{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T10:05:19Z","timestamp":1772013919274,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T00:00:00Z","timestamp":1763424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"SNSF Eccellenza Grant"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>A series of articles has tested von Neumann\u2019s minimax theory against behavioral evidence based on field data from professional sports. The evidence has been viewed and collectively cited as positive evidence that elite athletes in their familiar sports contexts mix well and behave in line with minimax. In this paper, based on open state-of-the-art tennis data and analytics, we shall uncover new and significant evidence against minimax at the very top of the game, where previously, such results had not been obtained. The kinds of behavioral deviations from minimax that we find become apparent, because we enrich the test strategy to take into account whether or not players face \u2018pressure\u2019 situations like break points and other decisive points. Our paper highlights that the prior literature\u2019s failure to reject minimax does not constitute positive behavioral evidence, as some of that literature argued, because it is not robust to data aggregations and separations that are psychologically natural given the relevant real-world context. In this case, this means separating serves into the serve types that players actually consider and separating situations by pressure levels, which leads to clear and sound rejection of minimax.<\/jats:p>","DOI":"10.3390\/g16060060","type":"journal-article","created":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T10:02:13Z","timestamp":1763546533000},"page":"60","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Minimax Under Pressure: The Case of Tennis"],"prefix":"10.3390","volume":"16","author":[{"given":"Ben","family":"Depoorter","sequence":"first","affiliation":[{"name":"Center for Internet & Society and Stanford Law School, Stanford, CA 94305, USA"},{"name":"University of California Law SF, San Franciso, CA 94102, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simon","family":"Jantschgi","sequence":"additional","affiliation":[{"name":"Zurich Center for Market Design and SUZ, University of Zurich, 8050 Zurich, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ivan","family":"Lendl","sequence":"additional","affiliation":[{"name":"Independent Researcher, San Diego, CA 92130, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miha","family":"Mlakar","sequence":"additional","affiliation":[{"name":"Pareto Lab, 1000 Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1261-8134","authenticated-orcid":false,"given":"Heinrich","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"568","DOI":"10.2307\/2525845","article-title":"Efficiency estimation of production functions","volume":"13","author":"Afriat","year":"1972","journal-title":"International Economic Review"},{"key":"ref_2","first-page":"11","article-title":"The effect of pressure on mixed-strategy play in tennis: The effect of court surface on service decisions","volume":"3","author":"Bailey","year":"2012","journal-title":"International Journal of Business and Social Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1111\/1468-0297.00645","article-title":"Does minimax work? An experimental study","volume":"111","author":"Binmore","year":"2001","journal-title":"Economic Journal"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1214\/aoms\/1177698513","article-title":"The big match","volume":"39","author":"Blackwell","year":"1968","journal-title":"The Annals of Mathematical Statistics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.2307\/2938300","article-title":"Testing the minimax hypothesis: A re-examination of O\u2019Neill\u2019s game experiment","volume":"58","author":"Brown","year":"1990","journal-title":"Econometrica"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.joep.2017.04.005","article-title":"Choking under pressure and gender: Evidence from professional tennis","volume":"61","author":"Krumer","year":"2017","journal-title":"Journal of Economic Psychology"},{"key":"ref_8","first-page":"47","article-title":"Recursive games","volume":"39","author":"Everett","year":"1957","journal-title":"Contributions to the Theory of Games"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gauriot, R., Page, L., and Wooders, J. (2016). Nash at Wimbledon: Evidence from half a million serves. SSRN Electronic Journal.","DOI":"10.2139\/ssrn.2850919"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"981","DOI":"10.3982\/QE1563","article-title":"Expertise, gender, and equilibrium play","volume":"14","author":"Gauriot","year":"2023","journal-title":"Quantitative Economics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1016\/j.jebo.2012.09.021","article-title":"Performing best when it matters most: Evidence from professional tennis","volume":"84","author":"Gossner","year":"2012","journal-title":"Journal of Economic Behavior & Organization"},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jeconom.2008.09.036","article-title":"The efficiency of top agents: An analysis through service strategy in tennis","volume":"148","author":"Klaassen","year":"2009","journal-title":"Journal of Econometrics"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.3982\/ECTA7405","article-title":"What happens in the field stays in the field: Exploring whether professionals play minimax in laboratory experiments","volume":"78","author":"Levitt","year":"2010","journal-title":"Econometrica"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s001990050020","article-title":"An experimental study of information and mixed-strategy play in the three-person matching-pennies game","volume":"15","author":"McCabe","year":"2000","journal-title":"Economic Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/BF01769259","article-title":"Stochastic games","volume":"10","author":"Mertens","year":"1981","journal-title":"International Journal of Game Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1006\/game.1994.1037","article-title":"Learning behavior in an experimental matching pennies game","volume":"7","author":"Mookherjee","year":"1994","journal-title":"Games and Economic Behavior"},{"key":"ref_18","first-page":"75","article-title":"Dynamic programming in tennis\u2014When to use a fast serve","volume":"36","author":"Norman","year":"1985","journal-title":"Journal of the Operational Research Society"},{"key":"ref_19","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_20","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_21","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_22","doi-asserted-by":"crossref","first-page":"100001","DOI":"10.1016\/j.serev.2022.100001","article-title":"Maradona plays minimax","volume":"1","year":"2023","journal-title":"Sports Economics Review"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1111\/j.0012-9682.2008.00818.x","article-title":"Experientia docet: Professionals play minimax in laboratory experiments","volume":"76","author":"Volij","year":"2008","journal-title":"Econometrica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/j.jebo.2023.05.042","article-title":"Gender differences in performance in competitive environments? Evidence from professional tennis players","volume":"212","author":"Paserman","year":"2023","journal-title":"Journal of Economic Behavior & Organization"},{"key":"ref_25","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_26","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02810663","article-title":"Two-player stochastic games I: A reduction","volume":"119","author":"Vieille","year":"2000a","journal-title":"Israel Journal of Mathematics"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF02810664","article-title":"Two-player stochastic games II: The case of recursive games","volume":"119","author":"Vieille","year":"2000b","journal-title":"Israel Journal of Mathematics"},{"key":"ref_28","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_29","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_30","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.geb.2010.04.011","article-title":"Equilibrium play in matches: Binary Markov games","volume":"71","author":"Walker","year":"2011","journal-title":"Games and Economic Behavior"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/6\/60\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T09:11:10Z","timestamp":1772010670000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/16\/6\/60"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,18]]},"references-count":30,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["g16060060"],"URL":"https:\/\/doi.org\/10.3390\/g16060060","relation":{},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,18]]}}}