{"id":353,"date":"2019-12-08T21:15:53","date_gmt":"2019-12-08T21:15:53","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=353"},"modified":"2019-12-08T22:33:56","modified_gmt":"2019-12-08T22:33:56","slug":"handicapping-pub-trivia","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/","title":{"rendered":"Handicapping pub trivia"},"content":{"rendered":"\n

Introduction<\/h3>\n\n\n\n

The following question was posted recently on Reddit’s statistics forum<\/a>:<\/p>\n\n\n\n

If there is a quiz of x<\/code> questions with varying results between teams of different sizes, how could you logically handicap the larger teams to bring some sort of equivalence in performance measure?<\/p>

[Suppose there are] 25 questions and a team of two scores 11\/25. A team of 4 scores 17\/25. Who did better […]?<\/p><\/blockquote>\n\n\n\n

One respondent suggested a binomial model, in which every player has the same probability of answering any question correctly.<\/p>\n\n\n\n

I suggested a model based on item response theory, in which each question has a level of difficulty, d<\/code>, each player has a level of efficacy e<\/code>, and the probability that a player answers a question is<\/p>\n\n\n\n

expit(e-d+c)<\/code><\/pre>\n\n\n\n
where c<\/code> is a constant offset for all players and questions and expit<\/code> is the inverse of the logit function.<\/p>\n\n\n\n
Another respondent pointed out that group dynamics will come into play.  On a given team, it is not enough if one player knows the answer; they also have to persuade their teammates.<\/p>\n\n\n\n
\"\"
Me (left) at pub trivia with friends in Richmond, VA.  Despite our numbers, we did not win.<\/figcaption><\/figure><\/div>\n\n\n\n
I wrote some simulations to explore this question.  You can see a static version of my notebook here<\/a>, or you can run the code on Colab<\/a>.<\/p>\n\n\n\n
I implement a binomial model and a model based on item response theory.  Interestingly, for the scenario in the question they yield opposite results: under the binomial model, we would judge that the team of two performed better; under the other model, the team of four was better.<\/p>\n\n\n\n
In both cases I use a simple model of group dynamics: if anyone on the team gets a question, that means the whole team gets the question.  So one way to think of this model is that “getting” a question means something like “knowing the answer and successfully convincing your team”.<\/p>\n\n\n\n
Anyway, I’m not sure I really answered the question, other than to show that the answer depends on the model.<\/p>\n","protected":false},"excerpt":{"rendered":"
Introduction The following question was posted recently on Reddit’s statistics forum: If there is a quiz of x questions with varying results between teams of different sizes, how could you logically handicap the larger teams to bring some sort of equivalence in performance measure? [Suppose there are] 25 questions and a team of two scores 11\/25. A team of 4 scores 17\/25. Who did better […]? One respondent suggested a binomial model, in which every player has the same probability…<\/p>\n
 Read More  Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[52,33,53,50,51],"class_list":["post-353","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-binomial","tag-cdf","tag-item-response-theory","tag-numpy","tag-trivia"],"yoast_head":"\nHandicapping pub trivia - Probably Overthinking It<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Handicapping pub trivia - Probably Overthinking It\" \/>\n<meta property=\"og:description\" content=\"Introduction The following question was posted recently on Reddit’s statistics forum: If there is a quiz of x questions with varying results between teams of different sizes, how could you logically handicap the larger teams to bring some sort of equivalence in performance measure? [Suppose there are] 25 questions and a team of two scores 11\/25. A team of 4 scores 17\/25. Who did better […]? One respondent suggested a binomial model, in which every player has the same probability... Read More Read More\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\" \/>\n<meta property=\"og:site_name\" content=\"Probably Overthinking It\" \/>\n<meta property=\"article:published_time\" content=\"2019-12-08T21:15:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-12-08T22:33:56+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/pbs.twimg.com\/media\/ELS2XD4XsAMoGzX?format=jpg&name=medium\" \/>\n<meta name=\"author\" content=\"AllenDowney\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@AllenDowney\" \/>\n<meta name=\"twitter:site\" content=\"@AllenDowney\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"AllenDowney\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\"},\"author\":{\"name\":\"AllenDowney\",\"@id\":\"https:\/\/www.allendowney.com\/blog\/#\/schema\/person\/4e5bfb2e9af6c3446cb0031a7bf83207\"},\"headline\":\"Handicapping pub trivia\",\"datePublished\":\"2019-12-08T21:15:53+00:00\",\"dateModified\":\"2019-12-08T22:33:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\"},\"wordCount\":331,\"publisher\":{\"@id\":\"https:\/\/www.allendowney.com\/blog\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/pbs.twimg.com\/media\/ELS2XD4XsAMoGzX?format=jpg&name=medium\",\"keywords\":[\"binomial\",\"CDF\",\"item response theory\",\"NumPy\",\"trivia\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\",\"url\":\"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/\",\"name\":\"Handicapping pub trivia - 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This workshop is based on the first session of the Applied Bayesian Modeling Workshop I teach along with my colleagues at PyMC Labs. If you would like to learn more, it\u2026","rel":"","context":"In \"bayesian statistics\"","block_context":{"text":"bayesian statistics","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian-statistics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/img.youtube.com\/vi\/PLGVZCDnMOq0qmerwB1eITnr5AfYRGm0DF\/0.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":609,"url":"https:\/\/www.allendowney.com\/blog\/2021\/05\/07\/founded-upon-an-error\/","url_meta":{"origin":353,"position":4},"title":"Founded Upon an Error","author":"AllenDowney","date":"May 7, 2021","format":false,"excerpt":"A recent post on Reddit asks, \"Why was Bayes' Theory not accepted\/popular historically until the late 20th century?\" Great question! As always, there are many answers to a question like this, and the good people of Reddit provide several. But the first and most popular answer is, in my humble\u2026","rel":"","context":"In \"bayesian\"","block_context":{"text":"bayesian","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":1233,"url":"https:\/\/www.allendowney.com\/blog\/2024\/02\/18\/the-gender-gap-in-political-beliefs-is-small\/","url_meta":{"origin":353,"position":5},"title":"The Gender Gap in Political Beliefs Is Small","author":"AllenDowney","date":"February 18, 2024","format":false,"excerpt":"In previous articles (here, here, and here) I've looked at evidence of a gender gap in political alignment (liberal or conservative), party affiliation (Democrat or Republican), and policy preferences. Using data from the GSS, I found that women are more likely to say they are liberal, and more likely to\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=700%2C400&ssl=1 2x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=1050%2C600&ssl=1 3x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2024\/02\/belief_gap3-1.png?resize=1400%2C800&ssl=1 4x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/comments?post=353"}],"version-history":[{"count":3,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/353\/revisions"}],"predecessor-version":[{"id":356,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/353\/revisions\/356"}],"wp:attachment":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/media?parent=353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/categories?post=353"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/tags?post=353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}