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By using the theory of dual games, we define dually-<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-essential coalitions and show that they characterize the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-prenucleolus of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-balanced games. Additionally, we demonstrate that the intersection of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-essential and dually-<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-essential coalitions also forms a characterization set for the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-prenucleolus, provided that the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-least-core is a proper subset of the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{u}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>u<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-core.\n<\/jats:p>","DOI":"10.1007\/s10479-025-06549-0","type":"journal-article","created":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T02:54:35Z","timestamp":1743562475000},"page":"1575-1607","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Characterizations of the $$\\textbf{u}$$-prenucleolus by dually-$$\\textbf{u}$$-essential coalitions"],"prefix":"10.1007","volume":"349","author":[{"given":"Zs\u00f3fia","family":"Dornai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6373-7550","authenticated-orcid":false,"given":"Mikl\u00f3s","family":"Pint\u00e9r","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,31]]},"reference":[{"key":"6549_CR1","series-title":"No.\u00a052 in Annals of Mathematical Studies","first-page":"443","volume-title":"Advances in game theory","author":"RJ Aumann","year":"1964","unstructured":"Aumann, R. 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