---
_id: '63731'
abstract:
- lang: eng
  text: The Banzhaf power index can be used to quantify the explanatory power of single
    conditions in a configurational analysis that aims at identifying whether combinations
    of conditions are sufficient for an outcome. The latter method is an integral
    part of the Qualitative Comparative Analysis (QCA) approach. It is widely used
    in the literature, e.g., in the field of International Management. Haake and Schneider
    (2023) give a rigorous description of the connection between the empirical and
    game theoretic modeling. To justify that the Banzhaf index is an appropriate tool
    to measure the influence of a condition, this paper discusses a novel axiomatization
    of the Banzhaf index that uses axioms that are directly linked to the QCA methodology.
    As a side result, we demonstrate that in our model the Banzhaf index can be reinterpreted
    as an average of Shapley-Shubik indices.
author:
- first_name: Claus-Jochen
  full_name: Haake, Claus-Jochen
  id: '20801'
  last_name: Haake
- first_name: Martin
  full_name: Schneider, Martin
  id: '471'
  last_name: Schneider
  orcid: https://orcid.org/0000-0002-6961-3716
citation:
  ama: Haake C-J, Schneider M. An Axiomatization of the Banzhaf Index to Measure Influence
    in Qualitative Comparative Analysis. <i>International Journal of Game Theory</i>.
    2026;55. doi:<a href="https://doi.org/10.1007/s00182-026-00978-2">10.1007/s00182-026-00978-2</a>
  apa: Haake, C.-J., &#38; Schneider, M. (2026). An Axiomatization of the Banzhaf
    Index to Measure Influence in Qualitative Comparative Analysis. <i>International
    Journal of Game Theory</i>, <i>55</i>. <a href="https://doi.org/10.1007/s00182-026-00978-2">https://doi.org/10.1007/s00182-026-00978-2</a>
  bibtex: '@article{Haake_Schneider_2026, title={An Axiomatization of the Banzhaf
    Index to Measure Influence in Qualitative Comparative Analysis}, volume={55},
    DOI={<a href="https://doi.org/10.1007/s00182-026-00978-2">10.1007/s00182-026-00978-2</a>},
    journal={International Journal of Game Theory}, publisher={Springer}, author={Haake,
    Claus-Jochen and Schneider, Martin}, year={2026} }'
  chicago: Haake, Claus-Jochen, and Martin Schneider. “An Axiomatization of the Banzhaf
    Index to Measure Influence in Qualitative Comparative Analysis.” <i>International
    Journal of Game Theory</i> 55 (2026). <a href="https://doi.org/10.1007/s00182-026-00978-2">https://doi.org/10.1007/s00182-026-00978-2</a>.
  ieee: 'C.-J. Haake and M. Schneider, “An Axiomatization of the Banzhaf Index to
    Measure Influence in Qualitative Comparative Analysis,” <i>International Journal
    of Game Theory</i>, vol. 55, 2026, doi: <a href="https://doi.org/10.1007/s00182-026-00978-2">10.1007/s00182-026-00978-2</a>.'
  mla: Haake, Claus-Jochen, and Martin Schneider. “An Axiomatization of the Banzhaf
    Index to Measure Influence in Qualitative Comparative Analysis.” <i>International
    Journal of Game Theory</i>, vol. 55, Springer, 2026, doi:<a href="https://doi.org/10.1007/s00182-026-00978-2">10.1007/s00182-026-00978-2</a>.
  short: C.-J. Haake, M. Schneider, International Journal of Game Theory 55 (2026).
date_created: 2026-01-26T12:30:50Z
date_updated: 2026-01-26T12:42:04Z
department:
- _id: '205'
- _id: '475'
- _id: '185'
doi: 10.1007/s00182-026-00978-2
intvolume: '        55'
keyword:
- Banzhaf index
- axiomatization
- QCA
- explanatory power
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://link.springer.com/article/10.1007/s00182-026-00978-2#citeas
oa: '1'
publication: International Journal of Game Theory
publisher: Springer
quality_controlled: '1'
status: public
title: An Axiomatization of the Banzhaf Index to Measure Influence in Qualitative
  Comparative Analysis
type: journal_article
user_id: '20801'
volume: 55
year: '2026'
...
---
_id: '34114'
abstract:
- lang: eng
  text: "Qualitative comparative analysis (QCA) enables researchers in international
    management to better understand how the impact of a single explanatory factor
    depends on the context of other factors. But the analytical toolbox of QCA does
    not include a parameter for the explanatory power of a single explanatory factor
    or “condition”. In this paper, we therefore reinterpret the Banzhaf power index,
    originally developed in cooperative game theory, to establish a goodness-of-fit
    parameter in QCA. The relative Banzhaf index we suggest measures the explanatory
    power of one condition averaged across all sufficient combinations of conditions.
    The paper argues that the index is especially informative in three situations
    that are all salient in international management and call for a context-sensitive
    analysis of single conditions, namely substantial limited diversity in the data,
    the emergence of strong INUS conditions in the analysis, and theorizing with contingency
    factors. The paper derives the properties of the relative Banzhaf index in QCA,
    demonstrates how the index can be computed easily from a rudimentary truth table,
    and explores its insights by revisiting selected papers in international management
    that apply fuzzy-set QCA. It finally suggests a three-step procedure for utilizing
    the relative Banzhaf index when the causal structure involves both contingency
    effects and configurational causation.\r\n"
article_type: original
author:
- first_name: Claus-Jochen
  full_name: Haake, Claus-Jochen
  id: '20801'
  last_name: Haake
- first_name: Martin
  full_name: Schneider, Martin
  id: '471'
  last_name: Schneider
  orcid: https://orcid.org/0000-0002-6961-3716
citation:
  ama: 'Haake C-J, Schneider M. Playing games with QCA: Measuring the explanatory
    power of single conditions with the Banzhaf index. <i>Journal of International
    Management</i>. 2024;30(2).'
  apa: 'Haake, C.-J., &#38; Schneider, M. (2024). Playing games with QCA: Measuring
    the explanatory power of single conditions with the Banzhaf index. <i>Journal
    of International Management</i>, <i>30</i>(2).'
  bibtex: '@article{Haake_Schneider_2024, title={Playing games with QCA: Measuring
    the explanatory power of single conditions with the Banzhaf index}, volume={30},
    number={2}, journal={Journal of International Management}, publisher={Elsevier},
    author={Haake, Claus-Jochen and Schneider, Martin}, year={2024} }'
  chicago: 'Haake, Claus-Jochen, and Martin Schneider. “Playing Games with QCA: Measuring
    the Explanatory Power of Single Conditions with the Banzhaf Index.” <i>Journal
    of International Management</i> 30, no. 2 (2024).'
  ieee: 'C.-J. Haake and M. Schneider, “Playing games with QCA: Measuring the explanatory
    power of single conditions with the Banzhaf index,” <i>Journal of International
    Management</i>, vol. 30, no. 2, 2024.'
  mla: 'Haake, Claus-Jochen, and Martin Schneider. “Playing Games with QCA: Measuring
    the Explanatory Power of Single Conditions with the Banzhaf Index.” <i>Journal
    of International Management</i>, vol. 30, no. 2, Elsevier, 2024.'
  short: C.-J. Haake, M. Schneider, Journal of International Management 30 (2024).
date_created: 2022-11-19T15:33:27Z
date_updated: 2024-04-24T12:58:51Z
department:
- _id: '205'
- _id: '475'
- _id: '178'
- _id: '185'
funded_apc: '1'
intvolume: '        30'
issue: '2'
keyword:
- Qualitative comparative analysis
- Banzhaf power index
- causality
- explanatory power
language:
- iso: eng
project:
- _id: '1'
  grant_number: '160364472'
  name: 'SFB 901: SFB 901'
- _id: '2'
  name: 'SFB 901 - A: SFB 901 - Project Area A'
- _id: '7'
  grant_number: '160364472'
  name: 'SFB 901 - A3: SFB 901 - Subproject A3'
publication: Journal of International Management
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: 'Playing games with QCA: Measuring the explanatory power of single conditions
  with the Banzhaf index'
type: journal_article
user_id: '20801'
volume: 30
year: '2024'
...