Professor Graham Roberts
email: g.c.roberts@soton.ac.uk.

Keywords: COSMIN, measurement properties, outcome measures, validity, severe asthma, systematic review

Background: Valid outcome measures are imperative to evaluate treatment response, yet the suitability of existing endpoints for severe asthma is unclear. This review aimed to identify outcome measures for severe asthma and appraise the quality of their measurement properties.

Methods:

A literature search was performed to identify 'candidate' outcome measures published between 2018 -2020 (PROSPERO, CRD42020204437). A modified Delphi exercise was conducted to select 'key' outcome measures within healthcare professional, patient, pharmaceutical, and regulatory stakeholder groups. Initial validation studies for 'key' measures were rated against modified quality criteria from COnsensus-based Standards for the selection of health Measurement Instruments (COSMIN). The evidence was discussed at multi-stakeholder meetings to ratify 'priority' outcome measures. Subsequently, four bibliographic databases were searched from inception to identify development and validation studies for these endpoints. Two reviewers screened records, extracted data, assessed their methodological quality, and graded the evidence according to COSMIN.

Results: 96 outcome measures were identified as 'candidates', 55 as 'key', and 24 as 'priority' for severe asthma; including clinical, healthcare utilisation, quality of life, asthma control, and composite. 32 studies reported measurement properties of 17 'priority' endpoints from the latter three domains. Only SAQ and C-ACT were developed with input from severe asthma patients. The certainty of evidence was 'low' to 'very low' for most 'priority' endpoints across all measurement properties, and none fulfilled all quality standards.

Conclusion:

Only two outcome measures had robust developmental data for severe asthma. This review informed development of core outcome measures sets for severe asthma.

BACKGROUND

Asthma is the most prevalent chronic respiratory condition affecting an estimated 339 million people worldwide1-3. Approximately 5-10% of these patients have severe asthma2,4. The American Thoracic Society / European Respiratory Society (ATS/ERS), define severe asthma as requiring treatment with high-dose inhaled corticosteroids plus additional controller medication(s) or oral corticosteroids (OCS) to maintain disease control, or which remains suboptimally controlled despite adherence once differential diagnoses have been addressed and any co-morbidities treated2. The impaired quality of life (QoL), adverse effects of long term OCS use, and disproportionately high healthcare cost are causes of concern in this group of patients5-10.

A more nuanced understanding of the pathophysiological mechanisms underlying severe asthma has facilitated the emergence of new-generation treatments, including 'biologics', which can reduce exacerbation rates11, OCS use12, and improve lung function13. In studies investigating these interventions the selection of efficacy endpoints is influenced by multiple aspects of trial design and analysis. However, it is imperative that the outcome measures have robust measurement properties to ensure good quality and appropriate interpretation of results. The selection of outcome measures should be grounded in whether they are valid, reliable, and responsive and sensitive to changes when evaluating differences in the treatment groups14.

The National Institutes of Health (NIH) recommendations15-20 for outcome measurement instruments for asthma are the status quo for clinical practice and research, yet there is no consensus on which outcome measures are important to patients with severe asthma and have robust psychometric properties in this population. Further, guidelines recommend the implementation of patient-reported outcome measures (PROMs) capturing domains such as QoL and asthma control to evaluate treatment response, but do not propose specific instruments21. On the other hand, outcomes that patients consider a treatment priority, such as QoL8,22, are not consistently reflected in physicians' approach to evaluating treatment response23. It is particularly important to use patient-centred outcome measures in effectiveness studies of biologics to justify the high economic24 and individual-level treatment burden6,8,25. A comprehensive view of the validity of outcome measures for severe asthma would also enable the selection of the most appropriate instruments for better disease management by patients and healthcare professionals in practice. This systematic review aimed to (i) identify patient-centred outcome measures considered to be a 'priority' for severe asthma by relevant stakeholders (patient representatives, which included patients, parents/carers, and patient advocacy organisations; healthcare professionals, pharmaceutical, and regulatory representatives) and, (ii) critically appraise, compare, and qualitatively summarize the quality of studies reporting the development and measurement properties of each 'priority' outcome measure using the COnsensus-based Standards for the selection of health Measurement INstruments (COSMIN) guidelines26,27. This systematic review is registered with the International Prospective Register of Systematic Reviews (PROSPERO registration: CRD42020204437). The Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) checklist has been used to guide the reporting of this systematic review28 (Appendix D.1). The methods are described in Figure 1 and full details are reported in the online supplement (Appendix A and B).

Steps 1 -3: Identifying 'priority' outcome measures for severe asthma

In Step 1, a literature search was performed to identify 'candidate' outcome measures (patient-reported, clinical, composite, and imaging) published in four bibliographic databases between May 7, 2018 -May 7, 2020. Subsequently, a modified Delphi exercise was conducted within four stakeholder groups recruited from across Europe (Step 2). This involved two consecutive rounds of surveys wherein stakeholders ranked the importance of 'candidate' outcome measures on a 9-point Likert scale (based on Grading of Recommendations Assessment, Development and Evaluation (GRADE)29). 'Candidate' outcome measures which ≥50% participants in each stakeholder group rated as critical (7-9 on 9-point Likert scale) were ratified as 'key' outcome measures; measures for which consensus was not achieved were discussed at a multi-stakeholder web-conference to obtain agreement on whether they should be classified as 'key'. Emphasis was placed on collating views about 'candidate' outcome measures ranked as 'critical' and 'important' by patient representatives in the round two survey.

Step 3 involved a literature search to identify initial validation studies for selected 'key' outcome measures, followed by appraisal of the quality of measurement property results against modified COSMIN quality criteria. The evidence was discussed at two multistakeholder meetings together with supporting patient-voice evidence30, followed by online voting to select 'priority' outcome measures for adult and paediatric severe asthma. For this voting process data were grouped by patient representatives, and 'academic' representatives inclusive of healthcare professionals, pharmaceutical, and regulatory representatives. Outcome measures which ≥33% participants in at least one of two stakeholder groups voted as a 'priority' were included.

Step 4: Development and measurement properties of 'priority' outcome measures

Search strategy

A search strategy was developed on EMBASE (OVID) and subsequently adapted for the following databases: MEDLINE (OVID), CINAHL (EBSCOhost, Cumulative Index to Nursing and Allied Health Literature), and ISI Web of Science (Thomson Web of Knowledge) (Appendix B.2). Databases were searched from the inception to 20 July 2020. The citation lists of identified studies and reference lists of systematic reviews were searched for additional articles.

Eligibility criteria

The inclusion criteria to identify eligible studies were as follows:

Population: adults (≥ 18 years) and/or children and adolescents (6-17 years) with a diagnosis of asthma. We collated data for all severities of asthma and accounted for the differing relevance for severe asthma in the modified GRADE assessment.

Intervention: any asthma therapy or no intervention.

Comparator: any comparator group including placebo or no comparator.

Study outcomes: outcome measures considered a 'priority' by key stakeholders (Appendix A.4). For patient-reported and composite outcome measures, studies were also eligible if they validated the 'priority' outcome measure with minor modifications according to cultural setting e.g. changing the list of activities to includes items that are more suited to tropical/cold climate.

Study designs: studies aiming to develop ('development' or 'inauguration' article), investigate the content validity (extent to which the outcome measure assesses the concept of interest) and/or one or more measurement properties of a 'priority' outcome measure (including studies with age and sociodemographic subgroups, and different modes of administration).

The exclusion criteria were as follows: systematic reviews and meta-analyses, narrative reviews, discussion papers, commentaries, non-research letters and editorials, abstracts only (e.g. conference paper), animal studies, non-asthma studies e.g. viral bronchiolitis, viral associated wheeze. Studies which reported on linguistic validity, translated, or modified versions of 'priority' patient-reported and composite outcome measures, or used 'priority' measures as outcome measurement instruments, such as clinical trials, were also excluded. Studies reporting correlations between a 'priority' clinical outcome measure and guidelines as 'gold standard' were excluded as they are susceptible to publication bias.

Study selection and data extraction

References were pooled and de-duplicated in Endnote version X9 (Thomson Reuters, Philadelphia, PA), and subsequently uploaded to Rayyan31 (rayyan.qcri.org). Study titles, abstracts, and full-text articles were screened independently by two reviewers (AR, EK) according to the predefined selection criteria. Data extraction from included articles was performed into a pilot-tested form based on the COSMIN guidelines26,27 by one reviewer (AR), and cross-checked by the second reviewer (EK). Any disagreements were resolved through discussion, and, if necessary, involvement of the third reviewer (GR).

Quality appraisal strategy

The methodological quality and certainty of evidence was evaluated in a three-step process by two independent reviewers (AR, EK). Any discrepancies were resolved through discussion, or arbitration by the third reviewer (GR). First, the Risk of Bias (RoB) of developmental, content validation, and studies reporting other measurement properties was assessed using the COSMIN checklist for PROMs26,32. For non-patient reported outcomes (including composites with clinician-performed components), a separate checklist33 was used. The definitions of measurement properties are provided in Table S1. Second, results of the measurement properties from each study were rated against quality criteria (Table S2) and qualitatively pooled to generate an overall rating of sufficient (+), insufficient (-), or indeterminate (?). Third, the certainty of evidence was determined using a modified GRADE approach26,32,34. The results were summarized using narrative synthesis.

RESULTS

Steps 1-3: Identifying 'priority' outcome measures for severe asthma

The detailed results for identifying 'priority' outcome measures for severe asthma can be found in Appendix C. Briefly, a total of 96 'candidate' outcome measures were retrieved from the literature search. 55 'key' outcome measures were selected following the modified Delphi exercise. Validation data were available for 32 'key' patient-reported and composite outcome measures, and one healthcare resource utilisation (HRU) measure, while no data were identified for clinical outcome measures. The 'priority' outcome measures selected following the multi-stakeholder meetings and online voting are presented in Figure 2.

Step 4: Appraisal of development and measurement properties of 'priority' outcome measures

Study characteristics

A total of 32 studies were included; 16 met the inclusion criteria for the adult population, 12 studies for the paediatric population, while 4 were eligible for both populations (Figure 3). The characteristics of included studies are presented in Tables S3 andS4 for adult and paediatric populations respectively. 12 and seven articles described the development and/or measurement properties of QoL measures for adults35-46, and children and adolescents47-53, respectively. Only one content validation study was identified54. Six articles reported the development and/or measurement properties of asthma control instruments for adults55-60, while eight articles were included for the paediatric population55,56,58,60-64. One study validated the Asthma Control and Communication Instrument (ACCI), a composite tool prioritised for adults65, and one reported the Composite Asthma Severity Index (CASI), a tool for patients older than 6 years66. Of the included studies, ten involved participants with severe asthma; but in most cases the percentage of this population was <50%, and differing definitions of asthma were used (including guidelines, physician diagnosis, and self-report).

Risk of bias and quality of evidence for 'priority' outcome measures

The developmental process for the 'priority' outcome measures can be found in Table S5-S6. The characteristics of the 'priority' QoL, asthma control, and composite outcome measures are presented in Table 1. An overview of the rating of measurement property results against COSMIN quality criteria, and the quality of evidence per 'priority' outcome measure for adult and paediatric severe asthma is presented in Table 2. Structural validity (for asthma control measures based on a reflective model) and measurement error were not assessed in any of the identified studies. The most common reason for downgrading was indirectness, as <50% of participants in multiple studies had severe asthma.

According to the COSMIN RoB checklist, the assessments for studies ranged from 'inadequate' to 'very good' (Tables S7 andS8) . For 'doubtful' or 'inadequate' ratings, frequent methodological limitations included lack of description of subgroup characteristics for test-retest reliability; small sample size; inappropriate statistical methods for evaluating responsiveness (or reporting of responsiveness index only), and lack of information on unidimensionality when assessing internal consistency.

'Priority' quality of life outcome measures for adult severe asthma

Content validity was rated most favourably for the Severe Asthma Questionnaire (SAQ) with sufficient 'moderate' quality evidence across all domains. The SAQ was the only 'priority' measure developed with input from adults with severe asthma. The Asthma Quality of Life Questionnaire (AQLQ) and its ad hoc standardised and mini versions had variable ratings with 'low' to 'very low' quality evidence.

For internal consistency there was sufficient 'very low' to 'moderate' quality evidence for the AQLQ, the Asthma Quality of Life Questionnaire Standardised (AQLQ-S) and Mini-AQLQ. Likewise, all measures had good to excellent test-retest reliability, but the quality of evidence was variable, with only the SAQ scored as 'moderate' quality evidence. The Mini-AQLQ showed strong correlation with AQLQ scores, indicating sufficient criterion validity. Only the Mini-AQLQ and SAQ had sufficient evidence for construct validityscores on both instruments had strong correlations with comparator QoL measures, and weak correlations with FEV1% predicted. The AQLQ and AQLQ-S had sufficient, but 'low' and 'very low' quality evidence for responsiveness to change; the change scores for patients who were unstable between study visits exceeded the Minimal Clinical Important Difference (MCID).

'Priority' asthma control outcome measures for adult severe asthma

The Asthma Control Test (ACT), Asthma Control Questionnaire 5 (ACQ-5), and ACQ-6 had insufficient to inconsistent evidence for content validity. The quality of evidence was 'very low' primarily because no patients were involved in the developmental process. However, all the questionnaires had good to excellent internal consistency and test-retest reliability to support the sufficient ratings. The shortened ACQ-6 and ACQ-5 showed strong correlations with the original ACQ-760, illustrating sufficient criterion validity. In contrast, there was insufficient 'low' to 'very low' quality evidence for construct validity and responsiveness to change for all asthma control instruments. There were studies reporting moderate correlations between scores on 'priority' instruments and comparators such as physician assessed asthma control and QoL, and weak correlations with FEV1. Similarly, some studies reported that patients with unstable asthma had meaningful changes in priority control instrument scores i.e. changes exceeding MCID. However, after qualitatively pooling the evidence the a priori threshold for hypotheses was not fulfilled for either of the measurement properties.

'Priority' composite outcome measures for adult severe asthma

The ACCI was awarded sufficient ratings for relevance and comprehensibility, as patients were involved in its development, but comprehensiveness was rated insufficient. Of the other measurement properties appraised, there was only sufficient, albeit 'low' quality evidence, for construct validity. ACCI scores had moderate to strong correlations with asthma control and asthma-specific QoL, and relatively weaker correlations with generic QoL.

'Priority' quality of life outcome measures for paediatric severe asthma

Content validity was rated inconsistent for all 'priority' QoL instruments except the Paediatric Asthma Quality of Life Questionnaire (PAQLQ), which had sufficient evidence for comprehensibility. The quality of evidence was 'very low' to 'low' as only patients with mildto-moderate asthma were involved in the development. There was sufficient evidence for other measurement properties appraised, except Mini-PAQLQ, which was rated insufficient for internal consistency. The quality of evidence was predominantly higher for the PAQLQ and Mini-PAQLQ compared to the PAQLQ-S.

'Priority' asthma control outcome measures for paediatric severe asthma

The Childhood Asthma Control Test (C-ACT) had the most robust developmental process; there was sufficient 'moderate' quality evidence for relevance and comprehensibility, with 28% of the patients in the developmental study diagnosed with severe asthma. The ACT, ACQ-5, ACQ-6, and ACQ-7 received inconsistent and insufficient ratings for relevance and comprehensiveness respectively. However, the ACQ versions performed better than ACT in comprehensibility as children and adolescents were consulted about the phrasing of items and response options. The 'priority' asthma control measures were rated sufficient for all measurement properties assessed, except ACT which had insufficient evidence for responsiveness to change. The latter was primarily because the threshold for a priori hypotheses as per COSMIN quality criteria was not achieved. In contrast, all ACQ versions had robust responsiveness to change whereby change scores on the instrument had moderate to strong correlations with QoL instrument change scores, and weak correlations with changes in FEV1% predicted. Further, ACQ versions' and C-ACT mean score changes exceeded the MCID for patients whose asthma was unstable between study visits.

'Priority' composite outcome measures for paediatric severe asthma

CASI received inconsistent to insufficient ratings for all components of content validity, primarily because no patients were involved in its development. The internal consistency and test-retest reliability were rated insufficient. As CASI is multidimensional, internal consistency should have been determined for each unidimensional subscale instead of the total score. There was sufficient 'low' quality evidence for construct validity, with moderate correlations between CASI and comparators such as symptom days and asthma control. Sufficient 'moderate' quality responsiveness to change was shown by an improvement in the treatment group captured by a higher standardized effect size for CASI compared to symptom days.

This systematic review used a multi-step approach to identify and determine the quality of the development and measurement properties of 'priority' outcome measures for severe asthma, in accordance with the COSMIN guidelines. The current study is unique as it synthesises evidence for outcome measures which are meaningful to patients with severe asthma, and acceptable to other stakeholders, including healthcare professionals, pharmaceutical, and regulatory representatives. From 96 'candidate' outcome measures identified by the literature search, stakeholders selected 55 'key' outcome measures in a modified Delphi exercise, and subsequently 11 and 13 'priority' outcome measures for adult and paediatric severe asthma respectively. The rationale for excluding outcome measures at each step included lack of relevance for effectiveness studies, poor feasibility in terms of cost and accessibility in pan-European healthcare systems, and redundancy with other outcome measures. Of the 'priority' outcome measures, the SAQ and C-ACT had the most robust developmental evidence for the severe asthma population. While other 'priority' patient-reported and composite outcome measures had data to support validity and reliability, the quality of evidence was limited primarily by the lack of studies with severe asthma participants. This review has been used as a basis for the development of Core Outcome Measures sets for adult and paediatric Severe Asthma (COMSA)67.

QoL achieved consensus as a 'priority' domain, yet it is used as a secondary outcome in most severe asthma studies68. Although QoL only moderately correlates with objective measures69, a higher frequency of exacerbations in patients with moderate-to-severe asthma has been associated with deterioration in QoL70. Additionally, biologics such as mepolizumab show promising improvements in QoL scores71,72. QoL is the most important domain for patients with severe asthma8, outperforming reduction in exacerbation rates. Pertinently, there is an indirect cost patients experience including the burden of daily symptoms resulting in impaired QoL73,74. Future trials should seek to capture this individual disease experience, and ultimately demonstrate QoL benefits beyond the efficacy of a treatment.

Most developmental studies for 'priority' QoL measures confounded the construct of QoL with asthma control (symptoms and/or functional status), and were therefore downgraded for comprehensiveness. Similarly, the NIH appraisal highlighted that questionnaires are more likely to assess symptom control instead of QoL impairments experienced due to the ongoing symptoms20. However, the SAQ was developed following the Food and Drug Administration (FDA) guidance14, and thus captures the impact of both asthma symptoms and treatment on QoL specific to adult severe asthma, illustrated by favourable ratings for content validity. A PROM that can capture the QoL limitations as experienced by children and adolescents with severe asthma remains an unmet need.

The majority of PROMs prioritised in this review are legacy measures. They were developed when different approaches to instrument design were accepted, and it was not compulsory to report the detailed methodological steps undertaken. These measures had inconsistent scores on the modern COSMIN appraisal tool. However, legacy PROMs such as the AQLQ continue to be implemented in research and practice68, amplified by their availability in a variety of languages and modes of administration. In contrast, the SAQ has not been utilised widely in research beyond the original studies, probably because it was recently published, so is only now being incorporated into study protocols. Identifying additional barriers impeding the use of modern instruments should enable their uptake by study sponsors across clinical programs.

The prioritisation of asthma control in this review reflects its positionality as an important treatment goal75. An appraisal conducted in 2011 also recommended the asthma control measures selected here as 'core' for asthma research16. Although reviews exist about the 'priority' control instruments76-78, none assess the methodological quality of measurement properties for severe asthma. The present synthesis highlights the shortcomings of traditional asthma control tools; the quality of evidence is limited for adult instruments due to the lack of patient involvement in their development55,58, while there is a dearth of studies exclusive to the paediatric severe asthma population validating measures designed for patients ≥ 12 years old55,58. However, these asthma control measures are used in the majority of severe asthma trials68, thus investigators should adjust for such limitations when interpreting scores derived from the questionnaires.

Responsiveness to change is a crucial measurement property for any trial endpoint, as poor responsiveness to change can result in false-negative conclusions for the effect of treatment79. Some of the priority outcome measures lacked data which could be assessed using the COSMIN toolkit. Of note, responsiveness to change data for the SAQ were published as a research letter after the search date, and thus not included in the current analysis80,81. If these data were included, the SAQ would have sufficient evidence for responsiveness to change, supporting its status as one of the only measures with robust evidence across the measurement properties assessed. The SAQ is also sensitive to detecting benefits of biologics in severe asthma, supporting its use as a primary endpoint in future clinical trials82. The MCID83 is not considered a measurement property as per COSMIN guidance, but rather an aspect of interpretability, and therefore its quality was not appraised for the 'priority' outcome measures. Nonetheless, MCIDs are imperative for endpoints in effectiveness studies, as they represent a clinically relevant change rather than just a statistically significant one.

Despite the lack of validation data for 'priority' clinical and HRU endpoints, they are a cornerstone of severe asthma management in practice. In addition, the European Medicine's Agency (EMA) recommends clinical outcomes such as lung function should be included at least as secondary endpoints in studies investigating treatments for asthma84. Future studies should therefore establish the validity of the 'priority' clinical and HRU outcome measures for severe asthma, with a focus on their relevance from a patient-centred standpoint.

Exacerbations are the primary endpoint in most phase III trials investigating new-generation therapies85-88, but the definitions used have immense variability89. A composite tool for severe asthma exacerbations90 developed in patients taking benralizumab was discussed by the multi-stakeholder panel in Step 3 of the review. However, it was not prioritised primarily because it was developed without any input from patients. The panel ratified the ATS/ERS definition of exacerbations2, but a validated tool to capture severe exacerbations acceptable to relevant stakeholders is a significant research gap.

Around 20-60% of adult patients with severe asthma91-93 are prescribed mOCS, and their associated adverse effects2,94 have been shown to impair QoL46. Reduction in OCS use is a treatment priority for these patients8, yet existing adult QoL tools apart from the SAQ fail to capture the unique treatment burden associated with long-term OCS use45. Although mOCS use was also prioritised for paediatric severe asthma, patient representatives voiced that for older adolescents, reduction in OCS use and its associated side effects are a 'priority' in some European countries. Healthcare professionals commented that the prioritisation of OCS may differ according to country or healthcare system; mOCS use is not part of the treatment regime for young children in most severe asthma specialist units across Europe. Investigations in paediatric severe asthma cohorts would be valuable in determining the importance of OCS use as a trial endpoint for this population. Some outcome measures validated in severe asthma, including those utilised for assessing response to biologics, were deprioritised early in the multi-step process. The rationale ranged from lack of patient-centredness and feasibility (poor accessibility in healthcare systems across Europe), to redundancy with other outcome measures. For example, the St George's Respiratory Questionnaire (SGRQ) has been used in severe asthma trials68. However, stakeholders commented the 50-item instrument was developed primarily for Chronic Obstructive Pulmonary Disease; thus, it may be burdensome for patients to complete, and fail to capture the breadth of experiences unique to patients with severe asthma. Likewise, generic QoL questionnaires are paramount as they enable cross-disease comparisons and calculations of health utility95. But none were prioritised as they are not as sensitive to impairment and change as asthma-specific QoL measures. Researchers and practitioners should adopt this lens of clinical utility and patient-relevance, as opposed to frequency of use, when selecting instruments for clinical trials and patient-monitoring in practice.

Strengths and limitations

Our study has several strengths. We used a rigorous multi-step process to identify outcome measures considered a 'priority' for severe asthma by four stakeholder groups. Higher weighting was placed on patient representatives' votes to ensure outcome measure selection at each stage of the study was grounded in a patient perspective. The rigorous COSMIN methodology enabled a standardised quality appraisal of included studies with minimal reviewer bias. Although most included studies had participants with a range of asthma severities, the quality of evidence was evaluated in the severe asthma context. This outlook on the evidence will enable readers to select the most appropriate or robust 'priority' outcome measure according to their required domain of assessment. Moreover, we engaged adult and paediatric patient and parent/carer representatives to inform the critical appraisal of content validity of outcome measures. The insights provided by patient representatives ranged from conceptual (e.g. importance of fatigue as a QoL deficit experienced by adults with severe asthma), to item design (e.g. preference for a global item to evaluate QoL or asthma control). We limited our review to studies of English language version of outcome measures published in English, and therefore cross-cultural validity (including measurement invariance) could not be appraised. Other studies have reported the validity of other language versions of 'priority' PROMs with a higher percentage of patients with severe asthma than those included in this review, such as QoL96-99 and asthma control99-102 domains. The data reported herein therefore should not be considered exhaustive, but rather an overview of the quality of the available evidence for the severe asthma subgroup. The literature was searched for contemporary 'candidate' outcome measures published in a two-year time frame with the safeguard that stakeholders could propose additional outcome measures where justified. The search for studies of 'priority' outcome measures was performed in 2020 as the review was undertaken to inform the development of the COMSA67. A worst case scenario was adopted for the overall quality of each study since COSMIN appraisal tools use a 'worst score counts' principle, whereby high quality scores may be overridden by the lowest scores returned. Lastly, we undertook a narrative synthesis because the heterogeneity in severe asthma definitions, some of the outcome measures such as exacerbations, and consequently study populations precluded a quantitative (meta-) analysis.

Implications for research and practice

The findings reported herein informed the COMSA consensus study67. The implementation of the COMSA will create a more consistent and transparent patient-relevant approach across regulatory, value assessment and market access institutions, as well as in clinical trial development. Currently, in most trials the sole selection criterion for selecting endpoints, especially PROMs, is whether they are widely used, with limited regard for the superiority of the quality of measurement properties. This evidence synthesis on the quality of measurement properties should aid researchers and practitioners to choose the most appropriate outcome measures to evaluate treatment response. Further, this review offers insights into the developmental limitations of historically used 'priority' QoL and asthma control measures for the severe asthma population. Poor content validity can indirectly hamper the quality of other measurement properties, including responsiveness to change. Regulatory bodies should therefore take caution when interpreting results of trials using legacy endpoints to investigate new-generation therapies for severe asthma. The lack of validation data for clinical and HRU endpoints warrants studies which implement standardised approaches to evaluate their measurement properties in participants with severe asthma.

Outcome measures with robust measurement properties and practical feasibility are a prerequisite for evidence-based healthcare. This review identified 96 'candidate' outcome measures and used a multi-stakeholder consensus process to select 24 'priority' outcome measures for adult and paediatric severe asthma. Clinical and HRU endpoints are the mainstay as per the regulatory demands for approval of medications, but the validity of these endpoints should be established in adult and paediatric severe asthma as a matter of some urgency. Of the included patient-reported and composite outcome measures, only the SAQ and C-ACT were developed with input from severe asthma patients, supported by predominantly sufficient 'moderate' quality evidence for the measurement properties assessed. Further validation in clinical trials, and translation to other languages should enable the SAQ to become a status quo QoL endpoint for phase III studies. Despite the 'low' to 'very low' quality developmental evidence for legacy QoL and asthma control measures, most had sufficient evidence for the measurement properties appraised except responsiveness to change. There is an urgent need to follow contemporary standards14,103,104 when developing new outcome measurement instruments. To that end, researchers should seek to develop a scale for paediatric QoL based on extensive qualitative input from patients with severe asthma and subsequently validate it in studies with good methodological quality. Overview of the multi-step process for identifying and appraising patient-centred priority outcome measures for severe asthma.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3. PRISMA flow diagram for study selection. PROM, Patient Reported Outcome Measure.

Acknowledgements:

We would like to acknowledge the support of the 3TR (Taxonomy, treatment, targets, and remission) consortium in funding this systematic review. The funder had no role in the development of the protocol, conduct and writing of the review or decision to publish. We would like to thank Paula Sands for her assistance in developing the search strategy and all 3TR Respiratory Work Package members for their contribution. We also acknowledge Thomy Tonia, a methodologist affiliated with ERS, for her constructive feedback on the study protocol. Special thanks to Professor Elizabeth Juniper for her valuable insights regarding outcome measures developed by her group. A part of this manuscript was submitted by AR for her second progression review report for her doctorate degree qualification (University of Southampton, UK).

ATS / ERS (

Funding:

AR and EK were supported by the European Union Innovative Medicines Initiative 2 (IMI2) 3TR project. GR is funded by University of Southampton and National Institute of Health Research via Southampton's Biomedical Research Centre. This project has received funding from the Innovative Medicines Initiative 2 Joint Undertaking (JU) under grant agreement No 831434 (3TR). The JU receives support from the European Union's Horizon 2020 research and innovation programme and EFPIA.

Authors' contributions:

Study concept and design: GR, AR. Conduct and analysis of modified Delphi exercise, multistakeholder meetings, and online voting: AR, EK. Literature searches, data extraction, and COSMIN assessments: AR, EK. Drafting of the original manuscript: AR, EK, GR. All authors provided critical review of the manuscript and approved the final version.

Disclaimer:

The content of this publication reflects only the authors' views and the JU is not responsible for any use that may be made of the information it contains.

Conflict of interest:

Anna Rattu and Ekaterina Khaleva declare funding from the European Commission's Innovative Medicines Initiative 2 Joint Undertaking (JU) under grant agreement No. 831434 (3TR) for this manuscript. Chris Brightling declares grants from GSK, AZ, Novartis, Chiesi, BI, Genentech, Roche, Sanofi, Mologic, 4DPharma; consulting fees from GSK, AZ, Novartis, Chiesi, BI, Genentech, Roche, Sanofi, Mologic, 4DPharma, TEVA and support from the 3TR project. Sven-Erik Dahlén declares 3TR IMI Grant; consulting fees for AZ, Cayman Co, GSK, Novartis, Regeneron, Sanofi and Teva; payment for lectures from AZ and Sanofi. Sven-Erik Dahlén declares 3TR IMI Grant; consulting fees for AZ, Cayman Co, GSK, Novartis, Regeneron, Sanofi and Teva; payment for lectures from AZ and Sanofi. Apostolos Bossios declares honoraria for lectures from GSK, AZ, Teva and Novartis; support for attending meetings from AZ and Novartis; honoraria for Advisory Board Meetings from GSK, AZ, Teva, Novartis and Sanofi; being a member of the steering Committee of SHARP, Secretary of Assembly 5 (Airway diseases, asthma, COPD, and chronic cough), European Respiratory Society; Vice-chair of Nordic Severe Asthma Network (NSAN). Louise J Fleming declares participation in advisory boards and honoraria for lectures from Sanofi, Respiri UK, AZ, Novartis and Teva outside of submitted work. All payments were made to her institution. Kian Fan Chung has received honoraria for participating in Advisory Board meetings of GSK, AZ, Roche, Novartis, Merck, and Shionogi regarding treatments for asthma, chronic obstructive pulmonary disease and chronic cough and has also been renumerated for speaking engagements for Novartis & AZ; received MRC grant on Precision Medicine for severe asthma, EPSRC grant on air pollution and asthma and GSK grant on mepolizumab and eosinophils in asthma. Erik Melén declares consulting fees from AZ, Chiesi, Novartis and Sanofi outside the submitted work. Rekha Chaudhuri has received lecture fees from GSK, AZ, Teva, Chiesi, Sanofi and Novartis; honoraria for Advisory Board Meetings from GSK, AZ, Teva, Chiesi, Novartis; sponsorship to attend international scientific meetings from Chiesi, Napp, Sanofi and GSK and a research grant to her Institute from AZ for a UK multicentre study. Ratko Djukanovic declares funding from ERS, TEVA, GSK, Novartis, Sanofi and Chiesi for the SHARP CRC; consulting fees for Synairgen; honorarium for a lecture from GSK; participation on a Data Safety Monitoring Board or Advisory Board for Kymab (Cambridge) and shares in Synairgen outside of the submitted work. Rekha Chaudhuri has received lecture fees from GSK, AZ, Teva, Chiesi, Sanofi and Novartis; honoraria for Advisory Board Meetings from GSK, AZ, Teva, Chiesi, Novartis; sponsorship to attend international scientific meetings from Chiesi, Napp, Sanofi and GSK and a research grant to her Institute from AZ for a UK multi-centre study. Andrew Exley declares being a minority shareholder in GlaxoSmithKline PLC. Gerard H Koppelman reports receiving research grants from Lung Foundation of the Netherlands, Ubbo Emmius Foundation, H2020 European Union, TEVA the Netherlands, GSK, Vertex, outside this work (money to institution); he reports memberships of advisory boards to GSK and PURE-IMS, outside this work (money to institution). Arnaud Bourdin declares unrestricted grants from AZ and Boeringher Ingelheim to his institute; consulting fees, honoraria for lectures, presentations, speakers bureaus, manuscript writing or educational events, and support for attending meetings and travel from AZ, Boeringher Ingelheim, GSK, Novartis, Sanofi; participation on a Data Safety Monitoring Board or Advisory Board for AZ, Boeringher Ingelheim, GSK, Novartis, Sanofi; Abs science. Celeste Porsbjerg declares grants (paid to institution), consulting fees (paid to institution and personal honoraria) and honoraria for lectures (paid to institution and personal honoraria) from AZ, GSK, Novartis, TEVA, Sanofi, Chiesi and ALK; participation in the Advisory Board (paid to institution and personal honoraria) for AZ, Novartis, TEVA, Sanofi and ALK.


Yukio Koriyama

Balázs Szentes

A Resurrection of the Condorcet Jury Theorem

This paper analyzes the optimal size of a deliberating committee where, (i) there is no con ‡ict of interest among individuals, and (ii) information acquisition is costly. The committee members simultaneously decide whether or not to acquire information, and then, they make the ex-post e¢ cient decision. The optimal committee size, k , is shown to be bounded. The main result of this paper is that any arbitrarily large committee aggregates the decentralized information more e¢ ciently than the committee of size k 2. This result implies that oversized committees generate only small ine¢ ciencies.

Introduction

The classical Condorcet Jury Theorem (CJT) states that large committees can aggregate decentralized information more e¢ ciently than small ones. Its origin can be traced to the dawn of the French Revolution when Marie Jean Antoine Nicolas Caritat le Marquis de Condorcet (1785, translation 1994) investigated the decision-making processes in societies. 1 A recent literature on committee design has pointed out that if the information acquisition is costly, the CJT fails to hold.

The reasoning is that if the size of a committee is large, a committee member realizes that the probability that she can in ‡uence the …nal decision is too small compared to the cost of information acquisition. As a result, she might prefer to avoid this cost and free-ride on the information of others. Therefore, larger committees might generate lower social welfare than smaller ones. These results suggest that in the presence of costly information acquisition, optimally choosing the size of a committee is both an important and delicate issue. In this paper, we characterize the welfare loss associated to oversized committees, and we show that this loss is surprisingly small in certain environments. Therefore, as long as the committee size is large enough, the careful design of a committee might not be as important as it was originally thought to be. In fact, if either the information structure is ambiguous, or the committee has to make decisions in various informational environments, it might be optimal to design the committee to be as large as possible.

Committee design receives a considerable attention by economists because, in many situations, groups rather than individuals make decisions. Information about the desirability of the possible decisions is often decentralized: individual group members must separately acquire costly information about the alternatives. A classical example is a jury trial where a jury has to decide whether a defendant is guilty or innocent. Each juror individually obtain some information about the defendant at some e¤ort cost (paying attention to the trial, investigating evidences, etc.). Likewise, when a …rm is facing the decision whether or not to implement a project each member of the executive committee can collect information about the pro…tability of the project (by spending time and exerting e¤ort). Yet another example is the hiring decisions of academic departments.

Each member of the recruiting committee must review the applications individually before making a collective decision. What these examples have in common is the fact that information acquisition is costly and often unobservable.

The exact setup analyzed in this paper is described as follows. A group of individuals has to make a binary decision. There is no con ‡ict of interest among the group members, but they have imperfect information about which decision is the best. First, k individuals are asked to serve in a committee. Then, the committee members simultaneously decide whether or not to invest in an informative signal. Finally, the committee makes the optimal decision given the acquired information. We do not explicitly model how the committee members communicate and aggregate information. Instead, we simply assume that they end up making the ex-post e¢ cient decision. 2The only strategic choice an individual must make in our model is the choice whether or not to acquire a signal upon being selected to serve in the committee.

The central question of our paper is the following: how does the committee size, k, a¤ect social welfare? First, for each k, we fully characterize the set of equilibria (including asymmetric and mixed-strategy equilibria). We show that there exists a k P (2 N) such that whenever k k P , there is a unique equilibrium in which each committee member invests in information with probability one. Furthermore, the social welfare generated by these equilibria is an increasing function of k. If k > k P , then there are multiple equilibria and many of them involve randomizations by the members. We also show that the social welfare generated by the worst equilibrium in the game, where the committee size is k, is decreasing in k if k > k P . The optimal committee size, k , is de…ned such that (i) if the committee size is k , then there exists an equilibrium that maximizes social welfare, and (ii) in this equilibrium, each member invests in information with positive probability. We prove that the optimal committee size, k , is either k P or k P + 1. This implies that the CJT fails to hold: large committees can generate smaller social welfare than smaller committees. Nevertheless, we show that if the size of the committee is larger than k , even the worst equilibrium generates higher social welfare than the unique equilibrium in the committee of size k 2. That is, the welfare loss due to an oversized committee is quite small.

Literature Review

Although the Condorcet Jury Theorem provides important support for the basis of democratic decision making, many of the premises of the theorem have been criticized. Perhaps most importantly, Condorcet assumes sincere voting. That is, each individual votes as if she were the only voter in the society. This means that an individual votes for the alternative that is best, conditional on her signal. Austen-Smith and Banks (1996) showed that in general, sincere voting is not consistent with equilibrium behavior. This is because a rational individual votes not only conditional on her signal, but also on her being pivotal. Feddersen and Pesendorfer (1998) have shown that as the jury size increases, the probability of convicting an innocent can actually increase under the unanimity rule.

A variety of papers have shown, however, that even if the voters are strategic, in certain environments the outcome of a voting converges to the e¢ cient outcome as the number of voters goes to in…nity. Feddersen and Pesendorfer (1997) investigate a model in which preferences are heterogeneous and each voter has a private signal concerning which alternative is best. They construct an equilibrium for each population size, such that the equilibrium outcome converges to the full information outcome as the number of voters goes to in…nity. The full information outcome is de…ned as the result of a voting game, where all information is public. [START_REF] Myerson | Extended Poisson Games and the Condorcet Jury Theorem[END_REF] has shown that asymptotic e¢ ciency can be achieved even if there is population uncertainty; that is, a voter does not know how many other voters there are.

In contrast, the Condorcet Jury Theorem might fail to hold if the information acquisition is costly. [START_REF] Mukhopadhaya | Jury Size and the Free Rider Problem[END_REF] has considered a model, similar to ours, where voters have identical preferences but information acquisition is costly. He has shown by numerical examples that mixedstrategy equilibria in large committees may generate lower expected welfare than pure-strategy equilibria in small committees. 3Martinelli ( 2006) also introduced cost of information acquisition and, he allows the precision of the signals to depend continuously on the amount of investment. [START_REF] Martinelli | Would Rational Voters Acquire Costly Information?[END_REF] proves that if the cost and the marginal cost of the precision are zero at the zero level of precision, then the decision is asymptotically e¢ cient. More precisely, if the size of the committee converges to in…nity, then there is a sequence of symmetric equilibria in which each member invests only a little, and the probability of a correct decision converges to one. 4We think that [START_REF] Martinelli | Would Rational Voters Acquire Costly Information?[END_REF] contributed substantially towards the understanding of the e¢ ciency properties of group decision making if there is no …xed cost associated to information acquisition. However, we believe that this …xed cost aspect is an essential feature of many environments. Indeed, one has to pay the price of a newspaper even if he decides to throw it away later.

The management of a company has to pay for a consultant even if the work of the consultant will be completely ignored. Similarly, a juror has to sit through the trial even if he decides not to pay any attention. Hence, we think that our paper is an important complementary result to [START_REF] Martinelli | Would Rational Voters Acquire Costly Information?[END_REF].

Numerous papers have analyzed the optimal decision rules in the presence of costly information. [START_REF] Persico | Committee Design with Endogenous Information[END_REF] discusses the relationship between the optimal decision rules and the accuracy of the signals. He shows that a voting rule that requires a strong consensus in order to upset the status quo is only optimal if the signals are su¢ ciently accurate. The intuition for the extreme case, where the decision rule is the unanimity rule, is the following: under the unanimity rule, the probability of being pivotal is small. However, this probability increases as the signals become more accurate. Therefore, in order to provide a voter with an incentive to invest in information, the signals must be su¢ ciently accurate.

Li (2001), [START_REF] Gerardi | Information Acquisition in Committees[END_REF], and Gershkov and Szentes (2004) have shown that the optimal voting mechanism sometimes involves ex-post ine¢ cient decisions. That is, the optimal mechanism might specify ine¢ cient decisions for certain signal pro…les. We believe that there are many situations where such a commitment device is not available. That is why we simply restrict attention to ex post e¢ cient decision rules. We believe that this is the appropriate assumption in the context of a deliberating committee in which there is no con ‡ict of interest among individuals.

Section 2 describes the model. The main theorems are stated and proved in Section 3. Section 4 concludes. Some of the proofs are relegated to the Appendix.

The Model

There is a population consisting of N (> 1) individuals. The state of the world, !, can take one of two values: 1 and 1. Furthermore, Pr [! = 1] = 2 (0; 1). The society must make a decision, d, which is either 1 or 1. There is no con ‡ict of interest among individuals. Each individual has a bene…t of u (d; !) if decision d is made when the state of the world is !. In particular,

u (d; !) = 8 > > < > > : 0 if d = !; q if d = 1 and ! = 1;
(1 q) if d = 1 and ! = 1;

support of the cost distribution is not bounded away from zero, asymptotic e¢ ciency can be achieved. On the other hand, if the cost is bounded away form zero and the number of voters is large, nobody acquires information in any equilibrium.

where q 2 (0; 1), indicates the severity of type-I error 5 . Each individual can purchase a signal at a cost c (> 0) at most once. Signals are iid across individuals conditional on the realization of the state of the world. The ex-post payo¤ of an individual who invests in information is u c. Each individual maximizes her expected payo¤.

There are two stages of the decision-making process. At Stage 1, k ( N ) members of the society are designated to serve in the committee at random. At Stage 2, the committee members decide simultaneously and independently whether or not to invest in information. Then, the e¢ cient decision is made given the signals collected by the members.

We do not model explicitly how committee members deliberate at Stage 2. Since there is no con ‡ict of interest among the members, it is easy to design a communication protocol that e¢ ciently aggregates information. Alternatively, one can assume that the acquired information is hard. Hence, no communication is necessary for making the ex-post e¢ cient decision. We focus merely on the committee members'incentives to acquire information.

Next, we turn our attention to the de…nition of social welfare. First, let denote the ex-post e¢ cient decision rule. That is, is a mapping from sets of signals into possible decisions. If the signal pro…le is (s 1 ; :::; s n ), where n is the number of acquired signals, then

(s 1 ; :::; s n ) = 1 , E ! [ u (1; !)j s 1 ; :::; s n ] E ! [ u ( 1; !)j s 1 ; :::; s n ] .
The social welfare is measured as the expected sum of the payo¤s of the individuals, that is,

E s1; ;sn;! [N u ( (s 1 ; :::; s n ) ; !) cn] ; (1)
where the expectation also takes into account the possible randomization of the individuals. That is, n can be a random variable.

If the committee is large, then a member might prefer to save the cost of information acquisition and choose to rely on the opinions of others. On the other hand, if k is too small, there is too little information to aggregate, and thus the …nal decision is likely to be ine¢ cient. The question is: What is the optimal k that maximizes ex-ante social welfare? To be more speci…c, the optimal size of the committee is k, if (i) the most e¢ cient equilibrium, in the committee with k members, maximizes the social surplus among all equilibria in any committee, and (ii) each member acquires information with positive probability in this equilibrium.

Since the signals are iid conditional on the state of the world, the expected bene…t of an individual from the ex post e¢ cient decision is a function of the number of signals acquired. We de…ne this function as follows:

(n) = E s1; ;sn;! [u ( (s 1 ; ; s n ) ; !)] .
We assume that the signals are informative about the state of the world, but only imperfectly.

That is, as the number of signals goes to in…nity, the probability of making the correct decision is strictly increasing and converges to one. Formally, the function is strictly increasing and lim n!1 (n) = 0. An individual's marginal bene…t from collecting an additional signal, when n signals are already obtained, is

g (n) = (n + 1) (n) :
Note that lim n!1 g (n) = 0. For our main theorem to hold, we need the following assumption.

Assumption 1 The function g is log-convex.

This assumption is equivalent to g (n + 1) =g (n) being increasing in n (2 N) : Whether or not this assumption is satis…ed depends only on the primitives of the model, that is, on the distribution of the signals and on the parameters q and . An immediate consequence of this assumption is the following.

Remark 1 The function g is decreasing.

Proof. Suppose by contradiction that there exists an integer, n 0 2 N, such that, g

(n 0 + 1) > g (n 0 ). Since g (n + 1) =g (n) is increasing in n, it follows that g (n + 1) > g (n) whenever n n 0 . Hence, g (n) > g (n 0 ) > 0 whenever n > n 0 . This implies that lim n!1 g (n) 6 = 0, which is a contradiction.
Next, we explain that Assumption 1 essentially means that the marginal value of a signal decreases rapidly. Notice that the function g being decreasing means that the marginal social value of an additional signal is decreasing. We think that this assumption is satis…ed in most economic and political applications. How much more does Assumption 1 require? Since g is decreasing and lim n!1 g (n) = 0, there always exists an increasing sequence fn m g

1 m=1
N, such that g (n m ) g (n m + 1) is decreasing in m. Hence, it is still natural to restrict attention to information structures where the second di¤erence in the social value of a signal, g (n) g (n + 1),

is decreasing. Recall that Assumption 1 is equivalent to (g (n) g (n + 1)) =g (n) being decreasing.
That is, Assumption 1 requires that the second di¤erence in the value of a signal does not only decrease, but decreases at an increasing rate.

In general, it is hard to check whether this assumption holds because it is often di¢ cult (or impossible) to express g (n) analytically. The next section provides examples where Assumption 1 is satis…ed.

Examples for Assumption 1

First, suppose that the signals are normally distributed around the true state of the world. The log-convexity assumption is satis…ed for the model where + q = 1. That is, the society would be indi¤erent between the two possible decisions if information acquisition were impossible. The assumption is also satis…ed even if + q 6 = 1 if the signals are su¢ ciently precise. Formally: Proposition 1 Suppose that s i N (!; ).

(i) If q + = 1 then Assumption 1 is satis…ed.

(ii) For all q; , there exists an " q; > 0, such that Assumption 1 is satis…ed if " q; > .

Proof. See the Appendix.

In our next example the signal is ternary, that is, its possible values are f 1; 0; 1g. In addition,

Pr (s i = !j!) = pr; Pr (s i = 0j!) = 1 r; and Pr (s i = !j!) = (1 p) r:
Notice that r (2 (0; 1)) is the probability that the realization of the signal is informative, and p is the precision of the signal conditional on being informative.

Proposition 2 Suppose that the signal is ternary. Then, there exists a p (r) 2 (0; 1) such that, if p > p (r), Assumption 1 is satis…ed.

Proof. See the Appendix.

Next, we provide an example where the log-convexity assumption is not satis…ed. Suppose that the signal is binary, that is, s i 2 f 1; 1g and

Pr (s i = !j!) = p; Pr (s i = !j!) = 1 p:
Proposition 3 If the signal is binary then Assumption 1 is not satis…ed.

Proof. See the Appendix.

Results

This section is devoted to the proofs of our main theorems. To that end, we …rst characterize the set of equilibria for all k (2 N). The next subsection shows that if k is small, the equilibrium is unique and each member incurs the cost of information (Proposition 4). Section 3.2 describes the set of mixed-strategy equilibria for k large enough (Propositions 5). Finally, Section 3.3 proves the main theorems (Theorems 1 and 2).

Pure-strategy equilibrium

Suppose that the size of the committee is k. If the …rst k 1 members acquire information, the expected gain from collecting information for the kth member is g(k 1). She is willing to invest if this gain exceeds the cost of the signal, that is, if

c < g (k 1) : (2)
This inequality is the incentive compatibility constraint guaranteeing that a committee member is willing to invest in information if the size of the committee is k. 6Proposition 4 Let k denote the size of the committee. There exists a k P 2 N; such that, there exists a unique equilibrium in which each member invests in a signal with probability one if and only if k min k P ; N . Furthermore, the social welfare generated by these equilibria is monotonically increasing in k min k P ; N .

Proof. Recall from Remark 1 that g is decreasing and lim k!1 g (k) = 0: Therefore, for any positive cost c < g (0) 7 , there exists a unique k P 2 N such that

g k P < c < g k P 1 : (3)
First, we show that if k < k P then there is a unique equilibrium in which each committee member invests in information. Suppose that in an equilibrium, the …rst k 1 members randomize according to the pro…le (r 1 ; :::; r k 1 ) ; where r i 2 [0; 1] denotes the probability that the ith member invests.

Let I denote the number of signals collected by the …rst (k 1) members. Since the members randomize, I is a random variable. Notice that I k 1, and

E r1;:::;r k 1 [g (I)] g (k 1)
because g is decreasing. Also notice that from k k P and (3), it follows that

g (k 1) > c.
Combining the previous two inequalities, we get

E r1;:::;r k 1 [g (I)] > c.
This inequality implies that no matter what the strategies of the …rst (k 1) members are, the kth member strictly prefers to invest in information. From this observation, the existence and uniqueness of the pure-strategy equilibrium follow. It remains to show that if k > k P , such a purestrategy equilibrium does not exist. But if k > k P , then g (k 1) < c: Therefore, the incentive compatibility constraint, (2), is violated and there is no equilibrium where each member incurs the cost of the signal.

Finally, we must show that the social welfare generated by these pure-strategy equilibria is increasing in k min k P ; N . Notice that since N > 1,

c < g (k 1) = (k) (k 1) < N ( (k) (k 1)) .
After adding N (k 1) ck, we get

N (k 1) c (k 1) < N (k) ck.
The left-hand side is the social welfare generated by the equilibrium in committee of size k 1, while the right-hand side is the social welfare induced by the committee of size k. The amount of purchased information in equilibrium is ine¢ ciently small. This is because when a committee member decides whether or not to invest, she considers her private bene…t rather than the society's bene…t. Since information is a public good, its social bene…t is bigger than the individual bene…t. Hence, the total number of signals acquired in an equilibrium is smaller than the socially optimal amount. This is why the social welfare is monotonically increasing in the committee size k, as long as k k P .

Mixed-strategy equilibrium

Suppose now that the size of the committee is larger than k P . We consider strategy pro…les in which the committee members can randomize when making a decision about incurring the cost of information acquisition. The following proposition characterizes the set of mixed-strategy equilibria (including asymmetric ones).

We show that each equilibrium is characterized by a pair of integers (a; b). In the committee, a members invest in a signal with probability one, and b members acquire information with positive but less than one probability. The rest of the members, a k (a + b) number of them, do not incur the cost. We call such an equilibrium a type-(a; b) equilibrium.

Proposition 5 Let the committee size be k > k P . Then, for all equilibria there is a pair (a; b), such that a members invest for sure, b members invests with probability r 2 (0; 1), and k (a + b) members do not invest. In addition

a k P a + b k; (4)
where the …rst two inequalities are strict whenever b > 0.

Proof. First, we explain that if, in an equilibrium in which one member invests with probability r 1 2 (0; 1) and another invests with probability r 2 2 (0; 1), then r 1 = r 2 . Since the marginal bene…t from an additional signal is decreasing, our games exhibit strategic substitution. That is, the more information the others acquire, the less incentive a member has to invest. Hence, if r 1 < r 2 then the individual who invests with probability r 1 faces more information in expectation and has less incentive to invest than the individual who invests with probability r 2 . On the other hand, since r 1 ; r 2 2 (0; 1) both individuals must be exactly indi¤erent between investing and not investing, a contradiction. Now, we formalize this argument. Let r i 2 [0; 1] (i = 1; ; k) be the probability that the ith member collects information in equilibrium. Suppose that r 1 , r 2 2 (0; 1), and r 1 > r 2 .

Let I 1 and I 2 denote the number of signals collected by members 2; 3; :::; k and by members 1; 3; :::; k, respectively. Notice that since r 1 > r 2 and g is decreasing,

E r2;r3;:::;r k [g (I 1 )] > E r1;r3;:::;r k [g (I 2 )] . (5)
On the other hand, a member who strictly randomizes must be indi¤erent between investing and not investing. Hence, for j = 1; 2

E rj ;r3;:::;r k [g (I j )] = c. (6)
This equality implies that (5) should hold with equality, which is a contradiction. Therefore, each equilibrium can be characterized by a pair (a; b) where a members collect information for sure, and b members randomize but collect information with the same probability.

It remains to show that there exists a type-(a; b) equilibrium if and only if (a; b) satis…es (4). First, notice that whenever k > k P , in all pure-strategy equilibria k P members invest with probability one, and the rest of the members never invests. In addition, the pair k P ; 0 satis…es (4). Therefore, we only have to show that there exists an equilibrium of type-(a; b) equilibrium where b > 0 if and only if (a; b) satis…es a < k P < a + b k:

Suppose that in a committee, a members invest in information for sure and b 1 invests with probability r. Let G(r; a; b) denote the expected gain from acquiring information for the (a + b)th

member. That is, G(r; a; b) = b 1 X i=0 b 1 i r i (1 r) b 1 i g (a + i) .
We claim that there exists a type-(a; b) equilibrium if and only if there exists an r 2 (0; 1) such that G (r; a; b) = c. Suppose …rst that such an r exists. We …rst argue that there exists a type-

Recall that k P satis…es g k P < c < g k P 1 :

Since g is decreasing, (8) holds if and only if a < k P and a + b > k P . That is, the two strict inequalities in [START_REF] Feddersen | Voting Behavior and Information Aggregation in Elections with Private Information[END_REF] are satis…ed. The last inequality in [START_REF] Feddersen | Voting Behavior and Information Aggregation in Elections with Private Information[END_REF], must hold because a + b cannot exceed the size of the committee, k. According to the previous proposition, there are several equilibria in which more than k P members acquire information with positive probability. A natural question to ask is: can these mixed-strategy equilibria be compared from the point of view of social welfare? The next proposition partially answers this question. We show that if one …xes the number of members who acquire information for sure, then the larger the number of members who randomize is, the smaller In order to prove this proposition we need the following results. Proof. See the Appendix.

Proof of Proposition 6. Suppose that a members collect information with probability one, and b members invest with probability r. Let f (r; a; b) denote the bene…t of an individual, that is,

f (r; a; b) = b X i=0 b i r i (1 r) b i (a + i): Clearly @f (r; a; b) @r = b X i=1 b i ir i 1 (1 r) b i (a + i) b 1 X i=0 b i r i (b i) (1 r) b i 1 (a + i) . Notice that b i i = b b 1 i 1 and b i (b i) = b b 1 i
. Therefore, the right-hand side of the previous equality can be rewritten as

b X i=1 b b 1 i 1 r i 1 (1 r) b i (a + i) b 1 X i=0 b b 1 i r i (1 r) b i 1 (a + i) .
After changing the notation in the …rst summation, this can be further rewritten:

b 1 X i=0 b b 1 i r i (1 r) b i 1 (a + i + 1) b 1 X i=0 b b 1 i r i (1 r) b i 1 (a + i) = b b 1 X i=0 b 1 i r i (1 r) b i 1 (a + i + 1) (a + i) .
This last expression is just bG (r; a; b), and hence, we have where the …rst inequality follows from [START_REF] Gerardi | Information Acquisition in Committees[END_REF], and the last one follows from part (ii) of Lemma 1.

The Proofs of the Theorems

First, we show that the optimal committee size is either k P or k P + 1. Second, we prove that if k > k then even the worst possible equilibrium yields higher social welfare than the unique equilibrium in the committee of size k 2:

Theorem 1 The optimal committee size, k , is either k P or k P + 1:

We emphasize that for a certain set of parameter values, the optimal size is k = k P , and for another set, k = k P + 1.

Proof. Suppose that k is the optimal size of the committee and the equilibrium that maximizes Again, from Proposition 4, it follows that k = k P or k P + 1.

Next, we turn our attention to the potential welfare loss due to oversized committees.

Theorem 2 In any committee of size k (> k ), all equilibria induce higher social welfare than the unique equilibrium in the committee of size k 2.

The following lemma plays an important role in the proof. We point out that this is the only step of our proof that uses Assumption 1.

Lemma 2 For all k 1 and i 2 N,

g (k 1) fg (i) g (k)g fg (k) g (k 1)g f (i) (k)g ; (10)
and it holds with equality if and only if i = k or k 1.

Proof of Theorem 2. Recall that S (a; b) denotes the expected social welfare generated by an equilibrium of type-(a:b). Using this notation, we have to prove that S (k 2; 0) < S (a; b). From Theorem 1, we know that k = k P or k P + 1. By Proposition 4, S k P 2; 0 < S k P 1; 0 .

Therefore, in order to establish S (k 2; 0) < S (a; b), it is enough to show that

S k P 1; 0 < S (a; b) ; (11)
for all pairs of (a; b) which satisfy [START_REF] Marquis De Condorcet | Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix[END_REF].

Notice that if a+i members invests in information, which happens with probability

b i r i a;b (1 r a;b ) b i in a type-(a; b) equilibrium, the social welfare is N (a + i) c(a + i). Therefore, S (a; b) = b X i=0 b i r a;b i (1 r a;b ) b i h N (a + i) c (a + i) i = ( b X i=0 b i r a;b i (1 r a;b ) b i h N (a + i) ci i ) ca = N ( b X i=0 b i r a;b i (1 r a;b ) b i (a + i) ) c (a + br a;b ) :
In the last equation, we used the identity

P b i=0 b i r a;b i (1 r a;b ) b i i = br a;b . Therefore, (11) can
be rewritten as

N k P 1 c k P 1 < N ( b X i=0 b i r i a;b (1 r a;b ) b i (a + i) ) c (a + br a;b ) .
Since a k P 1 by (4) and b N , the right hand side of the previous inequality is larger than

N ( b X i=0 b i r i a;b (1 r a;b ) b i (a + i) ) c k P 1 + N r a;b
Hence it su¢ ces to show that

N k P 1 c k P 1 < N ( b X i=0 b i r i a;b (1 r a;b ) b i (a + i) ) c k P 1 + N r a;b :
After adding c k P 1 to both sides and dividing through by N , we have

(k P 1) < ( b X i=0 b i r i a;b (1 r a;b ) b i (a + i) ) cr a;b : (12)
The left-hand side is a payo¤ of an individual if k P 1 signals are acquired by others, while the right-hand side is the payo¤ of an individual who is randomizing in a type-(a; b) equilibrium with probability r a;b . Since this individual is indi¤erent between randomizing and not collecting information, the right-hand side of ( 12) can be rewritten as

b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i (a + i):
Hence ( 12) is equivalent to

(k P 1) < b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i (a + i): (13)
By Lemma 2

g k P 1 ( b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i g(a + i) g k P ) (14)
> g k P g k P 1 ( b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i (a + i) k P ) : Notice that b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i g(a + i) = c < g k P 1 ; (15)
where the equality guarantees that a member who is randomizing is indi¤erent between investing and not investing, and the inequality holds by (3). Hence, from ( 14) and ( 15),

g k P 1 g k P 1 g k P > g k P g k P 1 ( b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i (a + i) k P ) :
Since g k P 1 g k P > 0; the previous inequality is equivalent to

g k P 1 > k P b 1 X i=0 b 1 i r i a;b (1 r a;b ) b 1 i (a + i).
Finally, since k P g k P 1 = k P 1 , this inequality is just [START_REF] Li | A Theory of Conservatism[END_REF].

The two graphs of Figure 3 show the social welfare in the worst equilibrium as a function of the committee size. In this numerical example, the prior is symmetric and the parameters are chosen such that N = 100; s i N (!; 1), = :3; p = :7; and c = 10 4 : In addition, k P = 11 and k = 12: The two graphs are indeed the same graph except that the scalings of the vertical axes are di¤erent. One can see that the welfare loss due to oversized committees is quite small.

Conclusion

In this paper, we have discussed the optimal committee size and the potential welfare losses associated with oversized committees. We have focused on environments in which there is no con ‡ict of interest among individuals but information acquisition is costly. First, we have con…rmed that the optimal committee size is bounded. In other words, the Condorcet Jury Theorem fails to hold, that is, larger committees might induce smaller social welfare. However, we have also showed that the welfare loss due to oversized committees is surprisingly small. In an arbitrarily large committee, Figure 3: Social Welfare, as a function of the committee size k even the worst equilibrium generates a higher welfare than an equilibrium in a committee in which there are two less members than in the optimal committee. Our results suggest that carefully designing committees might be not as important as it was thought to be.

Appendix

Lemma 3 Suppose that 2 C 1 (R + ) is absolutely continuous, strictly increasing and 0 (k + 1) = 0 (k) is strictly increasing for k > "; where " 0. Let g (k) = (k + 1) (k) for all k 0: Then

g (k + 1) =g (k) < g (k + 2) =g (k + 1) for k ":
Proof. Fix a k ( ") : Notice that 0 (k + 2) = 0 (k + 1) < 0 (t + 2) = 0 (t + 1) is equivalent to 0 (k + 2) 0 (t + 1) < 0 (k + 1) 0 (t + 2). Therefore,

0 (k + 2) Z k+1 k 0 (t + 1) dt < 0 (k + 1) Z k+1 k 0 (t + 2) dt , 0 (k + 2) [ (k + 2) (k + 1)] < 0 (k + 1) [ (k + 3) (k + 2)] . It follows that 0 (k + 2) 0 (k + 1) < (k + 3) (k + 2) (k + 2) (k + 1) = g (k + 2) g (k + 1) : (16)
Similarly, for all t 2 (k; k + 1), 0 (k + 2) = 0 (k + 1) > 0 (t + 1) = 0 (t) is equivalent to 0 (k + 2) 0 (t) > 0 (k + 1) 0 (t + 1). Therefore,

0 (k + 2) Z k+1 k 0 (t) dt > 0 (k + 1) Z k+1 k 0 (t + 1) dt , 0 (k + 2) [ (k + 1) (k)] > 0 (k + 1) [ (k + 2) (k + 1)] .
It follows that 0 (k + 2)

0 (k + 1) > (k + 2) (k + 1) (k + 1) (k) = g (k + 1) g (k) : (17)
From ( 16) and ( 17) it follows that

g (k + 1) g (k) < g (k + 2) g (k + 1)
for all k ":

Proof of Proposition 1. The sum of normally distributed signals are also normal;

P k i=1 s i N !k; p k . The density function of P k i=1 s i conditional on ! is 1 p k 0 @ P k i=1 s i !k p k 1 A
where (x) = (2 ) 1=2 exp x 2 =2 : The ex post e¢ cient decision rule is given by

(s 1 ; ; s k ) = 1 if s 1 + + s k ; and (s 1 ; ; s k ) = 1 if s 1 + + s k < ; where = 2 =2 log [(1 q) (1
) =q ] is the cut-o¤ value. Hence, for k 2 Nn f0g,

(k) = q Pr [ (s 1 ; ; s k ) = 1j! = 1] (1 q) (1 ) Pr [ (s 1 ; ; s k ) = 1j! = 1] = q k p k (1 q) (1 ) + k p k (18)
where is the cdf of standard normal distribution. If k = 0, (0) = max f q ; (1 q) (1 )g :

Notice that the right hand side of ( 18) converges to that of [START_REF] Myerson | Game Theory: Analysis of Con ‡ict[END_REF] as k goes to zero.

Part(i) If q + = 1; then q = (1 q) (1 ) and = 0. Hence

(k) = 2q p k ! and 0 (k) = q 1 p k p k ! for k > 0: Therefore, 0 (k + 1) 0 (k) = r k k + 1 exp 1 2 2
is increasing in k (> 0) : From Lemma 3, setting " to be zero, it follows that g (k + 1) =g (k) is increasing in k 2 N: Part(ii) First, we argue that for any " (> 0) ; 0 (k + 1) = 0 (k) is increasing for all k > " if is su¢ ciently small. For k > 0;

0 (k + 1) 0 (k) = r k k + 1 (k+1) p k+1 k p k = r k k + 1 exp 1 2 2 2 k+1 2 + k + 1 exp 1 2 2 2 k 2 + k = r k k + 1 exp 1 2 2 2 k (k + 1) 1 = r k k + 1 exp L 2 2 8k (k + 1) exp 1 2 2 (20)
where L = log f(1 q) (1 ) = (q )g : Now suppose that k > ". The last term in [START_REF] Myerson | Extended Poisson Games and the Condorcet Jury Theorem[END_REF] has no in ‡uence on whether 0 (k + 1) = 0 (k) is increasing or not. The second term converges to 1 as goes to 0. Obviously, the …rst term is strictly increasing in k. Hence, 0 (k + 1) = 0 (k) is increasing in k (> ") ; if is su¢ ciently small. By setting " 2 (0; 1) and using Lemma 3, we have shown that

g (k + 1) =g (k) < g (k + 2) =g (k + 1) for all k 1:
It remains to be shown that g (1) =g (0) < g (2) =g (1) : From the argument in the proof of Lemma 3, it follows that 0 (2) = 0 (1) < 0 (t + 1) = 0 (t) for all t 2 (1; 2) implies 0 (2) = 0 (1) < g (2) =g (1) : Hence, it is enough to show that g (1) =g (0) < 0 (2) = 0 (1) for su¢ ciently small :

Since lim !0 g (0) = (0) > 0; it is enough to show that lim !0 [g (1) = f 0 (2) = 0 (1)g] = 0.
In order to establish this equality, it is obviously enough to show that

lim !0 (k) 0 (2) = 0 (1) = 0 for k 2 f1; 2g : (21)
Remember L = log f(1 q) (1 ) = (q )g : By (18), for k > 0, which implies [START_REF] Persico | Committee Design with Endogenous Information[END_REF].

(k) = q ( L 2 p k p k ! + exp (L) L 2 p k p k !) which implies (k) 2 O E p k= as ! 0: 8 Using (20), 0 (2) = 0 (1) 2 O E exp 1=2 2 = O E ( ( 1=
Proof of Proposition 2. First, we claim that the ex post e¢ cient decision rule : f 1; 0; 1g k ! f 1; 1g is the following cut-o¤ rule:

(s 1 ; ; s k ) = 8 > > > > < > > > > : 1 if k X i=1 s i b ; 1 if k X i=1 s i < b ; (22) where b = log [(1 q) (1 ) =q ] = log [p= (1 p)] : Suppose that the signal sequence (s 1 ; ; s k ) is a permutation of 8 > < > : 1; ; 1 | {z } a ; 0; ; 0 | {z } k a b ; 1; ; 1 | {z } b 9 > = > ; : (23)
Then (s 1 ; ;

s k ) = 1 if E ! [u (!; 1) js 1 ; ; s k ] = (1 q) (1 ) Pr [s 1 ; ; s k j! = 1] Pr [s 1 ; ; s k ] > E ! [u (!; 1) js 1 ; ; s k ] = q Pr [s 1 ; ; s k j! = 1] Pr [s 1 ;
; s k ] :

In addition,

Pr [s 1 ; ; s k j! = 1] = (pr) a (1 r) k a b (r (1 p)) b and
Pr [s 1 ; ; s k j! = 1] = (pr) b (1 r) k a b (r (1 p)) a : Hence, (s 1 ; ; s k ) = 1 if (1 q) (1 ) p b (1 p) a > q p a (1 p) b ;
or equivalently,

a b > log (1 q)(1 ) q log p 1 p = b :
8 O E is a version of Landau's O, which describes the exact order of the expression. Formally, "f (x) 2 O E (g (x))

as x ! a" i¤ 9M > 0 s.t. limx!a jf (x) =g (x)j = M: Since P k i=1 s i = a b, (22) follows. Now, we shall consider the case where p converges to 1. Let " denote 1 p and let Pr [a; b] denote the probability of a signal sequence which is a permutation of (23). Then, 9 Notice that C k (a; b) is independent of " and symmetric with respect to a and b. We have 10

Pr [a; bj! = 1] = C k (a; b) (1 ") a " b ; Pr [a; bj! = 1] = C k (a; b) (1 ") b " a ; where C k (a; b) = [k!=(a!b! (k a b)!)] r a+b (1 r) k a b .
Pr [a b 1j! = 1] = C k (0; 1) " + O " 2 ; and

Pr [a b 0j! = 1] = C k (0; 0) + fC k (1; 0) + C k (1; 1)g " + O " 2 :
Observe that b < 1 if p is close enough to one. Without loss of generality, assume that q + 1: Then 1 < b 0: Hence,

(k) = q Pr [a b 1j! = 1] (1 q) (1 ) Pr [a b 0j! = 1] = q C k (0; 1) " (1 q) (1 ) [C k (0; 0) + fC k (1; 0) + C k (1; 1)g "] + O " 2 : Then g (k) = (k + 1) (k) = A (k) + B (k) " + O " 2 ;
where

A (k) = (1 q) (1 ) D k (0; 0) ; (24)
B (k) = q D k (0; 1) (1 q) (1 ) [D k (1; 0) + D k (
g (k + 1) g (k) = A (k + 1) + B (k + 1) " + O " 2 A (k) + B (k) " + O (" 2 ) = A (k + 1) 1 + B(k+1) A(k+1) " + O " 2 A (k) 1 + B(k) A(k) " + O (" 2 ) = A (k + 1) A (k) 1 + B (k + 1) A (k + 1) B (k) A (k) " + O " 2 : (25)
We want to show that g

(k + 1) =g (k) is increasing in k if " is su¢ ciently small. Since A (k + 1) =A (k) = 1 r; it is su¢ cient to show that B (k) =A (k) is convex in k: It is straightforward to see that D k (0; 1) D k (0; 0) = D k (1; 0) D k (0; 0) = (k + 1) r (1 r) k kr (1 r) k 1 (1 r) k+1 (1 r) k = (k + 1) r (1 r) kr (1 r) 2 (1 r)
is a polynomial of k with degree 1, hence it has no in ‡uence on the convexity of B (k) =A (k) : On the other hand, ). In such a case, the social decision changes from 1 to 1. Hence the loss is q if ! = 1; and the gain is (1 q) if ! = 1: Therefore, expected marginal bene…t is g (2m + 1) = q Pr [! = 1; piv o ] + (1 q) Pr [! = 1; piv o ] = q 2m + 1 m p m+1 (1 p) m+1 + (1 q) (1 ) 2m + 1 m p m+1 (1 p) m+1 = f q + (1 q) (1 )g 2m + 1 m p m+1 (1 p) m+1 :

D k (1; 1) D k (0; 0) = (k + 1) kr 2 (1 r) k 1 k (k 1) r 2 (1 r) k 2 (1 r) k+1 (1 r) k = (k + 1) kr 2 (1 r) k (k 1) r 2 (1 r) 3 ( 1
Recall that 0 < 1; which is equivalent to pq (1 p) (1 q) (1 ) > 0 and (1 q) (1 )

q 0: If > 0; g (2m + 2) g (2m + 1) = pq (1 p) (1 q) (1 ) (1 q) (1
) q ;

which is a constant function of m, and hence, Assumption 1 does not hold. If = 0; then g (2m + 1) = 0 and g (2m + 2) =g (2m + 1) is not well-de…ned. Proof of Lemma 2. The statement of the lemma is obvious if i 2 fk 1; kg. It remains to show that (10) hold with strict inequality whenever i = 2 fk 1; kg. First, notice that for any positive sequence, fa j g 1 0 , if a j+1 =a j < a j+2 =a j+1 for all j 2 N, then a k a k 1 > P k j=i+1 a j P k 1 j=i a j for all k > 1 and for all i 2 f0; :::; k 2g :

Assumption 1 allows us to apply this result for the sequence a j = g (j), and hence, for all k 1 g (k) g (k 1) > P k j=i+1 g (j) P k 1 j=i g (j) = (k + 1) (i + 1) (k) (i) for all i 2 f0; :::; k 2g :

Figure 1 :Figure 1

 11 Figure 1: Expected gain g (k) and the cost c

  (a; b) equilibrium in which b members invest with probability r. This means that the b members, those that are randomizing, are indi¤erent between investing and not investing. The a members, who invest for sure, strictly prefer to invest because the marginal gain from an additional signal exceeds G (r; a; b). Similarly, those members who don't invest, a k (a + b) number of them, are strictly better o¤ not investing because their marginal gains are strictly smaller than G (r; a; b). Next, we argue that if G (r; a; b) = c does not have a solution in (0; 1) then there exists no type-(a; b) equilibrium. But this immediately follows from the observation that if b members are strictly randomizing, they must be indi¤erent between investing and not investing and hence G (r; a; b) = c. Therefore, it is su¢ cient to show that G (r; a; b) = c has a solution in (0; 1) if and only if (7) holds. Notice that G(r; a; b) is strictly decreasing in r because g is strictly decreasing. Also observe that G(0; a; b) = g (a) and G(1; a; b) = g (a + b 1). By the Intermediate Value Theorem, G(r; a; b) = c has a solution in (0; 1) if and only if G(1; a; b) < c < G(0; a; b), which is equivalent to g (a + b 1) < c < g (a) :

Figure 2

 2 Figure 2 graphically represents the set of pairs (a; b) which satisfy (4).

Figure 2 :Proposition 6

 26 Figure 2: The set of mixed-strategy equilibria

Lemma 1 (

 1 i) G (r; a; b) > G (r; a; b + 1) for all r 2 (0; 1], and (ii) r a;b > r a;b+1 , where r a;b and r a;b+1 are the solutions for G (r; a; b) = c and G (r; a; b + 1) = c in r, respectively.

9 )Z r a;b 0 GZ r a;b 0 GZ r a;b+1 0 G

 9000 @f (r; a; b) @r = bG (r; a; b) . Next, we show that f (r a;b ; a; b) f (r a;b+1 ; a; b + 1) > b (r a;b r a;b+1 ) c. (Since f (0; a; b) = f (0; a; b + 1) = (a) f (r a;b ; a; b) f (r a;b+1 ; a; b + 1) = f (r a;b ; a; b) f (0; a; b) f (r a;b+1 ; a; b + 1) f (0; a; b + 1) = b (r; a; b) dr b Z r a;b+1 0 G (r; a; b + 1) dr. By part (i) of Lemma 1, this last di¤erence is larger than b (r; a; b) dr b (r; a; b) dr = b Z r a;b r a;b+1 G (r; a; b) dr. By part (ii) of Lemma, we know that r a;b+1 < r a;b . In addition, since G is decreasing in r, this last expression is larger than b (r a;b r a;b+1 ) G (r a;b ; a; b) . Recall that r a;b is de…ned such that G (r a;b ; a; b) = c and hence we can conclude (9). Let S(a; b) denote the social welfare in the type-(a; b) equilibrium, that is: S(a; b) = N f (r a;b ; a; b) c (a + br a;b ) . Then, S (a; b) S (a; b + 1) = N f (r a;b ; a; b) c (a + br a;b ) [N f (r a;b+1 ; a; b + 1) c (a + br a;b+1 )] > N b (r a;b r a;b+1 ) c cb (r a;b r a;b+1 ) = (N 1) cb (r a;b r a;b+1 ) > 0;

1 .

 1 social welfare is of type-(a; b). By the de…nition of optimal size, a + b = k . If b = 0, then all of the committee members invest in information in this equilibrium. From Proposition 4, k k P follows. In addition, Proposition 4 also states that the social welfare is increasing in k as long as k k P . Therefore, k = k P follows. Suppose now that b > 0. If there exists an equilibrium of type-(a; b 1), then, by Proposition 6, k is not the optimal committee size. Hence, if the size of the committee is k , there does not exist an equilibrium of type-(a; b 1). By Proposition 5, this implies that the pair (a; b 1) violates the inequality chain (4) with k = k . Since the …rst and last inequalities in (4) hold because there is a type-(a; b) equilibrium, it must be the case that the second inequality is violated. That is, k P a + b 1 = k This implies that k k P + 1.

  )) as ! 0. By l'Hôpital's Rule, for k 2 f1; 2g ;

  1; 1)] ; and D k (a; b) = C k+1 (a; b) C k (a; b) : Using these notations,

r) 2 has a positive coe¢ cient of k 2 :< 1 :Case 1 : 1 to 1 .Case 2 :

 2211112 Hence we conclude B (k) =A (k) is convex in k.Proof of Proposition 3. As in the proof of Proposition 2, the ex post e¢ cient decision rule is [(1 q) (1) =q ] = log [p= (1 p)] : By symmetry, we can assume 0 without loss of generality. First, suppose > 1: Then (k) = q for k < ; and the marginal bene…t from an additional signal is zero for k < 1. Therefore, g (k + 1) =g (k) is not well-de…ned. Second, suppose 0 We shall consider two di¤erent cases depending on whether k is even or odd. Suppose k = 2m; where m 2 N. Then, the (2m + 1)-st signal makes a di¤erence if and only if the …rst 2m signals have a tie between positive and negative signals and the (2m + 1)-st signal is positive (denote this situation as piv e ). In such a case, the social decision changes from Hence the gain is q if ! = 1; and the loss is(1 q) if ! = 1: Therefore, expected marginal bene…t is g (2m) = q Pr [! = 1; piv e ] (1q) Pr [! = 1; piv e ] Suppose k = 2m + 1; where m 2 N. Then, the (2m + 2)-nd signal makes a di¤erence if and only if the …rst (2m + 1) signals contains (m + 1) positive and m negative signals and the (2m + 2)-nd signal is negative (denote this situation as piv o

Proof of Lemma 1 . 1 X i=0 b 1 1 ,

 1111 Part (i). Notice thatG (r; a; b) = b i r i (1 r) b i 1 g (a + i) : Since r i (1 r) b i 1 = r i (1 r) b i + r i+1 (1 r) b i b i + r i+1 (1 r) b i 1 g (a + i) .Since g is decreasingG (r; a; b) > b i g (a + i) ,where the …rst equality holds because we have just rede…ned the notation in the second summation, and the second equality holds because, by convention, b i g (a + i) = G (r; a; b + 1) . Part (ii). By the de…nitions of r a;b and r a;b+1 , we have c = G (r a;b ; a; b) = G (r a;b+1 ; a; b + 1) ; and by part (i) of this lemma, G (r a;b+1 ; a; b + 1) < G (r a;b+1 ; a; b) : Therefore G (r a;b ; a; b) < G (r a;b+1 ; a; b) Since G (r; a; b) is strictly decreasing in r; r a;b > r a;b+1 follows.

Since there is no con ‡ict of interest among the individuals, it is easy to design a mechanism which is incentive compatible and e¢ ciently aggregates the signals. Alternatively, one can assume that the collected information is hard.

The results are quite di¤erent if the voting, rather than the information acquisition, is costly, see e.g. Borgers (2004).

In his accompanying paper,[START_REF] Martinelli | Rational Ignorance and Voting Behavior[END_REF] analyzes a model in which information has a …xed cost, voters are heterogeneous in their costs, and abstention is not allowed. On the one hand, the author shows that if the

In the jury context, where ! = 1 corresponds to the innocence of the suspect, q indicates how severe error it is to convict an innocent.

In what follows, we ignore the case where there exists a k 2 N such that c = g (k). This equality does not hold generically, and would have no e¤ect on our results.

If c > (1) (0); then nobody has incentive to collect information, hence k P = 0:

De…ne C k (a; b) = 0 if k < a + b. 1 0 f (x) 2 O (g (x)) as x ! 0 if and only if 9 > 0; M > 0 s.t. jxj < implies jf (x) =g (x)j < M:

Since (a) > (b) if a > b, this implies that for all i 2 f0; :::; k 2g

Similarly, for a positive sequence fa j g 1 0 if a j+1 =a j < a j+2 =a j+1 for all j 2 N, then

for all k 1 and for all i k + 1:

Again, by Assumption 1, we can apply this result to the sequence a j = g (j) and get

Multiplying through by g (k 1) ( (i) (k)), we get (26). That is, (26) holds whenever i = 2 fk 1; kg. After subtracting g (k 1) ( (i) (k)) from both sides of (26) we get [START_REF] Gerling | Information Acquisition and Decision Making in Committees: A Survey" European Central Bank[END_REF].