{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:14:55Z","timestamp":1760177695675,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,20]],"date-time":"2020-07-20T00:00:00Z","timestamp":1595203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["MTM2017-83455-P"],"award-info":[{"award-number":["MTM2017-83455-P"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A cooperative game represents a situation in which a set of agents form coalitions in order to achieve a common good. To allocate the benefits of the result of this cooperation there exist several values such as the Shapley value or the Banzhaf value. Sometimes it is considered that not all communications between players are feasible and a graph is introduced to represent them. Myerson (1977) introduced a Shapley-type value for these situations. Another model for cooperative games is the Owen model, Owen (1977), in which players that have similar interests form a priori unions that bargain as a block in order to get a fair payoff. The model of cooperation introduced in this paper combines these two models following Casajus (2007). The situation consists of a communication graph where a two-step value is defined. In the first step a negotiation among the connected components is made and in the second one players inside each connected component bargain. This model can be extended to fuzzy contexts such as proximity relations that consider leveled closeness between agents as we proposed in 2016. There are two extensions of the Banzhaf value to the Owen model, because the natural way loses the group symmetry property. In this paper we construct an appropriate value to extend the symmetric option for situations with a proximity relation and provide it with an axiomatization. Then we apply this value to a political situation.<\/jats:p>","DOI":"10.3390\/sym12071196","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1196","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Symmetric Banzhaf Cooperation Value for Games with a Proximity Relation among the Agents"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1088-3172","authenticated-orcid":false,"given":"In\u00e9s","family":"Gallego","sequence":"first","affiliation":[{"name":"Departamento de Did\u00e1ctica de las Matem\u00e1ticas, Universidad de Sevilla, 41004 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4814-2556","authenticated-orcid":false,"given":"Julio R.","family":"Fern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada II, Universidad de Sevilla, 41092 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7571-1240","authenticated-orcid":false,"given":"Andr\u00e9s","family":"Jim\u00e9nez-Losada","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada II, Universidad de Sevilla, 41092 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9804-3097","authenticated-orcid":false,"given":"Manuel","family":"Ord\u00f3\u00f1ez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada II, Universidad de Sevilla, 41092 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,20]]},"reference":[{"key":"ref_1","first-page":"307","article-title":"A value for N-Person Games","volume":"Volume 2","author":"Kuhn","year":"1953","journal-title":"Contributions to the Theory of Games"},{"key":"ref_2","first-page":"317","article-title":"Weighted voting doesn\u2019t work","volume":"19","author":"Banzhaf","year":"1965","journal-title":"Rutgers Law Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1287\/moor.4.2.99","article-title":"Mathematical Properties of the Banzhaf Power Index","volume":"4","author":"Dubey","year":"1979","journal-title":"Math. 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