{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T08:50:28Z","timestamp":1782291028645,"version":"3.54.5"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T00:00:00Z","timestamp":1764806400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Negative dependence in tournaments has received attention in the literature. The property of\n                    <jats:italic>negative orthant dependence<\/jats:italic>\n                    (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property,\n                    <jats:italic>negative association<\/jats:italic>\n                    (NA). For a knockout tournament with a non-random draw, they presented an example to illustrate that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100430_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold italic upper S<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">S<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>${\\boldsymbol{S}}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of\n                    <jats:italic>negative regression dependence<\/jats:italic>\n                    (NRD),\n                    <jats:italic>negative left-tail dependence<\/jats:italic>\n                    (NLTD), and\n                    <jats:italic>negative right-tail dependence<\/jats:italic>\n                    (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100430_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold italic upper S<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">S<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>${\\boldsymbol{S}}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is NA and NRTD, while\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100430_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">bold italic upper S<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">S<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>${\\boldsymbol{S}}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is, in general, not NRD or NLTD.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10043","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T09:41:56Z","timestamp":1764841316000},"page":"536-551","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Negative dependence in knockout tournaments"],"prefix":"10.1017","volume":"63","author":[{"given":"Yuting","family":"Su","sequence":"first","affiliation":[{"name":"University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-5774-726X","authenticated-orcid":false,"given":"Zhenfeng","family":"Zou","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9426-2348","authenticated-orcid":false,"given":"Taizhong","family":"Hu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"key":"S0021900225100430_ref24","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2015.0755"},{"key":"S0021900225100430_ref21","doi-asserted-by":"publisher","DOI":"10.1214\/15-STS525"},{"key":"S0021900225100430_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.insmatheco.2023.05.006"},{"key":"S0021900225100430_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176346079"},{"key":"S0021900225100430_ref6","first-page":"125","article-title":"Testing equality of players in a round-robin tournament","volume":"43","author":"Bruss","year":"2018","journal-title":"Math. 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