{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:39:33Z","timestamp":1760060373112,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T00:00:00Z","timestamp":1756080000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We investigate power indices for simple games with precoalitions which distribute power among players in an external and an internal step. We extend an existing approach which uses the Public Good index both on the external level in the quotient game as well as on the internal level for measuring the leverage of players to threaten their peers through departing the precoalition. We replace the Public Good index in that model by five other efficient power indices, i.e., the Shapley\u2013Shubik index, the Deegan\u2013Packel index, the Johnston index and two indices based on null player free winning coalitions. Axiomatizations of the novel power indices with threat partitions are presented. We also propose a slight modification to the existing framework for threat power indices which guarantees that null players are always assigned zero power. Numerical results for all power indices combined with different threat partitions are presented and discussed.<\/jats:p>","DOI":"10.3390\/g16050041","type":"journal-article","created":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T06:26:14Z","timestamp":1756189574000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Power Indices with Threats in Precoalitions"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0619-4606","authenticated-orcid":false,"given":"Jochen","family":"Staudacher","sequence":"first","affiliation":[{"name":"Fakult\u00e4t Informatik, Hochschule Kempten, 87435 Kempten, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Algaba, E., Fragnelli, V., and S\u00e1nchez-Soriano, J. (2019). The Shapley value, a paradigm of fairness. 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