tag:blogger.com,1999:blog-180498979832561696.post7365128780005229954..comments2017-01-03T07:38:35.379-08:00Comments on mlibbrecht: How likely are you to get 12 wins in the Hearthstone Arena, given your skill level?Maxwell Libbrechtnoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-180498979832561696.post-69054617954915229672016-07-13T05:12:40.535-07:002016-07-13T05:12:40.535-07:00This comment has been removed by a blog administrator.Herry jonsonhttps://www.blogger.com/profile/06336899634205525690noreply@blogger.comtag:blogger.com,1999:blog-180498979832561696.post-25785028977733918932014-02-11T21:39:27.251-08:002014-02-11T21:39:27.251-08:00Thanks for the comment! The logistic function is ...Thanks for the comment! The logistic function is a natural way to convert real numbers into probabilities, and it is used often in machine learning and statistics. You can read about it here:<br />http://en.wikipedia.org/wiki/Logistic_function<br /><br />The logistic function is the inverse of the log-odds function: log-odds(p) = log(p/(1-p)). If we imagine that the difference in score is proportional to the odds that the stronger player wins, we should use the logistic function to pick win rates.<br /><br />Of course, this is ultimately an assumption that can't be verified, so it's possible that the win rates would be better modeled with some other function.Max Libbrechthttps://www.blogger.com/profile/18331504912947748572noreply@blogger.comtag:blogger.com,1999:blog-180498979832561696.post-79840973110434278382014-02-11T21:32:20.477-08:002014-02-11T21:32:20.477-08:00Hello, would you mind telling us why you chose the...Hello, would you mind telling us why you chose the logistic function to describe the win rates in arena?Rhttps://www.blogger.com/profile/03363375604122572669noreply@blogger.comtag:blogger.com,1999:blog-180498979832561696.post-41144356963403227712014-02-11T11:29:11.882-08:002014-02-11T11:29:11.882-08:00TouchÃ©. The Gaussian prior might have those prope...TouchÃ©. The Gaussian prior might have those properties which may be less than realistic.<br /><br />While using a uniform prior provides a good baseline, I think we could could get a more accurate picture. For intuition, I'm not sure a .999 power player has the same likelihood to beat a .9 player as say a .299 player has to beat a .2 player.<br /><br />As you mentioned, the best way to get a hold of this kernel is with some real win statistics. But in their absence we must continue to theory-craft! I wonder how various differing Beta distributions for the prior would affect the final win distributions!?<br /><br />Anyway, thanks for adding some rigor the the /r/Hearthstone discussions!Dustin Maurerhttps://www.blogger.com/profile/11399306429943441305noreply@blogger.comtag:blogger.com,1999:blog-180498979832561696.post-39645792052423734362014-02-11T10:52:21.023-08:002014-02-11T10:52:21.023-08:00Thanks for the comment! Actually, the equal frequ...Thanks for the comment! Actually, the equal frequencies of players is given by the fact that we represent a player's power by the fraction of players they're stronger than. On the other hand, there could be a problem with the assumption that the advantage is given by the different in players' powers. If power was Gaussian-distributed, then we would expect the top-most players to have a huge advantage over players just slightly below them. Likewise, we would expect the worse players to have a huge disadvantage over those just slightly better than them. We would need win statistics to see if this were the case, both of these cases go against my intuition.Max Libbrechthttps://www.blogger.com/profile/18331504912947748572noreply@blogger.comtag:blogger.com,1999:blog-180498979832561696.post-29398300590346898382014-02-11T08:38:21.719-08:002014-02-11T08:38:21.719-08:00First of all, excellent post! Very solid analysis...First of all, excellent post! Very solid analysis. But I would expect no less of a PhD in CS with a Stanford background. =P<br /><br />I feel a valid critique of this analysis is the following assumption:<br /><br />"To do that, we start off with all the players at 0-0, with equal frequencies of players of all powers."<br /><br />This assumes that each power level is equally likely among arena entrants. A more valid assumption for starting powers might be to sample from a Gaussian distribution with mean 0.5 (and variance small, or perhaps a log-Normal distribution to avoid <0 values). This would make the extreme power levels less likely, and make the middling power levels more likely. The effect that this would likely have on your simulations is to make the extreme results (0 or 1 wins or 10+ wins) more likely due to more disparate power levels between opponents for those with extreme starting power levels being more likely.<br /><br />How do the results change (if at all) with a different prior distribution? You can use random.gauss(mu, sigma) to sample from a Gaussian.Dustin Maurerhttps://www.blogger.com/profile/11399306429943441305noreply@blogger.com