Karolin Rose Krause
email: karolin.krause.16@ucl.ac.uk

Abiodun Sophie Chung

Olugbenga Adewuya

Anne Marie Albano

Rochelle Babins-Wagner

Laura Birkinshaw

Peter G Brann

Catharine S Creswell

Kathleen Delaney

Bruno Falissard

Abiodun O Adewuya

Cathy Creswell

Christopher B Forrest

Jennifer L Hudson

Shin-Ichi Ishikawa

Meghna Khatwani

Christian Kieling

Judi Krause

Kanika Malik

Vania Martínez

Faraz Mughal

Thomas H Ollendick

Say How Ong

Evie Thomas

George C Patton

Ulrike Ravens-Sieberer

Peter Szatmari

Lucas Walters

Bryan Young

Yue Zhao

Miranda Wolpert Author

International consensus on a standard

Keywords: Anxiety, Depression, Stress Disorders, Post-Traumatic, Obsessive-Compulsive Disorder, Patient Reported Outcomes, Outcome Measures, Patient-Relevant Outcome, Quality Improvement, Reference Standards, Delphi Technique, Child Health, Adolescent Health, Core Outcome Set

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.

Measuring Response to Clinical Care in Children and Young People with Anxiety, Depression, OCD or PTSD: An International Standard Set of Outcome Measures

Introduction

Depression and anxiety affect an estimated 4.4% and 3.6% of the world's population, and rank as the first and sixth largest contributor to health-related disability, respectively [START_REF]Depression and Other Common Mental Disorders: Global Health Estimates[END_REF]. These disorders frequently emerge in childhood and adolescence, and unless treated early and effectively, commonly adversely affect mental health and psychosocial outcomes across the life course [START_REF] Kessler | Age of onset of mental disorders: A review of recent literature[END_REF][START_REF] Albano | Childhood anxiety disorders[END_REF][START_REF] Clayborne | Systematic Review and Meta-Analysis: Adolescent Depression and Long-Term Psychosocial Outcomes[END_REF]. Despite an increase in mental health care provision over recent decades, service systems have failed to reduce the prevalence of these disorders in children and young people (CYP; 5). The global response requires holistic strengthening, not only in specialist mental health services, but also in primary care, community health, child health, and school settings.

In addition to resourcing and training in evidence-based care, one essential element of service strengthening is the systematic monitoring of patient progress [START_REF] Boswell | Implementing routine outcome monitoring in clinical practice: Benefits, challenges, and solutions[END_REF][START_REF] Scott | Using measurement-based care to enhance any treatment[END_REF]. Valid data on treatment outcome is an essential facet in evaluating care effectiveness and can inform the setting of strategic targets for health systems, comparisons between systems and services, and clinical decision-making on a case-by-case basis [START_REF] Barkham | Bridging Evidence-Based Practice and Practice-Based Evidence: Psychological Therapies[END_REF][START_REF] Bickman | A Measurement Feedback System (MFS) Is Necessary to Improve Mental Health Outcomes[END_REF].

Currently there is neither agreement, nor global guidance on how best to track the response to clinical care for anxiety and depression in CYP. Uptake of routine outcome measurement remains low, and where it occurs, there is considerable variation in outcomes, instruments, and assessment time points, with a recent review recording 20 different measures used to assess primary outcome across 38 studies of routine treatment for youth anxiety and depression [START_REF] Bear | Systematic Review and Meta-analysis: Outcomes of Routine Specialist Mental Health Care for Young People With Depression and/or Anxiety [published online ahead of print[END_REF]. Resulting data gaps and inconsistencies severely limit the potential for comparing different models of clinical care, identifying good practice, and informing quality improvement efforts. This initiative aimed to address this challenge by devising a Standard Set, that is, a consensus-based standardised collection of treatment outcomes to be measured and reported as a minimum by all those providing relevant care [START_REF] Clarke | Core outcome sets and systematic reviews[END_REF].

To ensure that this standard is meaningful and acceptable to its intended users, the International Consortium for Health Outcomes Measurement (ICHOM) convened an international working group of service user representatives, practitioners, and researchers to build consensus on a set of outcome domains, measurement instruments, case-mix factors (i.e., case characteristics that should be considered when adjusting for differences in the composition of service user populations, or care provision across settings) and measurement timepoints to recommend.

Panel 1. The ICHOM Approach to Standard Set Development

ICHOM is a non-profit organisation specialised in the development of condition-specific Standard Sets for clinical practice. ICHOM has supported the development of more than 28 existing Standard Sets, including one for adult depression and anxiety [START_REF] Obbarius | Standardization of health outcomes assessment for depression and anxiety: recommendations from the ICHOM Depression and Anxiety Working Group[END_REF]. Outcome measurement is approached within a framework of personcentred and value-based healthcare, where value is defined as the health outcomes achieved, relative to the resources invested, rather than the volume of services delivered [START_REF] Porter | What Is Value in Health Care?[END_REF]. Within a person-centred framework, value should be defined around outcomes that matter to service users. All ICHOM Standard Sets are condition-specific, based on the understanding that service user needs and treatment options are at least partly shaped by the principal presenting problems [START_REF] Porter | What Is Value in Health Care?[END_REF][START_REF] Kazdin | The Meanings and Measurement of Clinical Significance[END_REF]. Service users are directly involved in defining the Standard Set, which must include patient-reported outcomes. The final set of outcomes should represent the end result rather than the process of care; balance a comprehensive approach to tracking outcome with a feasible recommendation that services can reliably implement; and be responsive to quality improvement efforts.

An existing ICHOM Standard Set for adult anxiety and depression covers young people from the age of 14 years (12; Appendix pg. 3). It includes a recommendation for tracking symptom change via the Generalized Anxiety Disorder 7-item Scale (GAD-7; 15), and the depression subscale of the Patient Health Questionnaire (PHQ-9; 16); as well as functioning through the World Health Organization Disability Schedule 2.0 (WHODAS 2.0) 12-item short form (17; the full recommendation is available from ichom.org). The present Standard Set aims to complement this effort by providing a set of recommendations specifically tailored for use with CYP. The combination of the two Sets will provide for transition from youth into adult services at any point between the ages of 14 and 24 years, allowing for local variation in transition and judgements about which Set is most suitable for different ages. This Standard Set is designed for CYP aged 6 to 24 years who access care for anxiety, depression, obsessivecompulsive disorder (OCD), or post-traumatic stress disorder (PTSD), as defined by standard diagnostic criteria. All are internalising disorders, typically characterised by high levels of negative affect, with OCD and PTSD long classified as anxiety disorders [START_REF] Regier | The DSM-5: Classification and criteria changes[END_REF]. It is recommended for use by all those providing care to the population in scope, worldwide, regardless of intervention setting or approach. The Working Group sought to combine self-, parent-, and clinician-reported outcome measures, to account for the different perspectives these reporters tend to provide [START_REF] De Los | The validity of the multi-informant approach to assessing child and adolescent mental health[END_REF][START_REF] Weisz | What five decades of research tells us about the effects of youth psychological therapy: A multilevel meta-analysis and implications for science and practice[END_REF][START_REF] Creswell | Practitioner Review: Anxiety disorders in children and young peopleassessment and treatment[END_REF]. Parent-report also serves as a proxy where CYP are unable to complete measures due to young age or developmental constraints [START_REF] Creswell | Practitioner Review: Anxiety disorders in children and young peopleassessment and treatment[END_REF]. A more detailed discussion of the scope of this Standard Set is provided in the Appendix (pg.

3).

Panel 2. Methodology

The Working Group comprised 27 experts from 13 countries, including service users, parents or carers (henceforth "parents"), mental health practitioners and researchers working within relevant disciplines (e.g., psychiatry, epidemiology, psychometrics; see Appendix pg. [START_REF] Albano | Childhood anxiety disorders[END_REF][START_REF] Clayborne | Systematic Review and Meta-Analysis: Adolescent Depression and Long-Term Psychosocial Outcomes[END_REF]. A central project team (KK, SC, MW) coordinated and facilitated the consensus-building process and completed supporting research tasks, but did not vote on the consensus recommendation. The Appendix (pg. [START_REF] Albano | Childhood anxiety disorders[END_REF][START_REF] Clayborne | Systematic Review and Meta-Analysis: Adolescent Depression and Long-Term Psychosocial Outcomes[END_REF][START_REF] Jorm | Has increased provision of treatment reduced the prevalence of common mental disorders? Review of the evidence from four countries[END_REF][START_REF] Boswell | Implementing routine outcome monitoring in clinical practice: Benefits, challenges, and solutions[END_REF][START_REF] Scott | Using measurement-based care to enhance any treatment[END_REF][START_REF] Barkham | Bridging Evidence-Based Practice and Practice-Based Evidence: Psychological Therapies[END_REF][START_REF] Bickman | A Measurement Feedback System (MFS) Is Necessary to Improve Mental Health Outcomes[END_REF][START_REF] Bear | Systematic Review and Meta-analysis: Outcomes of Routine Specialist Mental Health Care for Young People With Depression and/or Anxiety [published online ahead of print[END_REF][START_REF] Clarke | Core outcome sets and systematic reviews[END_REF][START_REF] Obbarius | Standardization of health outcomes assessment for depression and anxiety: recommendations from the ICHOM Depression and Anxiety Working Group[END_REF][START_REF] Porter | What Is Value in Health Care?[END_REF][START_REF] Kazdin | The Meanings and Measurement of Clinical Significance[END_REF][START_REF] Spitzer | A brief measure for assessing generalized anxiety disorder: The GAD-7[END_REF][START_REF] Spitzer | Validation and utility of a self-report version of PRIME-MD: The PHQ Primary Care Study[END_REF][START_REF] Üstün | Developing the World Health Organization disability assessment schedule 2.0[END_REF][START_REF] Regier | The DSM-5: Classification and criteria changes[END_REF] provides a detailed description of the process and methodology. A flow chart is provided in figure 1.

General process. Over 14 months, the Working Group completed a structured and evidence-informed consensusbuilding process (figure 1). The group convened for eight teleconferences, completed a three-round Delphi exercise to select outcome domains [START_REF] Linstone | The Delphi Method: Techniques and Applications[END_REF][START_REF] Prinsen | How to select outcome measurement instruments for outcomes included in a "Core Outcome Set" -a practical guideline[END_REF], and participated in iterative rounds of anonymous voting to arrive at recommendations for outcomes, measurement instruments, case-mix factors, and time points (Appendix pg. 4-5).

The process was informed by sequential research inputs performed by the central project team (Appendix pg. [START_REF] Jorm | Has increased provision of treatment reduced the prevalence of common mental disorders? Review of the evidence from four countries[END_REF][START_REF] Boswell | Implementing routine outcome monitoring in clinical practice: Benefits, challenges, and solutions[END_REF][START_REF] Scott | Using measurement-based care to enhance any treatment[END_REF][START_REF] Barkham | Bridging Evidence-Based Practice and Practice-Based Evidence: Psychological Therapies[END_REF][START_REF] Bickman | A Measurement Feedback System (MFS) Is Necessary to Improve Mental Health Outcomes[END_REF][START_REF] Bear | Systematic Review and Meta-analysis: Outcomes of Routine Specialist Mental Health Care for Young People With Depression and/or Anxiety [published online ahead of print[END_REF][START_REF] Clarke | Core outcome sets and systematic reviews[END_REF][START_REF] Obbarius | Standardization of health outcomes assessment for depression and anxiety: recommendations from the ICHOM Depression and Anxiety Working Group[END_REF][START_REF] Porter | What Is Value in Health Care?[END_REF][START_REF] Kazdin | The Meanings and Measurement of Clinical Significance[END_REF][START_REF] Spitzer | A brief measure for assessing generalized anxiety disorder: The GAD-7[END_REF][START_REF] Spitzer | Validation and utility of a self-report version of PRIME-MD: The PHQ Primary Care Study[END_REF][START_REF] Üstün | Developing the World Health Organization disability assessment schedule 2.0[END_REF][START_REF] Regier | The DSM-5: Classification and criteria changes[END_REF]. ). The instruments judged to best satisfy these criteria were shortlisted, and their relative strengths and shortcomings discussed. At this stage, other aspects such as content and construct validity were also considered. The final set of measures was then selected via consensus, through iterative rounds of anonymous voting. The Working Group aimed for a Standard Set that would be simple to use, impose a low burden on its intended users, and be applicable across different contexts.

Selection of case-mix factors and measurement time points.

Based on the systematic review and existing ICHOM Standard Sets, the central project team compiled a list of possible case mix factors, and conducted a rapid review of reviews examining predictors, mediators, and moderators of treatment response (Appendix pg. 17-18). The Working Group discussed this information and formed consensus on the case-mix factors to include.

Similarly, the central project team drew on existing ICHOM Standard Sets to present an initial proposal for measurement time points, which were discussed, refined, and voted on by the Working Group.

Open review of draft recommendations. To establish the generalisability and acceptability of the Working Group's recommendations, an Open Review web survey gathered external feedback from 463 practitioners, researchers, and policymakers from 45 countries, as well as 24 young people and parents from Denmark, the United States (US), and the United Kingdom (UK; see Appendix pg. 18).

Recommendations of the Working Group

Outcomes and Measures

As per Working Group consensus, the Standard Set recommends tracking response to treatment across the three outcome domains of symptoms, suicidal thoughts and behaviour, and functioning, using seven primarily selfreported instruments. These were selected for fulfilling most or all of the appraisal criteria (table 1). A detailed discussion of instrument properties, performance against appraisal criteria, and accessibility is provided in the Appendix (pg. [START_REF] De Los | The validity of the multi-informant approach to assessing child and adolescent mental health[END_REF][START_REF] Weisz | What five decades of research tells us about the effects of youth psychological therapy: A multilevel meta-analysis and implications for science and practice[END_REF][START_REF] Creswell | Practitioner Review: Anxiety disorders in children and young peopleassessment and treatment[END_REF][START_REF] Linstone | The Delphi Method: Techniques and Applications[END_REF]. During the Open Review, 75% of the 463 practitioners and researchers who provided feedback stated their overall confidence in the recommended outcomes and instruments and participants with lived experience confirmed the importance of outcomes included (75-100% of participants rated each included outcome domain as important) and the acceptability of the recommended measures (100% of participants confirmed the acceptability of the recommended measures for use in clinical practice). No additional outcome domains were consistently highlighted as missing from the recommended set during the Open Review.

Insert Table 1 Symptoms. The Set recommends measuring anxiety and depression symptoms through youth and parent-report for all CYP in scope. To minimise the length and complexity of the Standard Set, the Working Group chose to recommend a joint measure of anxiety and depression symptoms. The Set recommends the 25-item short form of the Revised Children's Anxiety and Depression Scale (RCADS-25; 26) for youth and parent-report. Based on the appraisal criteria, the group initially selected the scale's 47-item version [START_REF] Chorpita | Assessment of symptoms of DSM-IV anxiety and depression in children: A revised child anxiety and depression scale[END_REF], but decided to recommend the short form to reduce respondent burden. The RCADS-25 consists of 15 items tracking anxiety symptoms, and 10 items tracking symptoms of major depression, which can be summed to compute aggregate anxiety, depression, and total internalising symptom scores. Although less widely validated than the RCADS-47, the RCADS-25 met most inclusion criteria (table 1), although evidence of its sensitivity to change was not available at the time of the appraisal. Long or short versions of the RCADS have been applied in Africa, Europe, the Americas, and Asia [START_REF] Piqueras | The Revised Child Anxiety and Depression Scale: A systematic review and reliability generalization meta-analysis[END_REF].

Symptoms of OCD and PTSD should be tracked separately for CYP presenting with a diagnosis of OCD or PTSD, or with sub-threshold symptoms as appropriate. Symptoms of OCD and PTSD should be tracked via the selfreported 21-item Obsessive Compulsive Inventory for Children (OCI-CV; 29); and the Children's Revised Impact of Events Scale (CRIES; 30) for youth (CRIES-8) and parent report (CRIES-13). Both measures have been applied in Europe, Asia, and the Americas. A parent version of the OCI-CV is not currently available.

It is important to note that the above-mentioned symptom measures are recommended for the purpose of tracking treatment outcomes over time. They are not considered primarily for the purpose of diagnosing presenting problems, and are not intended to replace a thorough clinical assessment using state-of-the-art diagnostic tools. The latter require additional properties related to diagnostic validity (e.g., sensitivity and specificity), which the Working Group did not explicitly consider during measure appraisal.

Suicidal thoughts and behaviour. Consensus was reached on measuring suicidal thoughts and behaviour in all young people aged 10 years and older (unless considered inappropriate) using the Columbia Suicide Severity Rating Scale (C-SSRS) Recent Self-Report Screener (31)a short self-report version of the clinician-administered C-SSRS interview protocol. The measure consists of six items tracking the severity of suicidal ideation and behaviour in the previous month. The self-report screener has not been validated in CYP, but the clinician-rated C-SSRS has demonstrated good internal consistency, interrater reliability, and sensitivity to change in adolescent samples [START_REF] Posner | The Columbia-suicide severity rating scale: Initial validity and internal consistency findings from three multisite studies with adolescents and adults[END_REF].

Functioning. Functioning describes a child's ability to engage in typical activities and meet role demands in line with age-specific socio-cultural norms [START_REF] Hoagwood | Outcomes of mental health care for children and adolescents: I. A comprehensive conceptual model[END_REF][START_REF] Rapee | Annual research review: Conceptualising functional impairment in children and adolescents[END_REF]. None of the identified functioning measures fully satisfied the Working Group's requirements, and consensus was reached on mitigating this by tracking a broad concept of global functioning or health-related quality of life (HRQoL), as well as condition-specific functional impairment, through short dedicated measures of each concept. Generic measures allow for comparisons across conditions, while condition-specific measures may be more sensitive to change. As per group consensus, measures had to cover psychosocial functioning, peer relationships, and sleep functioning, at least at an item level, although sleep was eventually covered through the RCADS-25 (i.e., item 8: I have trouble sleeping; and item 9, I feel scared if I have to sleep on my own).

The KIDSCREEN-10 was selected as a generic measure of global functioning, to be completed by CYP and parents.

This unidimensional 10-item index of HRQoL tracks functioning in relation to physical health and energy levels, leisure activities, social and family relationships, and cognition [START_REF] Ravens-Sieberer | Reliability, construct and criterion validity of the KIDSCREEN-10 score: A short measure for children and adolescents' well-being and health-related quality of life[END_REF]. The more comprehensive KIDSCREEN-52

was originally developed through a process of cross-cultural harmonisation involving 13 European countries. Its 10item short form has been applied in Asia, Eastern Africa, Europe, North and Latin America. Although originally designed for epidemiological studies, the KIDSCREEN-10 has been shown to discriminate well between CYP with high and low levels of functioning, with few ceiling or floor effects [START_REF] Ravens-Sieberer | Reliability, construct and criterion validity of the KIDSCREEN-10 score: A short measure for children and adolescents' well-being and health-related quality of life[END_REF]. In addition to the KIDSCREEN-10, the Children's Global Assessment Scale (CGAS) was selected as a brief clinician-rated measure of global functioning [START_REF] Shaffer | A Children's Global Assessment Scale (CGAS)[END_REF]. On this widely used measure, clinicians preform a single rating by locating CYP on a scale from 1 to 100, placing them into one of ten categories, from 1 -10 (extremely impaired) to 91 -100 (doing very well).

The Children's Anxiety Life Interference Scale (CALIS; 36) was selected to measure condition-specific impairment via 9 items (10 for the parent version) that describe instances of anxiety impacting on functioning at home, at school, on social life, or on activities. The CALIS currently only covers anxiety-related impairment, and the Working Group did not identify any eligible measure that tracks depression-specific impairment, or captures impairment from both anxiety and depression. However, the CALIS author team recently revised the measure to cover impairment from anxiety and depression, and a validation study involving children, young people, and parents in community, school, and clinical settings is ongoing. As soon as validation results become available, the Working Group will consider replacing the original CALIS with the revised Children's Anxiety and Depression Life Interference Scale (CADLIS) in the Standard Set.

Case-Mix Factors

The Standard Set aims to facilitate comparisons and benchmarking of outcomes, which requires the collection of additional data to adjust for variation in populations and intervention settings. As per Working Group consensus, services should record demographic, clinical, complexity, and intervention factors, through a mix of self-, parent-, and clinician-report (table 2). Beyond the Working Group, the suggested case-mix factors were endorsed by more than 80% of practitioners and researchers who provided feedback during the Open Review. These factors represent a minimum that should be assessed by all those providing relevant care, and services may wish to add other indicators to meet local information needs.

Insert Services may wish to record additional detail on intervention characteristics to meet local, regional, or national information needs.

Measurement Time Points

Timelines for measurement are highly practical considerations, likely to vary substantially across services. The Standard Set makes a minimum recommendation (figure 2) for measuring outcomes over the full cycle of care, but encourages services to do so as often as is clinically helpful to inform decision-making, or to align with local or national data collection. The suggested timepoints were widely endorsed by over 80% of practitioners and researchers participating in the Open Review.

As per Working Group consensus, all case-mix factors and outcomes should be measured at assessment or intake (i.e., baseline), or as near to these time points as possible where services wish to collect data at second contact (figure 2). As a guideline, all outcome measures should be administered every three months following baseline.

Services are encouraged to consider more frequent intervals, including session-by-session measurement, to help embed monitoring into the clinical process [START_REF] Batty | Implementing routine outcome measures in child and adolescent mental health services: From present to future practice[END_REF], and reduce the risk of missing data due to drop-out prior to followup. The group recognises that effective session-by-session measurement requires well established systems and can otherwise be perceived as unduly burdensome.

To mark the end of an active treatment cycle, outcomes should be measured upon transition into adult services, into a different level of care (e.g., from outpatient to inpatient care), or upon completion when no further activities are planned (figure 2). Outcomes should then be measured again at a follow-up assessment, one year after baseline. This can enable important insights into longer-term outcomes, but may require significant adjustments to the way services are currently organised and funded.

Insert Figure 2.

Strengths of the Standard Set

Of the Working Group's 27 voting members, four were young adults between the ages of 18 and 24, and two were parents. This was the first ICHOM Working Group to involve young people with lived experience of service userather than just their parentsas full Working Group members at all stages of the consensus-building process, including the teleconferences, Delphi exercise, and subsequent surveys. While limited familiarity with outcome measurement has been described as a barrier to the meaningful participation of lived-experience experts ( 55), the teleconferences helped foster a common understanding within the group ahead of each round of voting. The Working Group chair (MW) solicited input from all call participants on all key discussion points, to promote equal participation and manage power imbalances. Additional feedback from a wider group of young people and parents was sought towards the end of the consensus-building process, through the Open Review. Variable requirements for ethical review for this stakeholder group meant that the survey was only accessible in three countries, while the professional survey was accessible globally. This led to a comparatively small sample of Open Review participants with lived experience (N = 24). However, separate analysis and review of feedback from the professional and livedexperience surveys meant that the Working Group was able to consider each feedback stream in its own right.

The Working Group included experts from low-and middle-income countries (LMIC, Appendix pg. 3), who have rarely been represented by similar initiatives to date [START_REF] Davis | Choosing important health outcomes for comparative effectiveness research: An updated systematic review and involvement of low and middle income countries[END_REF]. While 90% of CYP live in LMICs, 90% of research on youth mental health currently comes from high-income countries [START_REF] Kieling | Child and Adolescent Mental Health Research Across the Globe[END_REF]. Experts from LMIC were also consulted through the Open Review (Appendix pg. 18). A range of local needs, challenges, and cultural factors could thus be considered. The high endorsement of the Standard Set in the Open Review underscores its relevance and acceptability beyond the immediate Working Group.

Implementation

Working Group members will form a steering group to oversee the Standard Set's implementation, and to review recommendations in light of learning from pilot initiatives and new developments in the field. In health systems such as Australia, Canada, the Netherlands, the UK, or the US, the routine collection of outcome data is becoming increasingly embedded [START_REF] Trauer | Outcome Measurement in Mental Health: Theory and Practice[END_REF][START_REF] Delespaul | Routine outcome measurement in the Netherlands-A focus on benchmarking[END_REF], which should facilitate the adoption of the Standard Set, but can also cause issues of alignment with existing local or national systems (e.g., [START_REF]Mental Health National Outcomes and Casemix Collection: Overview of Clinician-Rated and Consumer Self-Report Measures V1[END_REF][START_REF]Measure Applications Partnership[END_REF]. As services adopt routine outcome monitoring, it is imperative that data is handled safely and securely, in accordance with relevant data protection frameworks, and that informed consent protocols are in place.

This Standard Set was designed for clinical practice, and may or may not be suitable for use in clinical trials depending on the requirements and the type of intervention tested. Since work on this Standard Set has ended, the Wellcome Trust and the National Institute of Mental Health (NIMH) have recommend the RCADS-25 as an outcome measure for research with children and adolescents experiencing depression or anxiety, along with the PHQ-9, GAD-7 and WHODAS for older youth. A Standard Set specifically for clinicial trials for adolescent depression is currently under development at the Hospital for Sick Children in Toronto [START_REF] Monga | Core Outcome Set Development for Adolescent Major Depressive Disorder Clinical Trials: A Registered Report[END_REF]. The United Nations

Children's Fund (UNICEF; 63) is leading another complementary initiative focusing on measuring health outcomes for adolescents with anxiety or depression in population surveys. While there is a unique opportunity to encourage further harmonisation, a certain degree of divergence may persist in light of different priorities, processes, and methodologies (e.g., outcome measures for clinical trials do not need to be freely available). The present work further adds to existing review efforts to identify meaningful, feasible, and acceptable outcome measures suitable for use with CYP in practice settings [START_REF] Myers | Ten-Year Review of Rating Scales. II: Scales for Internalizing Disorders[END_REF][START_REF] Winters | Ten-year review of rating scales, VII: Scales assessing functional impairment[END_REF][START_REF] Becker-Haimes | Evidence Base Update for Brief, Free, and Accessible Youth Mental Health Measures[END_REF][START_REF] Beidas | Free, brief, and validated: Standardized instruments for low-resource mental health settings[END_REF][START_REF] Mash | Evidence-based assessment of child and adolescent disorders: Issues and challenges[END_REF]. The steering group will consider opportunities for alignment with complementary initiatives, as well as with existing ICHOM Standard Sets, for example with a view to linking scores obtained from the CYP and adult Sets for the purpose of longitudinal analysis (e.g., 69).

Panel 3. How to Access the Standard Set Resources

A Standard Set reference guide, flyer, and data dictionary are available free of charge and can be accessed via ICHOM Connect (www.connect.ichom.org). The reference guide defines each outcome domain, and describes the recommended measures, case-mix factors, and time points.

Limitations and Areas for Future Research

The scope of this Standard Set is limited to anxiety, depression, OCD, and PTSD, and the Working Group has sought to make a minimum recommendation focusing on core outcomes for these target disorders. Clinical judgement is warranted in tracking additional symptoms over time that are not covered by the recommended outcome measures. Two complementary ICHOM Standard Sets are currently under development that will focus on psychotic disorders (covering bipolar disorder), personality disorders, and substance use and addictive behaviour disorders in young people and adults. The present Standard Set captures the presence of these and other comorbid presenting problems at baseline via the Current View screening tool, for the purpose of case-mix adjustment.

This Standard Set was developed by a relatively small Working Group of consistent membership, which convened at frequent intervals. There was not complete parity in representation across different strata of experts, including CYP and parents. In future efforts, large-scale Delphi-surveys may be more suitable for consulting equal numbers of different stakeholder groups, albeit at the expense of more in-depth and continuous group deliberation.

As per the nature of consensus-building, compromises had to be made and not all individual views and priorities could be reflected in the final Standard Set. For example, two additional outcome domains (i.e., coping and interference of treatment with daily life) reached consensus amongst the group's service user representatives, but not within the wider group. To promote person-centred care, services may want to consider tracking additional outcomes based on shared decision-making with CYP and families, either through suitable standardised measures, or by including personalised outcome measures that track progress in relation to idiosyncratic presenting problems or treatment goals [START_REF] Sales | Seeing the person in the patient: Making the case for individualized PROMs in mental health care[END_REF].

To promote uptake of this Standard Set, simplicity and feasibility were prioritised over detail and specificity. Brief and freely available instruments were prioritised over more complex and at times more established ones, and short forms over long versions. As the former tend to be less widely validated than the latter, it is hoped that the Standard Set will accelerate validation efforts and generate new data on their psychometric properties (e.g., 71). To maintain simplicity, the Set does not recommend separate measures for different age groups. While the selected measures have generally been validated in CYP aged 6-18 years, the Set is less specifically tailored to the experiences of 18-24-year-olds. As the Set moves into implementation, the steering group will consider whether on balance, the gains from increased feasibility can be seen to justify this design.

None of the recommended measures are perfect. The goal was to identify the best-possible suite of instruments within the Working Group's feasibility and psychometric criteria, based on the evidence available at the time, as a starting point for generating wider insights into how outcome measurement might be strengthened in the future.

While the Working Group considered sensitivity to change an essential measurement property to consider, its appraisal was limited by a general lack of data and objective appraisal guidelines. ISOQOL recommends that for longitudinal research patient-reported outcome measures "should have evidence of responsiveness, including empirical evidence of changes in scores consistent with predefined hypotheses" about the expected treatment outcome [START_REF] Reeve | ISOQOL recommends minimum standards for patient-reported outcome measures used in patient-centered outcomes and comparative effectiveness research[END_REF]. While the Working Group considered such evidence where available, no standard thresholds could be applied to determine whether sensitivity was sufficient.

An important area that could not be considered as part of the appraisal is the measurement invariance of the selected tools across languages and cultural backgrounds, and the extent to which cultural differences may impact on the validity of the selected measurement instruments [START_REF] Milfont | Testing measurement invariance across groups: applications in cross-cultural research[END_REF]. In the absence of invariance, comparisons between different groups are not fully meaningful, and the Group hopes that this initiative will enhance data availability and spur efforts to examine how consistently the recommended measures track their designated outcome concepts across cultural and language contexts.

The Working Group encountered challenges with identifying suitable measures of functioning, with common issues including overlap in item coverage between symptoms and functioning, a perceived overemphasis on bodily functions as opposed to psychosocial functioning, lack of validation across the full age span or in clinical populations, and cross-cultural validity. Overall, additional research is needed to understand how suitable the recommended measures are for capturing change over the course of treatment, and in different real-world clinical settings (e.g. primary versus specialist care), and how acceptable the full question battery is to services and service users.

Call for Action

This Standard Set provides the first global guideline for promoting the quality and consistency of routine outcome measurement for CYP with anxiety, depression, OCD, or PTSD. It is person-centred and devised specifically for use in clinical practice, with special attention given to acceptability and feasibility, including in resource-poor contexts.

It also has great relevance to the provision of mental health care outside the health service system, such as in school settings. It forms an essential step towards enhancing evidence on service effectiveness, enabling comparisons and benchmarking of results across care systems, and promoting care quality, in mobilising a comprehensive, forceful, and evidence-based response to the global burden from anxiety and depression. Future research should continue to expand the evidence base in relation to the sensitivity to change and measurement invariance of the included measures, and implementation initiatives should provide feedback on the relevance and acceptability of this recommendation to practitioners and service users. Both will be vital to ensure that this Standard Set makes a viable recommendation that meaningfully captures change for CYP across contexts. The RCADS-25 is currently available in four languages on the source website (see Table S12, Appendix pg. 22), but the 47-item long form is available in 16 languages, so that item-level translations are available in more than four languages for those items included in the RCADS-25. b These validation studies tested a parent-report version only. c These validation studies tested a parent-report version only. d Two initial questions serve as screeners, with the three remaining severity items administered only to those endorsing the first two; all young people are asked about suicidal behaviour. e To the authors' knowledge, the C-SSRS self-report screener had not been subject to a validation study in CYP at the time of writing. For the psychometric appraisal, studies assessing the psychometric properties of the severity subscale in the clinician-led C-SSRS semi-structured interview schedule were considered instead.

Figure 1 .

 1 Figure 1. Working Group Process

  Note. GAD = Generalized Anxiety Disorder; MDD = Major Depressive Disorder (MDD); OCD = Obsessive Compulsive Disorder; PD = Panic disorder; SAD = Separation Anxiety Disorder; SP = Social Phobia. a These thresholds are based on the ISOQOL minimum standards for patient-reported outcome measures (25). b

Selection of outcome domains. In

  line with methodological recommendations by the Core Outcome Measures in

	Effectiveness Trials (COMET) initiative (24), 70 possible outcomes and 507 measurement instruments were
	identified through multiple avenues, including a systematic review of 257 treatment outcome studies, a narrative
	review of supplemental sources (e.g., cohort studies; qualitative outcome research, instrument banks), and break-
	out groups with service user representatives (Appendix pg. 5-8)
	Selection of measurement instruments. Once the Group reached consensus on outcomes for inclusion, 107
	thematically relevant measures were systematically appraised to identify those most suitable for tracking the
	selected outcomes over time. Appraisal criteria included relevance; (i.e., comprehensive coverage of the selected
	outcome domain); feasibility and acceptability (i.e., completion within less than 20 minutes; free availability for
	use in clinical settings, including in paper-and-pencil format; prior validation in CYP; translation into at least a
	second language); and psychometric performance (i.e., interrater or retest reliability, and internal consistency
	above 0.70; evidence of sensitivity to change), in line with International Society for Quality of Life Research
	(ISOQOL) recommendations (25; Appendix pg 8-16

Table 2

 2 Demographic factors. The Set recommends recording age, gender, ethnic minority status, socio-economics status, and the child's living situation. Services should record the sex assigned at birth, and the gender reported by CYP.Socio-economic status should be measured by recording the highest level of education completed by any of the CYP's parents, as a widely accepted proxy that can be mapped onto the International Standard Classification of Education for international comparisons (ISCED; 37). The Set includes one question about CYP's living situation, and two questions capturing ethnic minority and marginalised group status via self-report (table2). the ICD-11 for pediatric populations[START_REF]International Classification of Diseases for Mortality and Morbidity Statistics[END_REF]. ICHOM standards are available for recording symptom duration and prior service use (table2).

	Clinical factors. Several studies suggest that symptom burden, symptom duration, and the presence of
	comorbidities affect treatment response in CYP with depression or anxiety (38-47; Appendix pg. 17-18). The group

recommends recording principal and comorbid presenting problems by administering the 30 problem descriptions of the Current View Tool

[START_REF] Jones | Current view tool: Completion guide[END_REF]

. While not equivalent with formal diagnoses, these broadly align with the diagnostic categories of Intervention Factors. Services should collect information about the intervention approach, including intervention focus (i.e., in terms of who is actively involved), treatment modality and prescribed medication (i.e., as per lists of options compiled by the Working Group and through Open Review feedback), and intervention setting (i.e., in terms of whether or not treatment was delivered via a digital platform or inpatient care, as opposed to other settings).

Table 1 . Overview of Recommended Outcome Measures and their Evaluation Criteria

 1

			Relevance	Feasibility & Acceptability			Psychometric Performance a
			Domain coverage	Short	Free	Validated	Translated	Reliability	Validity	Sensitivity to change
			Satisfactory domain	< 60 items /	No licencing	At least one	>1 language	Test-retest	Internal	Any evidence of
			coverage at subscale or	< 20 min	costs /	validation in a	version	reliability (TRT)	consistency (e.g.	sensitivity to change
			item level		restrictions	CYP sample	available	or inter-rater	Cronbach Alpha
								reliability (IRR) ≥	≥ 0.70
	Outcome	Measure						0.70
	Symptoms
	Anxiety &	RCADS-25	GAD; MDD; OCD; PD;	25 items	Yes	Ages 6-18	4 /16 b	TRT (71)	Yes (26,73)	No evidence
	depression		SAD; SP	(< 10 min)		cl. & non-cl.
	OCD	OCI-CV	Doubting/checking;	21 items	Yes	Ages 6-18	3+	TRT (29,74-76) Yes (29,74-78)	Some evidence (29)
			obsessing; hoarding;	(< 10 min)		cl. & non-cl.
			washing; ordering;
			neutralising
	PTSD	CRIES-8	Intrusion; avoidance	8 /13 items	Yes	Ages 7-18	27+	TRT (79)	Yes (79-81)	Some evidence (82) c
			(hyperarousal)	(< 5 min)		cl. & non-cl.
	Suicidal	C-SSRS	Severity of ideation;	3 or 6 items d	Yes	Ages 12-18 b	100+	IRR (83) e	Yes (31) e	Some evidence (31) e
	thoughts and		behaviour (attempts)	(< 5 min)		cl. & non-cl
	behaviour
	Functioning
	Global	KIDSCREEN-10	Physical activity &	10 items	Yes	Ages 8-18	22+	TRT (34,84)	Yes (34,84,85)	No evidence
			energy; emotions; leisure	(< 5 min)		non-cl.
			time and participation;
			relationships with parents
			& peers; cognition; school
	Global	CGAS	Global functioning	1 item	Yes	Ages 4-18	2+	IRR (34,86-88)	N.A.	Some evidence (89)
				(< 5 min)				TRT (35,87)
	Impact	CALIS	Enjoyable activities;	9/10 items	Yes	Ages 6-17	7	TRT (36)	Yes (36,90,91)	Some evidence (36)
			relationships with siblings,	(< 5 min)		cl. & non-cl.
			parents, friends, peers;
			sports; schoolwork;
			distress

Acknowledgments

The authors would like to gratefully acknowledge all individuals and organisations who took the time to share or complete the Open Review feedback survey towards improving the Standard Set recommendation. The authors would also like to thank Luz Sousa Fialho for providing administrative support, Katherine G. Young for providing research support, and Christiana Ilesanmi for providing both administrative and research support to this project, as well as Zofia Das-Gupta for providing guidance on the application and management of the ICHOM process until May 2019. The authors would like to thank one further service user representative who participated in the early stages of the process, including the Delphi process, but did not wish to be a named co-author.

. Administrative, technical, and organisational support was provided by SC. Declaration of interests AMA receives royalties from Oxford University Press for the Anxiety Disorders Interview Schedule (ADIS), public mental health. He chairs the National Mental Health Child and Adolescent Information Development Expert Advisory Group. There is an interest in supporting routine outcome measurement and benchmarking between organisations. This does not influence the content of the submitted work. SC was an employee of the International Consortium for Health Outcomes Measurement (ICHOM) during the conduct of this study. ICHOM received grants from the Government of New South Wales Agency for Clinical Innovation, Australia; Providence Health & Services, USA; Västra Götaland Regional Council, Sweden; and National Health Service, England, United Kingdom, in support of this work. CC has conducted several studies and ongoing evaluations of commonly used scales, including the Spence Children's Anxiety Scale (SCAS), Revised Children's The funders did support data collection by sharing the Open Review survey for practitioners within their networks. CC was supported by a National Institute for Health Research (NIHR) Research Professorship (RP_2014-04-018). FM was supported by an NIHR In-Practice Fellowship (IPF-2017-11-002), and is supported by a NIHR School for Primary Care Research GP Career Progression Fellowship. The views expressed in this manuscript of those of the authors and not necessarily those of the NHS, NIHR, or Department for Health and Social Care. VM received funding by the Millennium Science Initiative of the Ministry of Economy, Development and Tourism, grant Millennium Nucleus to Improve the Mental Health of Adolescents and Youths, Imhay, and the Innovation Fund for Competitiveness [Fondo de Innovación para la Competitividad, FIC], Project IS130005. CK is a Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) researcher and an Academy of Medical Sciences Newton Advanced Fellow.

Funding

This work has been funded by the National Health Service (NHS) England; Västra Götaland Regional Council, Sweden; the Government of New South Wales Agency for Clinical Innovation, Australia; and Providence Health & Services, USA. The funders had no role in the study design, data analysis, interpretation, or writing of the report.


Emiliano Lorini
email: emiliano.lorini@irit.fr

Equipe Lilac

A Logical Account of Social Rationality in Strategic Games

Keywords:

We propose a modal logic that enables to reason about different kinds of rationality in strategic games. This logic integrates the concepts of joint action, belief, individual preference and group preference. The first part of the article is focused on the notion of individualistic rationality assumed in classical game theory: an agent decides to perform a certain action only if the agent believes that this action is a best response to what he expects the others will do. The second part of the article explores different kinds of social rationality such as fairness and reciprocity. Differently from individualistically rational agents (alias self-interested agents), social rational agents also consider the benefits of their choice for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. In the article we also provide a complete axiomatization of our logic of joint action, belief, individual preference and group preference.

Introduction

In recent times economic theory has started to refute the standard assumption that people are individualistically rational, that is, that they are necessarily motivated by their material self-interest. Several experiments have demonstrated that people are often driven by otherregarding motives, and by social preferences. Consequently, the classical game-theoretic framework has been revised and extended in order to incorporate new important concepts such as fairness, inequity aversion, altruism and reciprocity. Several models of social rationality have been developed in recent years showing that the phenomena observed in experiments with humans can be explained in a rigorous and tractable manner. Following [START_REF] Fehr | Theories of fairness and reciprocity: Evidence and economic applications[END_REF], we can distinguish two families of models of social rationality in economics. Some models assume that players are only concerned about the distributional consequences of their acts but they do not care about the intentions that lead their opponents to choose these acts. For example, according to [START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF], players are inequity averse and may also care about how much material resources are allocated to other players. Other models assume that the behavior of a player during social interaction depends on the player's beliefs and expectations about the intention of the opponent. 1 For example, according to Rabin's model of reciprocity [START_REF] Rabin | Incorporating fairness into game theory and economics[END_REF], a person is willing to be kind with another person if she believes that the other has been kind to her. On the contrary, a person has a desire to retaliate, if she believes that the other person wanted to hurt her.

Although social preferences and social rationality have been extensively studied in experimental economics, no logical analysis of these concepts has been proposed up to now. In this article, we try to fill this gap by proposing a sound and complete modal logic which enables to model social preferences and to reason about different kinds of social rationality in strategic games. This logic is based on the primitive concepts of joint action, belief, individual preference and group preference.

We think that developing logical models of social interaction integrating social preferences is a promising research avenue especially for the area of multi-agent systems and for the area of social software. Indeed, existing logical models of social interaction proposed in these areas are still anchored to the basic assumption of classical game theory that agents are self-interested. As emphasized above, this assumption is unrealistic for human agents. Therefore, to take the presence of other-regarding preferences into account becomes extremely important when developing formal models of social procedures to be applied to human societies, and when developing logical models of artificial agents which have to interact with human agents (e.g. trading agents, recommender systems, tutoring agents, etc.).

The remainder of the article is organized as follows. In Section 2 we present a modal logic of epistemic strategic games called L integrating the concepts of joint action, individual preference and belief. Section 3 is devoted to the analysis in L of some basic gametheoretic concepts such as best response and Nash equilibrium, and of the notion of individualistic rationality assumed in classical game theory (i.e. agents are self-interested). In Section 4 the logic L is extended by operators of group preference. Section 5 explores two different kinds of social rationality: fairness and reciprocity. Differently from individual-istically rational agents (alias self-interested agents), social rational agents also consider the benefits of their choice for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. Section 6 discusses related works in the area of modal logics for game theory.

A logic of epistemic strategic games

We present in this section the modal logic L integrating the concepts of joint action, belief and individual preference. This logic supports reasoning about epistemic games in strategic form in which an agent might be being uncertain about the current choices of the other agents.

Syntax

The syntactic primitives of L are the finite set of agents Agt, the set of atomic formulas Atm, a nonempty finite set of atomic action names Act = {a 1 , a 2 , . . . , a |Act| }, a nonempty finite set of n integers I = {0, . . . , n}. Non-empty sets of agents are called coalitions or groups, noted C 1 , C 2 , . . .. We note 2 Agt * = 2 Agt \ ∅ the set of coalitions.

To every agent i ∈ Agt we associate the set Act i of all possible ordered pairs i:a, that is, Act i = {i:a | a ∈ Act}. Besides, for every coalition C we note ∆ C the set of all joint actions of this coalition, that is,

∆ C = i∈C Act i . Elements in ∆ C are C-tuples noted α C , β C , γ C , δ C , . . .. If C = Agt, we write ∆ instead of ∆ Agt .
Elements in ∆ are also called strategy profiles. Given δ ∈ ∆, we note δ i the element in δ corresponding to agent i. Moreover, for notational convenience, we write δ -i = δ Agt\{i} .

The language L of the logic L is given by the following BNF:

ϕ ::= p | ⊥ | ¬ϕ | ϕ ∨ ϕ | [δ C ]ϕ | ϕ | [B i ]ϕ | [≥ k i ]ϕ
where p ranges over Atm, i ranges over Agt, δ C ranges over C∈2 Agt * ∆ C , and k ranges over I. The classical Boolean connectives ∧, →, ↔ and (tautology) are defined from ⊥, ∨ and ¬ in the usual manner.

The formula [δ C ]ϕ reads "if coalition C chooses the joint action δ C then ϕ holds". Therefore, [δ C ]⊥ reads "coalition C does not choose the joint action δ C ".

The operator is an universal modality, which quantifies over all worlds of the current model. Since in our framework, there is a one-to-one correspondence between models and games (i.e. a model corresponds to a unique strategic game), and every world in a model corresponds to a unique strategy profile of a strategic game, is used to quantify over the strategy profiles of the current game. ϕ reads "ϕ holds for all strategies profiles of the current game", or simply "ϕ is necessarily true". The formula [B i ]ϕ is read as usual "agent i believes that ϕ", whereas the formula [≥ k i ]ϕ is read "ϕ is true in all states which are good for agent i at least to degree k". Operators [≥ k i ] are used in L to define agents' preference orderings over the strategy profiles of the current game.

Moreover, we have the following abbreviations:

δ C ϕ def = ¬[δ C ]¬ϕ, ♦ϕ def = ¬ ¬ϕ, B i ϕ def = ¬[B i ]¬ϕ, ≥ k i ϕ def = ¬[≥ k i ]¬ϕ.
♦ ϕ has to be read "ϕ is possibly true", and δ C ϕ has to be read "coalition C chooses the joint action δ C and ϕ holds". Therefore, δ C simply means "coalition C chooses the joint action δ C ". ≥ k i ϕ has to read "ϕ is true in at least one state which is good for agent i at least to degree k". B i ϕ has to be read "agent i thinks that ϕ is possible".

Semantics

Definition 1 (L-model). L-models are tuples M = W, R, B, P, π where:

• W is a nonempty set of possible worlds or states;

• R is a collection of total functions R C : W -→ ∆ C one for every coalition C ∈ 2 Agt * , mapping every world in W to a joint action of the coalition such that:

C1 δ C = R C (w) if and only if, for every i ∈ C δ i = R i (w), and
C2 if for every i ∈ Agt there is w i ∈ W such that δ i = R i (w i ) then there is a w ∈ W such that δ = R Agt (w), C3 if δ = R Agt (w) and δ = R Agt (v), then w = v; • B : Agt -→ 2 W ×W is
⊆ P k-1 i ; • π : Atm -→ 2 W is a valuation function.
For every coalition C, δ C = R C (w) means that at world w coalition C chooses the joint action δ C . Furthermore, if there exists w ∈ W such that C performs δ C at w then we say that δ C is possible or that δ C can be performed. Thus, δ can be performed if and only if δ is a strategy profile of the current game.

For every agent i, (w, v) ∈ B i means that at world w, according to agent i, world v is possible. Finally, the function P is used to specify the degree of goodness of a certain world according to a given agent. In particular, for every k ∈ I and i ∈ Agt, P k i is the set of worlds that are good for agent i at least to degree k.

Let us discuss the six semantic constraints C1-C6 in Definition 1.

According to Constraint C1, at world w coalition C chooses the joint action δ C if and only if, every agent i in C chooses the action δ i at w. In other words, a certain joint action is performed by a coalition if and only if every agent in the coalition does his part of the joint action. According to the Constraint C2, if every individual action in a joint action δ is possible, then their simultaneous occurrence is also possible. We moreover suppose determinism for the joint actions of all agents: different worlds in the same model correspond to the occurrences of different strategy profiles (Constraint C3).

Constraint C4 just says that agents believe what they are doing. This is a standard assumption in interactive epistemology and epistemic analysis of games (see [START_REF] Bonanno | A syntactic approach to rationality in games with ordinal payoffs[END_REF] for instance).

According to Constraint C5, every world is good for an agent i at least to minimal degree 0. This implies that an agent can always compare the goodness of two different worlds. According to the Constraint C6, for every integer k ∈ I \ {0}, the set of worlds that are good for agent i at least to degree k is a subset of of the set of worlds that are good for agent i at least to degree k-1. This means that worlds in a model are ordered according to their degrees of goodness for an agent and form a "system of spheres" [START_REF] Lewis | Counterfactuals[END_REF] for every agent i. In particular, an ordinal conditional function (OCF) κ i : W -→ I in Spohn's sense [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF] can be associated with every agent i, where κ i (w) corresponds to the exact degree of goodness of world w for agent i. 2Definition 2 (κ i ). For every i ∈ Agt and for every w ∈ W we define:

• κ i (w) = n if and only if w ∈ P n i ; • for every k ∈ I such that k < n, κ i (w) = k if and only if w ∈ P k i and w ∈ P k+1 i .
Example 1. By way of example, consider a two-player Prisoner Dilemma (PD) game [START_REF] Osborne | A course in game theory[END_REF]. We suppose Agt = {i 1 , i 2 } and ACT = {c, d } where c is the action of cooperating and d is the action of defecting. In order to model PD game we just need four different degrees of goodness I = {0, 1, 2, 3}. The elements W, R and P of the model M corresponding to the two-player PD game are defined as follows:

• W = {w 1 , w 2 , w 3 , w 4 }; • R {i 1 ,i 2 } (w 1 ) = i 1 :c, i 2 :c , R {i 1 ,i 2 } (w 2 ) = i 1 :d, i 2 :d , R {i 1 ,i 2 } (w 3 ) = i 1 :c, i 2 :d , R {i 1 ,i 2 } (w 4 ) = i 1 :d, i 2 :c ; • P 3 i 1 = {w 4 }, P 2 i 1 = {w 1 , w 4 }, P 1 i 1 = {w 1 , w 2 , w 4 }, P 0 i 1 = {w 1 , w 2 , w 3 , w 4 }; • P 3 i 2 = {w 3 }, P 2 i 2 = {w 1 , w 3 }, P 1 i 2 = {w 1 , w 2 , w 3 }, P 0 i 2 = {w 1 , w 2 , w 3 , w 4 }.
World w 3 (resp. w 4 ) belongs to P 0 i 1 (resp. P 0 i 2 ) and does not belong to P 1 i 1 (resp. P 1 i 2 ), hence it is a world that is good for agent i 1 (resp. agent i 2 ) exactly to degree 0. That is,

κ i 1 (w 3 ) = 0 (resp. κ i 2 (w 4 ) = 0).
World w 2 belongs to P 1 i 1 (resp. P 1 i 2 ) and does not belong to

P 2 i 1 (resp. P 2 i 2 ), hence it is a world that is good for agent i 1 (resp. agent i 2 ) exactly to degree 1. That is, κ i 1 (w 2 ) = 1 (resp. κ i 2 (w 2 ) = 1).
World w 1 belongs to P 2 i 1 (resp. P 2 i 2 ) and does not belong to

P 3 i 1 (resp. P 3 i 2 ), hence it is a world that is good for agent i 1 (resp. agent i 2 ) exactly to degree 2. That is, κ i 1 (w 1 ) = 2 (resp. κ i 2 (w 1 ) = 2).
World w 4 (resp. w 3 ) belongs to

P 3 i 1 (resp. P 3 i 2 ), hence it is a world that is good for agent i 1 (resp. agent i 2 ) exactly to degree 3. That is, κ i 1 (w 4 ) = 3 (resp. κ i 2 (w 3 ) = 3).
For every OCF function κ i , we define the corresponding preference order i over the worlds in W .

Definition 3 ( i ). For every i ∈ Agt and for every w, v ∈ W we have w i v if and only if κ

i (w) ≤ κ i (v).
Truth conditions for atomic formulas and the Boolean operators are entirely standard. The truth conditions for the operators of the logic L are:

• M, w |= [δ C ]ϕ ⇐⇒ if δ C = R C (w) then M, w |= ϕ; • M, w |= ϕ ⇐⇒ M, v |= ϕ for all v such that v ∈ W ; • M, w |= [B i ]ϕ ⇐⇒ M, v |= ϕ for all v such that (w, v) ∈ B i ; • M, w |= [≥ k i ]ϕ ⇐⇒ M, v |= ϕ for all v such that v ∈ P k i . A formula ϕ is true in an L-model M iff M, w |= ϕ for every world w in M . ϕ is L-valid (noted |= ϕ) iff ϕ is true in all L-models. ϕ is L-satisfiable iff ¬ϕ is not L-valid.

Axiomatization

All principles of classical propositional logic (CPL)

All principles of modal logic S5 for (S5 )

All principles of modal logic KD45 for every

[B i ] (S5 [B i ] ) All principles of modal logic K for every [≥ k i ] (K [≥ k i ] ) [δ C ]ϕ ↔ ( δ C → ϕ) (Def [δ C ] ) δ C ↔ i∈C δ i (JointAct) δ C ∈∆ C δ C (Active) δ C → [δ C ]⊥ if δ C = δ C (Single)   i∈Agt ♦ δ i   → ♦ δ (Indep) ( δ ∧ ϕ) → ( δ → ϕ) (JointDet) i:a ↔ [B i ] i:a (Aware) ϕ → [B i ]ϕ (Incl ,[B i ] ) ϕ → [≥ k i ]ϕ (Incl ,[≥ k i ] ) [≥ 0 i ]ϕ ↔ ϕ (Def [≥ 0 i ] ) [≥ k-1 i ]ϕ → [≥ k i ]ϕ (Incl [≥ k-1 i ],[≥ k i ] ) [≥ k i ]ϕ → [≥ k i ]ϕ (PNec [≥ k i ] ) ¬[≥ k i ]ϕ → ¬[≥ k i ]ϕ (NNec [≥ k i ] )
Table 1: Axiomatization of L

We call L the logic that is axiomatized by the principles given in Table 1, and we write

L ϕ if ϕ is a theorem of L.
Note that Axiom Indep is the L counterpart of the so-called axiom of independence of agents of STIT logic [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF]. This axiom enables to express the basic game theoretic assumption that the set of strategy profiles of a game in strategic form is the cartesian product of the sets of individual actions for the agents in Agt. Note also that Axiom Def [δ C ] could be replaced by the simpler Axiom ϕ → [δ C ]ϕ. We have chosen Def [δ C ] because we think that it better expresses the meaning of the operators [δ C ]. Furthermore, by the Axiom Def [δ C ] , we can prove that every operator [δ C ] satisfies all principles of the basic modal logic K and, in particular, Axiom K:

(1) L ([δ C ]ϕ ∧ [δ C ](ϕ → ψ)) → [δ C ]ψ,
and the rule of necessitation:

(2) if L ϕ then L [δ C ]ϕ.
The following property is also derivable by Axiom T for and Axiom

Def [δ C ] : L ϕ → [δ C ]ϕ.
Theorem 1. The logic L is sound and complete with respect to the class of L-models.

Proof. We here provide a sketch of the proof.

It is straightforward to show that all our axioms are valid and that the rules of inference preserve validity in the class of L-models. The other part of the proof is shown using two major steps.

Step 1. We provide an alternative semantics for L in terms of standard Kripke models whose semantic conditions correspond one-to-one to the axioms in Table 1. The definition of Kripke L-models is the following one.

Definition 4 (Kripke L-model). Kripke L-models are tuples M = W, ∼, R, B, P, π where:

• W is a nonempty set of possible worlds or states;

• ∼ is an equivalence relation on W ; • R : C∈2 Agt * ∆ C -→ 2 W ×W maps every joint action δ C to a transition relation R δ C ⊆ W × W between possible worlds such that: S1 R δ C (w) = ∅ if and only if, for every i ∈ C R δ i (w) = ∅, S2 if R δ C (w) = ∅ then R δ C (w) = {w}, S3 δ C ∈∆ C R δ C (w) = ∅, S4 if δ C = δ C then R δ C (w) = ∅ or R δ C (w) = ∅, S5 if for every i ∈ Agt there is v i such that w ∼ v i and R δ i (v i ) = ∅ then there is a v such that w ∼ v and R δ (v) = ∅, S6 if w ∼ v and R δ (w) = ∅ and R δ (v) = ∅, then w = v; • B : Agt -→ 2 W ×W is
S7 if (w, v) ∈ B i , then i:a = R i (w) if and only if i:a = R i (v), S8 if (w, v) ∈ B i then w ∼ v;
• P : I × Agt -→ 2 W ×W maps every integer k in I and agent i in Agt to a binary relation

P k i on W such that: S9 P 0 i (w) =∼(w), S10 P k i (w) ⊆∼(w), S11 if k ∈ I \ {0} then P k i (w) ⊆ P k-1 i (w), S12 if w ∼ v then P k i (w) ⊆ P k i (v), S13 if w ∼ v then P k i (v) ⊆ P k i (w); • π : Atm -→ 2 W is a valuation function.
As in the previous Constraint C2, accessibility relations on W can be viewed as functions from W to 2 W . Therefore, we write

∼(w) = {v | w ∼ v}, P k i (w) = {v | (w, v) ∈ P k i }, etc.
Truth conditions of L formulas in Kripke L-models are again standard for atomic formulas and the Boolean operators. The truth conditions for the other operators are:

• M, w |= [δ C ]ϕ ⇐⇒ M, v |= ϕ for all v ∈ R δ C (w); • M, w |= ϕ ⇐⇒ M, v |= ϕ for all v ∈∼(w); • M, w |= [B i ]ϕ ⇐⇒ M, v |= ϕ for all v ∈ B i (w); • M, w |= [≥ k i ]ϕ ⇐⇒ M, v |= ϕ for all v ∈ P k i (w).
It is a routine task to prove that the axiomatic system of the logic L given in Table 1 is sound and complete with respect to this class of Kripke L-models via the Sahlqvist theorem, cf. [6, Th. 2.42]. Indeed all axioms in Table 1 are in the so-called Sahlqvist class [START_REF] Sahlqvist | Completeness and correspondence in the first and second order semantics for modal logics[END_REF]. Thus, they are all expressible as first-order conditions on Kripke models and are complete with respect to the defined model classes.

Step 2. The second step shows that the semantics in terms of L-models of Definition 1 and the semantics in terms of Kripke L-models of Definition 4 are equivalent. As the logic L is sound and complete for the class of Kripke L-models and is sound for the class of L-models, we have that for every L formula ϕ, if ϕ is valid in the class of Kripke L-models then ϕ is valid in the class of L-models. Consequently, for every L formula ϕ, if ϕ is satisfiable in the class of L-models then ϕ is satisfiable in the class of Kripke L-models. Therefore, in this second step we just need to show that for every L formula ϕ, if ϕ is satisfiable in the class of Kripke L-models then ϕ is satisfiable in the class of L-models.

Suppose ϕ is satisfiable in the class of Kripke L-models. This means that there is a Kripke L-model M = W, ∼, R, B, P, π and world w such that M, w |= ϕ. We can now build a L-model M = W , R , B , P , π which satisfies ϕ. The construction of the model M is made in two steps. We first consider the submodel M w = W w , ∼ w , R w , B w , P w , π w generated from M and w: by the generated submodel property [START_REF] Blackburn | Modal Logic[END_REF] we have M w , w |= ϕ. M w is also a Kripke L-model, and ∼ w = W w × W w . The latter means that is interpreted as a universal modal operator. Finally, we can define M = W , R , B , P , π as follows:

• W = W w ; • for every C ∈ 2 Agt * and v ∈ W , R C (v) = δ C if and only if R δ C ,w (v) = ∅; • for every i ∈ Agt, B i = B i,w ;
• for every i ∈ Agt and k ∈ I, P k i = P k i,w (w);

• π = π w .
It is a routine task to check that M is indeed a L-model and, by induction on the structure of ϕ, that we have M , w |= ϕ.

Strategic games with individual rationality

This section is devoted to the analysis in the logic L of strategic games in which agents are self-interested. We first define preference operators. Then, we consider the basic gametheoretic concepts of best response and Nash equilibrium, and their relationships with the notion of individualistic rationality assumed in classical game theory.

Preferences over formulas

For every i ∈ Agt, we define a preference relation over formulas:

ψ ≤ i ϕ def = k∈I ( ≥ k i ψ → ≥ k i ϕ).
According to this definition "ϕ is for agent i at least as good as ψ" (noted ψ ≤ i ϕ) if and only if for every integer k ∈ I if ψ is true in at least one state which is good for agent i at least to degree k then ϕ is also true in at least one state which is good for agent i at least to degree k. The following are central properties of the operator ≤ i . For every i ∈ Agt we have:

L ψ ≤ i ψ (3) L ((ϕ 1 ≤ i ϕ 2 ) ∧ (ϕ 2 ≤ i ϕ 3 )) → (ϕ 1 ≤ i ϕ 3 ) (4) L (ϕ 1 ≤ i ϕ 2 ) ∨ (ϕ 2 ≤ i ϕ 1 ) (5) L ⊥ ≤ i (6) if L ϕ → (ψ 1 ∨ ... ∨ ψ s ) then L (ϕ ≤ i ψ 1 ) ∨ ... ∨ (ϕ ≤ i ψ s ) (7)
The L-theorems 3-5 highlight that ≤ i is a total preorder. The L-theorems 4-6 are exactly the three fundamental principles of Lewis's conditional logic (called VN ) [START_REF] Lewis | Counterfactuals[END_REF]. The language of Lewis's logic VN is that of classical logic augmented with a dyadic connective ≤. In this sense, the operators ≥ k i enables to reconstruct Lewis' dyadic connective. A similar result has been proved by del Cerro & Herzig [START_REF] Del Cerro | A modal analysis of possibility theory[END_REF] in the 90ies. The two authors have developed a modal logic of possibility called PL in which a normal operator of the form [x] for every parameter x in a set of parameters P is introduced (the dual of [x] is noted x ). In a way similar to the L formula [≥ k i ]ϕ, the PL formula [x] ϕ is read "the necessity of formula ϕ is at least degree x". PL have the standard axioms and rules of inference of system K for every operator [x] plus the following two axioms:

PL1 ¬ [1] ⊥ PL2 ([x] ϕ → [y] ϕ) ∨ ([x] ϕ → [y] ϕ) for every x, y ∈ P
where 1 is a particular element in the set of parameters P . The two authors reconstruct Lewis's preferential operators by means of the following translation from PL to Lewis's conditional logic: ψ ≤ ϕ def = x∈P ( x ψ → x ϕ). 3 We can easily prove that our logic L satisfies the previous two properties of del Cerro & Herzig's logic. Indeed, the following L-theorem is provable by Axiom Def [≥ 0 i ] and by Axiom T for : [START_REF] Boutilier | Modal logics for qualitative possibility and beliefs[END_REF] L ¬[≥ 0 i ]⊥. This L-theorem corresponds to Axiom PL1 of PL. Moreover, by the

L Axiom Incl [≥ k-1 i ],[≥ k i ] ,
the following L-theorem is also provable for every k, h ∈ I:

(9) L ([≥ k i ]ϕ → [≥ h i ]ϕ) ∨ ([≥ k i ]ϕ → [≥ h i ]ϕ)
This L-theorem corresponds to Axiom PL2 of PL. On the contrary in PL we cannot infer a theorem which corresponds to the L Axiom

Incl [≥ k-1 i ],[≥ k i ]
, that is, in PL we cannot infer as a theorem something like [x] ϕ → [y] ϕ if x < y. In this sense, the fragment of our logic L with preferential operators of type [≥ k i ] can be seen as a generalization of PL.

Before concluding let us define ψ <

i ϕ def = (ψ ≤ i ϕ) ∧ ¬(ϕ ≤ i ψ), δ ≤ i δ def = δ ≤ i δ and δ < i δ def = (δ ≤ i δ ) ∧ ¬(δ ≤ i δ).

Best response and Nash equilibrium

Some basic concepts of game theory can be expressed in L in terms of comparative goodness. We first consider best response. Agent i's action a is said to be a best response to the other agents' joint action δ -i , noted BR(i:a,δ -i ), if and only if i cannot improve his utility by deciding to do something different from a while the others choose the joint action δ -i , that is:

BR(i:a,δ -i ) def = b∈Act (( i:b ∧ δ -i ) ≤ i ( i:a ∧ δ -i )).
Given a certain strategic game, the strategy profile (or joint action) δ is said to be a Nash equilibrium if and only if for every agent i ∈ Agt, i's action δ i is a best response to the other agents' joint action δ -i :

Nash(δ) def = i∈Agt BR(δ i ,δ -i ).
Example In order to show that the previous formalization of Nash equilibrium is correct, we need to introduce the definition of strategic game in the game-theoretic sense, and then we need a notion of correspondence between strategic games and L-models. 3 An additional result proved by the two authors is that Lewis's conditional logic is nothing more that the logic of qualitative possibility developed by Dubois & Prade [START_REF] Dubois | Possibility theory: an approach to computerized processing of uncertainty[END_REF]. On this see also [START_REF] Boutilier | Modal logics for qualitative possibility and beliefs[END_REF].

Definition 5 (Strategic game). A strategic game is a tuple Γ = N, A, U where N is a finite set of agents and:

• A is a collection of sets of actions A i one for every agent i ∈ N ;

• U is a collection of total preorders ≤ U i over i∈Agt A i , one for every agent i ∈ Agt.

The strategy profile δ is a Nash equilibrium in the game Γ if and only if, for every i ∈ N and for every

δ i ∈ A i , (δ i , δ -i ) ≤ U i (δ i , δ -i ).
Definition 6 (Γ ≈ M ). We say that a strategic game Γ corresponds to a L-model M = W, R, B, P, π , noted Γ ≈ M , if and only if the following conditions are met:

• Agt = N ;
• for every i ∈ Agt, A i = {i:a | a ∈ Act and there is w ∈ W such that R i (w) = i:a};

• for every i ∈ Agt, and for every δ, δ

∈ i∈Agt A i , δ ≤ U i δ if and only if R -1 Agt (δ ) i R -1 Agt (δ);
where R -1 Agt is the inverse function of R Agt . 4Proposition 1. Suppose that Γ ≈ M and δ ∈ i∈Agt A i . Then, we have M, w |= Nash(δ) for every w ∈ W if and only if, δ is a Nash equilibrium in the game Γ.

Individualistic rationality

The following L formula characterizes a notion of individualistic rationality which is commonly supposed in the epistemic analysis of games (see, e.g., [START_REF] Battigalli | Recent results on belief, knowledge and the epistemic foundations of game theory[END_REF][START_REF] Van Benthem | Rational dynamics and epistemic logic in games[END_REF]):

a,b∈Act ( i:a → δ∈∆ ( B i δ -i ∧ ( δ -i , i:b ≤ i δ -i , i:a ))
). This means that an agent i is (individualistically) rational if and only if, if he chooses a particular action a then for every alternative action b, there exists a joint action δ -i of the other agents that he considers possible such that, playing a while the others play δ -i is for i at least as good as playing b while the others play δ -i . As in L δ ≤ i δ and [B i ](δ ≤ i δ ) are equivalent, the previous definition of rationality can be rewritten in the following equivalent form:

IRat i def = a,b∈Act i:a → δ∈∆ ( B i δ -i ∧ [B i ] ( δ -i , i:b ≤ i δ -i , i:a )) .
For every i ∈ Agt we have:

L IRat i ↔ [B i ]IRat i (10) L ¬IRat i ↔ [B i ]¬IRat i (11)
L-theorems 10 and 11 highlight that the concepts of individualistic rationality and irrationality are positively and negatively introspective. The following theorem specifies some sufficient conditions for guaranteeing that the chosen strategy profile is a Nash equilibrium: if all agents are individualistically rational (alias self-interested) and every agent knows the choices of the other agents, then the selected strategy profile is a Nash equilibrium. For every δ ∈ ∆ we have: [START_REF] Dubois | Possibility theory: an approach to computerized processing of uncertainty[END_REF] L ( i∈Agt

(IRat i ∧ [B i ] δ -i ) ∧ δ ) → Nash(δ)
A similar theorem has been stated for the first time by Aumann & Brandeburger [START_REF] Aumann | Epistemic conditions for Nash equilibrium[END_REF][START_REF] Brandenburger | Knowledge and equilibrium in games[END_REF]. The only difference is that Aumann & Brandeburger used knowledge instead of belief.

An extension with group preferences

In this section we extend the logic L by operators of group preferences of the form

[≥ k C ]. We call L + the extended logic. Operators [≥ k C ] are just a generalization of the operators of individual preference of the form [≥ k i ].
In the present approach a preference for a group C is given by the intersection of the preferences of the agents in the group.

Syntax

The language of the logic L + is obtained by extending the language of the logic L by formulas [≥ k C ]ϕ where C ranges over 2 Agt * and k ranges over I. The formula [≥ k C ]ϕ has to be read "ϕ is true in all states which are good for group C at least to degree k". We define

≥ k C ϕ def = ¬[≥ k C ]¬ϕ. ≥ k C ϕ
has to be read "ϕ is true in at least one state which is good for group C at least to degree k".

Semantics

Definition 7 (L + -model). L + -models are tuples M = W, R, B, P, GP , π where:

• M = W, R, B, P, π is a L-model; • GP : I × 2 Agt * -→ 2 W
is a function mapping every integer k in I and coalition C to a (possibly empty) set of worlds GP k C such that:

C7 GP k C = i∈C P k i .
The function GP is used to specify the degree of goodness of a certain world according to a given group of agents. In particular, for every k ∈ I and C ∈ 2 Agt * , GP k C is the set of worlds that are good for group C at least to degree k.

Let us generalize the functions κ i à la Spohn and the preference orders i of Section 2.2 to groups. Definition 8 (κ C ). For every C ∈ 2 Agt * and for every w ∈ W we define:

• κ C (w) = n if and only if w ∈ GP n C ; • for every k ∈ I such that k < n, κ C (w) = k if and only if w ∈ GP k C and w ∈ GP k+1 C .
Definition 9 ( C ). For every C ∈ 2 Agt * and for every w, v ∈ W we have

w C v if and only if κ C (w) ≤ κ C (v).
Definition 10 (Max C ). For every C ∈ 2 Agt * we define Max C = max w∈W κ C (w).

The following propositions highlight some interesting aspect of the present notion of group preference.

Proposition 2. κ C (w) ≤ κ C (v) if and only if min C κ i (w) ≤ min i∈C κ i (v).
According to Proposition 2, for the group of agents C the degree of goodness of world v is higher than the degree of goodness of world w if and only if, the lowest degree of goodness of world v for an agent in the group C is higher than the lowest degree of goodness of world w for an agent in the group C.

Proposition 3. w ∈ Max C if and only if w ∈ max v∈W min i∈C κ i (v).
Proposition 3 is about a maximin principle for group preferences: the degree of goodness of world w for group C is maximal if and only if, w belongs to the set of worlds whose lowest degree of goodness for an agent in the group C is maximal. Note that since [START_REF] Rawls | A theory of Justice[END_REF], social procedures based on a maximin principle have been considered appropriate for ensuring fairness and justice. Consequently, we think that maximin is also a good method for defining the concept of group preference.

• W = {w 1 , w 2 , w 3 , w 4 }; • R {i 1 ,i 2 } (w 1 ) = i 1 :c, i 2 :c , R {i 1 ,i 2 } (w 2 ) = i 1 :d, i 2 :d , R {i 1 ,i 2 } (w 3 ) = i 1 :c, i 2 :d , R {i 1 ,i 2 } (w 4 ) = i 1 :d, i 2 :c ; • P 3 i 1 = {w 4 }, P 2 i 1 = {w 1 , w 4 }, P 1 i 1 = {w 1 , w 2 , w 4 }, P 0 i 1 = {w 1 , w 2 , w 3 , w 4 }; • P 3 i 2 = {w 3 }, P 2 i 2 = {w 1 , w 3 }, P 1 i 2 = {w 1 , w 2 , w 3 }, P 0 i 2 = {w 1 , w 2 , w 3 , w 4 }.
Consider now group preferences. We have:

• GP 2 {i 1 ,i 2 } = {w 1 }, GP 1 {i 1 ,i 2 } = {w 1 , w 2 }, GP 0 {i 1 ,i 2 } = {w 1 , w 2 , w 3 , w 4 }. Therefore, w 1 ∈ Max {i 1 ,i 2 } .
In other words, the state in which both i 1 and i 2 cooperate is the best state for the group {i 1 , i 2 }. Indeed, this is the state whose lowest degree of goodness for an agent in {i 1 , i 2 } is maximal.

Before concluding we have to provide the truth condition for [≥ k

C ]ϕ:

M, w |= [≥ k C ]ϕ ⇐⇒ M, v |= ϕ for all v such that v ∈ GP k C .

Axiomatization

Every operator [≥ k C ] satisfies the principles of the basic modal logic K plus the following two additional principles:

[≥ k i ]ϕ ↔ [≥ k {i} ]ϕ (Def [≥ k i ] ) ≥ k C δ ↔ ( i∈C ≥ k i δ ) (GroupPref)
Note that Axiom GroupPref is the fundamental principle relating group preferences with individual preferences.

We call L + the logic axiomatized by the principles of the logic L given in Section 2.3 (Table 1) plus the axioms and the rules of inference of the basic system K for every operator [≥ k C ] and the previous two Axioms Def [≥ k i ] and GroupPref. Definitions of validity and theoremhood for the logic L + are just standard. We write |= L + ϕ if formula ϕ is valid in all L + -models. We write L + ϕ if formula ϕ is a theorem of L + .

The following Theorem 2 can be proved in a way similar to Theorem 1.

Theorem 2. The logic L + is sound and complete with respect to the class of L + -models.

The following are some properties of the operators of collective preference. For every B, C ∈ 2 Agt * such that B ⊆ C and for every k ∈ I we have:

L + [≥ k B ]ϕ → [≥ k C ]ϕ (13) L + [≥ 0 C ]ϕ ↔ ϕ (14) L + [≥ k-1 C ]ϕ → [≥ k C ]ϕ (15)

Strategic games with social rationality

In this section we explore strategic games in which agents are driven by other-regarding motives and social preferences. We consider two different kinds of social rationality: fairness and reciprocity.

Group preferences over formulas and fairness equilibrium

We can now generalize the notion of preference over formulas to groups. For every C ∈ 2 Agt * , we define ψ ≤ C ϕ ("ϕ is for group C at least as good as ψ") as follows:

ψ ≤ C ϕ def = k∈I ( ≥ k C ψ → ≥ k C ϕ).
As the following L + -theorems highlight the operators ≤ C are also total preorders. For every C ∈ 2 Agt * we have:

L + ψ ≤ C ψ (16) L + ((ϕ 1 ≤ C ϕ 2 ) ∧ (ϕ 2 ≤ C ϕ 3 )) → (ϕ 1 ≤ C ϕ 3 ) (17) L + (ϕ 1 ≤ C ϕ 2 ) ∨ (ϕ 2 ≤ C ϕ 1 ) (18) We define ψ < C ϕ def = (ψ ≤ C ϕ) ∧ ¬(ϕ ≤ C ψ), δ ≤ C δ def = δ ≤ C δ and δ < C δ def = (δ ≤ C δ ) ∧ ¬(δ ≤ C δ).
The notion of best response of Section 3.2 can also be generalized to groups. We say that agent i's action a is for the group C a best response to the other agents' joint action δ -i , noted BR C (i:a,δ -i ), if and only if i cannot improve the utility of group C by deciding to do something different from a while the others choose the joint action δ -i , that is:

BR C (i:a,δ -i ) def = b∈Act (( i:b ∧ δ -i ) ≤ C ( i:a ∧ δ -i )).
Note that the notion best response of Section 3.2 is just a special case of the previous definition, when C = {i}. In fact, BR {i} (i:a,δ -i ) is logically equivalent to BR(i:a,δ -i ).

Before concluding this section, we provide a notion of fairness equilibrium. Basically, fairness equilibrium is the collective counterpart of the individualistic notion of Nash equilibrium. Given a certain strategic game, the strategy profile (or joint action) δ is said to be a fairness equilibrium if and only if for every agent i ∈ Agt, i's action δ i is for the entire group Agt a best response to the other agents' joint action δ -i :

FairEq(δ) def = i∈Agt BR Agt (δ i ,δ -i ).
In other words, in a fairness equilibrium every agent chooses an action that, given what the others choose, maximizes the utility of the entire group of agents Agt.

Social rationality

It is time to look at social rationality. Differently from individualistically rational agents defined in Section 3.3, social rational agents also consider the benefits of their choice for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group.

We first define fairness. For every i ∈ Agt and C ∈ 2 Agt * such that i ∈ C:

Fair i,C def = a,b∈Act ( i:a → δ∈∆ ( B i δ -i ∧ ( δ -i , i:b ≤ C δ -i , i:a ))).
According to the previous definition, an agent i is fair with respect to his group C (noted Fair i,C ) if and only if, if he chooses action a then for every alternative action b, there exists a joint action δ -i of the other agents that he considers possible such that, playing a while the others play δ -i is for group C at least as good as playing b while the others play δ -i . Reciprocity can be defined from individualistic rationality and fairness. For every i ∈ Agt and C ∈ 2 Agt * such that i ∈ C: It has to be noted that the previous two notions of social rationality are also positively and negatively introspective. For every i ∈ Agt and C ∈ 2 Agt * we have:

Rec i,C def = ([B i ]( j∈C\{i} Fair j,C ) → Fair i,C ) ∧ (¬[B i ]( j∈C Fair j,C ) → IRat i ).
L + Fair i,C ↔ [B i ]Fair i,C (19) L + ¬Fair i,C ↔ [B i ]¬Fair i,C (20) L + Rec i,C ↔ [B i ]Rec i,C (21) L + ¬Rec i,C ↔ [B i ]¬Rec i,C (22)
The following L + -theorems specify some sufficient conditions for fairness equilibrium. For every δ ∈ ∆ we have:

L + ( i∈Agt (Fair i ∧ [B i ] δ -i ) ∧ δ ) → FairEq(δ) (23) L + ( i∈Agt (Rec i ∧ [B i ]( j =i Fair j ) ∧ [B i ] δ -i ) ∧ δ ) → FairEq(δ) (24) L + ( i∈Agt (Rec i ∧ ¬[B i ]( j =i Fair j ) ∧ [B i ] δ -i ) ∧ δ ) → Nash(δ) (25)
According to L + -theorem 23 if all agents are fair and every agent knows the choices of the other agents, then the selected strategy profile is a fairness equilibrium. According to L +theorem 24 if all agents are reciprocator, every agent believes that all other agents are fair, and every agent knows the choices of the other agents, then the selected strategy profile is a fairness equilibrium. According to L + -theorem 25 if all agents are reciprocator, every agent does not believe that all other agents are fair, and every agent knows the choices of the other agents, then the selected strategy profile is a Nash equilibrium.

Related works

As emphasized in the introduction, although social preferences and social rationality have been extensively studied in experimental economics, no logical analysis of these concepts has been proposed up to now. However, several logical systems exist which support reasoning about strategic games (see, e.g., [START_REF] Van Der Hoek | A logic for strategic reasoning[END_REF][START_REF] Troquard | A logic of games and propositional control[END_REF][START_REF] Lorini | A dynamic logic of agency II: deterministic DLA, Coalition Logic, and game theory[END_REF]) and epistemic strategic games (see, e.g., [START_REF] De Bruin | Explaining games: on the logic of game theoretic explanations[END_REF][START_REF] Roy | Thinking before acting: intentions, logic, rational choice[END_REF][START_REF] Bonanno | A syntactic approach to rationality in games with ordinal payoffs[END_REF][START_REF] Lorini | Epistemic Games in Modal Logic: Joint Actions, Knowledge and Preferences all together[END_REF]). Let us briefly discuss the latter.

De Bruin [START_REF] De Bruin | Explaining games: on the logic of game theoretic explanations[END_REF] has developed a very rich logical framework which enables to reason about the epistemic aspects of strategic games and of extensive games. His system deals with several game-theoretic concepts like the concepts of knowledge, rationality, Nash equilibrium, iterated strict dominance, backward induction. Nevertheless, de Bruin's approach differs from our logical approach to epistemic strategic games in several respects. First of all, our approach is minimalistic since it relies on few primitive concepts: joint action, belief, (individual and group) preference. All other notions such Nash equilibrium and (individual and social) rationality are defined by means of these three primitive concepts. On the contrary, in de Bruin's logic all those notions are atomic propositions managed by a ad hoc axiomatization (see, e.g., [10, pp. 61,65] where special propositions for rationality are introduced). Secondly, we provide a semantics and a complete axiomatics for our logic of epistemic games. De Bruin's approach is purely syntactic: no model-theoretic analysis of games is proposed nor completeness result for the proposed logic is given.

Roy [START_REF] Roy | Thinking before acting: intentions, logic, rational choice[END_REF] has recently proposed a modal logic integrating preference, knowledge and intention. In his approach every world in a model is associated to a nominal which directly refers to a strategy profile in a strategic game. This approach is however limited in expressing formally the structure of a strategic game. In particular, in Roy's logic there is no principle like the L Axiom Indep explaining how possible actions δ i of individual agents are combined to form a strategy profile δ of the current game. Another limitation of Roy's approach is that it does not enable to express the concept of individualistic rationality that we have been able to define in Section 3.3 (see [24, pp. 101]).

Bonanno [START_REF] Bonanno | A syntactic approach to rationality in games with ordinal payoffs[END_REF] integrates modal operators for belief, common belief with constructions expressing agents' preferences over individual actions and strategy profiles, and applies them to the semantic characterization of solution concepts like Iterated Deletion of Strictly Dominated Strategies (IDSDS). Although this logic enables to express the concept of individualistic rationality assumed in classical game theory, it is not sufficiently general to enable to express in the object language solution concepts like Nash equilibrium and IDSDS (note that the latter is defined by Bonanno only in the metalanguage).

Lorini et al. [START_REF] Lorini | Epistemic Games in Modal Logic: Joint Actions, Knowledge and Preferences all together[END_REF] have recently developed a modal logic of epistemic strategic games which overcomes the previous limitations. It enables to express in the object language solution concepts such as Nash equilibrium and IDSDS, and the concept of individualistic rationality. A complete axiomatization of this logic is given and its complexity is studied. However this logic does not have operators for group preference which support reasoning about different kinds of social rationality such as fairness and reciprocity.

Example 3 .

 3 REMARK. Note that the maximin principle also ensures Pareto optimality. In particular, if w ∈ Max C then w ∈ Pareto C , where Pareto C = {v|∀ u ∈ W, ∃i ∈ C: u i v}. Consider again the two-player Prisoner Dilemma (PD) game illustrated in the Example 1 of Section 2.2 with Agt = {i 1 , i 2 } and ACT = {c, d } where c is the action of cooperating and d is the action of defecting. The PD game just needs four different degrees of goodness I = {0, 1, 2, 3} and corresponds to the model M whose elements W, R, P are defined as follows:

Example 4 .

 4 In the Prisoner Dilemma there are two fairness equilibria: mutual cooperation and mutual defection. Hence, in the model illustrated in the Example 1 of Section 2.2 we have that formulas FairEq( i 1 :c, i 2 :c ) and FairEq( i 1 :d, i 2 :d ) are both true at each world w 1 , w 2 , w 3 , w 4 of the model M .

  2. It is well-known that in the Prisoner Dilemma the only Nash equilibrium is mutual defection. Hence, in the model illustrated in the Example 1 of Section 2.2 we have that formula Nash( i 1 :d, i 2 :d ) is true at each world w 1 , w 2 , w 3 , w 4 of the model M .

  That is, an agent i is a reciprocator with respect to his group C (noted Rec i,C ) if and only if, if he believes that the other agents in C \ {i} play fair then he plays fair, otherwise he plays egoistic. For notational convenience we write Fair i instead of Fair i,Agt and Rec i instead of Rec i,Agt .

These models adopt the concept of 'psychological game theory' introduced by[START_REF] Geanakoplos | Psychological games and sequential rationality[END_REF] and recently elaborated by[START_REF] Battigalli | Dynamic psychological games[END_REF]. In psychological game theory, utilities do not only depend on terminal-node payoffs but also on players' beliefs.

On this issue, see also[START_REF] Laverny | From knowledge-based programs to graded belief-based programs, part II: off-line reasoning[END_REF].

Note that this function exists, as Constraint C3 on L-models ensures that R Agt is a bijection.

KEYWORDS

Modal logic, game theory

ABSTRACT

We propose a modal logic that enables to reason about different kinds of rationality in strategic games. This logic integrates the concepts of joint action, belief, individual preference and group preference. The first part of the article is focused on the notion of individualistic rationality assumed in classical game theory: an agent decides to perform a certain action only if the agent believes that this action is a best response to what he expects the others will do. The second part of the article explores different kinds of social rationality such as fairness and reciprocity. Differently from individualistically rational agents (alias self-interested agents), social rational agents also consider the benefits of their choice for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. In the article we also provide a complete axiomatization of our logic of joint action, belief, individual preference and group preference.