Distributed Algorithm for Matching between Individuals and Activities

TL;DR

This paper presents a distributed, agent-based algorithm for matching individuals to activities, improving matching quality and speed compared to classical methods.

Contribution

It introduces two novel clearing-house mechanisms for coalition formation that optimize satisfaction and preferences, outperforming traditional search techniques.

Findings

Algorithms produce better matchings than classical methods.

Distributed approach speeds up runtime.

Mechanisms effectively balance satisfaction and preferences.

Abstract

In this paper, we introduce an agent-based model for coalition formation which is suitable for our usecase. We propose here two clearing-houses mechanisms that return sound matchings. The first aims at maximizing the global satisfaction of the individuals. The second ensures that all individuals are assigned as much as possible to a preferred activity. Our experiments show that the outcome of our algorithms are better than those obtained with the classical search/optimization techniques. Moreover, their distribution speeds up their runtime.

Tables

Table 1. Table 1: Welfares and runtimes of the matching algorithms with the real-world data.

Algorithm	Welfare	Runtime (ms)
Selec. with appr.	$U\left(M\right)=0.154$	$263$
Dis. selec. with appr.	$U\left(M\right)=0.156$	$210$
Hill-climbing	$U\left(M\right)=-0.066$	$49116$
Inclusive	$E\left(M\right)=0.0$	$22$
Dis. inclusive	$E\left(M\right)=0.0$	$44$
Hill-climbing	$E\left(M\right)=-0.433$	$20643$

Equations

$$\forall i\in I{v}_i(\theta )=0$$

$$\forall i\in I\forall a_j,a_k\in A\cup \{\theta \}(v_i(a_j)\ge v_i(a_k)\Rightarrow a_j{⊵}_i{a}_k)$$

$$w_i$$

$$\forall i\in I\forall g\in G(i)w_i(g)$$

$$\forall i\in I\forall g,g^{\prime }\in G(i)(w_i(g)\ge w_i(g^{\prime })\Rightarrow g{\succsim }_i{g}^{\prime })$$

$$u_i:G(i)\times A\cup \{\theta \}\to [-1,1]$$

$$\forall i\in I\forall g\in G(i)\forall a\in A\cup \{\theta \},u_i(g,a)=w_i(g)\oplus v_i(a)$$

$$\forall i\in I,a_M(i)\in A\cup \{\theta \}$$

$$\forall i\in I,i\in g_M(i)\subseteq I$$

$$\forall i\in I,a_M(i)=\theta \Rightarrow g_M(i)=\{i\}$$

$$\forall i\in I\forall j\in g_M(i),a_M(j)=a_M(i)$$

$$\forall i\in I\forall j\in I\setminus \{i\},a_M(i)=a_M(j)\ne \theta \Rightarrow g_M(i)=g_M(j)$$

$$p_M:A\cup \{\theta \}$$

$$p_M(a)$$

$$\forall a\in A\cup \{\theta \},⟨a,p_M(a)⟩\text{mbox}isacoalition$$

$$\bigcup _{a\in A\cup \{\theta \}}{p}_M(a)=I$$

$$\forall a_i\in A\cup \{\theta \}\forall a_j\in A\cup \{\theta \}\setminus \{a_i\},$$

$$p_M(a_i)\cap p_M(a_j)=\varnothing$$

$$o_M:A\cup \{\theta \}$$

$$o_M(a)=$$

$$\forall i\in I,u_i(g_M(i),a_M(i))\ge 0$$

$$C_M(i){\succsim }_i{C}_{M^{\prime }}(i)$$

$$⟨a,\{3,4\}⟩{\succ }_3⟨\theta ,\{3\}⟩{\succ }_3⟨b,\{3,4\}⟩$$

$$⟨a,\{3,4\}⟩{\succ }_4⟨\theta ,\{4\}⟩{\succ }_4⟨b,\{3,4\}⟩$$

$$\forall i\in g_C,C{\succ }_i{C}_M(i)$$

$$\forall i\in g_C,C{\succsim }_i{C}_M(i)\wedge \exists i\in g_C,C{\succ }_i{C}_M(i)$$

$$\forall i\in I\forall a\in A\cup \{\theta \}\setminus \{a_M(i)\}$$

$${\mathrm{full}}_M(a)\vee C_M(i){\succsim }_i⟨a,o_M(a)\cup \{i\}⟩$$

$$\forall i\in I\forall a\in A\cup \{a\}\setminus \{a_M(i)\}$$

$${\mathrm{full}}_M(a)$$

$$\vee$$

$$\displaystyle\exists j\in o_{M}(a),~{}C_{M}(j)\succ_{j}\langle a,o_{M}(a)\cup\{i\}\rangle\big{]}$$

$$\forall i\in I\forall a\in A\cup \{a\}\setminus \{a_M(i)\}$$

$${\mathrm{full}}_M(a)$$

$$\vee$$

$$\exists j\in o_M(a),C_M(j){\succ }_j⟨a,o_M(a)\cup \{i\}⟩$$

$$\displaystyle\vee\exists j^{\prime}\in g_{M}(i)\setminus\{i\}~{},C_{M}(i)\succ_{j^{\prime}}\langle a_{M}(i),g_{M}(i)\setminus\{i\}\rangle\big{]}$$

$$\begin{array}{rl}& \forall i\in I\forall a\in A\cup \{\theta \}\forall g\in G(i),C_M(i){\succsim }_i⟨a,g⟩\end{array}$$

$$\exists i\in I,C_{M^{\prime }}(i){\succ }_i{C}_M(i)$$

$$\wedge$$

$$U(M)=\frac{1}{m}\sum _{i\in I}{u}_i(g_M(i),a_M(i))$$

Full text

††journal: Engineering Applications of Artificial Intelligencemytitlenotemytitlenotefootnotetext: This work is part of the PartENS research project supported by the Nord-Pas de Calais region (researcher/citizen research projects).

1 Introduction

Multiagent systems (MAS) is a relevant paradigm for the analysis, the design and the implementation of systems composed of interacting autonomous entities. In order to design socio-technical systems for mediation or simulation, MAS allow to model the feedback loops between heterogeneous actors whose local decision-making leads to complex global phenomena. One of the challenges of the MAS community is to facilitate the elicitation of preferences.

Our work is part of a research project that aims to understand and model feedback dynamics that occur in a group interacting both in a social network and a real social network. The usecase we focus here consists of seniors sharing some activities. This scenario involves thousands of individuals. We aim at maximizing the activities shared in order to improve the social cohesion and avoid the isolation of seniors. We want to propose a social network such that the users form groups to practice together some activities. The purpose of this system is to suggest to each user people with whom to practice these activities.

In this paper, we present an agent-based model for coalition formation which is suitable for our usecase. A set of individuals must be matched together in order to share one of the activities with respect to the preferences of individuals for their peers and for the activities. We define decision rules for the elicitation of the individual and collective preferences with respect to the coalitions in terms of economic efficiency, fairness, stability and social cohesion. Based on the bilateral structure of the problem, we propose here two clearing houses mechanisms which return some matchings where none activity is oversubscribed. The first one, said ”selective”, aims at maximizing the global satisfaction of the individuals. The second one, said ”inclusive”, ensures that all individuals are assigned as much as possible to a preferred activity.

The agent-based modelling allows us to distribute the execution of these algorithms which are described as agent behaviours. Our experiments show that the outcomes reached are better than the outcome reached by mathematical optimization or classical local search techniques. Moreover, the decentralization speeds up the runtime (up to 5 times).

Section 2 motivates our approach and compares it with the related works. We introduce our matching problem in Section 3. We propose two matching algorithms in Section 4. The decentralization of these algorithms consists of agent behaviours described in Section 5. Section 6 and Section 7 exhibit our empirical results. Finally, we conclude with some directions for future work (cf Section 8).

2 Agent-based model of matching

Social choice theory aims at designing and analyzing collective decision-making processes which imply a set of agents selecting or classifying a subset of alternatives among the available ones. Contrary to Economy, Computer science is concerned in this field by the study of algorithms in order to propose operational methods. We focus here on a particular matching problem.

In order to illustrate our usecase, we present here a walk-through toy scenario.

Example 1** (walk-through scenario)**

Four individuals ($1$, $2$, $3$ and $4$) wants to performs passing with one person and six juggling props. Having six clubs and six balls, two activities are available: the clubs ($a$) and the balls ($b$). Each activity can be performed by at most two individuals. According to the interests of the individuals for the activities and their affinity for their peers (cf Fig.1), all the individuals prefer passing clubs rather than balls but not with anyone. If $1$ and $2$ like each other, it is not the case for all of them. For instance, we will see in this paper that $3$ and $4$ like passing each other clubs but they prefer staying alone rather than passing balls each other. Beyond the combinatorics of such a problem (63 matching in our toy example), the search of a harmonious outcome is a complex system.

In the problem of hedonic coalition formation, which has been formalized in [1], each player is endowed with a single preference relation over all the coalitions which contain this player. Our problem is a specialization of the hedonic coalition problem. We can represent our problem as a hedonic game. Therefore, the preferences are not additively separable as in [2], but it requires the generation of rational lists for coalitions (RLC) [3]. Conversely, the succint representation of the individual preferences (the preferences over the activities and over the peers are independent) and the two-sided structure of our problem, i.e. the fact that the activities are focal points [4], allows us to provide tractable clearing-houses mechanisms.

The problem of group activity selection has been proposed in [5]. Each player participates in at most one activity and its preferences over activities depend on the number of participants in the activity. This is a generalization of anonymous hedonic games. This problem has been extended [6] in order to take into account the relationships among the agents, the latter are encoded by a social network, i.e. an undirected graph where nodes correspond to agents and edges represent communication links between them. By contrast, in our problem, each individual is endowed with a preference relation over the activities and a preference relation over its peers.

The Hospital/Resident ($HR$) problem has been introduced in [7]. This problem is a specialization of the coalition formation game where a set of residents must be assigned to the hospitals in accordance with the preferences of the residents over the hospitals and the preferences of hospitals over the residents. The HR problem has many extensions [8]. To the best of our knowledge, no extension is suitable for our usecase.

How does an agent evaluate its preferences? In order to reduce the users’ effort for preference elicitation, we adopt here utility functions (cardinal preferences). Moreover, we assume some additively separable preferences and that the evaluation of activities and individuals are comparable. Even if their expressiveness is limited, the representation of our preferences is linear with respect to the number of individuals/activities.

What is the “best” solution for collective decision-making problem? In the literature, two kinds of rules derive the social choice from the individual preferences: the first ones are based on the desirable properties of the solution (e.g. stability or economic efficiency), while the latter are based on the aggregation of the individual preferences (i.e. social welfare), we consider here these two approaches.

How to reach a matching which maximizes the social welfare? It has been shown in [9] that DCOP (Distributed Constraint Optimization Problem) algorithms are not necessarily scalable for matching problems. We will see in this paper that mathematical optimization and local search techniques are not suitable here since the objective function consists of many local optima. That is the reason why we adopt a multiagent approach, in particular a multi-level model as recommended by [10] where each coalition agent represents a group of individuals.

3 Matching between individuals and activities

We abstract away from our practical application to introduce a formal framework for coalition formation which captures our usecase: the problem, the solutions and their evaluation from an individual/collective viewpoint.

3.1 Preferences over individuals and activities

We introduce here the individuals/activities (IA) problem. In such a problem, the individuals selects their favorite activities with the partners they prefer.

Definition 1** (IA Problem)**

An individuals/activities (IA) problem of size $(m,n)$, such that $m\geq 2$ and $n\geq 1$, is a couple $IA=\langle I,A\rangle$ with $m$ individuals and $n$ activities, where:

$A=\{a_{1},\ldots,a_{n}\}$* is a set of $n$ activities. Each activity $a_{j}$ has a maximal capacity $c_{j}\in\mathbb{N}^{+}$ ;*

• 2.

$I=\{1,\ldots,m\}$* is a set of $m$ individuals. Each individual $i$ is endowed with,*

1. (a)

an interest function over the activities $v_{i}:A\cup\{\theta\}\rightarrow[-1,1]$ 2. (b)

an affinity function over the peers $w_{i}:I\setminus\{i\}\rightarrow[-1,1]$

Intuitively, the void activity (denoted $\theta$), which corresponds to do nothing, has an infinite capacity : all the individuals can be assigned to it.

The interest/affinity functions allow to express the repulsion ($<0$), the attraction ($\geq 0$), and eventually the indifference ([math]) of an individual with respect to an activity/peer. By convention, the interest of an individual with respect to the void activity is neutral:

[TABLE]

We deduce from each interest function $v_{i}$ an individual preference relation over the activities, i.e. a reflexive, complete and transitive preference ordering over $A\cup\{\theta\}$. This preference relation (denoted $\unrhd_{i}$) is defined such that:

[TABLE]

The corresponding strict relation is denoted $\rhd_{i}$.

In order to evaluate the satisfaction of an individual $i$ to belong to a group $g$, we extend his affinity function over all the groups to which he belongs, denoted $G(i)=\{g\subseteq I\mid i\in g\}$. Thus, the satisfaction for a individual to belong to a group only depends on its peers in this group and not on the other groups formed by the remaining individuals. In other words, preferences over groups are purely hedonic preferences. For now, we abstractly define the affinity of an individual $i$ for a group in the following way:

[TABLE]

This affinity, which is normalized ($w_{i}(g)\in[1.1]$), evaluates the preferences of an individual over $2^{m-1}$ groups. The affinity function will be instancied in Section 6.

We deduce from each affinity function $w_{i}$ a preference relation over the groups of peers, i.e. a reflexive, complete and transitive preference ordering over $G(i)$. This preference relation (denoted $\succsim_{i}$) is defined such that:

[TABLE]

The corresponding strict relation is denoted $\succ_{i}$ and the equivalence one $\sim_{i}$.

The satisfaction of an individual $i$ to belong to a group $g$ for practice an activity $a$ depends on: (i) his or her interest in the activity; (ii) his affinity with the group. For now, we abstractly define the utility function in the following manner:

[TABLE]

We assume that the utility is normalized ($u_{i}(g,a)\in[-1,1]$). The utility function will be instancied in Section 6.

In the rest of the paper, we do not consider that the utilities are deduced from ordinal preferences as in [11] but we suppose a direct access to the utilities of agents since it is the case in our practical application (e.g. the preferences depicted in Fig. 1).

3.2 Matching

We can now define the possible solutions for the matching problem previously presented.

We aim at forming coalitions of individuals around the activities.

Definition 2** (Coalition)**

A coalition is a couple $C=\langle a,g\rangle$ where $a\in A\cup\{\theta\}$ and $g\subseteq I$. The capacity of the coalition (denoted $c_{C}$) corresponds to the capacity of the activity (denoted $a_{C}$) if $a_{C}\in A$ and $1$ otherwise. The size of the coalition is the cardinality of the group (denoted $g_{C}$). A coalition is said sound if the size does not exceed its capability: $\mathrm{card}(g_{C})\leq c_{C}$. A non-empty coalition $C$ is such that $g_{C}\neq\emptyset$. $C$ is for $i$ if $i\in g_{C}$.

We consider a problem instance such that the number of individuals is considerably larger than the number of available activities ($m>>n$) as in our practical application.

Definition 3** (Matching)**

*A matching $M$ is represented by the functions $a_{M}:I\rightarrow A\cup\{\theta\}$ and $g_{M}:I\rightarrow\mathcal{P}(I)$ such that: *

[TABLE]

The assignment of an individual is an activity, possibly the void activity (cf equation 6). In this case, the individual is said inactive. Each individual is associated with the group it belongs (cf equation 7). All the individuals which are assigned to the void activity are alone (cf equation 8). All the individuals which are associated with each others have the same activity (cf equation 9) and reciprocally all the individuals which are assigned to the same activity, excepted the void activity, are associated with each other (cf equation 10).

In a matching $M$, the participation function determines the set of individuals who practice an activity:

[TABLE]

The set of participants for an activity can be empty. If $a_{M}(i)=\theta$, $i$ is said inactive. In a matching $M$, an activity $a\in A\cup\{\theta\}$ is:

oversubscribed $\mathrm{card}(p_{M}(a))>c_{a}$ ; 2. 2.

undersubscribed $\mathrm{card}(p_{M}(a))<c_{a}$ ; 3. 3.

full (denoted $\mathrm{full}_{M}(a)$) otherwise.

A matching is said sound if no activity is oversubscribed.

A matching $M$ is a coalition structure, i.e. a partition of individuals:

[TABLE]

Indeed, the equations 12, 13 and 14 are deduced from the equation 11, the definition 2 and the definition 3. We denote $C_{M}(i)$ the coalition in $M$ which contains $i$. In order to simplify the definitions about stability in the next section, we introduce the target function which returns the newcomers by selecting the activity $a$ in the matching $M$:

[TABLE]

3.3 Individual rationality

We can now evaluate the satisfaction of the individuals with respect to the possible solutions.

Each individual assesses the coalition to which he belongs and thus the matching based on his group and his activity. An individual $i$ considers that a sound coalition $C$ for him ($i\in g_{C}$) is individually rational if and only if $u_{i}(g_{C},a_{C})\geq 0$. A sound matching $M$ is individually rational (denoted IR) if and only if:

[TABLE]

The existence of a sound IR matching is guaranteed, e.g. the trivial matching (denoted $M_{0}$) where all individuals are inactive.

We deduce from each utility function $u_{i}$ an individual preference relation over the coalitions, i.e. a reflexive, complete and transitive preference ordering over $A\cup\{\theta\}\times G(I)$. The individual $i$ prefers $C$ rather than $C^{\prime}$ (denoted $C\succsim_{i}C^{\prime}$) if $u_{i}(g_{C},a_{C})\geq u_{i}(g_{C^{\prime}},a_{C^{\prime}})$. An individual prefers a matching to another one if he prefers his coalition in the first. Let us consider $M$ and $M^{\prime}$ two matchings for our problem instance $IA$. The individual $i$ prefers $M$ rather than $M^{\prime}$ (denoted $M\succsim_{i}M^{\prime}$) if and only if

[TABLE]

The strict preference relation over the matchings/coalitions is denoted $\succ_{i}$.

Example 2** (Rationality)**

Let us consider the $IA$ problem depicted in Fig. 1, we focus on $3$ of $63$ possible sound matchings:

$M_{0}$* with $p_{M_{0}}(a)=\emptyset$, $p_{M_{0}}(b)=\emptyset$ and $p_{M_{0}}(\theta)=\{1,2,3,4\}$;*

• 2.

$M_{1}$* with $p_{M_{1}}(a)=\{1,2\}$, $p_{M_{1}}(b)=\{4\}$ and $p_{M_{1}}(\theta)=\{3\}$;*

• 3.

$M_{2}$* with $p_{M_{2}}(a)=\{1,2\}$, $p_{M_{2}}(b)=\{3,4\}$ and $p_{M_{2}}(\theta)=\emptyset$;*

Contrary to $M_{0}$ and $M_{1}$, $M_{2}$ is not individually rational since the sound matching $\langle b,\{3,4\}\rangle$ is neither rational for $3$, nor for $4$. These two individuals strictly prefer being inactve rather than being poorly accompanied for $b$. By contrast, they strictly prefer practice $a$ together rather than being inactive:

[TABLE]

3.4 Social evaluation

We can now evaluate the matchings from the collective viewpoint.

A first desirable property for matchings is stability. Intuitively, the individuals in a blocking coalition would like to separate and form their own coalition, which makes the underlying matching unstable.

A non-empty sound coalition $C$ strongly blocks a matching $M$ if and only if all the individuals of the coalition strictly prefer the latter to be assigned according to $M$:

[TABLE]

A sound matching is core stable (CS) if and only if no non-empty sound coalition which strongly blocks it. By definition, a matching which is core stable is individually rational.

A non-empty sound coalition $C$ weakly blocks a matching $M$ if and only if all the individuals of the coalition prefer the latter to be assigned according to $M$ and at least one individuals strictly prefers the coalition:

[TABLE]

A sound matching is strict core stable (SCS) if no non-empty sound coalition strongly blocks it. By definition, all the SCS matching are a fortiori CS.

A sound matching $M$ est Nash stable (NS) if and only if no individual can benefit by moving from its coalition toward another undersubscribed activity:

[TABLE]

Such a matching is immune to individual movements since the coalitions are rational for all the individuals and at least as good as the other coalitions which can be reached by a single move.

A sound matching $M$ is individually stable (IS) if and only if no individual can benefit by moving from his coalition to another undersubscribed activity or if so, the deviating individual is not unanimously appreciated by his newcomers partners:

[TABLE]

By definition, a matching IS is a fortiori NS.

A sound matching $M$ is contractually individually stable (CIS) if and only if no individual can benefit by moving from his coalition to another without decreasing the satisfaction of one side or the other:

[TABLE]

By definition, a CIS matching is a fortiori IS. It worth noticing that, contrary to a IS matching, a CIS matching is not necessarily IR.

We are now defining other desirable properties to evaluate the economic efficiency of a matching. Like the previous ones, theses properties are not based on inter-personal comparisons.

A sound $M$ is perfect (P) if the coalition for each individual is one of his preferred one:

[TABLE]

A matching $M^{\prime}$ Pareto-dominates $M$ if and only if $M^{\prime}$ is strictly preferred to $M$ for at least one individual and not worst for the others:

[TABLE]

A matching is Pareto-optimal (PO) if it is not Pareto-dominated. In other words, A matching is PO if there is no alternative in which all agents would be in an equivalent or better position.

The strict core stability is a sufficient condition for the Pareto-optimality.

Property 1** (Efficiency vs. Stability)**

$\mathrm{SCS}\subseteq\mathrm{PO}$**

Proof 1** (Efficiency vs. Stability)**

We prove by contradiction that a sound matching which is strict core stable is also Pareto-optimal. We assume that $M$ is a strict core stable matching which is not a Pareto-optimum. Then, there exists a matching $M^{\prime}$ which Pareto-dominates $M$. Let us consider the coalition $C_{M^{\prime}}(i)$ with $i$ satisfying the equations 27 and 27. The equation 21 is true. Therefore, we conclude that $C_{M^{\prime}}(i)$ is sound and weakly blocks $M$. This is in contradiction with our assumption.

Contrary to the Paretian approach, the social choice theory analyses the interpersonal comparisons [12]. This theory reconciles the utilitarianism of Jeremy Bentham and the distributive justice advocated by John Rawls [13].

Definition 4** (Social welfare)**

Let $M$ be a sound matching.

The utilitarian welfare of $M$ is:

[TABLE]

• 2.

The egalitarian welfare of $M$ is:

[TABLE]

The welfares are normalized ($U(M),E(M)\in[-1,1]$). The higher the welfare is, the better the matching is. A maximum utilitarian (respectively maximum egalitarian) - denoted MaxUtil (respectively MaxEgal)- matching maximizes the utilitarian (resp. egalitarian) welfare. It is worth noticing that, even all the individuals/activities are attractive, the egalitarian welfare is [math] if at least one individual is inactive. By definition, a sound MaxUtil matching is a fortiori PO. Similarly, a sound MaxUtil matching is a fortiori IR.

Finally, Social Psychology points out that cohesion is more closely linked to the interest of the activities rather than the interpersonal affinities [14]. Formally, a sound matching $M$ is socially cohesive (SC) if and only if all individuals are assigned as much as possible to the activities they prefer:

[TABLE]

The existence of a sound SC matching is guaranteed, e.g. the trivial matching when all activities are rebutting. Inspired by our practical application which aims at maximizing the activities shared (cf Section 7), this notion is orthogonal to the previous ones.

Inspired by [2], Fig. 2 depicts the inclusion relationships between the previous concepts as a lattice. For instance, the arc between SCS and PO indicates that the first concept is a necessary condition for the second but not a sufficient one as illustrated in the following example.

Example 3** (Social evaluation)**

Let us consider the $IA$ problem represented in Fig. 1, the sound MaxtUtil matching are :

$M_{1}$* with $p_{M_{1}}(a)=\{1,2\}$, $p_{M_{1}}(b)=\{4\}$ and $p_{M_{1}}(\theta)=\{3\}$;*

• 2.

$M^{\prime}_{1}$* with $p_{M^{\prime}_{1}}(a)=\{1,2\}$, $p_{M^{\prime}_{1}}(b)=\{3\}$ and $p_{M^{\prime}_{1}}(\theta)=\{4\}$.*

The sound MaxEgal matching are:

$M_{3}$* with $p_{M_{3}}(a)=\{1,4\}$, $p_{M_{3}}(b)=\{2,3\}$ and $p_{M_{3}}(\theta)=\emptyset$;*

• 2.

$M^{\prime}_{3}$* with $p_{M^{\prime}_{3}}(a)=\{3,4\}$, $p_{M^{\prime}_{3}}(b)=\{1,2\}$ and $p_{M^{\prime}_{3}}(\theta)=\emptyset$.*

The matchings $M_{1}$ and $M^{\prime}_{1}$ are PO but not SCS. Contrary to $M_{1}$ and $M^{\prime}_{1}$, the matchings $M_{3}$ and $M^{\prime}_{3}$ are SC but not IR.

In summary, if the stability is a desirable property, there is no necessarily such a solution. Moreover, Pareto-optimality doesn’t seem to be discriminating enough. By contrast, the concepts of social cohesion and social welfare are better candidate to assess the quality of a matching from the collective viewpoint.

4 Centralized matching mechanism

Based on the bilateral structure of the IA problem, we propose here two algorithms which return sound matchings.

Our algorithms are inspired by the student-proposing deferred acceptance algorithm [7]. Each individual $i$ is endowed with a list of concessions, i.e. rational activities in decreasing order of preference. On one side, the individual $i$ proposes itself to the activities he prefers, even if it must concede in case of rejection. On the other side, the coalition $\langle a,g\rangle$, as a host, apply its capacity constraint $c_{a}$ and the objective of the hosted individuals, i.e. a social decision rule which can be:

either a utilitarian rule

[TABLE]

• 2.

or an egalitarian rule

[TABLE]

The initial matching is the trivial one, i. e. all individuals are inactive. Each in turn, a free individual $i$ considers this prefers activity which attracts him $a$. If the coalition is empty, $i$ is assigned. Otherwise, the algorithms try to improve the satisfaction of this coalition, eventually by excluding the individuals whose presence penalizes the (utility or egalitarian) rule of the group. The individuals who are excluded, eventually $i$, must concede and so consider the next attractive activity.

In the selective algorithm,the social decision rule is evaluated for each proposal to a non-emtpy group. As long as the capacity of $a$ is not reached then the group can grow. Finally, an individual who is rejected by all his attractive activities is definitely inactive. Since a group of $k$ individuals contains $2^{k}-1$ non-emtpy subgroups, we propose an approximation algorithm which excludes no more than one individual at each step.

Our (approximation) algorithm always returns a sound matching.

Property 2** (Termination)**

Our (approximation) algorithm applied over an IA problem instance ends and returns a sound matching.

Proof 2** (Termination)**

We consider the following loop invariant: $\Sigma_{i\in I}|concessions(i)|+|free|$ with the list of the rational activities for $i$: $concessions(i)$ and the set of free individuals $free$. This invariant is positive. It strictly decreases after each loop since:

an individual, which is assigned, is removed from $Free$; 2. 2.

an individual, which is not assigned, concedes until it is definitely assigned to the void activity; 3. 3.

any individual, which is unassigned, concedes and at least one another individual is assigned (and so removed from $free$).

The resulting matching is sound since the activities are never oversubscribed.

Example 4** (Selective algorithm.)**

*Let us consider the $IA$ problem represented in Fig. 1. Whatever the social decision rule is, our selective algorithm allows to reach $M_{1}$ with $p_{M_{1}}(a)=\{1,2\}$, $p_{M_{1}}(b)=\{4\}$ and $p_{M_{1}}(\theta)=\{3\}$. This matching is Pareto-optimal but it is not socially cohesive. *

It is worth noticing that the selective algorithm does not not necessarily maximizes the utilitarian (respectively egalitarian) welfare by applying the utilitarian (respectively egalitarian) rule since the individuals give priority to the activities over the individuals. Despite this priority, the resulting matching is not necessarily socially cohesive. That is the reason why we propose the inclusive algorithm. In the latter, the social decision rule of a coalition, as a host, is assessed only if its capacity constraint is violated and the exclusion of a single individual is envisaged for each individual integration.

The inclusive algorithm always returns a sound match that is socially cohesive whatever the social decision rule is.

Property 3** (Termination)**

The inclusive algorithm applied over an IA problem instance ends and returns a sound matching.

Proof 3** (Terminaison)**

See proof 2.

Property 4** (Social cohesion)**

The inclusive algorithm applied over an IA problem instance returns a SC matching.

Proof 4** (Social cohesion)**

We prove by contradiction that the outcome $M$ of the inclusive algorithme is a SC matching. We assume that ther exists an individual $i$ and an activity $a\neq a_{M}(i)$ such that $v_{i}(a)\geq 0$, $v_{i}(a)>v(a_{M}(i))$ and $a$ is undersubscribed (cf equation 30). Since $i$ is attracted by $a$, this individual proposes himself to this activity which is in his list of concessions (line 4) before $a$. He has been excluded by this activity which is full since the size of the possible groups is $c_{a}$ (line 22). This is in contradiction with our assumption.

Example 5** (Inclusive algorithm)**

*Let us consider the $IA$ problem instance depicts in Fig. 1. whatever the social decision rule is, the inclusive algorithm allows to reach $M_{2}$ with $p_{M_{2}}(a)=\{1,2\}$, $p_{M_{2}}(b)=\{3,4\}$ and $p_{M_{2}}(\theta)=\emptyset$. This matching is socially cohesive but it is not individually rational. *

5 Agent behaviours

We describe here the previous matching mechanisms as agent behaviours in order to decentralize their execution.

We consider the asynchronous message-passing model of actor for concurrent programming introduced in [15]. In this framework, the primitives are agents and events: an agent is an independent program that runs on its own processor; an event corresponds to the creation of an agent or the utterance/reception of a message. The system is distributed since the underlying channels are assumed to be reliable (a message is delivered once and only once) and that the messages may arrive in different order from sending.

In order to propose a distributed solver based on this model, we distinguish 3 kinds of agents:

the solver agent which creates the other agents and records the assignments; 2. 2.

the individual agents which are endowed with the same behaviours but with different preferences; 3. 3.

the activity agents which are endowed with the same behaviours but with different capacity constraints and group objectives.

The behaviour of the solver agent consists of: i) creating other agents; ii) triggering the solving; iii) recording the assignments/unassignments; and iv) returning the matching when all the individual are assigned or definitely inactive.

Whether the mechanism is selective or inclusive, the behaviour of an individual agent is the same. An individual agent proposes himself to his preferred activity that attracts him. When the agent is assigned or excluded, he informs the solver agent. For an unassignment, the coalition agent waits for a confirmation before sending a new proposal such that a matching is not prematurely returned by the solver agent. If an individual agent becomes free, he concedes, then he proposes himself to the next attractive activity until being definitely inactive. In accordance with the principle of subsidiarity, the reconfiguration of a coalition may be subject to arbitration by the the coalition agent that requires the opinion of the individual agents which are concerned.

In order to describe the coalition agent behaviour, we consider the finite state machine design pattern for concurrent programming with 3 states:

the inital Disposing state for proposal processing; 2. 2.

the Casting state where the opinion of individual agents are requested in order to assess the group in conformance with the social decision rule; 3. 3.

the Firing state where the excluded individual agents are notified.

The state transitions between states are such as:

Event if conditions → actions.

They are triggered by an external event, e.g. the reception of a message. The automaton is deterministic, i. e. the conditions of triggering of outgoing transitions for the same state are mutually exclusive. This behavioral design pattern allows to manage multi-party dialogues. The proposals received by a coalition agent are processed each in turn, even if they are stashed to be evaluated later on, but their acceptance/rejection depend on the opinion of the other agents which are concerned.

For instance, Fig. 3 describes the behaviour of the agent representing the coalition $\langle a,g\rangle$ for the selective mechanism. According to this behaviour, an proposal from $i$ which is received in the Disposing state is accepted if the coalition is empty. Otherwise, the agent evaluates this proposal in the Casting by selecting the non-empty group according to the capacity constraint and the objective of the coalition, i.e. the social decision rule. Then, if the proposer is not a member of the selected group, the proposal is rejected. Otherwise, the proposal is accepted as soon as, if some members are excluded, the latter confirm the notification of exclusion by the solver agent (in the Firing state). In the other matching mechanisms, the behaviour of coalition agents is similar with the exception of the conditions and actions for the outgoing transitions of the Disposing state. For instance, the assessment of the coalition for the inclusive mechanism takes place only when the capacity is reached.

6 Experiments

Our experiments111Our experiments have been performed with a 8 core Intel (R) i7 2.8GHz and 16GB RAM. aim at evaluating: i) the benefits of the structural properties of the problem for our mechanisms; ii) the quality of the outcomes ; and iii) the speedup due to their distribution.

Our prototype ScaIA (https://github.com/maximemorge/ScaIA) has been implemented with the Scala progamming language and the Akka toolkit (http://akka.io). The latter, which is based on the actor model [16], allows us to fill the gap between the specification of our agent behaviours and their implementation.

Our mechanisms exploit the structure of the IA problem, which is, contrary to the hedonic game, two-sided. In order to evaluate its benefit, we translate an IA problem instance into:

a hedonic game by associating : i) one player for each activity such that all the sound coalitions are equally preferred; and ii) one player for each individual whose preferences are deduced from the preferences of the latter. Then, we compute a contractually individually stable (CIS) coalition; 2. 2.

a mixed integer quadratic programming (MIQP) problem (cf A) which is solved with IBM® ILOG® CPLEX®.

We consider some IA problem instances with $2$ activities and $m\in[2,20]$ individuals. For the sake of simplicity, all activities have the same capacity ($c=\lceil m/2\rceil$). We generate $100$ (pseudo-)random instances for each value of $m$.

First of all, we need to instanciate the definition of the affinity function and the definition of the utility function. Inspired by our practical application, we axiomatize the principle quo plus est eo hilarior, sed quo paucior eo melior cibus222In English the more the merrier, but the fewer the better fare. by computing the affinity of an individual $i$ for a group in the following way:

[TABLE]

This affinity, which is normalized ($w_{i}(g)\in[1.1]$), increases with the size of the group if $i$ is attracted by all the members of the group. This function, which evaluates the preferences of an individual over $2^{m-1}$ groups, is additively separable and so its representation is linear with respect to the number of individuals. For the sake of simplicity, we define the utility function in the following manner:

[TABLE]

Since $w_{i}(g),v_{i}(a)\in[-1,1]$, the utility is normalized ($u_{i}(g,a)\in[-1,1]$).

Figure 4 compares the utilitarian welfare reached by the selective algorithm with approximation, the CIS algorithm and the MIQP one. Figures 5 and 6 compare the runtimes of these algorithms with a logarithmic time scale and by distinguishing the runtime for the translation of: i) the IA problem in a hedonic game or in a mathematical program; and (ii) the solution. The runtime of our procedure benefits from the structural properties of the IA problem. Moreover, the CIS algorithm is penalized by the generation of RLC [3] which take exponential space. Even if the selective mechanism does not always maximize the utilitarian welfare computed by the MIQP, it is closed to. Moreover, the selective mechanism is ($25$ times) faster (for $20$ individuals).

Similarly, Figures 7 and 8 compare the runtimes and the egalitarian welfares reached by the inclusive algorithm and the approximation algorithm (MIQP). The egalitarian welfare reached by our algorithm is closed but better than the one approximated by MIQP while being ($120$ times) faster (for $20$ individuals).

Even if we do not have any theoretical results concerning the stability of the outcome for the selective algorithm, Figure 9 shows that $95\%$ of the outcomes of the selective algorithm with approximation are PO and $96\%$ of them are IR. We can note than we restrict ourselves to $n=2$ and $m\leq 13$ since the decision problem for the Pareto-optimality is not tractable.

Since the MIQP algorithms are not scalable, we have implemented a local search algorithm [17] to be compared with our algorithm. This hill-climbing algorithm, which starts with a sound and random matching (where all the individuals are assigned to an activity) and iteratively tries to improve the welfare. Two matchings are neighbours if they are identical with an exception for one individual which moves to another activity. If this new activity is full, then all the swaps of individuals with the members of that activity are considered.

We consider some IA problem instances with $n$ activities and $m$ individuals. For the sake of simplicity, all the activities have the same capacity ($c=\lceil m/n\rceil$).

Firstly, we focus on the utilitarian welfare of the outcome reached by the selective mechanism with approximation.

We can compare the utilitarian welfare of the matching reached by our selective algorithm with the one reached by local search. Fig. 10 shows the average utilitarian welfare of $100$ problem instances for each set of parameters ( $2\leq n\leq 10$ and $2\times n\leq m\leq 100$). The selective mechanism overreaches the local search. Indeed, the utilitarian welfare for a IA problem is a function with many local optima.

We can compare the runtime of the centralized version of our selective algorithm with the decentralized one. Fig. 11 shows the average runtime for each set of parameters (with $2\leq n\leq 10$ and $2\times n\leq m\leq 400$). While the centralized algorithm is faster when the number of individuals is low ($\sim 20$), its runtime quickly grows with the number of individuals ($13$ ms for $100$ individuals and $10$ activities) while the runtime of the distributed version is low ($4$ ms in the latter case). Moreover, the runtime of the hill-climbing algorithm is too high to be represented in Fig. 11 ($170,857$ ms for $100$ individuals and $10$ activities). We can expect a higher runtime if we adopt a local search method such as simulated annealing without any warranty about the optimality of the outcome.

Secondly, we focus on the egalitarian welfare of the outcome reached by the inclusive mechanism.

We can compare the egalitarian welfare of the matching reached by our algorithm with the one reached by local search. Fig. 12 shows the average egalitarian welfare of $100$ problem instances for each set of parameters ( $2\leq n\leq 10$ and $2\times n\leq m\leq 100$). The inclusive mechanism overreaches the local search. Indeed, the egalitarian welfare for a IA problem is a function with many local optima.

We can compare the runtime of the centralized version of our inclusive algorithm with the decentralized one. Fig. 13 shows the average runtime for each set of parameters (with $2\leq n\leq 10$ and $2\times n\leq m\leq 400$). While the centralized algorithm is faster when the number of individuals is low ($\sim 40$), its runtime quickly grows with the number of individuals ($7$ ms for $100$ individuals and $10$ activities) while the runtime of the distributed version is low ($2$ ms in the latter case). Moreover, the runtime of the hill-climbing algorithm is too high to be represented in Fig. 13 ($433,496$ ms for $100$ individuals and $10$ activities). We can expect a higher runtime if we adopt a local search method such as simulated annealing without any warranty about the optimality of the outcome.

In summary, the outcomes reached by our algorithms seem to be closed to the maximum utilitarian/egalitarian. Moreover, the distribution of our algorithms allows to speedup its runtime.

7 Practical application

Our practical application is concerned with a group of seniors who aim at sharing some holidays. The objective is to maximize the activities shared in order to improve the social cohesion and avoid the isolation of seniors.

According to the figure 14, which represents the entity-relation model of our database, each user is described by a set of features: gender, age, hobbies, etc. Thus, each user indicates (or not) the values which characterize him and he may inform its attraction/repulsion towards these values for his peers. From this knowledge we compute the affinities between users. In a way similar, tours are described by feature values on which users express some preferences. From this knowledge, we compute the interests of the users for the tours.

Even if this scenario involves thousands of users, we have tested our prototype with $102$ users and $10$ tours. Table 1 summarizes the runtimes and welfares reach by all the matching algorithms, except the ones based in mathematical optimization which are too slow. Whether we focus on the utilitarian welfare or the egalitarian one, our algorithms are faster and more efficient than hill-climbing. The runtimes of the distributed versions of our algorithms are faster when the number of individual integrations is sufficient.

8 Discussion

We have presented here a generic problem where some individuals must be assigned to the activities they enjoy with their favorite partners. Even if stability is desirable, it is not guaranteed. By contrast, Pareto-optimality seems not to be discriminative. This is the reason why we evaluated a matching from the collective viewpoint using concepts derived from social choice theory. Others concepts may be relevant, whether they come from Economy (envy-freeness, strategy-proofness, etc.) or inspired by social psychology such as the concept of social cohesion sketched here.

Even if a worst-case study is required here, our experiments show that social welfares are functions with many local optima. That is the reason why we have proposed here two heuristics, one for each of the notions of social welfare. They returns sound matchings in accordance with a social decision rule (either utilitarian or egalitarian) for the first and which is socially cohesive for the second. These mechanisms are based on the organizational aspect of agent-based modelling by explicitly introducing an intermediate level between the individual and the society, i.e. a coalitions of individuals who share a common interest for an activity. In accordance with the principle of subsidiarity, the assessment of a coalition is performed by an agent representing the objectives and constraints of a group of individuals.

Our empirical results, in particular with real-world data, illustrate that multiagent resolution methods can be more effective than general optimization methods which require problem reformulation. Beyond a simple decentralization that, as here, speedups its runtime, the modeling of agents behaviours allows to introduce a variety of strategies for implementing the constraints and the objectives of the different agent coalitions.

In future works, we plan to take into account the relational aspect of the agent-based modelling to distribute the negotiations between neighbours in the social network.