Mélanie Buy

William Digan

Xiaoyi Chen

Julien Husson

Mickael Ménager

Frédéric Rieux-Laucat

Nicolas Garcelon

Frédéric Rieux- Laucat

Atraction Members

A Multi-Omics Common Data Model for Primary Immunodeficiencies

Keywords: Cohort Studies, Databases, rare diseases

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

There are about 7000 different types of disorders and rare diseases which have an impact on a large population worldwide [START_REF] Griggs | Clinical research for rare disease: Opportunities, challenges, and solutions[END_REF]. In this context, patient data collection and analysis of rare and undiagnosed diseases have the potential to offer the opportunity to increase knowledge and discover new therapeutic approaches. In the project ATRACTion (Autoimmunity & inflammation Through RNAseq Analysis at the single Cell level for Therapeutic Innovation), we are interested in rare Primary Immunodeficiencies (PIDs). PIDs gather more than 400 rare monogenic diseases affecting the development, the function or the regulation of the immune response [START_REF] Fischer | Autoimmune and inflammatory manifestations occur frequently in patients with primary immunodeficiencies[END_REF]. Moreover, a given monogenic cause could be associated with variable expression ranging from no symptom (no penetrance) to a broad spectrum of severe and debilitating manifestations. As a national reference center of PIDs, Necker hospital registered more than 5000 PIDs. The variable expression observed in PIDs with autoimmunity/inflammation is therefore leading to diagnosis and therapeutics wandering. In this project, we will include 250 PIDs patients and 250 controls. Their clinical data and multi-omics data will be collected. As some clinical data are mandatory for omics partners for the results analysis and the determination of omic signature, it is important to combine clinical data and omics data into a common data model. Nevertheless, the combination of clinical and omics data remains challenging [START_REF] López De Maturana | Challenges in the Integration of Omics and Non-Omics Data[END_REF]. The first challenge comes from the complexity, heterogeneity and scale of non-omics data (i.e., clinical data). They can be structured (e.g., Body Mass Index (BMI), blood pressure) or unstructured (information extracted from clinical narratives), in various data types (qualitative and quantitative), and from different providers. Moreover, some phenotypes can be absent from a patient record, due to the absence of a medical test. The second challenge lies in combining microbiota data with other omics data [START_REF] Chu | Integration of Metabolomic and Other Omics Data in Population-Based Study Designs: An Epidemiological Perspective[END_REF]. In fact, the metabolome state is a back-and-forth process with the immune system. On one hand, metabolites captured the end products of biochemical reactions, which lead to a patient phenotype. On the other hand, metabolites shape the immune response and therefore impact transcriptomics and proteomics. The third challenge comes from the relation between omics and non-omics data. Omics data show an ascertainment bias due to the experiment itself (case versus control), which can be specific to a patient condition (e.g., age, gender, drugs).

State of the art

Many efforts have been made for solutions to combine clinical data and omics data. In the medical informatics community, two approaches have been observed. The Observational Medical Outcomes Partnership Common Data Model (OMOP-CDM) [START_REF] Makadia | Transforming the Premier Perspective® Hospital Database into the Observational Medical Outcomes Partnership (OMOP) Common Data Model[END_REF] is dedicated to the integration of heterogeneous hospital data such as administrative data, clinical results, information extraction from electronic health records. Shin SJ, et al 2019 [START_REF] Shin | Genomic Common Data Model for Biomedical Data in Clinical Practice[END_REF], [START_REF] Shin | Genomic Common Data Model for Seamless Interoperation of Biomedical Data in Clinical Practice: Retrospective Study[END_REF] proposed the Genome Common Data model as an extension of the OMOP-CDM to include clinical sequencing data (gene name, variant type and actionable mutation). They integrated 114 lung cancer patients [START_REF] Shin | Genomic Common Data Model for Seamless Interoperation of Biomedical Data in Clinical Practice: Retrospective Study[END_REF] from the Ajou University Hospital, Suwon, Republic of Korea and 1060 patients from the Cancer Genome Atlas. To do so, they added four tables linked to the OMOP-CDM, namely Genomic_Test, Target_Gene, Variant_Occurrence and Variant_Annotation. Nevertheless, they did not deliver a general way to integrate other omics data. Carnival [START_REF] Birtwell | Carnival: A Graph-Based Data Integration and Query Tool to Support Patient Cohort Generation for Clinical Research[END_REF] The LinkedOmics [START_REF] Vasaikar | LinkedOmics: analyzing multi-omics data within and across 32 cancer types[END_REF] database contains multi-omics data (genomics, epigenomics, transcriptomics and mass spectrometry) for 32 cancer types associated with 11,158 patients from The Cancer Genome Atlas (TCGA) project.

Goals

In this work, according to the experimental design of the ATRACTion project, we propose an integrative model of clinical data and multi-omics data, which can be used to obtain standardized information necessary for the characterization of PIDs patients and the identification of molecular signatures. Furthermore, it will allow machinelearning based unsupervised approaches such as network inferences and could facilitate the visualization of the complex data via multi-layer networks.

Methods

We here defined a minimal dataset to integrate both clinical data and multi-omics data. Regarding clinical data, the OMOP data model offers a comprehensive patient-centered description for clinical trial purposes. In this work, we have built our data model by selecting only the tables of OMOP relevant for our studies. Indeed, we have included the Person, Observation, Visit_occurrence, Condition_occurrence, Drug_exposure, Measurement, Procedure_exposure, Survey_conduct and all the tables related to the concepts and vocabularies. We have excluded tables like Cost, Care_site, Device_exposure, which provide a specific level of detail but are not relevant for our studies.

ATRACTion multimodal database

As each data type needs a specific storage format, we have looked for ad hoc ontologies and thesaurus.

Samples collection

The cohort includes 250 patients and 250 healthy relatives followed in the Necker hospital. The required clinical data for the project will be collected from the clinical data warehouse, as well as specific interviews driven by medical staff. Plasma, peripheral blood cells, feces and urine samples will be collected from all individuals and be registered in the laboratory information management system, in which the link between the samples and the analyses will be kept.

Clinical data

To compute the analyses and determine omic signatures, some metadata are mandatory for the omics partners, (e.g. breastfeeding information for metagenomic analyses). For that purpose patients will be interviewed by clinicians within the use of RedCap and collected information then transferred in the ATRACTion database. We have defined the minimal dataset based on both administrative information and clinical data needed for further analyses. We also included for each patient, the medical history from the Necker hospital database to complete the clinical data. We mapped the data with the OMOP data model.

Ontologies and knowledge databases

To ensure data interoperability, symptoms will be described using the Medical Subject Headings (MeSH), the Logical Observation Identifiers Names & Codes (LOINC), the Human Phenotype Ontology (HPO), genes with HGNC ontology, and diagnoses with Orphanet Rare Disease Ontology (ORDO). These ontologies and thesaurus will be stored in the OMOP Concept table. Molecules and biological compounds like proteins or lipids will be linked to KEGG databases.

Omics data

Beyond clinical information, six different types of omics will be analyzed from patient samples (Figure 1): transcriptomics, epigenomics, proteomics, lipidomics, metabolomic, and metagenomic. In order to build a well-designed data model for each omic dataset, we have organized meetings with project partners to give insights from omics field specialists and to collect example data files. We were then also able to highlight some data types inherent issues. In our data model we compare cases versus controls patients in order to compute fold changes. More details are provided in the following for each omic data.

Transcriptomics

These data will allow us to identify differential gene expression between patients. The data files will be generated from single-cell experiments performed by using 10x Genomics CITE-seq protocol on peripheral blood mononuclear cells (PBMCs) extracted from blood samples. Epigenomics Epigenomics data will inform us about chromatin accessibility which is required to allow binding of transcription factors and subsequent gene expression. Data files will be generated through 10X Genomics ATAC-seq protocols performed on nuclei extracted from PBMCs Proteomics CYTOF mass cytometry and Olink experiments will generate a large dataset of differential expression for inflammatoryrelated proteins in blood samples. These experiments will be performed at single-cell level, which could constrain the group comparison, as needed for others single-cell results.

Metabolomics

The data files will be generated from analyses via chromatography and dosage of LPS activity, to investigate the impact of disease on over 300 metabolites belonging to biological pathways. The results data files will give the quantity of a large number of molecules involved in biological pathways linked with the inflammation process.

Metagenomics

Sequencing process will be performed on fecal samples to find which bacterial genes are present in these samples and infer the corresponding bacteria species. The presence of some species could be linked with specific metabolites, allowing the link between metagenomic and metabolomic data.

Lipidomics

Lipidomics gives the concentration of molecules and can be linked with the presence of bacterial species for specific metabolites. This allows lipidomics and metagenomics data to be correlated. Lipidomics data files will be obtained from analyses performed using high-resolution mass spectrometry on blood et urine samples.

Implementation

Regarding the technical aspect, the database is based on an object-relational database implemented in PostgreSQL (available at https://framagit.org/imagine-plateforme-bdd/pub-rhu4-atraction.) and queried via a Django web interface. The relation model of the database stores each type of omic data in specific tables. For each omic, a table suffixed with '_exp' describes metadata about the experience (device, culture medium, statistic model, …) and a table suffixed by '_result' contains numeric results, including p-value, fold change or other types of metrics. For results obtained via group comparisons, the group identifier is stored into the attribute '[omic]_group_id'.

Results

Figure 2 shows the Unified Modeling Language (UML) diagram of the multi-omics ATRACTion OMOP-CDM.

Clinical data set

This dataset consists of 6 parts:

-Patient information and administrative data (e.g. birth date, gender, origin):

Discussion

Technical significance

To the best of our knowledge, it is the first description of a clinical and multi-omics (including 6 types of omics data) database implementation, which extended the OMOP-CDM for PIDs diseases. This model can be generalized to the studies of other types of rare diseases, with a focus of case versus control comparison.

Significance for secondary use of clinical data

This data representation approach could enhance reproducibility and sharing of models that are learned from this data. Such OMOP-CDM models could be deployed in other reference PIDs centers and researchers will be able to perform distributed queries across them.

Perspectives

In the near future, we will have access to all omics data files to complete and further improve the model. Such a multi-omics database can help identify specific molecular signatures for PIDs, facilitate the visualization of complex data, allow network inferences, and finally build a multi-layer disease network.

Conclusions

In this work, we present a multi-omics ATRACTion OMOP-CDM, which expands the OMOP-CDM in order to include both clinical and multi-omics data. The data definition language is available at https://framagit.org/imagineplateforme-bdd/pub-rhu4-atraction.

Figure 1 :

 1 Figure 1: data flow of the ATRACTion data process

  International Medical Informatics Association (IMIA) and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License 4.0 (CC BY-NC 4.0). doi:10.3233/SHTI220031 (age, sex), vital signs (BMI, smoking status, blood pressure), ICD9/10 codes and the loss of genes function. In the bioinformatics community, we had a look at three propositions: Moped 2.5, Aging atlas, LinkedOmics. Moped 2.5 [9] is an Integrated Multi-Omics Resource which includes approximately 5 million of transcriptomics and proteomics expression records from over 250 experiments coming from four reference organisms: human, mouse, worm, and yeast. Those data contain protein absolute and relative expression and gene relative expression. Relative expression data in Moped are displayed in terms of pairwise comparisons of conditions (case vs control) using the expression ratio for each protein. Moped provides both p-values and false discovery rate (FDR) estimates. Furthermore, to support the goal of reproducible science they provide a metadata checklist information about experimental design, instrument details, sample preparation, data processing, and analysis.

				MEDINFO 2021: One World, One Health -Global Partnership for Digital Innovation
				P. Otero et al. (Eds.)
	© 2022 Another step is reached by Aging Atlas [10], which is a
	curated aging biology database which integrates multi-omics
	data	(transcriptomics,	single-cell	transcriptomics,
	epigenomics, proteomics, and pharmacogenomics) from
	heterogeneous sources.
				is a graph database model inspired by the Open
				Biological and Biomedical Ontology (OBO) Foundry
				ontologies. This database contains over 60,000 patients from
				the University of Pennsylvania Health System and includes
				both clinical and whole-exome genomics data (blood and
				tissue samples). For the clinical data it contains demographics

table PERSON -

 PERSON Diagnosis and medical condition at the time of the sample (e.g. inflammation state of the patient, BMI, Figure 2: UML diagram of ATRACTion OMOP-CDM, we represent the main key concepts (the complete diagram is in the git project). All the concept_id are linked to the CONCEPT tableOmicsThe omic_experiment table is used to group samples sequenced together (batch). Each table with suffix exp groups sequenced samples processed together by computational biologist (*_group_id). Each table with a suffix _results contain analysed data and proposed a Fold Change and adjusted pvalue of an experiment.TranscriptomicsWith the help of the raw and normalized files, we build 6 information about the bacteria taxonomy. The Bacterial_gene table linked genes to species and biological function of these genes.LipidomicsWe include 2 tables: Lipidomic_results and Lipid. Lipid contains the lipid name and function.

	breastfeeding, diet): tables OBSERVATION,
	CONDITION_OCCURRENCE,
	SURVEY_CONDUCT, OBSERVATION _PERIOD
	-Biological results (creatininemia, autoantibodies
	etc.): Table MEASUREMENT
	-Drugs, one month before the samples: Table
	DRUG_EXPOSURE
	-The type of omic analysis performed can be stored in
	the	PROCEDURE_OCCURRENCE	(MeSH
	Procedure).
	-The sample description (type of sample, quantity,
	etc.): table SPECIMEN, TRANSFORM_SAMPLE.
	-In addition, the Vocabulary table stores the different
	used ontologies, each ontology term being listed in
	the Concept table, with corresponding entry code.

tables: Single_cell_exp, Coordinates, Sequence_annot, Raw_counts, Single_cell_results, and Gene. Single_cell_exp contains metadata about the experiment. Coordinates gathers the sequence reads coordinates for each gene. Sequence_annot contains metadata about sequences, like cellular type, the read sequence in nucleotides. Raw_counts contains raw sequence count associated with an experiment. At last, Gene presents information like gene name and chromosomal location. Epigenomics The Modification table relates the genes accessibility by transcription factors associated with peaks of accessible chromatin. Proteomics The Protein table details protein information like name, ontology entry, function, and coding gene linked to an experiment. Metabolomics 3 tables described Metabolomics: Metabolite, Pathway, Metabolomics_Results. The Metabolite table associates a metabolite to a pathway. The Pathway table details pathways within the use of ontology. Metabolomics_Results contains analysed results of an experiment. Metagenomics We include 3 tables: Bacterial_gene, Species, and Metagenomic_result. Species table contains relevant

M. Buy et al. / A Multi-Omics Common Data Model for Primary Immunodeficiencies

Acknowledgements

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investments for the Future programme (PIA) under grant agreements No ANR-10-IAHU-01 and ANR-18-RHUS-0010. We thank the ATRACTion members: Zsolt Bocskei, Florence Castelli, Nadine Cerf-Bensussan, Alain Fischer, Nicolas Pons, Anne Remaury, Michel Didier, Francesco Carbone, Mohammad Afshar, Christophe Junot, Dominique Rainteau, Anita Burgun, Josh Waterfall.

Address for correspondence

Nicolas Garcelon

Institut Imagine, 24 boulevard du Montparnasse, 75015 Paris, France nicolas.garcelon@institutimagine.org


Florian Navarro

"One person, one n th !": The incentive effects of egalitarianism in cooperative environments 1

Keywords: Cooperative game theory, shapley value, equal division, egalitarian value, role structure, incentives

This article introduces a new axiom of intersecting population monotonicity for TU-games as well as a new model. The axiom, alongside efficiency, characterizes the egalitarian value. The new model addresses situations where a number of roles have to be performed by players. Each player have a specific role and coalitions are feasible only if it contains at most one player for each role. We use the new characterization of the egalitarian value for TU-games to obtain the following result in our class of problem: the egalitarian value is the only sharing rule that ensures that each player of the most productive group is better off joining this most productive group. Thus, we characterize the egalitarian value without fairness requirement and show that, for this new class of problem, egalitarianism can provide incentives towards optimal cooperation.

Introduction

In the present article we develop a model of cooperative games in which a number of roles have to be filled by players. Each role can be perform by a precise set of players, and players can only perform one role. The role structure restricts the feasibility of coalitions: a coalition is feasible if it contains at most one player performing each role. This means we have situations in which the grand coalition cannot form. Such a situation can arise when members of an organization have to form a subgroup to tackle a specific project (see application 2). Several groups having to choose a representative to participate in a committee can also be a situation covered by our model. A central planer having to manage the use of a public resource between different actors can be a third (see application [START_REF] Béal | Efficient extensions of the Myerson value[END_REF]. The problem raised by our model is thus different than the usual problem of cooperative games (sharing the worth of the grand coalition).

A solution ϕ to this model consists of two mechanisms: a function µ that selects a group of players specifying, for each role, a single player being able to perform the role, and a sharing rule ψ that shares the worth produced by the group amongst its members. We are interested in solution that fosters the emergence of the most productive group that fills all role. To this end, we propose an axiom of optimality (the solution selects the best group) and an axiom of no outside options (no member of the selected group can be better off in another group) that apply to the solution ϕ. Together, these axioms imply requirements on the allocation problem for the chosen group. We introduce an axiom of intersecting population monotonicity for the sharing rule ψ. This axiom asks that no member of a population can be worse off by choosing to associate with a group of a given size with which it has the best synergy. We show that if µ satisfies optimality, then the axiom of no outside option on ϕ and of intersecting population monotonicity on ψ are equivalent. As an intermediary result, we offer a characterization of the egalitarian value (which shares the total worth equally between all players) that relies on the intersecting population monotonicity axiom alongside the standard axiom of efficiency. As such we characterize the egalitarian principle without any fairness criteria. Formally, our model is related to the Aumann-Drèze model [START_REF] Aumann | Cooperative Games with Coalition Structures[END_REF] as it also involves coalition structure. In a coalition structure, players are part of pre-existing groups that can not communicate with each others. This structure has an effect on the worth of coalitions and alter the characteristic function of the underlying transferable utility game (TU-game). [START_REF] Owen | Values of games with a priori unions[END_REF] also relies on coalition structures (called a-priori unions) although they directly impact the payoff of the players. In the present article we take yet another direction: this structure restricts the formation of coalitions and therefore their feasibility. We can cite for instance the Position-value [START_REF] Meessen | Communication games[END_REF], the Average-Tree Value [START_REF] Herings | The average tree solution for cycle-free graph games[END_REF], the Mean value [START_REF] Hamiache | A mean value for games with communication structure[END_REF], the F-value [START_REF] Hamiache | A value with incomplete communication[END_REF], the Hamiache-Navarro value [START_REF] Hamiache | Associated consistency, value and graphs[END_REF] and numerous variations of the Myerson value3 . Additionally, different forms of hierarchical constraints have been explored by [START_REF] Gilles | Games with permission structures: The conjunctive approach[END_REF], [START_REF] Faigle | The Shapley value for cooperative games under precedence constraints[END_REF] and more recently by [START_REF] Béal | The priority value for cooperative games with a priority structure[END_REF]. The distinction between the framework of [START_REF] Aumann | Cooperative Games with Coalition Structures[END_REF], [START_REF] Owen | Values of games with a priori unions[END_REF] and ours can similarly be made for, respectively, the framework of [START_REF] Gilles | Games with permission structures: The conjunctive approach[END_REF], [START_REF] Béal | The priority value for cooperative games with a priority structure[END_REF] and [START_REF] Faigle | The Shapley value for cooperative games under precedence constraints[END_REF] However, to the best of our knowledge few models have been proposed to enrich the individual characteristics of the players. Another approach to roles, or more precisely differently skilled players, has been proposed by [START_REF] Bachrach | Computing cooperative solution concepts in coalitional skill games[END_REF] with coalitional skill games (CSG). However, in a CSG framework, certain players can perform certain tasks and the worth of a coalition depends on the nature or number of tasks that can be performed by the coalition. The model is thus similar to the standard model of cooperative game theory but restricts itself to a certain type of characteristic functions defined by the players' "skills". Thus, our model tackles issues not addressed by existing research and contributes to the economic theory literature on several levels. For the standard model of TU-games, we provide a new characterization of the egalitarian value.

In addition, we use a formal framework of coalition structure to develop a novel class of unexplored problems as of yet. Within this class, we investigate the effect of sharing rules on the spontaneous emergence of coalitions, making it somewhat endogenous to the model. We give a solution for these problems and show that, for this class of problem, egalitarianism can provide incentives whereas approaches based on the evaluation of marginal contributions can be detrimental for cooperation.

In section 2 of this paper we introduce the canonical model of cooperative games with transferable utility and present the axiom of intersecting population monotonicity with which we build a new characterization of the egalitarian value. In Section 3 we present our new model of games with roles. In section 4

we propose axioms for a desirable solution to this new type of problems as well as present our main results, which make use of the axiom and result presented in section 2. We study the robustness of our results in section 5. We offer applications of the new model and the results in section 6 before adding some concluding remarks in section 7.

Egalitarian value and Intersecting Population Monotonicity

Let U be a non-empty and infinite set of players. A coalition is a non-empty and finite subset of U. A coalitional game with transferable utility (also called a cooperative TU-game) is a pair (N, v) where N is a coalition and v is a function satisfying v : 2 N → R and v(∅) = 0. We note Γ the set of these games. Given a coalition S ⊆ N we write s = |S| its cardinality. For any S ⊊ N we will write (S, v |S ) the sub-game restricted to coalition S with v |S (T ) = v(T ), ∀T ⊆ S.

A player i ∈ N is a null player in the game (N, v) if v(S ∪ {i}) = v(S) for all S ⊆ N \ {i}. A player i ∈ N is a nullifying player in the game (N, v) if v(S) = 0,
for all S ⊆ N such that i ∈ S. Two players i, j ∈ N are equals in the game

(N, v) if v(S ∪ {i}) = v(S ∪ {j}) for all S ⊆ N \ {i, j}.
A TU-game v is said to be a convex game if it satisfies v(S ∪ {i}) -v(S) ≤ v(T ∪ {i}) -v(T ) for all i ∈ N and all S ⊆ T ⊆ N \ {i}. Convex game have been shown by [START_REF] Shapley | Cores of convex games[END_REF] to have a non-empty core 4 .

A sharing rule on Γ is a function ψ which associates with each game (N, v) ∈ Γ a vector ψ(N, v) ∈ R N . The Shapley value, first presented by [START_REF] Shapley | A value for n-person games. Contributions to the Theory of Games II[END_REF], rewards players based on their marginal contributions, weighted by all the possible orders of entry of players in coalitions. It is noted Sh and defined as

Sh i (N, v) = S S⊆N i∈S (s -1)!(n -s)! n! [v(S) -v(S \ {i})], ∀i ∈ N.
The Egalitarian value divides equally the worth of coalition N between all players. It is noted EG and defined as

EG i (N, v) = v(N ) n , ∀i ∈ N.
We introduce the following classical axioms for sharing rules over TU-games.

Efficiency. For each game (N, v) ∈ Γ it holds that i∈N ψ i (N, v) = v(N ).
Linearity. For each couple games (N, v), (N, w) ∈ Γ and each real number

α ∈ R, it holds that ψ(N, v + αw) = ψ(N, v) + αψ(N, w).
Null player. For each game (N, v) ∈ Γ and each null player i ∈ N , it holds

that ψ i (N, v) = 0.
Nullifying player. For each game (N, v) ∈ Γ and each nullifying player i ∈ N , it holds that ψ i (N, v) = 0.

Equal treatment of equals. For each game (N, v) ∈ Γ and each couple of players i, j ∈ N who are equals in (N, v), it holds that

ψ i (N, v) = ψ j (N, v).
As shown by [START_REF] Shubik | Incentives, decentralized control, the assignment of joint costs and internal pricing[END_REF] We introduce a new axiom of intersecting population monotonicity by taking another direction from [START_REF] Thomson | Problems of Fair Division and the Egalitarian Solution[END_REF]. Let us consider a given population P offered with the choice of cooperating with two different groups of the same size. Our axiom states that no member of population P can be worse off if the population chooses the group with whom it has the best synergy. In the context of cooperative games, given two coalition of players whose intersection is non-empty, we ask that no member of the intersection is rewarded less when considering the coalition with the better worth. Formally,

Intersecting population monotonicity. For each game (N, v) with n ≥ 3 and each couple of coalitions S, T ⊆ N with s = t, S ∩ T ̸ = ∅ if v(S) ≥ v(T ) it holds that ψ i (S, v | S ) ≥ ψ i (T, v | T ), ∀i ∈ S ∩ T .
We give an alternative interpretation for this axiom. If a subgroup P of S decides to replace the other members of S by an identical number of outsiders, then the members of P should not be worse off if the newly formed coalition T is at least as productive as the original coalition S. Note that this axiom puts no requirement on any payoffs in the game (N, v). Moreover, for the two sub-games considered, it says nothing about the payoffs of players outside the intersection of coalitions S and T . As a result, this axiom is not as strong as it can first appear.

The axiom of intersecting population monotonicity is also quite different from the axiom of grand coalition monotonicity used by [START_REF] Casajus | Weakly monotonic solutions for cooperative games[END_REF] to characterize the egalitarian value. Grand coalition monotonicity compares the same population involved in two different games, whereas we compare two different populations. Nonetheless some proximity can be seen in the sense that our axiom also provides a monotonic approach to the egalitarian principle, as we will see with our first results.

Lemma 1 For every game (N, v) ∈ Γ such that all players i ∈ N are equal in (N, v), the sharing rule ψ(N, v) satisfies efficiency and intersecting population monotonicity if and only if ψ i (N, v) = ψ j (N, v), ∀i, j ∈ N .

Proof . We have a game (N, v) such that all players are equal in v. Let p ∈ U \ N , and let us consider a bigger game (N + , w) such that N + = N ∪ {p} and w |N (S) = v(S) for all S ⊆ N and such that all players of N + are equal in w. This means that w(S) = c s , ∀S ⊆ N + , with (c s ) s=1,...,n+1 a vector of real numbers. Since U is infinite, such a construction is always possible 5 .

We consider coalition N and we will prove that

ψ i (N, w |N ) = ψ j (N, w |N )
for all i, j ∈ N . For any two i, j ∈ N , let us consider two additional coalitions S, T ⊊ N + of size n in which, respectively, players j and i are replaced with player p. We have

S = N \ {j} ∪ {p}, T = N \ {i} ∪ {p}, s = t = n, S ∩ T = N \ {i, j} ∪ {p}, S ∩ N = N \ {j}, T ∩ N = N \ {i}.
By definition of w, we have w(S) = w(T ) = v(N ). Such a construction of S and T is always feasible if n ≥ 3. Since the worths of coalitions S, T and N are equal, we use intersecting population monotonicity in both directions and obtain the following useful conditions on ψ:

ψ k (S, w |S ) = ψ k (T, w |T ), ∀k ∈ S ∩ T, ψ i (S, w |S ) = ψ i (N, w |N ), ψ j (T, w |T ) = ψ j (N, w |N ).
By efficiency we have the following condition on ψ:

k∈S ψ k (S, w |S ) = w(S) = w(T ) = k∈T ψ k (T, w |T ).
Therefore we obtain that

k∈S∩T ψ k (T, w |T ) = k∈S∩T ψ k (S, w |S ) ⇔ k∈S∩T ψ k (T, w |T ) + ψ i (S, w |S ) = k∈S∩T ψ k (S, w |S ) + ψ i (S, w |S ) ⇔ k∈S∩T ψ k (T, w |T ) + ψ i (S, w |S ) = w(S) ⇔ k∈S∩T ψ k (T, w |T ) + ψ i (S, w |S ) = k∈S∩T ψ k (T, w |T ) + ψ j (T, w |T ) ⇔ ψ i (S, w |S ) = ψ j (T, w |T ) ⇔ ψ i (N, w |N ) = ψ j (N, w |N ).
This proves that for any two players i, j ∈ N we have

ψ i (N, w |N ) = ψ j (N, w |N )
which, by definition of w, is equivalent to

ψ i (N, v) = ψ j (N, v). □ Theorem 1 6
For every game (N, v) ∈ Γ, the sharing rule ψ(N, v) satisfies efficiency and intersecting population monotonicity if and only if ψ = EG.

Proof . We know that EG satisfies efficiency. It is trivial that it satisfies intersecting population monotonicity. Let us prove that, for games (N, v) ∈ Γ if a sharing rule ψ satisfies efficiency and intersecting population monotonicity then it is the egalitarian value.

Let us consider a game (N, v) and a player i ∈ N . We can build a game

(N i , w i ) with N i such that N ⊊ N i , ∃T ⊊ N i with N ∩ T = {i} and n = t.
We also put the following conditions on w i :

w i (P ) = v(P ), ∀P ⊆ N w i (T ) = v(N ), w i (P ) = v({i}), ∀P ⊊ T, P ̸ = {i}.
Such a construction is always possible since U is infinite. 7 By definition we

have (w i ) | N (S) = v(S) for all S ⊆ N hence ψ i (N, (w i ) | N ) = ψ i (N, v).
Using efficiency and lemma 1 we have

ψ i (T, (w i ) | T ) = (w i ) | T (T ) t = v(N ) n
since all players j ∈ T are equals in (w i ) |T . As we have N, T ⊆ N i , n = t and w i (T ) = w i (N ) using intersecting population monotonicity we obtain that

ψ i (T, (w i ) | T ) ≥ ψ i (N, (w i ) | N )
and

ψ i (N, (w i ) | N ) ≥ ψ i (T, (w i ) | T )
which means

ψ i (N, (w i ) | N ) = ψ i (T, (w i ) | T ) = v(N ) n = ψ i (N, v).
Since U is an infinite set it is possible to build a game (N i , w i ) that satisfies our above conditions for every i ∈ N . Hence we have that

ψ i (N, v) = v(N )
n , ∀i ∈ N , which concludes our proof.

□

We now prove that our three axioms are logically independent.

Efficiency

Consider the sharing rule f 1 (N, v) = αEG(N, v) with α ̸ = 1. It obviously satisfies all axioms but efficiency.

Intersecting population monotonicity

Consider the Shapley value. It satisfies efficiency and equal treatment of equals.

Application 2 in section 6 shows it does not satisfy intersecting population monotonicity.

Games with roles

We develop here a new model of cooperative games in which players have a specific role they can play in the coalitions.

Consider an entity (a firm for instance) is setting up a project that asks for specific roles to be performed. It has several players (candidates, employees, other firms) that can fill these roles. Each player can only assume a specific role but several players are able to play the same role. Once the team is put together, the player that are parts of the team will work on the project which implies working all together as well as in subgroups and alone.

Formally, we consider k roles with 1 < k < n. Each player i ∈ N can perform only one specific role and we note K α ⊊ N the set of players able to perform the role α ∈ {1, . . . , k}. We write K = {K 1 , K 2 , ..., K k } the set of those sets.

We write K i the set of players able to perform the same role as player i. Noting P(N ) the partitions of the player set N , the set K ∈ P(N ) can be understood as a coalition structure as introduced by Aumann and Dreze (1974). We have

1≤α≤k K α = N and K α1 K α2 = ∅ for any α 1 , α 2 ∈ {1, . . . , k} and α 1 ̸ = α 2 .
We name full coalition a coalition where each role is filled. We note F = Π 1≤α≤k K α the set of full coalitions, defined by the cartesian products of the sets that partition players into roles. Therefore, the size of each full coalition is k.

We also note F i = {F l ∈ F | i ∈ F l } the set of full coalitions to which player i belongs. We assume that sub-coalitions of a full coalitions are still feasible but that coalitions with excess players are not 8 . Hence the set of feasible coalitions F N,K is the union of the subsets of full coalitions.

We consider that a value is generated from the cooperation of players. This production is modelled by a characteristic function v defined over 2 N . Hence, we define a game with roles as a triplet (N, v, K) with N ⊆ U , (N, v) ∈ Γ and K ∈ P(N ). We note Γ role the set of those games. This model naturally raises three issues: the formation of one of the full coalitions, the allocation of its worth, and the influence of this allocation on the stability of the formation process of the selected full coalition. To address these issues, a solution on Γ role is a pair ϕ = (µ, ψ). The function µ is a mechanism that selects for each (N, v, K) ∈ Γ role one of its full coalitions µ(N, v, K) and ψ is a sharing rule on Γ.

Optimality and outside options

A solution ϕ = (µ, ψ) on Γ role consists of a coalition selector µ and a sharing rule ψ and is therefore two-dimensional. As such we will put requirements on both dimensions separately as well as a requirement on their interaction.

One desirable and reasonable requirement for a mechanism µ would be that it selects an optimal full coalition. A full coalition F l ∈ F is optimal if and only

if v(F l ) = max Fp∈F v(F p ). We note F opt = {F l ∈ F | v(F l ) = max Fp∈F v(F p )}
the set of optimal full coalitions. Hence, we ask that the mechanism µ selects a coalition that is optimal 9 . This gives us a first axiom:

Optimality. For every games with roles (N, v, K) ∈ Γ role it holds that µ(N, v, K) ∈ F opt .

Next we invoke the standard axiom of efficiency (as defined in section 2)

for the sharing rule ψ on Γ.

Finally, we introduce an axiom that concerns the interaction between µ and ψ. Given a solution ϕ = (µ, ψ), can we make sure that µ(N, v, K) is stable ? That is to say, given ψ, can we make sure that no players in µ(N, v, K) would be better off associating with other players ? We consider the payoff a player can obtain with another coalition as a player's outside options. Hence the following question : does the solution ϕ = (µ, ψ) is such that no player that belongs to µ(N, v, K) has better outside options ?10 Formally:

No outside options. For every game with roles

(N, v, K) ∈ Γ role it holds that ϕ = (µ, ψ) is such that ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp ), ∀F p ∈ F i , ∀i ∈ µ(N, v, K).
In the following two results we explore the relationship between optimality, no outside options and intersecting population monotonicity.

Theorem 2

Let us have a game with role (N, v, K) ∈ Γ role and a solution ϕ(µ, ψ). If µ satisfies optimality then ϕ satisfying no outside options is equivalent to ψ satisfying intersecting population monotonicity.

Proof . We first prove that if µ satisfies optimality and ψ satisfies intersecting population monotonicity then ϕ satisfies no outside options.

Let us consider a solution ϕ = (µ, ψ). If µ satisfies optimality then µ(N, v, K) ∈

F opt which implies that v(µ(N, v, K)) ≥ v(F p ) for all F p ∈ F . By definition we have that |µ(N, v, K)| = |F p |, for all F p ∈ F . Using intersecting population monotonicity we know then that ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp ) for all F p such that µ(N, v, K) ∩ F p ̸ = ∅, ∀i ∈ µ(N, v, K) ∩ F p . By definition, if i ∈ µ(N, v, K) ∩ F p then F p ∈ F i . Hence ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp )
for all F p ∈ F i , for all i ∈ µ(N, v, K). This coincides with the requirement for no outside options.

We now prove that if µ satisfies optimality and ϕ satisfies no outside options then ψ satisfies intersecting population monotonicity.

Let us consider a solution ϕ = (µ, ψ). If µ satisfies optimality then µ(N, v, K) ∈

F opt which implies that v(µ(N, v, K)) ≥ v(F p ) for all F p ∈ F . Using no out- side options we know that ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp ), ∀F p ∈ F i , ∀i ∈ µ(N, v, K). All F p ∈ F i for a given i ∈ µ(N, v, K) are all F p ∈ F such that i ∈ µ(N, v, K) ∩ F p .
Hence the requirement for no outside options be- N,v,K). This requirement must be true ∀K ∈ P(N ). Whatever K ∈ P(N ) is considered we always have

comes ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp ), ∀F p ∈ F , F p ∩ µ(N, v, K) ̸ = ∅, ∀i ∈ F p ∩ µ(
F p ⊆ N , µ(N, v, K) ⊆ N , |F p | = |µ(N, v, K)| for all F p ∈ F , F p ∩ µ(N, v, K) ̸ = ∅.
This being true for all K means that, given a certain TU-game (N, v) ∈ Γ, for every two coalitions S, T ⊆ N with S ∩ T ̸ = ∅, s = t we have the following requirement on ψ:

ψ i (S, v | S ) ≥ ψ i (T, v | T ), ∀i ∈ S ∩ T, whenever v(S) ≥ v(T ).
This coincides with intersection population monotonicity for ψ.

□ Theorem 3

For every game with role (N, v, K) ∈ Γ role such that k ≤ n+1 2 . Consider the solution ϕ(µ, ψ) with µ satisfying optimality and ψ satisfying efficiency. The solution ϕ satisfies no outside options if and only if ψ = EG.

Proof . We first prove the existence part. We consider a solution ϕ(µ, EG) with µ satisfying optimality. We know that EG satisfies efficiency and intersecting population monotonicity. From theorem 2 we conclude that ϕ(µ, EG) satisfies no outside options. Now the unicity part. We consider a solution ϕ(µ, ψ) with µ satisfying optimality and ψ satisfying efficiency. Since ϕ satisfies no outside options, from theorem 2 we know that ψ satisfies intersecting population monotonicity. Given the game (N, v, K), every full coalition is of size k. Since k ≤ n+1 2 we can use theorem 1 to conclude that ψ = EG.

□

Robustness

In this section we study the robustness of our results by looking at different variations of the no outside options axiom.

The definition of the axiom of no outside option can appear quite restrictive. We previously defined the no outside option axiom on the following idea of coalitional stability: a coalition can only form if each player consents to its formation. This conception is in line with the model 1 of stability presented in [START_REF] Hart | Endogenous formation of coalitions[END_REF]. Additionally, this conception can also be interpreted as a requirement for a deviation. Therefore, in this sense, coalitional stability can be thought as the inability for players to form another coalition than the one they are currently in. This second approach would give us the following axiom:

No outside options (weak). For every game with roles

(N, v, K) ∈ Γ role , it holds that the solution ϕ(N, v, K) = (µ, ψ) is such that ∃i ∈ µ(N, v, K) ∩ F p such that ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ i (F p , v |Fp ), ∀F p ∈ F.
An alternative coalition F p can form only if no member of µ(N, v, K) ∩ F p object to its formation. Now, let us consider the following axiom defined for sharing rule over TUgames:

Intersecting population monotonicity (weak). For each game (N, v) ∈ Γ with n ≥ 3 and each couple of coalitions S, T ⊆ N with

s = t, S ∩ T ̸ = ∅ if v(S) ≥ v(T ) it holds that ∃i ∈ S ∩ T such that ψ i (S, v | S ) ≥ ψ i (T, v | T ).
This axiom is sufficient to replace the intersecting population monotonicity in the proof of theorem 1. Theorem 2 establishes a link between the axiom of intersection population monotonicity for sharing rule on TU-games and no outside options for solutions on games with role, when µ satisfies optimality.

Similarly we can write the following theorem:

Theorem 4
Let us have a game with role (N, v, K) ∈ Γ role and a solution ϕ(µ, ψ). If µ satisfies optimality then ϕ satisfying no outside options (weak) is equivalent to ψ satisfying intersecting population monotonicity (weak).

This result draws the same link between no outside options (weak) and intersecting population monotonicity (weak). The proofs for this theorem is omitted as it is quite similar to its counterpart. This implies that theorem 3 still holds when we replace no outside options by this weaker version.

Going in another direction, we can build a more restrictive version of the no outside option axiom. We require that the payoff obtained by each player with the chosen full coalition µ(N, v, K) should be at least as large as the payoff that any other player able to perform the same role can obtain in any other full coalition. Application 1 in section 6 illustrate a situation in which this requirement is relevant.

Therefore, we introduce the following axiom:

No outside options (strong). For every game with roles (N, v, K) ∈ Γ role it holds that ϕ = (µ, ψ) is such that ψ i (µ(N, v, K), v |µ(N,v,K) ) ≥ ψ j (F p , v |Fp ),

∀F p ∈ F j , ∀j ∈ K i , ∀i ∈ µ(N, v, K).
In this stronger version we compare the payoff of a player i of the selected full coalition µ(N, v, K) with every payoff of every other players able to perform the same role as i in every full coalitions. It is clear that this axiom is a stronger version of no outside options and that no outside options (strong) implies no outside options. We now prove the following result.

Theorem 5

For every game with role (N, v, K) ∈ Γ role such that k ≤ n+1 2 . Consider the solution ϕ(µ, ψ) with µ satisfying optimality and ψ satisfying efficiency. The solution ϕ satisfies no outside options (strong) if and only if ψ = EG.

Outside of tournaments and organized competitions where professional teams oppose each other, players that want to play a game join a queue of waiting players. These players are then matched by a match-making algorithm. These algorithms match players based on various informations, one of which is a skill rating. These ratings are updated after each games. Usually, a win leads to an increase in rating and a loss to a decrease in rating.

The design of skill ratings is the subject of an active literature (see for In the context of our model, each player in the video-game can be represented by a role K α . Each player can pick a different character, represented in our model by a player j ∈ K α . We are interested in a rating system that shares the value of the outcome of a match between team members in such a way that players are encouraged to form the best team of characters. Dota 2 rating system used to be a modified ELO system until april 2023 of finding a sharing rule for this bonus that gives the manager the incentives to put together the team that produce the most valuable work. Formally we have N = {1, 2, 3, 4, 5} and the role structure K = {{1}, {2, 3}, {4, 5}}. The full coalitions are F = {{1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}}. The set of feasible coalition is given by

F N,K = {{1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}}.
Imagine that 2 has poor synergy with both 4 and 5 while 3 has a really good synergy with 5 (only amplified under 1 management) but can't stand working with 4. However, as a manager, 1 is able to be at its best when bridging the divide between 4 and 3. The following worths for the feasible coalitions express this situation 13 . The worths express the value of intermediary works as well as the end product. We have v({i}) = 0 for all i ∈ N and S 12 [START_REF] Gómez | Centrality and power in social networks: a game theoretic approach[END_REF] Across the organization, using a solution ϕ(µ, EG) with µ satisfying optimality ensures that the most effective team will be formed willingly.

Concluding remarks

In this paper we developed a model of cooperative environments in which multiple roles have to be performed. This model is built on the canonical model of cooperative TU-games. We showed that, if the number of players is sufficiently larger than the number of roles, an egalitarian approach to the allocation of the outcome of cooperation is the only efficient approach that always provide incentive for the best coalition to form. We showed that this result is still true when a weaker or stronger definition of stability is considered.

In our approach, the full domain of cooperative games in considered. Application 2 shows that the solution ϕ = (µ, Sh) does not satisfy no outside options even for convex games. On specific classes of games, it would be interesting to determine whether other sharing rule than the egalitarian value are compatible with the axiom of no outside options

In addition, it is well known that the payoff vector given by the egalitarian value often lies outside the core, even for convex games. Consequently the selected full coalition can offer no outside options in the sense that all players involved prefer working together than within another full coalition but not be stable in the sense that this cooperation might fail. As such our result could also be framed as an impossibility result: there are no solution ϕ = (µ, ψ) with µ satisfying optimality and ϕ satisfying no outside options such that ψ(µ, v |µ ) lies in the core of (µ, v |µ ). However, the fact that ψ might not be in the core can possibly be irrelevant if we consider situations where the cooperation is based on free association but, once established, is subject to binding agreements. The applications we have presented show that these situations can exist in different contexts, on which our result can provide insights.

In conclusion, on the basis of the main result of this paper, we argue that egalitarianism is the only approach that provides incentives towards optimality for every cooperative situations that satisfies the following conditions: cooperation is based on free association; the numbers of players needed is sufficiently smaller than the numbers of candidates.

  More generally, this paper joins a line of works enriching the standard model of cooperative game theory by considering new structures of cooperation. A large part of this research focuses on structures that restrict cooperation, by modelling either asymmetries in communication or hierarchical constraints. In two papers, Myerson modelled communication by, respectively a graph (1977) and an hyper-graph (1980), that both represent a restriction of communication between players. Various solution concepts have been proposed for such games.

  Béal et al. (2021) offer a deeper discussion on how the same mathematical model (in this case, of hierarchical constraint) can affect the overall game in different ways. In this sense, our work is in line with this trend of research.

  , the Shapley value is the unique sharing rule satisfying efficiency, linearity, null player and equal treatment of equals. The egalitarian value has been shown by van den Brink (2007) to be the unique sharing rule satisfying efficiency, linearity, nullifying player and the equal treatment of equals. In 1983, Thomson introduced the axiom of population monotonicity in the context of fair division. This axiom stated that when facing a given problem of fair division, no player should be better off by having to share with more players than less. Chun (1986) and later Chun and Park (2012) expanded on this axiom on, respectively, fair division problem and allocation problems in cooperative games. This weakened version stated that when a population was joined by newcomers, all members of the original population are all either better off or worse off.

instance

  Chen et al. (2017), Minka et al. (2018)) especially in the context of games opposing teams of players. According to Minka et al. (2018), incentives for cooperative behaviours are a concern of developers when designing their rating system, mainly because the rating is an evaluation of a player's level of competency and, as such, often becomes a long term objective for players. One popular genre of team-based competitive video games are the multiplayer online battle arena games (or MOBA games, for short). The MOBA game Dota 2, developped by Valve Corporation, is currently the biggest video game in terms of competitive earnings 11 . In this game, two teams of five players must defend their own base while destroying the base of the other team. The game is played on a symmetrical map. Every player independently controls a character. Every character has specific abilities and a different play style. There are currently 124 different characters to pick from. One of the key strategic element is therefore the synergy between the different characters and their style of play.

Application 3 (

 3 Water sharing)Finally we propose an application of the model to a public economy problem.Let us consider a water source with N potential consumers. The consumption time is broken into k time periods 1, ..., k. Each consumer only need water in one period. We note K α the set of agents wanting to consume water at periodα for α ∈ {1, 2, . . . , k}. K = {K 1 , K 2 , . . . , K k } is the induced partition on N .Every potential consumer i ∈ N has a need b i > 0 and each period k has a maximum capacity of C K k . We want to choose the set of consumers such that there is no overconsumption of the resource while the least amount of water goes unused. Hence the worth of a coalition S ⊆ N is given by v(S) = α∈{1,...,k} max i∈S∩Kα {min(b i ; C Kα )}, and is the feasible total consumption of coalition S that leaves the least amount of water unused.

  The solution ϕ(µ, Sh) with µ satisfying optimality does not satisfy no outside options. The full coalition giving the highest worth is {1, 3, 5} but 1 would prefer that 3 and 4 accept to work on her team. Similarly, 4 would rather work with 2 than with 3. In each full coalitions, at least one player has an outside options when the solution ϕ uses the Shapley value.

		14 15 24	25	34 35 124 125 134 135
	v(S) 0.5 0	1	1 0.5 0.75 0	2	2	2	3	4
	Let us first consider a sharing rule based on marginal contribution: the
	Shapley value. It gives the following results for each full coalitions
	Full coalition F p ∈ F 124	125		134	135
	Sh 1 (F p , v |Fp )	0.75 ∼ 0.666 ∼ 1.166 ∼ 0.833
	Sh 2 (F p , v |Fp )	0.5 ∼ 0.541		-	-
	Sh 3 (F p , v |Fp )	-	-		∼ 0.666 ∼ 1.333
	Sh 4 (F p , v |Fp )	0.75	-		∼ 0.666	-
	Sh 5 (F p , v |Fp )	-	∼ 0.791		-	∼ 1.833
	We can see that the Shapley value does not satisfy intersecting population mono-
	tonicity.

I would like to thank Sylvain Béal who was always available to provide insightful feedbacks and comments at various stages of this paper. I also thank Stéphane Gonzalez for seeing some merits in the original draft, which gave me the confidence to pursue this idea.

Email address: floriannavarro.unipro@gmail.com Preprint submitted to Elsevier December 5, 2023

For instances two efficient extensions of the Myerson value were proposed by Hamiache (2012) and Béal et al. (2015). Examples of variations on the theme can be found by Gómez et al. (2003) and González-Arangüena et al. (2003).

Quoting this perfect definition from[START_REF] Shapley | Cores of convex games[END_REF], "the core of [a TU-game] is the set of feasible outcomes that cannot be improved upon by any coalition of players".

If U is finite, this construction is always possible for any N such that n ≤ u -1.

If we consider a finite universe of player U , this result holds for any N such that n ≤ u+1 2 .

If U is finite, such a construction is possible for any N such that n ≤ u+1 2 .

As actors in a play have scenes with each other as well as monologues and a full-house grand finale, our players working on a project have tasks to perform in subgroups or alone.

Note that there can be multiple optimal full coalitions. We only ask that µ selects one of these. Hence it is possible that µ is not anonymous.

The concept of outside options considered here is in line with the one used by[START_REF] Casajus | Outside options, component efficiency, and stability[END_REF] in the context of communication graph games.

Dota 2 cumulated a total prize pool of 32,954,188.52$ in 2022, across

competitive events according to the website esportsearnings.com. Source: https://www.esportsearnings.com/history/2022/games, consulted on the 11/7/2023.

Note that in this example, for any Fp ∈ F the sub-game v |Fp is convex.

Proof . We first prove the existence part. Consider a solution ϕ(µ, EG) with µ satisfying optimality. We know that EG satisfies efficiency. We have

as well as

Since µ satisfies optimality we have

Hence, ϕ(µ, ED) satisfies no outside options (strong).

Now the unicity part. We consider a solution ϕ(µ, ψ) with µ satisfying optimality and ψ satisfying efficiency. Since ϕ satisfies no outside options (strong), it satisfies no outside options. From theorem 2 we know that ψ satisfies intersecting population monotonicity. Given the game (N, v, K), every full coalition is of size k. Since k ≤ n+1 2 we can use theorem 1 to conclude that ψ = EG. □

Applications

In this section we illustrate our results by applying our model to rating systems in online gaming (application 1), to organizational settings (application 2) and to resource management (application 3). We intend to show by these applications that our model can tackle widely different issues. Application 1 illustrate both the model and our results in an already used mechanism. Application 2 provide another context where the model can be used, and show how the Shapley value would behave in this context. Application 3 only describe how the model could be used to describe a water sharing problem.

Application 1 (Team Matchmaking Rating (MMR))

In modern team-based competitive video games, players from over the world team up with strangers to compete against other strangers.

where it was replaced by a modified Glicko system (a bayesian variation of the ELO system). These two systems were initially developed for chess rankings.

Compared to other rating system used in competitive video games such as the TrueSkill algorithm [START_REF] Minka | TrueSkill 2: An improved Bayesian skill rating system[END_REF], they do not measure the individual performance of a player. When a team wins (or loses), every member of a team is given the same bonus (or malus) to its rating.

The way these two systems used for Dota 2 allocate the value of a match's outcome between team members is egalitarian since every member gets the same change in rating. This system is not always well received by players 12 .

Players can have a tendency to overestimate their skills, and blame a loss on their teammates. However, theorem 5 shows that this egalitarian approach is the only one where players always have an incentive to form the most efficient team (or what they think would be the most efficient team). In particular, a system relying on a measure of individual performance would lead players to try and maximize their performance according to this measure, possibly to the detriment of the one obvious common goal: a win for the team.

Hence, while the system that Valve Corporation adopted for Dota 2 is already easier to design and compute than a complex system that would try to capture a individual performance within a team effort, our results show that it also has the benefit of a strong incentive to behave in line with the common goal.

Application 2 (Team project)

Consider the following situation: a business trusts one of its managers to take on a new project that needs two other specialists. She is given freedom to choose other employees to fill these roles. The other employees are free to reject the proposition. Once the project is done, the team will get a bonus proportional to the estimated value of the work produced. This raises the problem


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Relay oral therapy in

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Febrile urinary tract infections (FUTIs) are the most common proven bacterial infections among infants and young children presenting fever without a source [START_REF] Dayan | Urinary Tract Infections Caused by Community-Acquired Extended-Spectrum β-Lactamase-Producing and Nonproducing Bacteria: A Comparative Study[END_REF][START_REF] Roberts | Urinary tract infection: clinical practice guideline for the diagnosis and management of the initial UTI in febrile infants and children 2 to 24 months[END_REF][START_REF] Shaikh | Prevalence of urinary tract infection in childhood: a metaanalysis[END_REF]. FUTIs are most frequently due to infection with Enterobacteriaceae, mainly Escherichia coli [START_REF] Hanna-Wakim | Epidemiology and characteristics of urinary tract infections in children and adolescents[END_REF][START_REF] Zorc | Diagnosis and Management of Pediatric Urinary Tract Infections[END_REF]. The emergence of ESBL-producing Enterobacteriaceae (ESBL-E) as a cause of FUTI presents a serious threat to public health given that therapeutic options are limited [START_REF] Pitout | Extended-spectrum β-lactamase-producing Enterobacteriaceae: an emerging public-health concern[END_REF][START_REF] Livni | Treatment of resistant bacterial infections in children: thinking inside and outside the box[END_REF][START_REF] Fan | Rise of community-onset urinary tract infection caused by extended-spectrum β-lactamase-producing Escherichia coli in children[END_REF].

The main international guidelines recommend prescription of an oral antibiotic for FUTI [START_REF]Urinary tract infection in children[END_REF][START_REF] Roberts | American Academy of Pediatrics Subcommittee on Urinary Tract Infection, Steering Committee on Quality Improvement and Management. Urinary tract infection: clinical practice guideline for diagnosis and management of the initial UTI in febrile infants and children 2 to 24 months[END_REF][START_REF] Stein | European Association of Urology and European Society for Pediatric Urology. Urinary tract infections in children: EAU/ESPU guidelines[END_REF]. Parenteral treatment is required only for children who are severely ill or unable to retain oral intake. These guidelines suggest that FUTI treatment last 7 to 14 days. This broad range is due to the lack of sufficient data identifying the optimal treatment duration [START_REF] Strohmeier | Antibiotics for acute pyelonephritis in children[END_REF]. However, because ESBL strains also frequently harbor resistance genes to cotrimoxazole and quinolones, no oral compound is suitable to treat FUTI due to ESBL-E. Thus, a sequence treatment is recommended in these children. Because an intravenous antibiotic treatment active for ESBL-PE is prescribed in many centres, oral antibiotic relay is necessary afterward to reduce the total duration of hospitalization after checking the strain's antibiotic susceptibility. All the more so as most children are apyretic after a mean duration of antibiotic treatment of 2 to 4 days. This observation underscores the major value of using early oral treatment for this type of infection.

Indeed, children with a diagnosis of FUTI receive empirical treatment with broad-spectrum cephalosporins such as ceftriaxone or cefotaxime intravenously or cefixime orally. Some teams use aminoglycosides as first-line treatment [START_REF] Poey | Aminoglycosides Monotherapy as First-Line Treatment for Febrile Urinary Tract Infection in Children[END_REF]. After receiving the drug susceptibility testing (DST) results on urine culture, the treatment is switched to antimicrobials to which the causative organism is susceptible [START_REF] Roberts | Urinary tract infection: clinical practice guideline for the diagnosis and management of the initial UTI in febrile infants and children 2 to 24 months[END_REF].

In a previously published prospective cohort, cotrimoxazole and ciprofloxacin were the most frequently used antibiotics for oral relay therapy in FUTI due to ESBL-E in children [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF]. The non-orthodox amoxicillin-clavulanate and cefixime (AC-cefixime) combination was given to 86 (31.3%) children [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF]. The short-term evolution was similar whatever the efficacy of the empirical treatment [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF]. We sought to describe the risk of recurrence at 1 month after the end of treatment of the infectious episode according to the oral relay used (cotrimoxazole, ciprofloxacin or the non-orthodox AC-cefixime combination) in that same cohort. In a second step, we studied the susceptibility of ESBL-E strains on DST results to the AC-cefixime combination and to a large number of other antibiotics.

Materials and methods

Study design and population

We retrospectively identified children from the hospital-based active surveillance of FUTI due to ESBL-E created by the GPIP/ACTIV network. These children were included in the first part of a prospective observational study between March 2014 and March 2017 on behalf of the FUTI National Observatory due to ESBL-E in children, which was previously published [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF]. Inclusion and non-inclusion criteria, demographic features of patients and clinical outcomes were previously described [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF].

For this second analysis, all children with FUTI (as defined in the French recommendations) [START_REF] Cohen | Socie ´te ´de pathologie infectieuse de langue franc ¸aise. [Management of urinary tract infections in children. Recommendations of the Pediatric Infectious Diseases Group of the French Pediatrics Society and the French-Language Infectious Diseases Society[END_REF] due to ESBL-E were enrolled if the oral relay treatment (after empirical and definitive treatment) was cotrimoxazole, a fluoroquinolone or the non-orthodox AC-cefixime combination. Patients were excluded from the final analysis if their parent's email, address or telephone number could not be found or with no response to the interviewer's emails and calls.

We sent parents an email or a letter describing and explaining the reasons and expected results of our study and informing them of an upcoming phone call. Patients were given the opportunity to decline to participate in the study prior to the phone call, by return email to dpo@chicreteil.fr or letter. All legal guardians of included children provided oral informed consent.

Parents were then invited to a telephone interview of up to 10 min, during which they were asked "Did a recurrence occurred within the first month following the end of treatment of the initial FUTI due to ESBL-E?" If the answer was no, the interviewer ended the call. If the answer was yes, the investigator sought additional information: the precise diagnosis of the recurrence (cystitis or FUTI), which bacteria was isolated (with or without ESBL resistance mechanism), and whether the recurrence required antibiotic treatment, and if so, which one. In the event the parents did not remember the information mentioned above, the interviewer asked them to search for it in their child's "health record". If they did not have the health record at the time of the interview, a new telephone appointment was scheduled. Following the telephone call, the patient's participation in the study is documented in his or her medical record by the investigator.

Microbiology

ESBL-E strains identified during the first published study were sent to the E. coli National Reference Center (NRC) laboratory at Robert Debre ´Hospital (Paris, France) for further phenotype characterization. In brief, one colony of each morphologic type growing on the medium was identified by using the API20E system (bio Me ´rieux, Marcy l'Etoile, France) or with the Bruker Biotyper Matrix-Assisted Laser Desorption Ionization-Time of Flight Mass Spectrometer. Antibiotic susceptibility was determined by using the disc diffusion method on Mueller-Hinton agar and interpreted as specified by the European Committee on Antimicrobial Susceptibility Testing (EUCAST; http://www.eucast.org/). ESBL production was defined as synergy between clavulanic acid and at least one of the extended-spectrum cephalosporins (ceftazidime, cefotaxime, or cefepime) or aztreonam [START_REF]Performance Standards for Antimicrobial Susceptibility Testing[END_REF]. The minimum inhibitory concentration (MIC) of the AC-cefixime combination was determined by the E-test method (AB bio Me ´rieux, Solna, Sweden) as described [START_REF] Drieux | Phenotypic detection of extended-spectrum beta-lactamase production in Enterobacteriaceae: review and bench guide[END_REF]. The breakpoint used for the AC-cefixime combination was the one for cefixime (1 mg/L) as specified by EUCAST.

Statistical analysis

We compared categorical variables with chi-square or Fisher exact test and continuous variables with Mann-Whitney U tests. P<0.05 was considered statistically significant for all tests. The analysis was performed with STATA 14 (Stata Corp LP, College Station, TX).

Ethical approval

The data collection was approved by the French National Data Protection Commission (CNIL, no. A01886-33), the Committee on the Processing of Research Information (CCTIRS, no. 13.341) and the Ethics Committee of Creteil Intercommunal Hospital. Because this is a noninterventional study, our local ethics committee and the CCTIRS approved obtaining oral consent from parents for this retrospective part of the study. Written consent was not required. The study was registered at ClinicalTrials.gov (NCT02832258).

Results

Baseline demographics

Among the 283 children included in our previously published study, we identified 255 who received cotrimoxazole, ciprofloxacin or the AC-cefixime combination as the oral relay therapy. After excluding 56 patients for the reasons mentioned above, 199 were included in the analysis: 72 (36.2%) received cotrimoxazole, 38 (19.1%) ciprofloxacin and 89 (44.7%) ACcefixime combination (Fig 1, study flow chart).

Baseline characteristics of our study population are described in Table 1. Urine was collected using a collector bag (n = 87, 43.7%), midstream collection (n = 40, 20.1%) or urethral catheterization (n = 67, 33.7%). The bacteria involved in the initial FUTI were E. coli (n = 180; 90.5%), Klebsiella pneumoniae (n = 16, 8%), Klebsiella oxytoca (n = 1, 0.5%) and others (n = 2, 1%). Only 3 (1.5%) children had a positive blood culture, with the same bacteria as in urine samples.

Antimicrobial susceptibility testing data

In total, 249 strains of ESBL-E identified during the first published study were sent to the NRC for further phenotypic characterization. Were included 135 of the 199 (67.8%) strains involved in this clinical study, because some centers did not send the strains to the NRC and others sent strains from 2014 up to 2019. Overall, 97.6% of the 249 strains analyzed by the NRC were susceptible to the AC-cefixime combination (MIC � 1 with the E-test method). By comparison, only 33.9% of the strains were susceptible to cotrimoxazole and 52.2% to ciprofloxacin (38.2% to nalidixic acid) (Fig 2).

Among the 6 strains with a MIC > 1 mg/L to the AC-cefixime combination, all were susceptible to amikacin, and all had an effective intravenous therapeutic alternative including one of 3 drugs: cefoxitin, piperacillin-tazobactam and temocillin, which allows for carbapenems sparing. Furthermore, among the strains susceptible to the AC-cefixime combination, 96% were also susceptible to cefoxitin, 94.4% to temocillin and 73.5% to piperacillin-tazobactam.

Initial treatment and short outcome

Overall, 177 (88.9%) children received empirical treatment with a parenteral therapy including the following antibiotics (alone or in combination): ceftriaxone/cefotaxime (n = 117, 58.8%), amikacin (n = 106, 53.3%), other aminoglycosides (n = 19, 9.5%), penems (n = 3, 1.5%) (Table 2). The others received empirical treatment with an oral antibiotic therapy including cefixime alone (n = 18; 9%), ciprofloxacin (n = 1; 0.5%), amoxicillin-clavulanate (n = 1; 0.5%) and a combination of cefixime with amoxicillin (n = 1; 0.5%) or cefixime with AC (n = 1; 0.5%). After recovery of the strain's DST, treatment was changed to an effective intravenous antibiotic (EIA) therapy followed by the oral relay or directly to the oral relay therapy. The mean length of hospital stay (LOS) was 2.05 days (range 0-13), but about half of the patients (n = 99; 49.7%) received treatment on an outpatient basis. Among the 199 children included, 159 (79.9%) received an EIA therapy before oral relay: 138 (69.3%) received amikacin or another aminoglycoside, 19 (9.5%) penems and 2 (1%) piperacillin-tazobactam. The mean EIA therapy duration was 2.58 days (range 0-11) overall but differed in the 3 groups of oral relay therapy: 1.99 days (range 0-9) with cotrimoxazole, 2.82 (0-9) with ciprofloxacin and 2.98 days (0-11) with the AC-cefixime combination (Table 2). The other 40 (20.1%) patients received a non-effective oral or parenteral therapy before oral relay: 21 in the cotrimoxazole group, 15 in the AC-cefixime combination group and 4 in the ciprofloxacin group.

Recurrence at 1 month after treatment for a FUTI due to ESBL-E

Among the 199 children included, 9 (4.5%) reported a recurrence within the first month after the end of treatment: 4 (5.6%) with cotrimoxazole, 2 (5.3%) with ciprofloxacin and 3 (3.4%) with the AC-cefixime combination (Table 2). The 3 groups did not differ in recurrence rate (p = 0.8). The recurrences were 8 FUTIs and 1 cystitis and were caused mainly by an ESBL-E: E. coli in 6 cases and K. pneumonia in 1 case. The remaining recurrences were caused by a non-ESBL producing E. coli in 1 case and an unknown pathogen in the last case (missing data). We were not able to retrieve the information on the antibiotics used to treat these recurrence because in most cases the parents did not remember it and it was not clearly recorded in the child's "health record".

Patients who did or did not receive an EIA therapy before oral relay did not differ in FUTI recurrence rate at 1 month after the end of treatment (7 recurrences [4.4%] vs 2 recurrences [5.0%], respectively) (Table 3).

Notably, 4 of the 9 (44.4%) patients with a recurrence had an underlying urinary tract malformation as compared with 42 (21.1%) in the overall population: 18 (20.2%) of the AC-cefixime group, 9 (12.5%) of the cotrimoxazole group and 15 (39.5%) of the ciprofloxacin group. In addition, 4 of the 9 recurrences (44.4%) were related to an initial FUTI caused by Klebsiella (pneumoniae and oxytoca) strains, the rest to E. coli.

Discussion

To our knowledge, this is the first study comparing oral antibiotic relay in FUTI due to ESBL-E in children in terms of recurrence at 1 month after the end of treatment. We found no differences in the recurrence rate whatever the oral antibiotic relay used (cotrimoxazole, ciprofloxacin or the AC-cefixime combination). In addition, 97.6% of the ESBL-E strains analyzed by the NRC were susceptible to the AC-cefixime combination. This combination could be an interesting choice for oral relay, all the more so as few therapeutic options are available and only half of the same strains were susceptible to ciprofloxacin (and nalidixic acid) and a third to cotrimoxazole.

Our study used data from a large national prospective study that described FUTI due to ESBL-E in children over a 3-year period. It represents 199 of the 283 (70%) patients analyzed and previously published [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF]. The short-term evolution of these infections was similar when they were treated with effective or ineffective empirical therapy [START_REF] Madhi | Febrile urinary-tract infection due to extended-spectrum beta-lactamase-producing Enterobacteriaceae in children: A French prospective multicenter study[END_REF], but we showed a similar outcome in recurrence at 1 month after the end of treatment whatever the oral antibiotic relay used.

In our cohort of children, 159 (79.9%) received an EIA therapy as first-line treatment before oral relay (adapted to DST); of these, 19 (9.5%) received carbapenems. Patients who received an EIA therapy before oral relay and those who did not, had a similar recurrence rate at 1 month after the end of treatment. Hence, the positive outcome of FUTI due to ESBL-E that we report here, regardless of the antibiotic therapy received, supports a spare of carbapenems in treating these infections.

We also found that the mean duration of the EIA therapy before oral relay significantly differed among the 3 treatment groups. This finding may have affected the outcome, given that the AC-cefixime combination group received an EIA therapy for longer than the 2 other groups. However, in our opinion, this result only reflects the time needed to test the strain's susceptibility to the AC-cefixime combination (with the E-test method) after receiving the initial antibiogram results showing resistance to cotrimoxazole and ciprofloxacin.

Furthermore, the choice of the oral relay antibiotic does not seem to be the main determinant for recurrence because significantly more patients had an underlying urinary tract malformation in the recurrence group than in the overall population (44.4% vs 21.1%). These malformations are known to be major contributors for recurrent FUTI. Likewise, we noted a significant higher percentage of ESBL-producing Klebsiella (pneumoniae and oxytoca) strains causing the initial FUTI in patients with recurrence than in the overall population (44.4% vs 8.5%). Klebsiella sp. strains are usually less uropathogenic than E. coli strains but are more frequently involved in FUTI in children with urinary tract malformations.

The 2 major strengths of our study are the large number of patients with FUTI due to ESBL-E included from multiple centers throughout France and the analysis by the NRC of a large representative sample of the strains involved in this study (135/199; 67.8%).

Other studies have investigated the clinical effectiveness of an oral third-generation cephalosporin combined with AC and found high clinical cure rates but in a smaller number of adults [START_REF] Stuart | Ceftibuten plus amoxicillin-clavulanic acid for oral treatment of urinary tract infections with ESBL producing E. coli and K. pneumoniae: a retrospective observational case-series[END_REF][START_REF] Al-Tamimi | Effective Oral Combination Treatment for Extended-Spectrum Beta-Lactamase-Producing Escherichia coli[END_REF]. This situation emphasizes the need for larger prospective studies in children.

The main limitation of our study is that the diagnosis of pyelonephritis was not certain. Indeed, urine was collected by bladder probing or a collector bag as allowed by French recommendations. To have a definite diagnosis, a renal scintigraphy is recommended but limits the feasibility of such studies.

Another limitation might be the use of the E-test method for the DST of the ESBL-producing strains to the AC-cefixime combination because it is not standardized. However, several other studies, using various non-standardized methods [START_REF] Al-Tamimi | Effective Oral Combination Treatment for Extended-Spectrum Beta-Lactamase-Producing Escherichia coli[END_REF][START_REF] Livermore | Orthodox and unorthodox clavulanate combinations against extended-spectrum beta-lactamase producers[END_REF][START_REF] Campbell | Detection of favorable oral cephalosporin-clavulanate interactions by in vitro disk approximation susceptibility testing of extended-spectrum-Beta-lactamase-producing members of the enterobacteriaceae[END_REF], including the E-test [START_REF] Bingen | In vitro interaction between cefixime and amoxicillin-clavulanate against extended-spectrum-beta-lactamase-producing Escherichia coli causing urinary tract infection[END_REF], have shown in vitro synergy between third-generation cephalosporins and clavulanate. We chose the E-test because it can detect in vitro synergy between cefixime and clavulanate and for its ease of use in clinical laboratory settings [START_REF] Bingen | In vitro interaction between cefixime and amoxicillin-clavulanate against extended-spectrum-beta-lactamase-producing Escherichia coli causing urinary tract infection[END_REF]. In fact, our study reports a large number of strains (n = 249) tested by the NRC with the E-test method for susceptibility to the AC-cefixime combination, which was also used by many centers throughout France before treatment with this combination.

Although this was a retrospective study whose results need to be confirmed by prospective studies, our results support the use of the AC-cefixime combination for oral relay therapy of FUTI due to ESBL-E in children, especially if the strain is resistant to cotrimoxazole or fluoroquinolones. This solution could help many physicians in an era of increasing resistance to usual antibiotics, allowing for shorter parenteral treatment time and shorter hospitalizations.

Conclusions

The AC-cefixime combination represents an interesting therapeutic option for oral relay treatment of FUTI due to ESBL-E as the recurrence rate at 1 month after the end of treatment was the same when compared to cotrimoxazole and ciprofloxacin, and most ESBL-E strains remained susceptible to this unorthodox combination.

Fig 1 .

 1 Fig 1. Study flow chart. https://doi.org/10.1371/journal.pone.0257217.g001

Fig 2 .

 2 Fig 2. Susceptibility drug testing of the ESBL producing strains analyzed by the National Reference Center. Susceptible, resistant and intermediate results were analyzed following the latest EUCAST guidelines. https://doi.org/10.1371/journal.pone.0257217.g002

Table 1 . Baseline characteristics of study population. Characteristics Overall population (n = 199) Children receiving cotrimoxazole (n = 72) Children receiving ciprofloxacin (n = 38) Children receiving AC-cefixime combination (n = 89)

 1

	Median age, years (range)	1 (0.04-15.93)	0.83 (0.08-15.93)	1.38 (0.05-15.63)	1 (0.04-10.53)
	Girls, n (%)	118 (59.3)	44 (61.1)	23 (60.5)	51 (57.3)
	Previous FUTI, n (%)	43 (21.6)	10 (26.3)	12 (16.7)	21 (23.6)
	Urinary tract malformation, n	42 (21.1)	9 (12.5)	15 (39.5)	18 (20.2)
	(%)
	Antibiotic prophylaxis, n (%)	20 (10.1)	3 (7.9)	7 (9.7)	10 (11.2)
	Median CRP level, mg/L	79 (1-448)	80.5 (1-448)	98 (1.8-406)	69.5 (1-312)
	(range)
	Bacteria involved in initial
	episode, n (%)
	Escherichia coli	180 (90.5)	67 (93.1)	33 (86.8)	80 (89.9)
	Klebsiella	17 (8.5)	4 (5.6)	5 (13.2)	8 (9.0)

AC: amoxicillin-clavulanic acid; FUTI: febrile urinary tract malformation; CRP: C reactive protein https://doi.org/10.1371/journal.pone.0257217.t001

Table 2 . Treatment and clinical evolution, according to oral antibiotic relay. Characteristics Overall population (n = 199) Children receiving cotrimoxazole (n = 72) Children receiving ciprofloxacin (n = 38) Children receiving AC-cefixime combination (n = 89) Empirical antibiotic therapy, n (%)

 2

	Parenteral 3GC	117 (58.8)	47 (65.3)	24 (63.2)	46 (51.7)
	Amikacin	106 (53.3)	32 (44.4)	21 (55.3)	53 (59.6)
	Oral antibiotic alone	22 (11.1)	8 (11.1)	2 (5.3)	12 (13.5)
	Other aminoglycosides	19 (9.5)	5 (6.9)	4 (10.5)	10 (11.2)
	Penems	3 (1.5)	-	-	3 (3.4)
	EIA, n (%)	159 (79.9)	51 (70.8)	34 (89.5)	74 (83.1)
	Time to apyrexia, days, mean	1.73	1.52	2.00	1.80
	LOS, days, mean	2.05	1.73	2.42	2.15
	Duration of treatment (oral and/or	3.86	3.27	4.24	4.20
	intravenous) before oral relay, days, mean
	Duration of EIA, days, mean	2.58	1.99	2.82	2.98
	Duration of oral relay, days, mean	8.20	8.47	8.32	7.92
	Total duration of treatment, days, mean	12.06	11.74	12.55	12.11
	Recurrence, n (%)	9 (4.5)	4 (5.6)	2 (5.3)	3 (3.4)

AC: amoxicillin-clavulanic acid; 3GC: third-generation cephalosporin; EIA: effective intravenous antibiotic therapy before and after receiving drug susceptibility testing; LOS: length of hospital stay https://doi.org/10.1371/journal.pone.0257217.t002

Table 3 . Clinical evolution comparison between children who received and those who did not receive an effective intravenous antibiotic (EIA) therapy before oral relay.

 3

	No EIA before oral relay	EIA before oral relay
	(n = 40)	(n = 159)

PLOS ONE | https://doi.org/10.1371/journal.pone.0257217September 16, 2021

Acknowledgments

We are very grateful to Laura Smales for her contributions to the report.

We would like to acknowledge the help of Maxime Brussieux, Me ´lanieVassal, Claire Prieur and Elsa Sobral and ESBL-E Group.

ESBL-E Group: Agnès Ferroni (Necker), Jean Gaschignard, Marie Desmarest (Robert Debre ´), Sandra Biscardi, Said Aberrane (Cre ´teil), Hoang Vu-Thien, Didier Moissenet (Trousseau), Laetitia Beraud (Lyon), Ferielle Zenkhri, Gaèlle Cuzon (Bicêtre), Isabelle Poilane (Jean-Verdier), Alain Martinot, Rodrigue Dessein (Lille), Emmanuel Cixous, Anne Vachee (Roubaix), Be ´atrice Pangon (Versailles), Me ´lanie Cochez, Florence Doucet-Populaire (Be ´clère), Nevena Danekova, Bogdan Cojocaru, Catherine Branger (Louis Mourier), Vale ´rie Sivadon-Tardy (Ambroise Pare ´), Christèle Gras-Leguen, Jocelyne Caillon (Nantes), Sophie Boyer (Rouen), Anne Chace, Jack Breuil (Villeneuve-Saint George), Christine Orzechowski, Ximena Sanchez, He ´lène Garrec (Saint Camille), Isabelle Breant, Aure ´lia Pitsch (Melun), Philippe Traore, Ce ´dric Tahiri, Ce ´cile Farrugia (Sud-Essonne, site Dourdan), Sarah Ducrocq, Anne Farges-Berth (Nord-Essonne, site Longjumeau), Herve ´Haas, Benoit Starck (Nice), Alain Fiacre (Meaux), Abdelmalek Belgaid, Franck Labbe (Le Havre), Marion Decobert, Marie Noe ¨lle Adam (Nord-Essonne, site Orsay).

The authors received no specific funding for this work.

There are ethical and legal restrictions on sharing a de-identified dataset from our study. Our local ethics committee and CCTIRS do not allow our data to be made public. As with our first study published in PLOS ONE (from which the data was taken), our local committee and the CCTIRS did not allow the data to be made public. We added in the text: The protocol of this Interested researchers may send data access requests

Author Contributions

Conceptualization: Gabriel Lignieres, Camille Jung, Corinne Levy, Robert Cohen, Fouad Madhi.

Data curation: Gabriel Lignieres, Fouad Madhi.

Formal analysis: