{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:24:19Z","timestamp":1772252659196,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,7,26]],"date-time":"2017-07-26T00:00:00Z","timestamp":1501027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>This paper formulates the classic Monty Hall problem as a Bayesian game. Allowing Monty a small amount of freedom in his decisions facilitates a variety of solutions. The solution concept used is the Bayes Nash Equilibrium (BNE), and the set of BNE relies on Monty\u2019s motives and incentives. We endow Monty and the contestant with common prior probabilities (p) about the motives of Monty and show that, under certain conditions on p, the unique equilibrium is one in which the contestant is indifferent between switching and not switching. This coincides and agrees with the typical responses and explanations by experimental subjects. In particular, we show that our formulation can explain the experimental results in Page (1998), that more people gradually choose switch as the number of doors in the problem increases.<\/jats:p>","DOI":"10.3390\/g8030031","type":"journal-article","created":{"date-parts":[[2017,7,26]],"date-time":"2017-07-26T11:45:01Z","timestamp":1501069501000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Monty Hall Problem as a Bayesian Game"],"prefix":"10.3390","volume":"8","author":[{"given":"Mark","family":"Whitmeyer","sequence":"first","affiliation":[{"name":"Department of Economics, University of Texas at Austin, Austin, TX 78712, USA"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,26]]},"reference":[{"key":"ref_1","first-page":"67","article-title":"A problem in probability","volume":"29","author":"Selvin","year":"1975","journal-title":"Am. Stat."},{"key":"ref_2","first-page":"134","article-title":"On the Monty Hall problem","volume":"29","author":"Selvin","year":"1975","journal-title":"Am. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1257\/jep.1.2.157","article-title":"Puzzles","volume":"1","author":"Nalebuff","year":"1987","journal-title":"J. Econ. Perspect."},{"key":"ref_4","unstructured":"Gardner, M. (1961). The 2nd Scientific American Book of Mathematical Puzzles and Diversions, Simon and Schuster."},{"key":"ref_5","unstructured":"Vos Savant, M. (Parade Magazine, 1990). Ask Marilyn, Parade Magazine."},{"key":"ref_6","unstructured":"Vos Savant, M. (Parade Magazine, 1990). Ask Marilyn, Parade Magazine."},{"key":"ref_7","first-page":"347","article-title":"Marilyn vos Savant\u2019s reply","volume":"45","year":"1991","journal-title":"Am. Stat."},{"key":"ref_8","unstructured":"Tierney, J. (The New York Times, 1991). Behind Monty Hall\u2019s doors: Puzzle, debate and answer, The New York Times."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1080\/13546780244000114","article-title":"Reasoning in the Monty Hall problem: Examining choice behaviour and probability judgements","volume":"9","author":"Derks","year":"2003","journal-title":"Think. Reason."},{"key":"ref_10","first-page":"933","article-title":"Monty Hall\u2019s three doors: Construction and deconstruction of choice anomaly","volume":"88","author":"Friedman","year":"1998","journal-title":"Am. Econ. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1297","DOI":"10.1177\/0146167211410245","article-title":"Learning inhibition in the Monty Hall Problem: The role of dysfunctional counterfactual prescriptions","volume":"37","author":"Petrocelli","year":"2011","journal-title":"Personal. Soc. Psychol. Bull."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1080\/13546783.2014.918562","article-title":"Inhibitory control in a notorious brain teaser: The Monty Hall dilemma","volume":"21","author":"Saenen","year":"2015","journal-title":"Think. Reason."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1007\/s11858-015-0667-6","article-title":"A randomized Monty Hall experiment: The positive effect of conditional frequency feedback","volume":"47","author":"Saenen","year":"2015","journal-title":"ZDM Math. Educ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"353","DOI":"10.3389\/fpsyg.2015.00353","article-title":"Reasoning and choice in the Monty Hall Dilemma (MHD): Implications for improving Bayesian reasoning","volume":"6","author":"Tubau","year":"2015","journal-title":"Front. Psychol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/BF00540060","article-title":"Cognitive models and problem spaces in \u2018purely random\u2019 situations","volume":"23","author":"Lecoutre","year":"1992","journal-title":"Educ. Stud. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"561","DOI":"10.2307\/1423551","article-title":"Further exploration of two stage decision making in the Monty Hall dilemma","volume":"111","author":"Granberg","year":"1998","journal-title":"Am. J. Psychol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1037\/a0017703","article-title":"Are birds smarter than mathematicians ? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma","volume":"12","author":"Herbranson","year":"2010","journal-title":"J. Comp. Psychol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1177\/0146167295217006","article-title":"The Monty Hall dilemma","volume":"21","author":"Granberg","year":"1995","journal-title":"Personal. Soc. Psychol. Bull."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1080\/0025570X.1999.11996732","article-title":"Should she switch ? A game-theoretic analysis of the Monty Hall problem","volume":"72","author":"Fernandez","year":"1999","journal-title":"Math. Mag."},{"key":"ref_20","unstructured":"Mueser, P., and Granberg, D. (1999). The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making, Mimeo, The University of Missouri."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1080\/0025570X.2000.11996821","article-title":"Monty Hall uses a mixed strategy","volume":"73","author":"Bailey","year":"2000","journal-title":"Math. Mag."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1111\/j.1467-9574.2010.00474.x","article-title":"The Monty Hall problem is not a probability puzzle (It\u2019s a challenge in mathematical modelling)","volume":"65","author":"Gill","year":"2011","journal-title":"Stat. Neerl."},{"key":"ref_23","unstructured":"Fudenberg, D., and Tirole, J. (1991). Game Theory, MIT Press."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0165-1765(98)00158-X","article-title":"Let\u2019s Make a Deal","volume":"61","author":"Page","year":"1998","journal-title":"Econ. Lett."},{"key":"ref_25","unstructured":"Segal, L. (2017, July 25). Letters to the editor. Available online: http:\/\/www.nytimes.com\/1991\/08\/11\/opinion\/l-suppose-you-had-100-doors-with-goats-behind-99-of-them-624691.html."},{"key":"ref_26","unstructured":"Chen, W., and Wang, J.T. (2010). Epiphany Learning for Bayesian Updating: Overcoming the Generalized Monty Hall Problem, Mimeo, National Taiwan University. Available online: http:\/\/homepage.ntu.edu.tw\/~josephw\/EpiphanyMonty_20101207.pdf."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Miller, J.B., and Sanjurjo, A. (2015). A Bridge from Monty Hall to the (Anti-)Hot Hand: Restricted Choice, Selection Bias, and Empirical Practice, IGIER. Available online: https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=2709837.","DOI":"10.2139\/ssrn.2709837"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/3\/31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:43:59Z","timestamp":1760208239000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/3\/31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,26]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["g8030031"],"URL":"https:\/\/doi.org\/10.3390\/g8030031","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201705.0187.v1","asserted-by":"object"}]},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,7,26]]}}}