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A mathematical discussion of pong. Mlayth: PLEASE READ!

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austin Joined: 01.01.1991 Age: 32 Total Games: 122 W/L %: 77.87 Avg CD: 1.85 Full Stats	Posted: Aug 7th 2007, 05:47 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! mLayth So perhaps you can explain something to me. I know that the skill curve is mathematically solid, and judging by the tests preformed by austin and yourself it appears that better players are not given the kind of advantage one would be lead to believe. You've shown me the numbers.. but whats the reason. Why not? I was thinking about this on my drive home yesterday. It isn't very obvious to me, I could only come up with one possible answer: When you look at something discrete like a win or loss of a beer pong game, it doesn't matter what the final score is. The 60% team may win by an average of 5 cups and the 50% may only win by an average of 1. Maybe this accounts for the results which weren't what you expected. I've noticed this in actual game play. Just because you win by 5 cups one game doesn't mean the next game won't go into overtime. The team that loses by 5 doesn't have to hit those 5 before the next game starts; they get to start over after they lose. If you think about it that way it almost makes sense. With no bring backs you are getting closer to pure shooting percentage and you'd expect the better team to win more. When you add the bring-backs, you allow for the other team to squeak in some wins with hot streaks. I don't really have time to figure out for sure, that was just one idea that I had. I think we just need someone with more time than I have to figure it out.
prusch Joined: 08.22.2006 Age: 30 Total Games: 145 W/L %: 77.24 Avg CD: 1.70 Full Stats	Posted: Aug 7th 2007, 08:02 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! skinny In theory, every game should be independent of every other game. Therefore, the probability distribution for 1000 games should be binomial, with n=1000 and p = the expected winning percentage for each game. In this case, say p=71%. Then the variance for 1000 trials should be np(1-p) = 1000.71(1-.71) =205.9, giving a s.d. of 14.3. As p gets closer to 50%, the s.d should go a little bit higher (max of 15.8 or so). You're definitely right here. We should have a p for each game, but we didn't before starting this discussion. And since an experiment was run, I figured those results should be presented with a standard deviation. The story would be a little different if the probability were derived. Rather than set up a way to run a simulated game, it should be possible (if not a little bit of a chore) to figure out how to calculate a distinct probability for each scenario. Since you ran your program for 500,000 games, you would think that a percent of games won would match a calculated probability. I'm just a little curious. Would that be unnecessary tinkering?
skinny Joined: 01.01.1991 Age: 31 Total Games: 1 W/L %: 100 Avg CD: 1.00 Full Stats	Posted: Aug 7th 2007, 08:20 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! prusch Would that be unnecessary tinkering? absolutely not. I'm working on it now...hope to have something by this weekend (although I am currently busy as hell with a whole slew of things)....
mlayth Joined: 01.01.1991 Total Games: 0 W/L %: 0 Avg CD: 0 Full Stats	Posted: Aug 8th 2007, 12:36 am Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! edit: see next post
mlayth Joined: 01.01.1991 Total Games: 0 W/L %: 0 Avg CD: 0 Full Stats	Posted: Aug 8th 2007, 02:57 am Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! skinny prusch Would that be unnecessary tinkering? absolutely not. I'm working on it now...hope to have something by this weekend (although I am currently busy as hell with a whole slew of things).... Skinny, my initial pov was that rollbacks will help better teams in the preliminaries (I was not debating for the final tournament) so in that sense I think it would be relevant if you could post the overall cup differential for each trial run as well.
mlayth Joined: 01.01.1991 Total Games: 0 W/L %: 0 Avg CD: 0 Full Stats	Posted: Aug 8th 2007, 05:44 am Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! I am proposing a theory that with rollbacks other teams may get lucky and win a couple of extra times, but overall you will win by more cups more often... In other words, it seems like the unlimited rollbacks will have a more statisticallys significant effect on the cup differential than the winning percentage. That and game time, but in a system where everyone plays the same number of games it won't come into effect. It seems to make sense, because without any rollbacks the higher game-percentaged team is always going to win as billy showed us last year. With rollbacks it will allow them to win by a lot more, but at times it becomes possible under lucky scenarios for the higher game-percentaged team to lose. I see now that on a large scale with unlimited rollbacks, the door has been opened for lower percentaged teams to necessarily win a little bit more often, during those lucky games. Thanks for helping me to see this skinny, I was wrong and not connecting the dots correctly. The effect is pretty interesting, but it looks like the numbers are close enough together that it doesn't have a ground breaking impact either way. It will be interesting to see if my theory is right about cup differential and, if so, how significant the impact is. As an aside.. with my experience tracking stats as a player, there are definite hot and cold streaks. I have some suspicions that this sort of statistical oddity may be underplayed in the computer generated stats, but I don't know if that is factored out or not in the big picture.
skinny Joined: 01.01.1991 Age: 31 Total Games: 1 W/L %: 100 Avg CD: 1.00 Full Stats	Posted: Aug 8th 2007, 05:45 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! mLayth Skinny, my initial pov was that rollbacks will help better teams in the preliminaries (I was not debating for the final tournament) so in that sense I think it would be relevant if you could post the overall cup differential for each trial run as well. I will a) remove redemptions/overtimes from the code, and then b) report on the distribution of the cup differentials. mLayth I am proposing a theory that with rollbacks other teams may get lucky and win a couple of extra times, but overall you will win by more cups more often... In other words, it seems like the unlimited rollbacks will have a more statisticallys significant effect on the cup differential than the winning percentage. That and game time, but in a system where everyone plays the same number of games it won't come into effect. It seems to make sense, because without any rollbacks the higher game-percentaged team is always going to win as billy showed us last year. With rollbacks it will allow them to win by a lot more, but at times it becomes possible under lucky scenarios for the higher game-percentaged team to lose. I see now that on a large scale with unlimited rollbacks, the door has been opened for lower percentaged teams to necessarily win a little bit more often, during those lucky games. without running the numbers first, i will not begin to seriously hypothesize on this. my gut feeling tells me that the rollbacks will simply increase the standard deviation of the cup differential, but will not significantly effect the average cup differential (though I would not be surprised if the average cup differential did sway towards the better team with rollbacks). mLayth As an aside.. with my experience tracking stats as a player, there are definite hot and cold streaks. I have some suspicions that this sort of statistical oddity may be underplayed in the computer generated stats, but I don't know if that is factored out or not in the big picture. This is a point that I have made MANY times. It is one of the reasons that I do not put all that much stock in my own numbers/programs. There is no question in my mind that the effect of streaks factors in, but I have no way to quantify it, because I simply do not have enough (well...any) player data. If you do have actual shot-by-shot data, I would love to see it (it would have to be a fairly large amount of data to be useful for this purpose...)
austin Joined: 01.01.1991 Age: 32 Total Games: 122 W/L %: 77.87 Avg CD: 1.85 Full Stats	Posted: Aug 8th 2007, 05:51 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! austin mLayth So perhaps you can explain something to me. I know that the skill curve is mathematically solid, and judging by the tests preformed by austin and yourself it appears that better players are not given the kind of advantage one would be lead to believe. You've shown me the numbers.. but whats the reason. Why not? I was thinking about this on my drive home yesterday. It isn't very obvious to me, I could only come up with one possible answer: When you look at something discrete like a win or loss of a beer pong game, it doesn't matter what the final score is. The 60% team may win by an average of 5 cups and the 50% may only win by an average of 1. Maybe this accounts for the results which weren't what you expected. I've noticed this in actual game play. Just because you win by 5 cups one game doesn't mean the next game won't go into overtime. The team that loses by 5 doesn't have to hit those 5 before the next game starts; they get to start over after they lose. If you think about it that way it almost makes sense. With no bring backs you are getting closer to pure shooting percentage and you'd expect the better team to win more. When you add the bring-backs, you allow for the other team to squeak in some wins with hot streaks. I don't really have time to figure out for sure, that was just one idea that I had. I think we just need someone with more time than I have to figure it out. I did a little more work and have come up with evidence to support what I said in my post yesterday (quoted above). Here are the results when adding in the average number of cups you win by. Note that this is not pure cup differential since I don't subtract cups when a team loses. For the 60% team: 0: 2.72 cups 1: 3.38 cups 2: 3.79 cups 3: 4.35 cups For the 50% team: 0: 1.85 cups 1: 2.36 cups 2: 2.77 cups 3: 3.39 cups So for better teams, as the number of roll-backs increases, the cup differential of wins increases, but win percentage goes down.
mlayth Joined: 01.01.1991 Total Games: 0 W/L %: 0 Avg CD: 0 Full Stats	Posted: Aug 8th 2007, 06:15 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! Very cool to see. Thanks Austin. Skinny, I do have a fairly large amount of actual shooting data, but I don't know if the format would be useful to you or not. I assign a character to hit or miss for each player on each team, and then assign a character to represent the end of a game. So the data ends up looking like: 10qp29wo10qp29wo10qp29wo10qpw510qowp291oq02pw0...etc
prusch Joined: 08.22.2006 Age: 30 Total Games: 145 W/L %: 77.24 Avg CD: 1.70 Full Stats	Posted: Aug 8th 2007, 07:18 pm Post subject: A mathematical discussion of pong. Mlayth: PLEASE READ! skinny If you do have actual shot-by-shot data, I would love to see it (it would have to be a fairly large amount of data to be useful for this purpose...) We've just been taking hits/misses in our league. I'm not sure how shot-by-shot score keeping would go. Most people just don't care. As for me though - I'm willing to give it a try for the sake of having more data.