td>ν(w<sub>2</sub>), o(w<sub>2</sub>), ρ(w<sub>2</sub>)</td><td>ν(w<sub>3</sub>), o(w<sub>3</sub>)</td></tr><tr><td>ρ(f<sub>1</sub>), ξ(f<sub>1</sub>)</td><td>o(f<sub>2</sub>), ν(f<sub>2</sub>)</td><td></td><td></td><td>ν(w<sub>1</sub>), o(w<sub>1</sub>)</td><td></td><td>ξ(w<sub>3</sub>), ρ(w<sub>3</sub>)</td></tr></tbody></table>

Let $\varphi$ be a matching rule such that $\varphi = \{\mu\}$. In the following, we illustrate that the conditions of mutual preferences and optimality provided in Definition 1 are satisfied for firms $f_1$ and $f_2$.

* Mutual preferences

$$ \nu(f_1) = \{w_3\} P_{f_1} \xi(f_1) = \emptyset, $$

$$ \text{i.e., } \nu = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{array} \right) P_{f_1} \xi = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{array} \right) $$

and

$$ \xi(f_2) = \{w_1\} P_{f_2} \nu(f_2) = \emptyset, $$

$$ \text{i.e., } \xi = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{array} \right) P_{f_2} \nu = \left( \begin{array}{ccccc} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{array} \right). $$

* $\varphi$-optimality

- For firm $f_1$:
$$ \mu(f_2)R_{f_2}\nu(f_2) = \emptyset, \text{ i.e., } \mu R_{f_2}\nu = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{pmatrix}. $$

$$ \mu(f_3)R_{f_3}\nu(f_3) = \emptyset, \text{ i.e., } \mu R_{f_3}\nu = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{pmatrix}. $$

$$ \mu(w_1)R_{w_1}\nu(w_1) = \emptyset, \text{ i.e., } \mu R_{w_1}\nu = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{pmatrix}. $$

$$ \mu(w_2)R_{w_2}\nu(w_2) = \emptyset, \text{ i.e., } \mu R_{w_2}\nu = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{pmatrix}. $$

$$ \mu(w_3)R_{w_3}\nu(w_3) = f_1, \text{ i.e., } \mu R_{w_3}\nu = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \{w_3\} & \emptyset & \emptyset & \{w_2\} & \{w_1\} \end{pmatrix}. $$

– For firm $f_2$:

$$ \mu(f_1)R_{f_1}\xi(f_1) = \emptyset, \text{ i.e., } \mu R_{f_1}\xi = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{pmatrix}. $$

$$ \mu(f_3)R_{f_3}\xi(f_3) = \emptyset, \text{ i.e., } \mu R_{f_3}\xi = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{pmatrix}. $$

$$ \mu(w_1)R_{w_1}\xi(w_1) = f_2, \text{ i.e., } \mu R_{w_1}\xi = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{pmatrix}. $$

$$ \mu(w_2)R_{w_2}\xi(w_2) = \emptyset, \text{ i.e., } \mu R_{w_2}\xi = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{pmatrix}. $$

$$ \mu(w_3)R_{w_3}\xi(w_3) = \emptyset, \text{ i.e., } \mu R_{w_3}\xi = \begin{pmatrix} f_1 & f_2 & f_3 & \emptyset & \emptyset \\ \emptyset & \{w_1\} & \emptyset & \{w_2\} & \{w_3\} \end{pmatrix}. $$

A similar explanation will be provided for the case of workers $w_1$ and $w_3$, firm $f_1$ and worker $w_1$, and firm $f_2$ and worker $w_3$. Thus, the agents $f_1$, $f_2$, $w_1$, and $w_3$ each have the following separately exclusive matchings: $a_{\text{sea}}^{f_1} = \nu$, $a_{\text{sea}}^{f_2} = \xi$, $a_{\text{sea}}^{w_1} = \rho$, and $a_{\text{sea}}^{w_3} = o$. That is,

$$ A_{\text{sea}}^{f_1} = \{\nu\}, \quad A_{\text{sea}}^{f_2} = \{\xi\}, \quad A_{\text{sea}}^{w_1} = \{\rho\}, \quad \text{and} \quad A_{\text{sea}}^{w_3} = \{o\}. $$

Therefore,
$$N_{sea} = \{f_1, f_2, w_1, w_3\}.$$

Based on the existence of agents with separately exclusive alternatives, we demonstrate in the following sections how it is possible to circumvent the problem of Maskin's impossibility (1979), outlined in Theorem 4 above. This issue arises because the no-veto power property becomes incompatible with the strong version of Nash equilibrium. We also address the challenges of non-existence of appropriate sufficient conditions for strong implementation without Holocaust alternatives (Korpela (2013)), and provide positive and permissible results that can be applied to interesting economic domains.

# 4 The Main Results

We now present the sufficient conditions that characterize the family of SCCs that can be implemented in strong equilibria, assuming there are at least three agents. To begin, we require the following assumption on the set of agents:

**Assumption 1** There exists at least one pair of agents with separately exclusive alternatives.

Next, we introduce a strong version of Maskin monotonicity, referred to here as Y-monotonicity<fnref n="7" />. We show that the presence of at least one pair of agents with separately exclusive alternatives makes any weakly Pareto optimal SCC satisfying Y-monotonicity strongly implementable, provided there are at least three agents in the society.

**Definition 2** (Y-monotonicity)

An SCC F satisfies Y-monotonicity, if for any R, R' ∈ ℝ, any a ∈ F(R) and any c ∈ $\bigcup_{i \in N} L_i(a, R)$, $L_i(a, R) \subseteq L_i(c, R')$ for all i ∈ N, then c ∈ F(R').

In simpler terms, Y-monotonicity ensures that if a is selected under the preference profile R, and c is a potential alternative such that for some agent i, c is dominated by a under R,

<footnote n="7">⁷In Yi's (2012) work, Y-monotonicity is denoted by condition α. Yi demonstrated that, with three or more agents, condition α, combined with no-veto power, is sufficient for double implementability in both Nash and M-Nash equilibria, where M represents coalitions containing fewer than half of the total agents.</footnote>
and the options available to each agent that are dominated by $a$ in $R$ are included in the options dominated by $c$ in $R'$, then $c$ must also be selected under the profile $R'$.

**Lemma 1** (Yi, 2012)

*Y-monotonicity implies Maskin monotonicity.*

**Remark 1** (Yi, 2012)

*Y-monotonicity is not a necessary condition for double implementation in Nash and M-Nash equilibria.*

**Observation 1** *Y-monotonicity is not a necessary condition for strong implementation.*

The proof of Observation 1 can be derived from the fact that Nash and M-Nash equilibria
are weaker than a strong equilibrium.

In the next, we show that Y-monotonicity together with weak Pareto optimality implies
unanimity.

**Definition 3** (*Unanimity*)

An SCC $F$ satisfies unanimity if for any $a \in A$ and any $R \in \Re$: $L_i(a, R) = A$ for all $i \in N$
implies $a \in F(R)$.

Unanimity requires that, for any alternative *a* and any preference profile *R*, if every agent
in the group prefers *a* over all other alternatives, then *a* must be selected by *F(R)*. In other
words, if all agents agree that *a* is the best option, *a* will always be chosen.

**Lemma 2** If an SCC *F* satisfies weak Pareto optimality and *Y*-monotonicity, then it
satisfies unanimity.

*Proof.* Assume that *F* satisfies weak Pareto optimality and *Y*-monotonicity, but it
violates unanimity. That is, there exist *a* ∈ *A*, *R* ∈ *D* such that for all *i* ∈ *N*, *L*(*a*, *R*<sub>*i*</sub>) = *A*
(1), but *a* ∉ *F*(*R*). As *F*(*R*) ≠ ∅, there exists *c* ∈ *F*(*R*). From (1) we have *aR*<sub>*ic*</sub> for all
*i* ∈ *N* (2). Since *F* satisfies weak Pareto optimality, it follows that there exists *j* ∈ *N* such
that $cR_j a$ (3). It follows from (2) and (3) that $a \sim_j c$ (4). By (1), (4), and Y-monotonicity we obtain $a \in F(R)$, a contradiction. Q.E.D

The following result constitutes the main findings of the paper. Theorem 6 offers an alternative to the existing results in the literature concerning the implementation in strong Nash equilibrium.

**Theorem 6** Let $n \ge 3$ and suppose Assumption 1 holds. Then, if a weakly Pareto optimal SCC F satisfies Y-monotonicity, F is implementable in strong equilibria.

*Proof of Theorem 6.* We construct a mechanism form $G = (S_1, ..., S_n; g)$ that implement F. Let $i \in N$ and $S_i = \mathbb{R} \times A \times \{F, NF\} \times \mathbb{N}$. The generic element of strategic space $S_i$ is noted by: $s_i = (R^i, a^i, x^i, m^i)$. Each agent announces a preference profile $R^i \in \mathbb{R}$, an alternative $a^i \in A$, raises a ‘flag’ ($x^i = F$) or not ($x^i = NF$) and announces a nonnegative integer $m^i \in \mathbb{N}$. The outcome function $g$ is divided into four cases which we denote as rules:

**Rule 1:** If for each $i \in N$, $s_i = (R, a, NF, m)$ and $a \in F(R)$, then $g(s) = a$.

**Rule 2:** If for some coalition $C$ with $\emptyset \ne C \subseteq N$, $s_i = (R, a, NF, m)$ for all $i \in N \setminus C$ where $a \in F(R)$, and:

2.1) If $s_j = (R^j, a^j, F, m^j)$ for all $j \in C$ and $C$ contains at least two agents, then:

$$g(s) = \begin{cases} a^l & \text{where } l = \min\{\argmax_{j \in C} m^j / a^j \in \bigcup_{k \in C} L_k(a, R)\}, \\ a & \text{otherwise.} \end{cases}$$

2.2) If $C = \{j\}$ ($C$ is a singleton coalition), and $s_j = (R^j, a^j, x^j, m^j)$ independently of the state of the flag, then:

$$g(s) = \begin{cases} a^j & \text{if } a^j \in L_j(a, R), \\ a & \text{otherwise.} \end{cases}$$

**Rule 3:** If $x^i = F$ for all $i \in N$, then $g(s) = a^i$ where $i = \min\{\argmax_{j \in N} m^j\}$.
**Rule 4:** In all other cases, $g(s) = a_{sea}^l$ where $l = \min\{\argmax_{t \in N_{sea}} m^t\}$.

Before proving our Theorem, let us some useful remarks.

1. There cannot be a strong equilibrium in which rule 4 applies. Assuming the opposite, any agent with separately exclusive alternatives could reverse the strong equilibrium by announcing an alternative from their set of separately exclusive alternatives.

2. Consequently, inside an equilibrium state, any two agents who do not raise their flags have necessarily the same strategy.

3. In an equilibrium state only Rule 1, Rule 2 (with the raising ‘flag’ option) or Rule 3 can apply.

Now, we show in two steps that for all $R \in \mathbb{R}$, $F(R) = g(SE(S, g, R))$.

**Step 1.** Show that for all $R \in \mathbb{R}$, $F(R) \subseteq g(SE(S, g, R))$;

Let $R \in \mathbb{R}$ and $a \in F(R)$. For each $i \in N$, let $s_i = (R, a, NF, m^i)$. By rule 1, $g(s) = a$. Show that $s \in SE(S, g, R)$. The grand coalition $N$ cannot deviate profitably since $F$ is weakly Pareto optimal. Let $C \subsetneq N$, we discuss according to the size of $C$, denoted by $|C|$: If $2 \le |C| \le n-1$, two sub-cases appear: i) all members of $C$ raise flag, then we use rule 2.1; ii) the deviation induces the remaining case (rule 4): in this case $g(\tilde{s}_C, s_{-C}) = a_{sea}^l$ where $l = \min\{\argmax_{t \in N_{sea}} m^t\}$. As $a \in F(R)$, and $2 \le |C|$, there exists $j \in C$, $j \ne l$ - by Assumption 1 - such that $aR_jg(\tilde{s}_C, s_{-C}) = a_{sea}^l$. If $|C| = 1$, rule 2.2 applies and ensures the conclusion.

**Step 2.** Show that for all $R \in \mathbb{R}$, $g(SE(S, g, R)) \subseteq F(R)$;

Let $s \in SE(S, g, R)$. Show that $g(s) \in F(R)$.

**Case a:** If Rule 1 applies. Let $s = (s_1, s_2, ..., s_n)$ be a strategy profile. Assume there exists $(R', a, NF, m) \in \mathbb{R} \times A \times \{F, NF\} \times \mathbb{N}$, with $a \in F(R')$, such that $s$ is defined by $s_i = (R', a, NF, m)$ for all $i \in N$. By rule 1, $g(s) = a$.
Let $i \in N$ and $b \in L_i(a, R')$. Any agent who proposes $\tilde{s}_i = (\tilde{R}, b, F, \tilde{m})$ can obtain by rule 2.2 his choice $b = g(\tilde{s}_i, s_{-i}) \in L_i(a, R)$ as $s \in SE(S, g, R)$. Therefore, $L_i(a, R') \subseteq L_i(a, R)$ for each $i$ for each $i'$. Maskin monotonicity ensures that $a \in F(R)$.

**Case b:** If Rule 2 applies. Let $s = (s_1, s_2, ..., s_n)$ be a strategy profile. For some coalition $C$ with $\emptyset \neq C \subseteq N$, let $s_i = (R', a, NF, m)$ for all $i \in N \setminus C$ where $a \in F(R')$, and:

2.1) If $s_j = (R'^j, a^j, F, m^j)$ for all $j \in C$ and $C$ contains at least two agents, then:

$$g(s) = \begin{cases} a^l & \text{where } l = \min\{\argmax_{j \in C} m^j / a^j \in \bigcup_{k \in C} L_k(a, R')\}, \\ a & \text{otherwise.} \end{cases}$$

2.2) If $C = \{j\}$ ($C$ is a singleton coalition), and $s_j = (R'^j, a^j, x^j, m^j)$ independently of the state of the flag, then:

$$g(s) = \begin{cases} a^j & \text{if } a^j \in L_j(a, R'), \\ a & \text{otherwise.} \end{cases}$$

In this case, we should demonstrate that for all $R \in \Re$, $g(s) \in F(R)$ for all $s \in SE(S, g, R)$.

**Subcase b.1:** $a^l = g(s)$. As $a^l \in \bigcup_{k \in C} L_k(a, R') \neq \emptyset$, then there exists $j \in C$ such that $a^l \in L_j(a, R')$. Let $b \in L_j(a, R') \subset \bigcup_{k \in C} L_k(a, R')$, do a deviation with agent $j$, such that $\tilde{s}_j = (\tilde{R}, b, F, \tilde{m})$ such that $\tilde{m}$ is the higher over players in $C$, then $g(\tilde{s}_j, s_{-j}) = b$. As $s \in SE(S, g, R)$, then $a^l = g(s)R_jg(\tilde{s}_j, s_{-j}) = b$. We get for $j \in C$, $a^l \in L_j(a, R') \subset L_j(a^l, R)$ (1).

Let $j' \in C \setminus \{j\}$, and let $b \in L_{j'}(a, R') \subset \bigcup_{k \in C} L_k(a, R')$, do a deviation with agent $j'$, such that $\tilde{s}_{j'} = (\tilde{R}', b, F, \tilde{m}')$ such that $\tilde{m}'$ is the higher over players in $C$, then $g(\tilde{s}_{j'}, s_{-j'}) = b$.
As $s \in SE(S, g, R)$, then $a^l = g(s)R_jg(\tilde{s}_j, s_{-j'}) = b$. We get for each $j' \in C \setminus \{j\}$,
$L_{j'}(a, R') \subset L_{j'}(a^l, R)$ (2).

Now, let $k \in N/C$ and let $b \in L_k(a, R')$. Then, in this case we proceed as before and
we do a unilateral deviation on $N \setminus C$, such that $\tilde{s}_k = (\tilde{R}', b, F, \tilde{m}')$ with $\tilde{m}'$ is the higher
over players in $C \cup \{k\}$, then, by rule 2 (if $|N/C| \ge 2$) or by rule 3 (if $|N/C| = 1$),
$g(\tilde{s}_k, s_{-k}) = b$. As $s \in SE(S, g, R)$, then $a^l = g(s)R_k g(\tilde{s}_k, s_{-k}) = b$. We get for each
$k \in N \setminus C$, $L_k(a, R') \subset L_k(a^l, R)$ (3). Hence, it follows from (1), (2), and (3) that
$L_j(a, R') \subset L_j(a^l, R)$ for each $j \in N$. From Y-monotonicity we get the result: $a^l \in F(R)$.

**Subcase b.2:** $a^j = g(s)$. By definition, we have $a^j \in L_j(a, R') \ne \emptyset$. Let $b \in L_j(a, R')$,
do a deviation with agent $j$, such that $\tilde{s}_j = (\tilde{R}, b, \dots, \tilde{m})$ such that $b \ne a^j$ and $\tilde{m} > m^j$, then,
by rule 2.2, $g(\tilde{s}_j, s_{-j}) = b$. As $s \in SE(S, g, R)$, then $a^l = g(s)R_j g(\tilde{s}_j, s_{-j}) = b$. We get,
$a^j \in L_j(a, R') \subset L_j(a^j, R)$ (4).

For any $k \in N \setminus \{j\}$, let $b \in L_k(a, R')$. Then, in this case we proceed as before and
we do a unilateral deviation on $N \setminus \{j\}$, such that $\tilde{s}_k = (\tilde{R}', b, F, \tilde{m}')$ such that $\tilde{m}'$ is the
higher over players in $\{j, k\}$, then, by rule 2, $g(\tilde{s}_k, s_{-k}) = b$. As $s \in SE(S, g, R)$, then
$a^j = g(s)R_k g(\tilde{s}_k, s_{-k}) = b$. We get for each $k \in N \setminus \{j\}$, $L_k(a, R') \subset L_k(a^j, R)$ (5). Hence,
it follows from (4), (5), and Y-monotonicity that $a^j \in F(R)$.

**Subcase b.3:** If $a = g(s)$, we proceed as in the previous cases using Maskin monotonicity.

**Case c:** If Rule 3 applies, let $s = (s_1, s_2, \dots, s_n)$ be a strategy profile. When $s$ constitutes
a strong equilibrium under Rule 3 for a profile $R$, $g(s) = a^i$ for some $i \in N$. We claim that
$a^i \in F(R)$. To see this, let observe that for any agent $i$, proposing unilaterally any alternative
$b \in A$, $\tilde{s}_i = (\tilde{R}', b, F, \tilde{m}')$ such that $\tilde{m}'$ is the higher over all players we obtain $b \in L_i(a^i, R)$
by the mechanism, and hence $A \subseteq L_i(a^i, R)$. By unanimity $a^i \in F(R)$. Q.E.D
Next, we will discuss some important implications that will help in analyzing the implementability of certain social choice correspondences (SCCs) in Section 6. In Proposition 2 below, we demonstrate the strong implementability of the non-empty intersection of SCCs. The proof of this result follows from Theorem 6 above, along with Lemmas 3 and 4 below, which establish the stability by intersection of the two properties: weak Pareto optimality and Y-monotonicity.

**Definition 4** (Stability by Intersection)

A condition $\gamma$ is said to be stable by intersection if, for any SCCs $F$ and $G$ satisfying $\gamma$, the correspondence defined by $F \cap G$ (which is non-empty) also satisfies $\gamma$.

**Lemma 3** If for any SCCs $F$ and $G$ satisfying weak Pareto optimality, then their non-empty intersection $F \cap G \neq \emptyset$ also satisfies weak Pareto optimality.

*Proof.* Assume, for contradiction, that there exists a profile $R \in \Re$ such that $a \in (F \cap G)(R)$ is not weakly Pareto optimal. This implies that there exists an alternative $b \in A$ such that $bP_ia$ for all $i \in N$. Since $F$ satisfies weak Pareto optimality, $a$ must be weakly Pareto optimal in $F(R)$, meaning no alternative $b$ can strictly dominate $a$. This contradicts the assumption that $bP_ia$ for all $i \in N$. Similarly, since $G$ satisfies weak Pareto optimality, $a$ must be weakly Pareto optimal in $G(R)$, which again contradicts the assumption that $bP_ia$ for all $i \in N$. Therefore, our assumption is false, and $(F \cap G)(R)$ must select only weakly Pareto optimal alternatives. Q.E.D

**Lemma 4** If for any SCCs $F$ and $G$ satisfying Y-monotonicity, then their non-empty intersection $F \cap G \neq \emptyset$ also satisfies Y-monotonicity.

*Proof.* Assume, for contradiction, that the intersection $F \cap G$ does not satisfy Y-monotonicity. This means that there exists a preference profile $R$, an outcome $a \in (F \cap G)(R)$, and an outcome $c \in \bigcup_{i \in N} L_i(a, R)$, as well as another profile $R'$ such that $L_i(a, R) \subseteq L_i(c, R')$ for all $i \in N$, and $c \notin (F \cap G)(R')$. Since $a \in (F \cap G)(R)$, we know that $a \in F(R)$ and $a \in G(R)$. Because $F$ satisfies Y-monotonicity, $c \in F(R')$. Similarly, since $G$ satisfies Y-monotonicity, $c \in G(R')$. Thus, $c \in F(R') \cap G(R')$, which implies $c \in (F \cap G)(R')$.
This contradicts the assumption that $c \notin (F \cap G)(R')$. Therefore, the intersection $F \cap G$ must satisfy Y-monotonicity. Q.E.D

**Proposition 2** Let $n \ge 3$ and suppose Assumption 1 holds. If two SCCs $F$ and $G$ satisfy weak Pareto optimality and $Y$-monotonicity, and their intersection $F(R) \cap G(R)$ is non-empty for all $R \in \mathbb{R}$, then $F(R) \cap G(R)$ is strongly implementable.

We now provide useful characterization for strong implementation. To this end, we begin by presenting the following property.

**Definition 5 (Citizen Sovereignty)**

An SCC $F$ satisfies citizen sovereignty if for every $a \in A$, there exists a profile $R \in \mathbb{R}$ such that $a \in F(R)$.

**Proposition 3** Let $n \ge 3$, suppose Assumption 1 holds, and let $F$ be an SCC that satisfies $Y$-monotonicity and citizen sovereignty. Then, $F$ is strongly implementable if and only if it satisfies weak Pareto optimality.

*Proof.* The first part of the proof follows from theorem 6. For the second part, assume that an SCC $F$ is strongly implementable, but does not satisfy weak Pareto optimality. That is, there exists $a \in F(R) = g(SE(S,g,R))$ for some $R \in \mathbb{R}$, such that there exists $b \in A$ for which $bP_ia$ for all $i \in N$. For $b$, it follows from citizen sovereignty that there exists a preference profile $R'$ such that $b \in F(R')$. Since $F$ is strongly implementable, we have $b \in F(R') = g(SE(S,g,R'))$, and hence there exists $s' \in SE(S,g,R')$ such that $g(s') = b$ and $s'$ is obtained from $s$ by a deviation of a coalition $C$ (C can be $N$). As $a = g(s)$, where $s \in SE(S,g,R)$, it follows that there exists $i \in C$ such that $a = g(s)R_ig(s') = b$, a contradiction. Q.E.D.

In comparison to similar literature, this result circumvents Maskin's impossibility (1979), as outlined in Theorem 4 above, where the no-veto power property becomes incompatible with the strong version of Nash equilibrium. We generalize Maskin's result from the strong implementability of weakly Pareto optimal SCCs satisfying individual rationality
to the strong implementability of weakly Pareto optimal SCCs satisfying Y-monotonicity.
Additionally, we extend Yi's (2012) result from double implementation in Nash and M-Nash
equilibria to implementation in the strong equilibrium of Aumann (1959), by combining a
strong version of monotonicity with weak Pareto optimality in environments containing at
least two agents with separately exclusive alternatives.

We also revisit Korpela's (2013) result, which used the strong equilibrium of Aumann (1959). Korpela introduced a stronger variant of monotonicity, called the axiom of sufficient reason<fnref n="8" />, and showed that it, along with weak Pareto optimality and the Holocaust alternative<fnref n="9" />, ensures implementability in strong equilibrium when there are at least three agents. However, the use of the Holocaust alternative (HA) is restrictive. In the presence of at least one pair of agents with separately exclusive alternatives, our result complements Korpela's theorem. Our approach enables the implementation of an SCC through Aumann-strong equilibrium by combining weak Pareto optimality and Y-monotonicity, without requiring the Holocaust alternative.

For instance, in Example 1, SCC *F* is implementable in strong equilibrium via Theorem 6.
This example shows no Holocaust alternative, and the axiom of sufficient reason is violated,
meaning Korpela's theorem does not apply to the strong implementability of *F*. Thus, we
propose an alternative theorem to Korpela's (2013) that is more widely applicable.

# 5 Several-to-One Matching Markets

## 5.1 Model and Definitions

In this subsection, we assume that all agents have linear orderings, i.e., strict preferences.
We consider a finite set of firms $\mathcal{F} = \{f_1, \dots, f_n\}$ and a finite set of workers $\mathcal{W} = \{w_1, \dots, w_m\}$.
Let $q = (q_{f_1}, \dots, q_{f_n})$ be a vector of positive natural numbers, where $q_{f_i}$ is the capacity of
firm $f_i \in \mathcal{F}$. Both firms and workers have their preferences over agents (or over potential

<footnote n="8">⁸An SCC *F* satisfies the axiom of sufficient reason if for all *R*, *R'* ∈ ℜ, *a* ∈ *F*(*R*), and *c* ∈ *A*, *L*<sub>*i*</sub>(*a*, *R*) ⊆ *L*<sub>*i*</sub>(*c*, *R'* ) for all *i* ∈ *N* implies *c* ∈ *F*(*R'*).</footnote>

<footnote n="9">⁹An SCC *F* has a Holocaust alternative if there exists *a*<sub>*H*</sub> ∈ *A* such that for all *R* ∈ ℜ, *a*<sub>*H*</sub> ∉ *F*(*R*), and for all *R* ∈ ℜ, for all *a* ∈ *A* \ {*a*<sub>*H*</sub>}, *a* ∉ *L*<sub>*i*</sub>(*a*<sub>*H*</sub>, *R*) for all *i* ∈ *N*.</footnote>
matches) of the other kind. To ensure that the set of stable matchings is non-empty, we assume that the firm's and workers' preferences have linear orderings. Thus, let $P_w$ be the preference relation of worker $w \in W$, which is a linear order on the set $\mathcal{F} \cup \{\emptyset\}$, where $\emptyset$ is interpreted as $w$ not being matched with any $f \in \mathcal{F}$. Let $P_f$ be the preference relation of firm $f \in \mathcal{F}$, which is a linear order on the set $W_f = \{G \mid G \subseteq W \text{ and } |G| \le q_f\}$, where $\emptyset$ is also interpreted as $f$ not being matched with any $w \in W$. As Roth (1985), Kara and Sönmez (1996) showed, we assume that $P_f$ is responsive for all $f \in \mathcal{F}$. This means that for all $G \subseteq W$ such that $|G| < q_f$, the following two properties are checked:

1) For all $w, w' \in W \setminus G$, $G \cup \{w\}P_fG \cup \{w'\}$ if, and only if $wP_fw'$.<fnref n="10" />

2) For all $w \in W \setminus G$, $G \cup \{w\}P_fG$ if, and only if $wP_f\emptyset$.

A several-to-one matching $\mu$ is a mapping from the set $\mathcal{F} \cup W$ into $2^{\mathcal{F} \cup W}$ such that:

i) for all $w \in W$, $|\mu(w)| \le 1$, and $\mu(w) \ne \emptyset$ implies $\mu(w) \in \mathcal{F}$,

ii) for all $f \in \mathcal{F}$, $0 \le |\mu(f)| \le q_f$, and $\mu(f) \ne \emptyset$, implies $\mu(f) \in W$,

iii) for all $(f, w) \in \mathcal{F} \times W$, $\mu(w) = f$ if and only if $w \in \mu(f)$.

For any $i \in \mathcal{F} \cup W$, we define $\mu(i)$ as the assignment of $i$ under the matching $\mu$. The set of all possible matchings is denoted by $\mathcal{M}$. We extend the preference relation $P_f$ of a firm $f \in \mathcal{F}$, which is originally defined on the class of subsets $2^{W \cup \{\emptyset\}}$, to the set of matchings $\mathcal{M}$ as follows: firm $f$ prefers the matching $\mu$ over $\mu'$ if and only if it prefers its assignment under $\mu$ to its assignment under $\mu'$. Similarly, we extend the preference relation for workers.

*All strict preference relations $(P_i)_{i \in \mathcal{F} \cup W}$ over agents naturally induce weak preferences over matchings. Therefore, we introduce a new relation $(R_i)_{i \in \mathcal{F} \cup W}$ to denote this extension.*

To illustrate this, let $\mathcal{F} = \{f_1, f_2, f_3\}$, $W = \{w_1, w_2, w_3\}$, $q_{f_1} = q_{f_2} = q_{f_3} = 1$, and $\mathcal{M} = \{\mu, \nu\}$, where

$$ \mu = \begin{pmatrix} f_1 & f_2 & f_3 \\ \{w_1\} & \{w_2\} & \{w_3\} \end{pmatrix}, \quad \nu = \begin{pmatrix} f_1 & f_2 & f_3 \\ \{w_1\} & \{w_3\} & \{w_2\} \end{pmatrix}. $$

In this case, we have $\mu(f_1) = w_1$ and $\nu(f_1) = w_1$. Therefore, we must have **indifference**

<footnote n="10">¹⁰We write $w$ instead of $\{w\}$.</footnote>
between $\mu(f_1)$ and $\nu(f_1)$. Note that a firm $f_i$ can only be indifferent between $\mu(f_i) = w_j$ and $\nu(f_i) = w_k$ if $j = k$, since all preferences are strict. A similar situation holds for the preferences of workers. We summarize this in Observation 3 below.

**Observation 2** For all $i \in \mathcal{F} \cup \mathcal{W}$ and for all $\mu, \nu \in \mathcal{M}$, $\mu R_i \nu$ if and only if $\mu(i) P_i \nu(i)$ or $\mu(i) = \nu(i)$.

Let $\Re_w$ be the class of all such preferences for a worker $w \in \mathcal{W}$ over matchings. Let $\Re_f$ be the class of all such preferences for a firm $f \in \mathcal{F}$ over matchings. Let $\Re = \Re_f^n \times \Re_w^m$. As usual, $R = (R_i)_{i \in \mathcal{F} \cup \mathcal{W}} \in \Re$ denotes a preference profile of firms and workers.

We define the choice $H$ of a firm $f$ from a group of workers $G \subseteq \mathcal{W}$ under the preference $R_f$ as follows: $H_f(G; R_f) = \{G' \subseteq G : |G'| \le q_f, G' R_f G'' \text{ for all } G'' \subseteq G \text{ such that } |G''| \le q_f\}$; that is $H_f(G; R_f)$ denotes firm $f$'s most-preferred subset of $G$ according to its preference relation $R_f$.

A matching $\mu$ is blocked by a worker $w \in \mathcal{W}$ under a preference profile $R$ if $\emptyset P_w \mu(w)$. A matching $\mu$ is blocked by a firm $f \in \mathcal{F}$ under a preference profile $R$ if $\mu(f) \ne H_f(\mu(f); R_f)$. Under responsive preferences, this statement is equivalent to the following: A matching $\mu$ is blocked by a firm $f \in \mathcal{F}$ under a preference profile $R$ if there is a worker $w \in \mu(f)$ such that $\emptyset P_f w$.

A matching $\mu$ is *individually rational* under a preference profile $R$ if it is not blocked by a worker or a firm under $R$; i.e., if $\mu(w) R_w \emptyset$ for all $w \in \mathcal{W}$ and $\mu(f) = H_f(\mu(f); R_f)$ for all $f \in \mathcal{F}$.

A matching $\mu$ is blocked by a firm-worker pair $(f, w) \in \mathcal{F} \times \mathcal{W}$ under a preference profile $R$ if $f P_w \mu(w)$ and $\mu(f) \ne H_f(\mu(f) \cup \{w\}; R_f)$.

An individually rational matching $\mu$ is blocked by a firm-worker pair $(f, w) \in \mathcal{F} \times \mathcal{W}$ under a preference profile $R$ if:

(i) $f P_w \mu(w)$ and at least one of the following situations occurs:

(ii - a) $w P_f w'$ for some $w' \in \mu(f)$ or

(ii - b) $|\mu(f)| < q_f$ and $w P_f \emptyset$.

A matching $\mu$ is *stable* under a preference profile $R$ if it is both individually rational and not
blocked by any firm-worker pair.

We denote the set of all stable matchings under $R$ by $\mathcal{S}(R)$. The stable several-to-one matching correspondence is a rule $\chi : \mathbb{R} \to 2^M$ such that $\chi(R) = \mathcal{S}(R)$ for all $R \in \mathbb{R}$. An improper sub-solution of the stable many-to-one matching rule is any improper sub-correspondence $\phi : \mathbb{R} \to 2^M$ such that $\phi(R) \subseteq \mathcal{S}(R)$ for all $R \in \mathbb{R}$ and $\phi(R) = \mathcal{S}(R)$ for some $R \in \mathbb{R}$. We summarize the results of Gale and Shapley (1962) and Roth (1984) on the existence of stable matchings in the following corollary.

**Corollary 1** For any profile of linear orderings, a several-to-one matching problem has at least one stable matching.

To define the set of stable matchings that are not blocked by coalitions of bigger size rather than the size of one or two, we proceed as Kara and Sönmez (1997). A matching $\mu \in \mathcal{M}$ is blocked by a coalition $G \subseteq \mathcal{F} \cup \mathcal{W}$ at $R$, if there is $\mu' \in \mathcal{M}$ such that for each $w \in G$ and each $f \in G$,

(1) $\mu'(w) \in G$ and $\mu'(w)P_w\mu(w)$, and

(2) $\mu'(f) \subseteq (G \cup \mu(w))$ and $\mu'(f)P_f\mu(f)$.

A matching $\mu$ is group stable at $R$ if it is not blocked by any $G \subseteq \mathcal{F} \cup \mathcal{W}$ at $R$. We denote the set of all group stable matchings under $R$ by $\mathcal{S}^G(R)$. The group stable several-to-one matching correspondence is a rule $\psi : \mathbb{R} \to 2^M$ such that $\psi = \mathcal{S}^G(R)$ for all $R \in \mathbb{R}$. An improper sub-solution of the group stable many-to-one matching rule is any improper sub-correspondence $\varphi : \mathbb{R} \to 2^M$ such that $\varphi(R) \subseteq \mathcal{S}^G(R)$ for all $R \in \mathbb{R}$ and $\varphi(R) = \mathcal{S}^G(R)$ for some $R \in \mathbb{R}$. Under the responsiveness of the preferences of the firms, Roth (1985b) shows that group stability is equivalent to stability for several-to-one matching markets.

**Theorem 7** (Roth (1985b)) A matching is group stable if and only if it is stable.

To give some applications in matching markets, we need to reformulate some properties of implementation theory, presented in general setup in Section 2. We begin by the following set.
We denote by $L_i(\mu, R) = \{\nu \in \mathcal{M} \mid \mu(i)R_i\nu(i)\}$ the lower contour set for agent $i$ at matching $\mu$ for a profile $R$. An improper sub-solution to the group stable matching correspondence $\varphi : \Re \to 2^{\mathcal{M}}$ satisfies weak Pareto optimality if for any $R \in \Re$, $\varphi(R) \subseteq P_w(R)$, where $P_w(R) = \{\mu \in \mathcal{M} \mid \nexists \nu \in \mathcal{M} \text{ such that } \forall i \in (\mathcal{F} \cup \mathcal{W}), \nu(i)P_i\mu(i)\}$. An improper sub-solution to the group stable matching correspondence $\varphi : \Re \to 2^{\mathcal{M}}$ satisfies $Y$-monotonicity, if for any $R, R' \in \Re$, any $\mu \in \varphi(R)$ and any $\nu \in \bigcup_{i \in (W \cup \mathcal{F})} L_i(\mu, R)$, $L_i(\mu, R) \subseteq L_i(\nu, R')$ for all $i \in (W \cup \mathcal{F})$, then $\nu \in \varphi(R')$.

## 5.2 Main Result for Matchings

As discussed in Example 3 in Section 3, several-to-one matching markets can operate under the condition that there are agents, such as firms, workers, or both, who have separate exclusive matchings. Based on this requirement, this subsection aims to examine the robustness of $Y$-monotonicity in order to investigate the strong implementability of improper sub-solutions to group stable matching correspondences (Theorem 7). To achieve this, we present the following lemma.

**Lemma 5** Any improper sub-solution to the group stable matching correspondence $\varphi : \Re \to 2^{\mathcal{M}}$ satisfies weak Pareto optimality.

*Proof.* Let $\varphi$ be a sub-solution of the group stable matching correspondences, i.e., $\varphi(R) \subseteq S^G(R)$ for all $R \in \Re$. Assume that $\varphi$ does not satisfy the property of weak Pareto optimality. That is, there exists a preference profile $R \in \Re$ and a matching $\mu \in \varphi(R)$, but $\mu \notin P_w(R)$. This means that there exists $\nu \in \mathcal{M}$ such that $\nu(i)P_i\mu(i)$ for all $i \in (\mathcal{F} \cup \mathcal{W})$. Also, it is trivial that $\nu(i) \subseteq (\mathcal{F} \cup \mathcal{W})$ for all $i \in (\mathcal{F} \cup \mathcal{W})$. Thus, $\mu$ is blocked by the coalition $\mathcal{F} \cup \mathcal{W}$ under $R$, and so it is not group stable, which leads to a contradiction. Q.E.D.

Now, using Lemma 5 and Theorem 6, we derive a sufficient condition for the implementation of improper sub-solutions to the group stable matching correspondences in strong equilibrium for matching markets.
**Theorem 8** Let $|W \cup F| \ge 3$ and suppose Assumption 1 holds. If an improper sub-solution of the group stable matching correspondence $\phi : \Re \to 2^M$ satisfies Y-monotonicity, then it is implementable in strong equilibria.

## 5.3 Implementation of the Strong Core

In this subsection, we present the strong core of several-to-one matching markets, which we propose as a solution for assigning groups of workers to firms, and explore its implementability in strong equilibria.

**Definition 6** The strong core, that we denote by $C_{strong}(R)$, is the set of matchings $\mu$ for which there is no $F' \subseteq F$, $W' \subseteq W$ with $F' \cup W' \ne \emptyset$, and $\mu' \in M$ such that : (i) For all $w \in W'$ and $f \in F'$, $\mu'(w) \subseteq F'$ and $\mu'(f) \subseteq W'$. (ii) For all $w \in W'$ and $f \in F'$, $\mu'(w)R_w\mu(w)$ and $\mu'(f)R_f\mu(f)$. (iii) For some $i \in F' \cup W'$, $\mu'(i)P_i\mu(i)$.

Roth (1985b) determines the relationship between the set $S(R)$ of stable matchings and the strong core $C_{strong}(R)$ as follows:

**Theorem 9** (Roth (1985b)) When preferences are responsive, the set of stable matchings coincides with the strong core for each matching problem, i.e., $S(R) = C_{strong}(R)$.<fnref n="11" />

From Theorems 7 and 8 we give the next corollary.

**Corollary 2** When preferences are responsive, the set of group stable matchings coincides with the strong core for each matching problem, i.e., $S(R)^G = C_{strong}(R)$.

**Proposition 4** The strong core satisfies Y-monotonicity.

*Proof.* Assume that the strong core does not satisfy Y-monotonicity. This means that for some $R, R' \in \Re$, $\mu \in C_{strong}(R)$ and $\nu \in M$, $L_i(\mu, R) \subseteq L_i(\nu, R')$ for all $i \in (W \cup F)$ and $\nu \in L_j(\mu, R) \subseteq L_j(\nu, R')$ for some $j \in (W \cup F)^*$, but $\nu \notin C_{strong}(R')$. Therefore, there

<footnote n="11">¹¹Roth and Sotomayor (1990) extend this result to the case of “substitutable” preferences of the firms and show that matching is in the core if and only if it is stable. Echenique and Oviedo (2004) use a fixed-point approach to characterize the core. They show that the core is a subset of the stable matching.</footnote>
is $\mathcal{F}' \subseteq \mathcal{F}$, $W' \subseteq W$ with $\mathcal{F}' \cup W' \neq \emptyset$, and $\nu' \in \mathcal{M}$ such that : (i) For all $w \in W'$ and $f \in \mathcal{F}'$, $\nu'(w) \subseteq \mathcal{F}'$ and $\nu'(f) \subseteq W'$. (ii) For all $w \in W'$ and $f \in \mathcal{F}'$, $\nu'(w)R'_{w}\nu(w)$ and $\nu'(f)R'_{f}\nu(f)$. (iii) For some $i \in \mathcal{F}' \cup W'$, $\nu'(i)P'_{i}\nu(i)$. From (*), (ii), (iii), and strictness of preferences we obtain $\nu'(i)R_{i}\mu(i)$ for all $i \in \mathcal{F}' \cup W'$ and $\nu'(i)P_{i}\mu(i)$ for some $i \in \mathcal{F}' \cup W'$. That is, $\mu \notin C_{\text{strong}}(R)$, a contradiction. Q.E.D

From Theorems 7 and 8, and Proposition 4 we obtain the next result.

**Corollary 3** Let $|W \cup \mathcal{F}| \ge 3$ and suppose Assumption 1 hold. The strong core is implementable in strong equilibrium.

In this section, like most of the literature on two-sided matching markets, we have maintained the assumption that firms and workers are never indifferent between two members of the other kind. In practice, however, ties in preferences appear naturally and are widespread. The market design needs to handle ties carefully, because, in the presence of indifferences, stability may not exist and some algorithms cannot be applied to produce stable matchings. In this context, some authors explored this problem. For example, in one-to-one matching markets, Irving (1994) allowed ties in the preference lists and generalized the notion of stability by providing appropriate algorithms to find it. In several-to-one matching markets, Erdil and Ergin (2017) also introduced indifference and designed some algorithms to compute stable matchings. Thus, it would be interesting to investigate whether our results can be generalized in allowing ties in the preference lists of two members on the opposite side. This is a topic we leave for future research.

# 6 Infinite Fair Allocation Problems with Single-Peaked Preferences

## 6.1 Model and Single-Peaked Domain

We are given an amount $\Omega > 0$ of certain infinitely divisible goods to be allocated among a set $N = \{1, \dots, n\}$ of $n$ agents. We begin by introducing the terminology and notations
required for our results. The preference of each agent $i \in N$ is represented by a continuous relation $R_i$ over $[0, \Omega]$.<fnref n="12" />

For any two $x_i, y_i \in [0, \Omega]$, $x_i R_i y_i$ means that it is as good for agent $i$ to consume a share $x_i$ as it is to consume $y_i$. The asymmetrical part of the relation is denoted by $P_i$, and the symmetrical part is denoted by $\sim_i$.

**Single-Peaked Preferences:** A preference relation $R_i$ is said to be single-peaked if there exists a number $p(R_i) \in [0, \Omega]$ such that for all $x_i, y_i \in [0, \Omega]$, if $y_i < x_i \le p(R_i)$ or $p(R_i) \le x_i < y_i$, then $x_i P_i y_i$. We call $p(R_i)$ the peak of $R_i$. The class of all single-peaked preference relations is denoted by $\mathfrak{R}_{sp_i} \subseteq \mathfrak{R}_i$, and $\mathfrak{R}_{sp} = \mathfrak{R}_{sp_1} \times \cdots \times \mathfrak{R}_{sp_n}$ represents the domain of single-peaked preferences.

For $R \in \mathfrak{R}_{sp}$, a feasible allocation for the economy $(R, \Omega)$ is a vector $x = (x_i)_{i \in N} \in \mathbb{R}_+^n$ such that $\sum_{i \in N} x_i = \Omega$. Let $X$ be the set of feasible allocations. Note that the set of feasible allocations $X \subseteq [0, \Omega]^n$. For all $R_i \in \mathfrak{R}_{sp}$ and all $x_i \in [0, \Omega]$, the lower contour set for agent $i$ at consumption $x_i$ is denoted by $L_i(x_i, R) = \{y_i \in [0, \Omega] \mid x_i R_i y_i\}$. Moreover, for any two feasible allocations $x = (x_i)_{i \in N}$ and $y = (y_i)_{i \in N}$ in the set of feasible allocations $X$, the expression $x R_i y$ implies $x_i R_i y_i$ for all $i \in N$.

Thus, the agent's preferences are defined over individual consumption spaces, not over the allocation space. The properties of implementation theory, presented in the general setup in Section 2, become as follows. An SCC $F$ is a multi-valued mapping from $\mathfrak{R}_{sp}$ to $X$. An SCC $F$ satisfies weak Pareto optimality if, for any $R \in \mathfrak{R}_{sp}$, $F(R) \subseteq P_w(R)$, where $P_w(R) = \{x \in X \mid \nexists y \in X \text{ such that } \forall i \in N, y_i P_i x_i\}$. An SCC $F$ satisfies Y-monotonicity if for all $R, R' \in \mathfrak{R}_{sp}$, for any $x \in F(R)$ and $y \in X$, we have that if $L_i(x_i, R) \subseteq L_i(y_i, R')$ for all $i \in N$, and $y_j \in L_j(x_j, R)$ for some $j \in N$, then $y \in F(R')$. Note that the free disposability of the good is not assumed.

Before proceeding to the next results through examples, we recall, as mentioned in Example 2 in Section 3, that models of infinite fair allocations can be structured to guarantee the existence of agents with separately exclusive alternatives.

<footnote n="12">¹²A continuous preference relation means that if $[a, b[ \cup [b, c] \subseteq L(x_i, R'_i)$ for some $a, b, c, x_i, R'_i$, then $[a, c] \subseteq L(x_i, R'_i)$.</footnote>
## 6.2 Examples of Non-Strongly/Strongly Implementable Solutions

In this subsection, we examine the strong implementability of a family of well-known solutions to the problem of infinite fair allocation with single-peaked preferences. We consider the individually rational correspondence from equal division, the no-envy correspondence, and their intersection, as well as Pareto selection correspondences, the core correspondence of equal division, and the group no-envy correspondence.

### 6.2.1 Individually Rational Correspondence from Equal Division

The individually rational correspondence from equal division ($I_{ed}$) selects the feasible allocations for which each agent prefers their share to the average. Formally, it is defined as follows:

$$I_{ed}(R) = \{x \in X \mid \forall i \in N, x_i R_i(\Omega/n)\}.$$

**Proposition 5** In infinite fair allocation problems with single-peaked preferences, the individually rational correspondence from equal division does not satisfy the property of weak Pareto optimality.

*Proof:* Let $N = \{1, 2, 3\}$, $R \in \Re_{sp}$, and $x = (x_1 = 2, x_2 = 3, x_3 = 7)$, $y = (y_1 = 1.5, y_2 = 2.5, y_3 = 8) \in X$. As depicted in Figure 2, we have $x_i R_i(\Omega/3)$ for all $i \in N$, so $x \in I_{ed}(R)$.

However, for all $i \in N$, $y_i P_i x_i$, and so $I_{ed}$ violates the property of weak Pareto optimality. Q.E.D

### 6.2.2 No-Envy Correspondence

The no-envy correspondence (NE) selects the feasible allocations for which each agent prefers their share over the shares of other agents. Formally, it is defined as follows:

$$NE(R) = \{x \in X \mid \forall \{i, j\} \subseteq N, x_i R_i x_j\}.$$

**Proposition 6** In infinite fair allocation problems with single-peaked preferences, the no-envy correspondence does not satisfy the property of weak Pareto optimality.
<figure><img src="image_6.png" /><figcaption>Figure 1: The individually rational correspondence from equal division does not satisfy the property of Pareto optimality.</figcaption></figure>

*Proof:* Let $N = \{1, 2, 3\}$, $R \in \mathbb{R}_{sp}$, and $x = (x_1 = 2, x_2 = 4, x_3 = 6)$, $y = (y_1 = 10, y_2 = 5, y_3 = 7) \in X$. As depicted in Figure 3, we have $x_i R_i x_j$ for all $\{i, j\} \subseteq N$, so $x \in NE(R)$.

<figure><img src="image_7.png" /><figcaption>Figure 2: The no-envy correspondence does not satisfy the property of weak Pareto optimality.</figcaption></figure>

However, for all $i \in N$, $y_i P_i x_i$, and so NE violates the property of weak Pareto optimality.

Q.E.D
According to Thomson (2010), as pointed out in Remark 2 below, the intersection of the individually rational correspondence from equal division and the no-envy correspondence is nonempty. In the next observation, we prove that these correspondences satisfy citizen sovereignty.

**Observation 3** The individually rational allocation from equal division, the no-envy allocation, and their intersection all satisfy citizen sovereignty.

*Proof:* Assume that the individually rational allocation from equal division does not satisfy citizen sovereignty. This would imply that for an allocation $x \in X$, $x \notin I_{ed}(R)$ for all $R \in \Re_{sp}$. Due to single-peakedness, there exists a preference profile $R' \in \Re_{sp}$ such that $x_i R'_i(\Omega/n)$ for all $i \in N$ (for example, choose $x_i$ to be the peak of $R'_i$). Consequently, $x$ is an element of $I_{ed}(R')$, leading to a contradiction. The same reasoning applies to the no-envy allocation and their intersection. Q.E.D

Through Observation 3, and Propositions 1, 3, 5, and 6, we obtain the following corollary.

**Corollary 4** In infinite fair allocation problems with single-peaked preferences, neither the individually rational correspondence from equal division, nor the no-envy correspondence, nor their intersection are implementable in strong equilibria.

Now, we focus on the following solutions: Pareto selection correspondences, the core correspondence of equal division, and the group no-envy correspondence.

### 6.2.3 Pareto Selection Correspondences

The Pareto correspondence ($P$) associates each economy with its feasible allocation set such that no other feasible allocation exists where all agents prefer it weakly, and at least one agent prefers it strictly. Formally, it is defined as follows:

$$P(R) = \{x \in X : \nexists y \in X \text{ such that for all } i \in N, y_i R_i x_i, \text{ and for some } j \in N, y_j P_j x_j\}.$$
Our set of Pareto selection correspondences is constructed from the intersections of the Pareto solution (P), the no-envy solution (NE), and the individually rational correspondence from equal division ($I_{ed}$). We refer to these by juxtaposing the names of the components. Specifically, we present our set of Pareto selection correspondences as follows: the Pareto correspondence (P), the Pareto and no-envy correspondence (PNE), the Pareto and individually rational correspondence from equal division ($PI_{ed}$), and the Pareto, no-envy, and individually rational correspondence from equal division ($PNEI_{ed}$). Following Thomson (2010), all these intersections are well-defined, as they contain the uniform rule. Thus, we have the following remark.

**Remark 2** In infinite fair allocation problems with single-peaked preferences, the solutions PNE, $PI_{ed}$, and $PNEI_{ed}$, derived from the intersections of the Pareto correspondence with the individually rational correspondence from equal division and the no-envy correspondence, are always non-empty.

Although the correspondences $I_{ed}$, NE, and $NEI_{ed}$ are not strongly implementable (as noted in Corollary 4), since they violate weak Pareto optimality, the solutions derived from the intersections of these correspondences with the Pareto correspondence have the potential to be implemented in strong equilibria. This follows from the fact that the Pareto correspondence P and its intersections PNE, $PI_{ed}$, and $PNEI_{ed}$ satisfy weak Pareto optimality. Therefore, in order to study their possible implementability, we need only check if they satisfy Y-monotonicity. To simplify this process, we will use Remark 2 and Lemma 4. In Proposition 7, we examine whether these solutions satisfy Y-monotonicity separately. Specifically, we will check if the correspondences P, NE, and $I_{ed}$ satisfy Y-monotonicity. Then, in Proposition 8, we will analyze whether the correspondences resulting from the intersections with the Pareto correspondence P - namely, PNE, $PI_{ed}$, and $PNEI_{ed}$ - satisfy Y-monotonicity.

**Proposition 7** In infinite fair allocation problems with single-peaked preferences, the correspondences P, NE, and $I_{ed}$ all satisfy Y-monotonicity.
**Proof:** Let $R, R' \in \mathbb{R}_{sp}, x \in P(R), y \in X$, and suppose $L_i(x_i, R) \subseteq L_i(y_i, R')$ for all $i \in N$, and $y_j \in L_j(x_j, R)$ for some $j \in N$ (1). We need to show that $y \in P(R')$. Since $x \in P(R)$, it follows that for all $z \in X$, there exists $i \in N$ such that $x_i P_i z_i$ (2), or for all $j \in N$, $x_j R_j z_j$. From (1), (2), and the single-peakedness of the preferences, we have $y_i P'_i z_i$, and hence $y \in P(R')$. The same reasoning applies to the NE and $I_{ed}$ correspondences. Q.E.D.

The proof of the next proposition follows from Remark 2, Lemma 4, and Proposition 7.

**Proposition 8** *In infinite fair allocation problems with single-peaked preferences, the correspondences PNE, $PI_{ed}$, and $PNEI_{ed}$ satisfy Y-monotonicity.*

Now, we turn to the next set of solutions, which are the core correspondence from equal division and the group no-envy correspondence, and will examine whether they satisfy the two properties: weak Pareto optimality and Y-monotonicity.

### 6.2.4 Core Correspondence from Equal Division, $C_{ed}$

The core correspondence from equal division generalizes the notion of the individually rational correspondence from equal division. It associates each economy with its feasible allocation set in such a way that no group of agents can improve the situation of all its members by redistributing a portion of the social endowment based on their size, without making each of its members at least as well off as they were initially, and at least one of them better off. Formally, it is defined as follows:

$$C_{ed}(R) = \{x \in X : \nexists C \subseteq N \text{ and } (y_i)_{i \in C} \text{ such that } (i) \sum_{i \in C} y_i = |C| \mid \Omega/n, (ii) \text{ for all } i \in C, y_i R_i x_i \text{ and for some } i \in C, y_i P_i x_i \}.$$

**Proposition 9** *In infinite fair allocation problems with single-peaked preferences, the core correspondence from equal division satisfies weak Pareto optimality.*

**Proof:** Let $R \in \mathbb{R}_{sp}, x \in C_{ed}(R)$, but suppose $x \notin P_w(R)$. Then, there exists an allocation $y \in X$ such that $y_i P_i x_i$ for all $i \in N$. Therefore, there exists $C = N$ and $(y_i)_{i \in N}$ such that
(i) $\sum_{i \in N} y_i = |N|\Omega/n$, (ii) for all $i \in N$, $y_i R_i x_i$, and for some $i \in N$, $y_i P_i x_i$. This means that $x \notin C_{ed}(R)$, which is a contradiction. Q.E.D

**Proposition 10** *In infinite fair allocation problems with single-peaked preferences, the core correspondence from equal division and the group no-envy correspondence satisfy $Y$-monotonicity.*

The proof of Proposition 10 follows the same reasoning as the proof of Proposition 7.

### 6.2.5 Group No-Envy Correspondence, GNE

The group no-envy correspondence generalizes the concept of the no-envy correspondence. It associates each economy with its feasible allocation set in a way that prevents any group of agents from redistributing the resources assigned to another group of the same size to make each member at least as well off as before, while also ensuring that at least one member is better off. Formally, it is defined as follows:

$GNE(R) = \{x \in X: \nexists C, C' \subseteq N \text{ with } |C| = |C'| \text{ and } (y_i)_{i \in C} \text{ such that } \sum_C y_i = \sum_{C'} x_i, \text{ for all } i \in C, y_i R_i x_i, \text{ and for some } i \in C, y_i P_i x_i\}$.

**Proposition 11** *In infinite fair allocation problems with single-peaked preferences, the group no-envy correspondence satisfies weak Pareto optimality.*

*Proof:* Let $R \in \Re_{sp}$, $x \in GNE(R)$, but suppose $x \notin P_w(R)$. Then, there exists an allocation $y \in X$ such that $y_i P_i x_i$ for all $i \in N$. Therefore, there exist $C, C' = N$ with $|C| = |C'| = |N| = n$ and $(y_i)_{i \in N}$ such that, by feasibility, $\sum_{i \in N} y_i = \sum_{i \in N} x_i = \Omega$, for all $i \in N$, $y_i R_i x_i$, and for some $i \in N$, $y_i P_i x_i$. This means that $x \notin GNE(R)$, which is a contradiction. Q.E.D.

**Proposition 12** *In infinite fair allocation problems with single-peaked preferences, the group no-envy correspondence satisfies $Y$-monotonicity.*

The proof of Proposition 12 follows the same reasoning as the proof of Proposition 7.

Since Pareto selection correspondences satisfy weak Pareto optimality, we can derive the following corollary from Theorem 6 and Propositions 7, 8, 9, 10, 11, and 12.
**Corollary 5** Let $n \ge 3$ and suppose Assumption 1 holds. In infinite fair allocation problems with single-peaked preferences, the Pareto selection correspondences, the core correspondence from equal division, and the group no-envy correspondence are implementable in strong equilibria.

# 7 Strong Nash Implementation of Weak Efficient Sharing Rules for a Commonly Owned Technology

For the notation of the model, we adopt the framework of Corchón and Puy (2002). A single consumption good is produced using a publicly owned technology that depends on a vector of potentially heterogeneous inputs. The economy consists of $n$ individuals, indexed by $i$, where the set of individuals is defined as $N = \{1, \dots, n\}$. Each individual is endowed with $\bar{l} \in \mathbb{R}^+$, representing units of input. The consumption set for each individual is defined as:

$$X = \{(x_i, l_i) : x_i \in \mathbb{R}^+, l_i \in [0, \bar{l}]\},$$

where $x_i$ represents agent $i$'s consumption, and $l_i$ represents their input contribution. The individual preferences $R_i$ of the agents are assumed to be complete and transitive, then they can be represented by utility function $u_i$, $u_i : X \to \mathbb{R}$. Specifically, for all $(x, l), (x', l') \in X$, $(x, l)R_i(x', l')$ if and only if $u_i(x, l) \ge u_i(x', l')$. Let $u = (u_1, \dots, u_n)$.

The lower contour set for agent $i$ at allocation $(x_i, l_i) \in X$ is given by:

$$L_i((x_i, l_i), u) = \{(x'_i, l'_i) \in X \mid u_i(x_i, l_i) \ge u_i(x'_i, l'_i)\}.$$

The upper contour set for agent $i$ at allocation $(x_i, l_i) \in X$ is given by:

$$U_i((x_i, l_i), u) = \{(x'_i, l'_i) \in X \mid u_i(x_i, l_i) \ge u_i(x'_i, l'_i)\}.$$

The utility function $u$ is assumed to be:

* Continuous if $L_i((x_i, l_i), u)$ and $U_i((x_i, l_i), u)$ are both closed for any $(x_i, l_i) \in X$.
* *Strict concave* on X if and only if $L_i((x_i, l_i), u)$ is a concave set for every $(x_i, l_i) \in X$. That is if, for any two distinct points $(x'_i, l'_i)$ and $(x''_i, l''_i)$, and for any $\lambda \in ]0, 1[$, we have:

$$u_i(\lambda(x'_i, l'_i) + (1-\lambda)(x''_i, l''_i)) > \lambda u_i(x'_i, l'_i) + (1-\lambda)u_i(x''_i, l''_i).$$

The production process is defined by a publicly owned production function $f: \mathbb{R}^n \to \mathbb{R}^+$. The function $f$ is continuous, increasing in each component, concave, and satisfies $f(0, \dots, 0) = 0$. The feasible set is given by:

$$\bar{X} = \{(x_1, l_1, \dots, x_n, l_n) \in X^n : \sum_{i=1}^{n} x_i \le f(l_1, \dots, l_n)\}.$$

A feasible allocation is $(x, l) \in \bar{X}$, where $x = (x_1, \dots, x_n)$ and $l = (l_1, \dots, l_n)$. We assume that $\bar{X}$ is fixed, while the utility functions vary. Thus, an economy is denoted by $u = (u_1, \dots, u_n)$. The set of admissible economies is denoted by $\mathbb{E}$.

The *Pareto efficient solution*, denoted by $\phi^E: \mathbb{E} \to \bar{X}$, maps each economy within the domain to its corresponding set of Pareto efficient allocations.

$$\phi^E(u) = \{(x, l) \in \bar{X} : \exists (x', l') \in \bar{X} \text{ such that } u_i(x'_i, l'_i) \ge u_i(x_i, l_i) \forall i \in N \text{ and } u_j(x'_j, l'_j) > u_j(x_j, l_j) \text{ for at least one } j \in N\}$$

A sharing rule is an agreement that determines consumption allocations based on input contributions. Formally, a sharing rule $P = (P_1, \dots, P_n)$ is a collection of functions such that $P_i: [0, \bar{l}]^n \to \mathbb{R}^+$ for $i = 1, \dots, n$, with the condition that $\sum_{i=1}^n P_i(l) = f(l)$ for each $l \in [0, \bar{l}]^n$. Each $P_i$ determines the consumption of agent $i$ as a function of $l$. Moreover, $P$ distributes the total output. Some examples of solutions that can be described using sharing rules include the following:

1. *The Proportional Solution*

The sharing rule is:

$$P_i(l) = \frac{l_i}{\sum_{j \in N} l_j} \cdot f(l), \quad \forall i \in N, \qquad (1)$$
Under this rule, each agent receives a share of the total output that is proportional to the amount of input they contributed. This rule ensures efficiency (all output is distributed) and input-based fairness (more input means more reward).

## 2. The Equal Benefit Solution

The sharing rule is defined as:

$$P_i(l) = \frac{\partial f(l)}{\partial l_i} l_i + \frac{1}{n} \left[ f(l) - \sum_{j \in N} \left( \frac{\partial f(l)}{\partial l_j} l_j \right) \right], \quad \forall i \in N \quad (2)$$

This rule aims to allocate the total output in a way that balances efficiency and fairness. It is derived from a Walrasian (competitive) equilibrium in which all agents receive equal profits, regardless of differences in their input contributions.

## 3. The Equal Sharing Solution

The sharing rule is:

$$P_i(l) = \frac{f(l)}{n}, \quad \forall i \in N, \quad (3)$$

This rule represents the simplest and most egalitarian form of allocation. It divides the total output equally among all individuals, regardless of how much each person contributed to the production process.

## 4. The Aumann-Shapley Prices (1974)

For a vector of heterogeneous input contributions, the sharing rule is defined as:

$$P_i(l) = \int_0^1 \frac{\partial f(tl)}{\partial l_i} dt, \quad \forall i \in N, \quad (4)$$

This rule allocates output to each agent based on their *average marginal contribution* to the production process, as their input is gradually increased from zero to its final value. When inputs are *homogeneous*, this rule coincides with the proportional solution.
In addition, Moulin (1987) proposed a family of sharing rules that also fit within this framework. Examples include:

5. Threshold-Based Rule

Let $m \in \mathbb{R}^+$ be a fixed threshold. The sharing rule is defined as:

$$ P_i(l) = \begin{cases} \frac{l_i}{\sum_{j \in N} l_j} \cdot f(l), & \text{if } \frac{f(l)}{\sum_{j \in N} l_j} \le m, \\ l_i m + \frac{1}{n} \left( f(l) - m \sum_{j \in N} l_j \right), & \text{otherwise.} \end{cases} \quad \forall i \in N. \quad (5) $$

This rule strikes a balance between efficiency and fairness in the allocation of output from a commonly owned technology. The key idea is to adjust the distribution of output depending on the average productivity of inputs.

* Low Productivity: If the average output per unit of input is less than or equal to the threshold $m$, the rule applies a proportional allocation. In this case, each agent receives a share of the output proportional to their contribution. This approach promotes efficiency and aligns rewards with effort.

* High Productivity: If the average productivity exceeds the threshold, the rule caps the returns from individual inputs at the rate $m$ and redistributes the remaining output equally among all agents. Thus, each agent receives a fixed return based on their input, plus an equal share of the surplus. This ensures fairness by limiting excessive individual rewards in high-output scenarios.

6. Parameterized Redistribution Rule

Let $\lambda \in [0, 1]$. The sharing rule is given by:

$$ P_i(l) = \frac{f(l)}{n} + \left( l_i - \frac{1}{n} \sum_{j \in N} l_j \right) \left( \left( 1 + \frac{f(l)}{\sum_{j \in N} l_j} \right)^{\lambda} - 1 \right), \quad \forall i \in N. \quad (6) $$

This rule introduces the parameter $\lambda$ to control the degree of redistribution between
equal and input-based sharing. It blends egalitarian and contribution-based principles in a flexible manner:

* The first term, $\frac{f(l)}{n}$, ensures that each agent starts with an equal share of the total output.

* The second term redistributes additional output based on how much each individual's input deviates from the average input $\frac{1}{n} \sum_{j \in N} l_j$.

* The factor $(1 + \frac{f(l)}{\sum_{j \in N} l_j})^{\lambda} - 1$ adjusts the strength of this redistribution according to the parameter $\lambda$.

When $\lambda = 0$, the rule coincides with equal sharing. As $\lambda$ increases, the influence of individual input contributions becomes stronger. In the limit as $\lambda \to 1$, the rule approximates a more input-sensitive distribution. Thus, the parameter $\lambda$ allows the designer to interpolate smoothly between equity and efficiency.

## 7. Power Mean Form Rule

For each $\lambda \in [0, 1]$, the rule is:

$$P_i(l) = \frac{l_i^{\lambda}}{\sum_{j \in N} l_j^{\lambda}} \cdot f(l), \quad \forall i \in N. \quad (7)$$

This rule defines a continuum of sharing mechanisms based on a parameter $\lambda$ that modulates the sensitivity of the allocation to differences in input levels.

* When $\lambda = 0$, all inputs are treated equally regardless of their size, and the rule yields the equal sharing solution, where each agent receives $\frac{f(l)}{n}$.

* When $\lambda = 1$, the rule becomes the proportional sharing solution, where output is allocated in direct proportion to individual contributions.

* For values $0 < \lambda < 1$, the rule interpolates between these two extremes, giving relatively more weight to higher contributors while still maintaining some egalitarian smoothing.
Thus, this rule offers a flexible trade-off between fairness and incentive compatibility by tuning the parameter $\lambda$. The higher the value of $\lambda$, the more weight is placed on individual input contributions.

## 8. Implicit Nonlinear Form Rule

For each $\lambda \in [0, 1)$, the sharing rule is defined as:

$$P_i(l) = (l_i^{1-\lambda} + a)^{\frac{1}{1-\lambda}} - l_i, \quad \forall i \in N, \qquad (8)$$

where $a$ is the unique solution to the equation:

$$\sum_{j \in N} (l_j^{1-\lambda} + a)^{\frac{1}{1-\lambda}} = \sum_{j \in N} l_j + f(l).$$

This rule distributes output by applying a nonlinear transformation to each agent's input level before assigning rewards. The parameter $\lambda$ controls the degree of inequality aversion in the allocation process.

* The expression $(l_i^{1-\lambda} + a)^{\frac{1}{1-\lambda}}$ represents a nonlinear transformation of each agent's input, incorporating a common adjustment term $a$, which is chosen so that the total allocation equals the total output plus total input.

* Each agent's share is the difference between this transformed value and their original input, i.e., they receive the “net gain” from this transformation.

* The parameter $\lambda$ shapes the allocation:

- When $\lambda$ is close to 0, the transformation is nearly linear, and the rule tends to treat agents more equally.

- As $\lambda$ increases toward 1, the transformation becomes more sensitive to differences in input, giving more weight to higher contributors, thus favoring efficiency over equality.

* The rule balances fairness and incentive: it rewards contributions, but also includes a redistributive component through the shared parameter $a$.
We note that any convex combination of the above sharing rules is also a valid sharing rule.

Now, let us introduce a more general efficient sharing rule, denoted by $\phi_G^{PE}$, which represents the intersection of the Pareto efficient allocations and those that satisfy the sharing rule $P$. Formally, we define it as:

$$ \phi_G^{PE}(u, P) = \left\{ (x_1, l_1, \dots, x_n, l_n) \in \phi^E(u) : \sum_{i=1}^{n} (x_i - P_i(l)) \le 0 \right\}. $$

**Lemma 6** Assume that $u \in \mathbb{E}$ and that $P$ is continuous. Then, $\phi_G^{PE}(u, P)$ is non-empty.

*Proof.* From Theorem 1 of Corchón and Puy (2002), the efficient sharing rule where each $x_i$ equals $P_i(l)$ is non-empty. Therefore, the rule $\phi_G^{PE}(u, P)$ is also non-empty.

In the following definition, we adapt the property of Y-monotonicity to the model of efficient sharing rules for a commonly owned technology presented in this section.

**Definition 7** (Y-monotonicity)

A general efficient sharing rule $\phi_G^{PE}(u, P)$ satisfies Y-monotonicity if, for any two economies $u, u' \in \mathbb{E}$, and for any feasible allocation $(x, l) \in \phi_G^{PE}(u, P)$, whenever $(\hat{x}_i, \hat{l}_i) \in \bigcup_{i \in N} L_i((x_i, l_i), u)$, and for all $i \in N$, we have $L_i((x_i, l_i), u) \subseteq L_i((\hat{x}_i, \hat{l}_i), u')$, then $(\hat{x}, \hat{l}) \in \phi_G^{PE}(u', P)$.

**Proposition 13** A general efficient sharing rule satisfies Y-monotonicity.

*Proof.* Assume, for the sake of contradiction, that $\phi_G^{PE}(u, P)$ does not satisfy Y-monotonicity. That is, for some $u, u' \in \mathbb{E}$, and for a feasible allocation $(x, l) \in \phi_G^{PE}(u, P)$, there exist allocations $(\hat{x}, \hat{l})$ such that:

* $(\hat{x}_i, \hat{l}_i) \in \bigcup_{i \in N} L_i((x_i, l_i), u)$, meaning $(\hat{x}_i, \hat{l}_i)$ is weakly dominated by $(x_i, l_i)$ for some $i \in N$,

* $L_i((x_i, l_i), u) \subseteq L_i((\hat{x}_i, \hat{l}_i), u')$ for all $i \in N$ (Condition 1),
• $(\hat{x}, \hat{l}) \notin \phi_G^{PE}(u', P)$.

The condition $(\hat{x}, \hat{l}) \notin \phi_G^{PE}(u', P)$ implies that one of the following must be true:

(1) $\sum_{i=1}^{n}(\hat{x}_i - P_i(\hat{l})) > 0$, or

(2) $(\hat{x}, \hat{l}) \notin \phi^E(u')$.

**Case 1:** $\sum_{i=1}^{n}(\hat{x}_i - P_i(\hat{l})) > 0$

If $\sum_{i=1}^{n}(\hat{x}_i - P_i(\hat{l})) > 0$, then $\sum_{i=1}^{n}\hat{x}_i > f(\hat{l})$. This contradicts the feasibility condition, which requires that $\sum_{i=1}^{n}\hat{x}_i \le f(\hat{l})$. Therefore, this case cannot occur.

**Case 2:** $(\hat{x}, \hat{l}) \notin \phi^E(u')$

If $(\hat{x}, \hat{l}) \notin \phi^E(u')$, then by the definition of Pareto efficiency, there exists a feasible allocation $(x', l') \neq (\hat{x}, \hat{l}) \in \bar{X}$ such that:

$$u'_{k}(x'_{k}, l'_{k}) \geq u'_{k}(\hat{x}_{k}, \hat{l}_{k}) \quad \text{for all } k \in N,$$

and

$$u'_{j}(x'_{j}, l'_{j}) > u'_{j}(\hat{x}_{j}, \hat{l}_{j}) \quad \text{for at least one } j \in N.$$

a) If the strict inequality in utility holds for all agents, that is, if $u'_{k}(x'_{k}, l'_{k}) > u'_{k}(\hat{x}_{k}, \hat{l}_{k})$ for all $k \in N$, then from condition (1), we know that $L_k((x_k, l_k), u) \subseteq L_k((\hat{x}_k, \hat{l}_k), u')$ for all $k \in N$. This implies:

$$u_k(x'_k, l'_k) > u_k(x_k, l_k) \quad \text{for all } k \in N.$$

This leads to a contradiction.

b) If the inequality in utility is strict or equal for some agents, we consider two subsets of agents, I and J, with $I \cup J = N$, such that:

• b.1) J is the subset of agents for which:

$$u'_{j}(x'_{j}, l'_{j}) > u'_{j}(\hat{x}_{j}, \hat{l}_{j}) \quad \text{for all } j \in J \quad (\text{Condition 2}),$$
• b.2) I is the subset of agents for which $u'_i(x'_i, l'_i) = u'_i(\hat{x}_i, \hat{l}_i)$ for all $i \in I$.

For case b.2), since the utility function $u'_i$ is strictly concave, we have that for any convex combination $\lambda \in (0, 1)$ of the two bundles $(x'_i, l'_i)$ and $(\hat{x}_i, \hat{l}_i)$, the utility of the convex combination is strictly greater than the convex combination of their utilities:

$$u'_{i}(\lambda(x'_{i}, l'_{i}) + (1 - \lambda)(\hat{x}_{i}, \hat{l}_{i})) > \lambda u'_{i}(x'_{i}, l'_{i}) + (1 - \lambda)u'_{i}(\hat{x}_{i}, \hat{l}_{i}) \quad \text{for all } i \in I \quad (\text{Condition 3}).$$

Since $u'_i(x'_i, l'_i) = u'_i(\hat{x}_i, \hat{l}_i)$, this gives:

$$u'_{i}(\lambda(x'_{i}, l'_{i}) + (1 - \lambda)(\hat{x}_{i}, \hat{l}_{i})) > u'_{i}(\hat{x}_{i}, \hat{l}_{i}) \quad \text{for all } i \in I \quad (\text{Condition 4}).$$

Now, consider the convex combination of the two bundles $(x'_i, l'_i)$ and $(\hat{x}_i, \hat{l}_i)$, which gives the new bundle:

$$(\lambda x'_{i} + (1 - \lambda)\hat{x}_{i}, \lambda l'_{i} + (1 - \lambda)\hat{l}_{i}).$$

The utility of this convex combination is:

$$u_i(\lambda(x'_i, l'_i) + (1 - \lambda)(\hat{x}_i, \hat{l}_i)) = u_i(\lambda x'_i + (1 - \lambda)\hat{x}_i, \lambda l'_i + (1 - \lambda)\hat{l}_i).$$

Thus, inequality (4) becomes:

$$u'_{i}(\lambda x'_{i} + (1 - \lambda)\hat{x}_{i}, \lambda l'_{i} + (1 - \lambda)\hat{l}_{i}) > u'_{i}(\hat{x}_{i}, \hat{l}_{i}) \quad \text{for all } i \in I \quad (\text{Condition 5}).$$

We know that $L_k((x_k, l_k), u) \subseteq L_k((\hat{x}_k, \hat{l}_k), u')$ for all $k \in N$. From conditions (1), (2), and (5), we obtain:

$$u_k(\lambda x'_k + (1 - \lambda)\hat{x}_k, \lambda l'_k + (1 - \lambda)\hat{l}_k) > u_k(x_k, l_k) \quad \text{for all } k \in N.$$

As the allocation $\lambda(x', l') + (1 - \lambda)(\hat{x}, \hat{l}) \in \bar{X}$ ($\bar{X}$ is a convex subset by the concavity of
$f$), we obtain that $(x, l) \notin \phi_G^{PE}(u, P)$, leading to a contradiction.

Since both Case 1 and Case 2 lead to contradictions, the assumption that $\phi_G^{PE}(u, P)$ does not satisfy Y-monotonicity must be false. Therefore, $\phi_G^{PE}(u, P)$ satisfies Y-monotonicity. Q.E.D.

In the following, we adapt the definition of unanimity in the context under consideration.

**Definition 8 (Unanimity)**

An efficient sharing rule $\phi_G^{PE}(u, P)$ satisfies unanimity if, for any economy $u \in \mathbb{E}$ and any feasible allocation $(x, l) \in \bar{X}$, if for all $i \in N$, $L_i((x_i, l_i), u) = X$, then $(x, l) \in \phi_G^{PE}(u, P)$.

**Proposition 14** *A general efficient sharing rule satisfies unanimity.*

*Proof.* Assume, for the sake of contradiction, that the general efficient sharing rule $\phi_G^{PE}(u, P)$ does not satisfy unanimity. That is, for some economy $u \in \mathbb{E}$ and a feasible allocation $(x, l) \in \bar{X}$, suppose that for all $i \in N$, $L_i((x_i, l_i), u) = X$, but yet $(x, l) \notin \phi_G^{PE}(u, P)$.

By the definition of $\phi_G^{PE}(u, P)$, the condition $(x, l) \notin \phi_G^{PE}(u, P)$ implies that either:

$$ (1) \sum_{i=1}^{n} (x_i - P_i(l)) > 0, \text{ or} $$

$$ (2) \quad (x, l) \notin \phi^E(u). $$

**Case 1:** $\sum_{i=1}^{n} (x_i - P_i(l)) > 0$

If $\sum_{i=1}^{n} (x_i - P_i(l)) > 0$, then $\sum_{i=1}^{n} x_i > f(l)$, which contradicts the feasibility condition since the total amount of goods cannot exceed the total feasible supply. Therefore, this case cannot occur.

**Case 2:** $(x, l) \notin \phi^E(u)$

If $(x, l) \notin \phi^E(u)$, then by the definition of Pareto efficiency, there exists a feasible allocation $(x', l') \neq (x, l)$ such that:
$$u_k(x'_k, l'_k) \geq u_k(x_k, l_k) \quad \text{for all } k \in N,$$

and

$$u_j(x'_j, l'_j) > u_j(x_j, l_j) \quad \text{for at least one } j \in N. (*)$$

Since $L_i((x_i, l_i), u) = X$ for all $i \in N$, it follows that $u_j(x_i, l_i) \geq u_j(x'_i, l'_i)$ for all $i \in N$, which contradicts (*). Therefore, both cases lead to contradictions, and the assumption that $\phi_G^{PE}(u, P)$ does not satisfy unanimity must be false. Hence, $\phi_G^{PE}(u, P)$ satisfies unanimity. Q.E.D.

Now, let us assume Assumption 1: There are two agents with separately exclusive alternatives. In this model, we consider two allocations. In the first allocation, agent $i$ achieves a high level of consumption with minimal effort, while agent $j$ faces the opposite scenario. In the second allocation, the roles are reversed, with agent $j$ benefiting from high consumption and minimal effort, while agent $i$ faces the opposite situation.

We have proven that a general efficient sharing rule satisfies both Y-monotonicity and unanimity. These properties, along with Assumption 1, are crucial for strong implementation as stated in Theorem 6. Therefore, we can conclude that the implementability of efficient sharing rules is as follows:

**Corollary 6** Let $n \geq 3$, and assume that Assumption 1 holds. The general weak efficient sharing rule $\phi_G^{WPE}$ is implementable in strong equilibria.

# 8 Implementation of the Weak Core in Coalitional Games

A coalitional game consists of a finite set $N = \{1, \dots, n\}$ of agents with $n \geq 3$, a nonempty set $A$ of alternatives, a preference profile $R \in \mathbb{R}$, and a characteristic function $\nu: 2^N \setminus \{\emptyset\} \to 2^A$. A coalition, denoted $T$, is simply a set of agents, i.e., a subset of $N$ such that $T \subset N$. An
outcome $a \in A$ is strongly blocked by a coalition $T$ if there exists an outcome $b \in \nu(T)$ such that $bP_ia$ for all $i \in T$. The nonempty weak core of a coalitional game is the set of all outcomes that are not strongly blocked by any coalition $T$. Formally, the nonempty weak core of a coalitional game environment is represented by:

$$C_w(R) = \{x \in \nu(N) : \nexists T \subseteq N \text{ and } \nexists y \in \nu(T) \text{ such that } yP_ix, \forall i \in T\}.$$

In this section, we examine the implementation of the weak core within the context of coalitional games, a cooperative solution concept suited for environments where agents can form coalitions. Unlike individualistic solution concepts, such as Nash equilibrium or subgame perfect equilibrium, the weak core captures the cooperative nature of coalitional games, making it more suitable for our purposes.

The admissible set of preferences, $\Re = \Re_1 \times \dots \times \Re_n$, is such that $C_w(R) \neq \emptyset$ for each $R \in \Re$. This ensures that the weak core is nonempty for any preference profile, guaranteeing the existence of stable outcomes under this framework.

We assume that Assumption 1 is satisfied. In the context of a voting game (for example), we assume there are two candidates, $a$ (far-right) and $b$ (far-left). There are two voters, 1 and 2, where voter 1 strictly prefers $a$ over $b$, and voter 2 strictly prefers $b$ over $a$. Assumption 1 is trivially verified.

It is shown by Bouchta et al. (2025) that the weak core satisfies Y-monotonicity and unanimity. Thus, the requirements of Theorem 6 are satisfied, and the following result holds:

**Corollary 7** Let $n \ge 3$, and assume that Assumption 1 holds. The weak core is implementable in strong equilibria.

By Corollary 7, we extend the findings of Shinotsuka and Takamiya (2003) on the implementability of the weak core, moving from its implementability in Nash equilibrium in simple games to its implementability in strong equilibrium in coalitional games. We also extend the result of Doghmi and Ziad (2008b) on the implementation of the weak core in coalitional games, from Nash equilibrium to strong equilibrium. Additionally, we improve
the result of Bouchta et al. (2025) by generalizing it from simple games to general coalitional
games, which may or may not be monotone, proper, or satisfy inverse consistency.<fnref n="13" />

9 Conclusion

In this paper, we have revisited Maskin’s (1979) results on the implementability of SCCs in
strong equilibria. His first result, in contrast to his findings on Nash implementation, was
negative. It states that if there are at least three agents and an SCC satisfies the no-veto
power condition, it cannot be implemented in strong equilibria. The second result, which is
positive, provides a sufficient condition based on the two properties of weak Pareto optimality
and individual rationality, but it applies only to a specific family of SCCs.

In contrast to most of the literature on strong implementation following Maskin's (1979) work, which relied on Holocast mechanisms considered overly restrictive, we have proposed a new approach by considering agents with specific alternatives, which we have called *separately exclusive alternatives*—a concept that appears in several applications. We have provided a new sufficient condition for strong implementation when there are at least three agents. This condition is based on a strong variant of Maskin monotonicity, introduced by Yi (2012), which we refer to as *Y-monotonicity*, in combination with weak Pareto optimality.

We have illustrated our results through various applications. First, in several-to-
one matching markets, we have demonstrated that Y-monotonicity alone ensures strong
implementation, generalizing the results of Shin and Suh (1996) and Korpela (2013) from
one-to-one to several-to-one matching rules, and extending Doghmi and Ziad's (2015) work
from Nash implementation to strong implementation. However, Maskin monotonicity, which
fully characterizes Nash implementation for the improper sub-solution to the stable matching
rules, does not apply to strong implementation.

Second, in private goods economies with single-peaked preferences, we examined the
implementability of several solutions to fair allocation problems, extending the work of

<footnote n="13">¹³Monotonicity of the coalitions ensures that adding players to a winning coalition does not make it lose. A proper simple game means that if a coalition is winning, its complement is a losing coalition. Inverse consistency (or a strong simple game) stipulates that if a coalition is losing, its complement is a winning coalition.</footnote>
Thomson (2010) and Doghmi and Ziad (2008b) from Nash implementation to strong
implementation. We showed that certain Pareto selection correspondences, the core
correspondence from equal division, and the group no-envy correspondence are implementable
in strong equilibria, while the individually rational correspondence from equal division, the
no-envy correspondence, and their intersection are not.

Third, we have applied our results to efficient sharing rules for commonly owned
technologies. We have demonstrated that the general efficient sharing rule is implementable
in strong equilibria, thereby extending Corchón and Puy's (2002) result from Nash equilibria
to strong equilibria. Importantly, this extension eliminates the need for strict monotonicity<sup>14</sup>
in strictly concave utility functions.

Finally, in coalitional games, we have proved that the weak core is implementable
in strong equilibria, extending the findings of Shinotsuka and Takamiya (2003) on Nash
implementability. We also have extended Doghmi and Ziad's (2008b) result from Nash
equilibrium to strong equilibrium and improved Bouchta et al.'s (2025) result by generalizing
it from simple games to general coalitional games.

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