raspel Posted Wednesday at 03:00 PM Share Posted Wednesday at 03:00 PM (edited) Intro The game of "Blood Bowl" contains significant luck element, therefore it often raises question of assessment said luck. In this work I try to develop a method for estimating coaches' luck, based on their dice rolls. Input Data "Blood Bowl" is a game with a lot of dice rolling. Throughout the match each coach makes over a hundred individual independent dice rolls with variable success, failure and neutral outcome chances. For starting analysis, we have to gather all these rolls for both coaches; their chances, outcomes and amounts. This can be done manually, automation of this process is possible, but far from my competention. As the result, we get a list of individual dice rolls for each coach, looking like: [[chance_1, weight_1], [chance_2, weight_2], ... [chance_n, weight_n]], where chance - probability of individual roll, weight - whether its outcome was positive (1), neutral (0) or negative (-1). Sum of possible weights for each individual roll should be 0, e.g. you can't have a roll without positive or negative outcome (if you make such rolls - don't include them into list, they don't affect your luck anyway). Sum of possible probabilities for each individual roll should be 1, by definition of probability. Exact assumptions I've made in this work: Your Armour and Injury Rolls are your opponent's rolls. Kicking team makes Kick-Off Roll. Both coaches can roll dice at kick-off event, if needed. Starting coin roll I also assigned to both coaches (this is questionable: 1 roll appears in both lists). Each individual dice roll counts separately (Armour and Injury Rolls separately, before and after RR - separately etc). Modificators change your probabilities. If MB or DP are not used at Armour Roll - they are not taken into account, if they didn't help or armour is pierced using their modificator - they are. For Injury Rolls success - KO or better, failure - stun. Casualty Rolls are not taken into account - they are always successful. So far I don't count chances of expulsion during fouls. Scatters and bounces are not counted. Only fighting rolls have neutral outcome probability; that's usually Pushes or BD with Block or Wrestle. Success and failure probabilities of fighting rolls depend on situation. For example, during one-turn Push may be a success and DD - a failure. If you need at least Push - Pushes, DS and DD will be success. Need to evaluate situations like this manually is the main obstacle before an automation of this method. Working Hypothesis For each roll we add new metric luck = weight/chance It has several useful properties: sign, showing whether outcome was positive or negative; its absolute value is greater, the more unlikely the dice roll was; expected sum of luck for "fair" dice rolls is 0. Let's prove the last statement. Imagine us rolling dice n times with chance of positive outcome p, chance of negative outcome q, chance of neutral outcome (1-p-q). Then with "fair" dice you expect to get positive outcome np times, neutral outcome n(1-p-q) times and negative outcome nq times. Your luck sum would be (1/p)np+(0/(1-p-q))n(1-p-q)+(-1/q)nq=n-n=0 So should be sum of luck with "fair" luck distribution for the whole match (later "cumulative luck", CL). If you get more positive outcomes, than expected, your CL will be above 0. If you get more negative - below 0. Luck Comparision For each coach you get a list of luck scores of his rolls, looking like [[luck_1], [luck_2], ... [luck_n]]. Count each coaches cumulative luck as sum of his rolls' luck scores. You get 2 numbers (likely different from 0 and from each other). Now let's try to estimate significance of their differences statistically. I suggest to take as the null hypothesis equality of coaches' real CLs, then evaluate its p-value by comparing confidence intervals for them (using this article), build with bootstrap method. Now we are able to answer following questions: whether each coach's CL significantly differs from expected (0); whether coaches' CL significantly differ from each other. That gives us statistic-based method for assessing coaches' luck in "Blood Bowl" match. Example You may find an example of using my method in ipynb-file here (it is a Python code for JupyterNotebook). It produces nice pictures Spoiler Discussion and Controversial Issues I do my first steps in statistics, not sure I fully understand it. Your ideas for improving this method are welcomed. I'd like to point some controversies myself: Luck metric is invented by me, not sure, whether it has physical or mathematical sense. Weights -1, 0 and 1 are arbitrary. Probably, they should be different for different actions. Right now 1 for 2+ dodge with a ball has the same luck, as 1 for unimportant Bone Head, or 1 for useless GFI. Double Skull at block has luck -36, series 1-RR-1 at dodge has luck -12. If you own or want to make a dice-meter site, you can take my method for it, just let me know. Sources Bootstrap. Karpov.courses Inference by Eye: Confidence Intervals and How to Read Pictures of Data Edited Friday at 10:45 AM by raspel added picture 2 Link to comment Share on other sites More sharing options... Awards 2 5 2
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