{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T05:15:25Z","timestamp":1765430125992,"version":"3.46.0"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T00:00:00Z","timestamp":1765152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001711","name":"SNSF","doi-asserted-by":"publisher","award":["Markets and Norms"],"award-info":[{"award-number":["Markets and Norms"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>Von Neumann\u2019s minimax theorem defines optimal strategic unpredictability in zero-sum games. Empirical evidence from professional sports has been interpreted as positive behavioral evidence for minimax. In this article, we analyze the strategic optimality of offensive plays in the basketball endgame when a team has a final possession and trails by no more than a single basket. This final moment of the game most closely approximates the simultaneous-move conditions of a game where minimax theory applies. Using comprehensive NBA data from 2010 to 2025, we test for equality of success rates across shooter types (star vs. non-stars) and shot selection (two-point vs. three-point). Our analysis reveals systematic violations of minimax play that have intensified with basketball\u2019s shift to three-pointers and higher expected points. In the final decisive moment of the game, we find that teams systematically overuse three-point shots even though the two-point attempt yields higher field goal percentages. In addition, teams over-rely on star players for the final shot; non-star two-point shots have been the top-performing endgame option in 2022\u20132025.<\/jats:p>","DOI":"10.3390\/g16060064","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T15:30:07Z","timestamp":1765207807000},"page":"64","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Decisions in the Basketball Endgame: A Downside of the Three-Point Revolution"],"prefix":"10.3390","volume":"16","author":[{"given":"Luka","family":"Secilmis","sequence":"first","affiliation":[{"name":"Computer Science, ETH Zurich, 8092 Zurich, Switzerland"}]},{"given":"Teo","family":"Secilmis","sequence":"additional","affiliation":[{"name":"Life Sciences Engineering, EPFL, 1015 Lausanne, Switzerland"}]},{"given":"Simon","family":"Jantschgi","sequence":"additional","affiliation":[{"name":"Zurich Center for Market Design and SUZ, University of Zurich, 8050 Zurich, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1261-8134","authenticated-orcid":false,"given":"Heinrich H.","family":"Nax","sequence":"additional","affiliation":[{"name":"Zurich Center for Market Design and SUZ, University of Zurich, 8050 Zurich, Switzerland"},{"name":"Behavioral Game Theory, ETH Zurich, 8092 Zurich, Switzerland"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2202\/1559-0410.1030","article-title":"Optimal end-game strategy in basketball","volume":"2","author":"Annis","year":"2006","journal-title":"Journal of Quantitative Analysis in Sports"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1111\/1468-0297.00645","article-title":"Does minimax work? An experimental study","volume":"119","author":"Binmore","year":"2001","journal-title":"Economic Journal"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1257\/00028280260344678","article-title":"Testing mixed-strategy equilibria when players are heterogeneous: The case of penalty kicks in soccer","volume":"92","author":"Chiappori","year":"2002","journal-title":"American Economic Review"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Depoorter, B., Jantschgi, S., Lendl, I., Mlakar, M., and Nax, H. H. (2023). Minimax under pressure: The case of tennis. Games, 16, (6).","DOI":"10.3390\/g16060060"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s13235-018-0252-z","article-title":"Minmax via replicator dynamics","volume":"8","author":"Hofbauer","year":"2018","journal-title":"Dynamic Games and Applications"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1257\/aer.97.1.517","article-title":"Minimax play at wimbledon: Comment","volume":"97","author":"Hsu","year":"2007","journal-title":"American Economic Review"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.ejor.2023.04.009","article-title":"A review of attacker-defender games: Current state and paths forward","volume":"313","author":"Hunt","year":"2023","journal-title":"European Journal of Operational Research"},{"key":"ref_8","unstructured":"Jantschgi, S., Nax, H. H., Secilmis, L., and Secilmis, T. (2023). Should I stay or should I go? Over-switching in penalty shootouts, DRAFT."},{"key":"ref_9","unstructured":"Kenter, F. H. J. (, January February). An analysis of the basketball endgame: When to foul when trailing and leading. 9th MIT Sloan Sports Analytics Conference, Available online: https:\/\/www.stat.berkeley.edu\/~aldous\/157\/Papers\/kenter.pdf."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kovash, K., and Levitt, S. D. (2009). Professionals do not play minimax: Evidence from Major League Baseball and the National Football League, National Bureau of Economic Research. Available online: http:\/\/www.nber.org\/papers\/w15347.","DOI":"10.3386\/w15347"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2195","DOI":"10.1111\/ecin.12565","article-title":"The optimal defense of networks of targets","volume":"56","author":"Kovenock","year":"2018","journal-title":"Economic Inquiry"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"40","DOI":"10.5539\/jmr.v13n2p40","article-title":"Star players in the NBA\u2013decoys or game-changers?","volume":"13","author":"Liu","year":"2021","journal-title":"Journal of Mathematics Research"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1006\/game.1995.1030","article-title":"Games with unique, mixed strategy equilibria: An experimental study","volume":"10","author":"Ochs","year":"1995","journal-title":"Games and Economic Behavior"},{"key":"ref_14","first-page":"2106","article-title":"Nonmetric test of the minimax theory of two-person zerosum games","volume":"84","year":"1987","journal-title":"Proceedings of the National Academy of Sciences of the United States of America"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/1467-937X.00249","article-title":"Professionals play minimax","volume":"70","year":"2003","journal-title":"The Review of Economic Studies"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/0899-8256(92)90019-O","article-title":"Mixed strategies in strictly competitive games: A further test of the minimax hypothesis","volume":"4","author":"Rapoport","year":"1992","journal-title":"Games and Economic Behavior"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF01448847","article-title":"Zur Theorie der Gesellschaftsspiele","volume":"100","year":"1928","journal-title":"Mathematische Annalen"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1257\/aer.91.5.1521","article-title":"Minimax play at Wimbledon","volume":"91","author":"Walker","year":"2001","journal-title":"American Economic Review"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1177\/15270025251328264","article-title":"The effect of basketball analytics investment on National Basketball Association (NBA) Team Performance","volume":"26","author":"Wang","year":"2025","journal-title":"Journal of Sports Economics"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/6\/64\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T05:14:09Z","timestamp":1765430049000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/6\/64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,8]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["g16060064"],"URL":"https:\/\/doi.org\/10.3390\/g16060064","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2025,12,8]]}}}