{"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_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_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},"jetpack_post_was_ever_published":false},"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? 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