A mathematical discussion of pong. Mlayth: PLEASE READ!
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August 5, 2007 at 5:04 am #5873
For those of you who read these forums often, you may have noticed that a guy by the name of Mlayth posts here on a fairly regular basis (1148 posts to date). Mlayth, like the rest of us, takes the game of Pong very seriously, and apparently spends a lot of time thinking about it.
As you all may know, a lot of his time on here is spent making mathematical arguments on how certain playing variations affect game outcomes. He exhibits a great deal of confidence when posting, and generally assumes that his ability to understand the mathematical basis of pong is extremely strong (and better than everyone elses on here).
I have read many of the posts that Mlayth has made over the past. I am, in general, shocked with the conclusions that he comes up with, as he is, in my opinion, does not employ proper logic
If someone calls him out on being full of shit, he says things like "math is a perfect science", or "the numbers don’t lie". These types of comments are extremely frustrating to me personally, because I fully appreciate the power of math, and realize that math itself is not flawed; however, the person using/distorting math often is…
In one recent thread, jjm5750 made a point that pretty much sums up most people’s opinions: "Your stats are awesome and impressive, but, to me, are just that—numbers and diagrams". Well, even though I don’t personally believe that his stats are impressive, awesome, or even meaningful, I do believe that most people are not all too concerned with the mathematical effects of various rule changes – we play pong because its just fun. We are not usually as concerned with how rule changes effect the better teams chance of winning as much as we are concerned with creating excitement and fun in games.
One of the problems that I see in all of this, is that most of the people posting on this board are not strong enough in probability theory to know that the arguments that Mlayth makes are not logical. Most people on here love the game of pong, and are interested to see what an ‘expert’ has to say about the math behind it. Since Mlayth comes on here pretending to be an expert, people simply believe that he is making logical arguments and take his conclusions at face value. I am here to say that Mlayth is not an expert.
I will start by saying that I happen to be pretty good at math myself ; Honestly, my professional and academic history would generally certify me as an ‘expert’ in math. Having said this, I would still not (normally) come on this board with the type of intellectual elitism that Mlayth brings – I generally view that type of attitude as uncalled for. However, for this post, I will be a tad bit elitist, because….well…fuck it.
While it will be hard for me to convince most people on this board of the unreasonableness of Mlayth’s arguments, I will dedicate the rest of this post to debunking at least one of the arguments that Mlayth has claimed (over and over again). If you are not interested in reading a mathematical treatment of beer pong, I suggest you stop reading now and start (or continue) drinking – which is what you (and I) should be doing anyway….
One thing that Mlayth invokes that gets me absolutely furious is his beer pong skill curve. He posts a simple graph, and then says things like "well, because of the exponentiality of the beer pong skill curve, we can clearly determine"… whatever argument Mlayth feels like making at the time.
So what does the beer pong skill curve say exactly? It says that, in the case of unlimited rollbacks, the number of shots that a player makes per turn on average is exponentially related to the players "skill". For this purpose, skill is defined as a players shooting percentage.
Now, let me first start by saying that I do not believe that players have "shooting percentages". I believe that the streaky nature of pong combined with the effects of alcohol make the beer pong skill curve absolutely meaningless in practice – the shooting distributions for the average player do not in any way follow a purely random process. Mlayth: Contrary to what you normally claim, these facts MUST be considered before even beginning a real mathematical treatment of pong.
HOWEVER: for the sake of this discussion, I will let all of that slide and assume that every player has a shooting percentage (call it P), and hits cups with complete independence (i.e, the probability that a player hits a cup on a shot is always P, regardless of whether they hit their last shot, or their partner hit their shot, etc…)
Given this assumption, I agree then that the number of shots a player hits on an average turn is exponential on the players shooting percentage, assuming unlimited rollbacks. Note: the beer pong skill curve, as Mlayth has presented it, is only exponential when there are UNLIMITED rollbacks. It is only a function of X^2 when there is a maximum of 1 rollbackperplayerperturn in effect.
For some reason though, Mlayth has claimed that this means that having rollbacks benefits better players (he has repeated this about 10 million times). He has drawn this conclusion outright, and justified it by saying something like "well, as we all know the skill curve is exponential, so this means that rollbacks help better players on average".
I have always been really, really annoyed by this argument, because it is completely abstracting many very important concepts in probability theory (some of which are basic concepts, some of which are not). However, the truth is, I was never really sure how rollbacks would actually effect the more "skilled" players.The more I thought about it, the more I realized that it is an extremely difficult determination to make just based on pure logic….the only way to really know and be sure, I thought, was to run trials.
SO…….I decided to run trials. I’ve written code in C++ that generates random pong games, and have observed the results under various rollback scenarios.
My Assumptions/Specs:
For each team, both players have the same shooting percentage. I will eventually alter this and allow for 4 separate shooting percentages, but for now, a team consists of of two players that are exactly equal….
The game is 10Cup, with redemptions. As soon as one team hits last cup, the other team shoots until they miss. There are no LockOuts; i.e., even if both players on a team hit the last cup, the other team still has the opportunity for Redemption.
I have considered each game that goes into overtime a draw. I think this is ok for the purpose of this study, because in the context of the overall stats, it means that overtime is a new 10cup game (this is, of course, different than reality, where we have 3cup overtime, but I think that any conclusion with regards to rollbacks will hold true with a 3cup overtime as opposed to a 10cup overtime).
For each game, the team shooting first is chosen randomly. First team gets 1 ball. After that, each team shoots twice.
I ran this under 4 different rollback rule sets:
1) No Rollbacks at all (WSOBP I).
2) 1 ball Rollback (WSOBP II): If both players hit their shots, the team gets 1 extra ball. In this case, remember that each player on a team has the same shooting pct, so it doesnt matter which player shoots it….
3) 2 ball Rollback: If both players hit their shots, the team gets 2 extra balls. This is equivalent to the "4cupperturn Maximum" described in the forums…
4) Unlimited Rollbacks: If both players hit their shots, they get the balls back. If they hit both again, the get the balls back..this keeps going until one player misses….
I then chose sample shooting percentages for each team. For each set of shooting percentages, I generated 500,000 games of pong under each of the above scenarios….
I started by having each player shoot 50%. Naturally, under all 4 scenarios,
each team won 50% of the time.I then had Team 1 players shooting 60%, and Team 2 players shooting 50%. The winning percentages for Team 1 under the 4 scenarios were:
(1) 77.41%
(2) 76.63%
(3) 74.15%
(4) 74.99%As you can clearly see (the numbers don’t lie ), the better team (Team 1), won the most games when there were no rollbacks. The end result is that the addition of rollbacks created more volatility in the games which helps close the gap between players of different caliber (by increasing the effects of hot streaks for the weaker players).
You will notice that the win pct jumped a bit under scenario 4 (unlimited rollbacks). This did surprise me a bit, but I think it is reasonable, in that the really long hitting streaks will provide more benefit to the better players (i.e., Mlayth’s skill curve does come into effect a little bit….but keep in mind that the unlimited rollback scenario still has a lower win pct than the no rollback scenario).
I ran these with a bunch of different sets of shooting percentages for each team
(50%/40%, 90%/80%,90%/50%, etc…) and the basic result was universal: The inclusion of rollbacks made the games more even. Note: The relationship between unlimited rollbacks and max4perturn was more volatile. In some cases, the unlimited rollbacks actually yielded a lower win pct than scenario 3.In all cases, however, the No Rollback situation created the highest win percentage.Please realize, that I did not conduct this study looking for a certain result (Please compare that the methodology that Mlayth generally uses). I was really just curious as to how rollbacks would affect the outcome of games. While I did not know what the outcome of my experiment would be, I did know for sure that Mlayth’s arguments made absolutely no sense. I think that this is a general truth for Mlayth: he is always drawing conclusions without proper logic.
I am more than happy to provide my source code to anyone who wants to check my results. In addition, I welcome anyone who would like to try and independently verify my results. I do ask that if you do that, please use a fairly standard programming language, so that if we do disagree, we can resolve our differences in an academic manner….
August 5, 2007 at 9:06 pm #5874:standing applause:!!!!!!!!!!!
i too was very confused how the numbers would work out with a varied skill curve and infinite rollbacks. it seemed to me like it needed a good trial. great experiment and well done.
lets hope mlathe isn’t too defensive but instead has some interesting counter ideas. that was the best argument ever stated on here. except maybe when we tried to decide how many boobs could be shown at a time in one turn. now that was a tough discussion.
mmm… beer pong
August 6, 2007 at 3:50 pm #5875I’m pretty good at math but I suck at reading so I think I only made it through 23.29% of your post.
That was enough for me to realize that skinny definitely drank his HATERADE this morning.
August 6, 2007 at 3:59 pm #5876FLAWLESS VICTORY!!!
August 6, 2007 at 5:25 pm #5877Brilliant! (c) guy from Guiness ad
August 6, 2007 at 6:49 pm #5878Thanks for clearing this up Skinny– I’m interested to hear what mLayth has to say.
You should publish that in a statistical journal, I’m going to have to read it again and then post some thoughts. Very interesting results, for sure.
btw Pope, I spent like a half hour looking for an "oh snap" graphic, thanks for coming through!
August 6, 2007 at 7:08 pm #5879I read the whole post.
It’s not too surprising to see that a team will win more of their games with no rollbacks simply because of a conclusion you stated: rollbacks make the game more volatile, allowing lessskilled teams to go on streaks.
Logically one can come to this conclusion, but I think it’s interesting to see the mathematical exercise put to the test.
One thing I must say though. Mlayth makes a pretty cool ball rollback system for practicing on
August 6, 2007 at 8:19 pm #5880Mlayth makes a pretty cool ball rollback system for practicing on
haha – good conclusion to a rollback discussion.
August 6, 2007 at 11:06 pm #5881Skinny – I went ahead and wrote a quick program to verify your results.
Everything looked about the same except I never saw the jump in win % for unlimited rollbacks. That means there is a bug in either your code or mine. I ran a ton of test runs and didn’t see any errors but that doesn’t mean they aren’t there.
Either way, here are my results (30,000 games each):
0: 77.35%
1: 76.41%
2: 74.94%
3: 74.57%I also added a 3 cup overtime into the mix.
It didn’t effect the trends but it did change the results slightly:
0: 74.84%
1: 74.68%
2: 73.84%
3: 73.22%So, I definitely think you’re right.
If you want to play with my version, you can find it here:http://mdbeerpong.com/sim/game_simulator.php
Keep in mind I made this in 34 hours so I’m sure their might be some problems with it still.
August 6, 2007 at 11:36 pm #5882thanks for taking the time austin….
without having access to your backend code, it will be hard for me to figure out which one of us has the bug. I certainly would not be surprised if it were me…
your results implementing the 3 cup overtime make sense to me – a 3cup overtime is much more volatile than a 10cup overtime….
at this point, I am going to change directions and focus on a more rigorous study using Bayesian math rather than simply running trials. While at first this seemed overly daunting, I’m thinking I can put some stuff together in excel (basically using an equivalent of recursion…). I’ve already got the simplistic version done, which is 1on1 pong, and the results are interesting: I see a small drop with the inclusion of a single rollback, and then the win pct does come back for unlimited rollbacks. I will try and extend this to regular pong, and let you know how it turns out…
Skinny – I went ahead and wrote a quick program to verify your results.
Everything looked about the same except I never saw the jump in win % for unlimited rollbacks. That means there is a bug in either your code or mine. I ran a ton of test runs and didn’t see any errors but that doesn’t mean they aren’t there.
Either way, here are my results (30,000 games each):
0: 77.35%
1: 76.41%
2: 74.94%
3: 74.57%I also added a 3 cup overtime into the mix.
It didn’t effect the trends but it did change the results slightly:
0: 74.84%
1: 74.68%
2: 73.84%
3: 73.22%So, I definitely think you’re right.
If you want to play with my version, you can find it here:http://mdbeerpong.com/sim/game_simulator.php
Keep in mind I made this in 34 hours so I’m sure their might be some problems with it still.
August 6, 2007 at 11:37 pm #5883Well done Austin… I love the web interface (I need to learn php is all I can say about that).
August 7, 2007 at 1:35 am #5884Good job skinny!
August 7, 2007 at 3:18 am #5885I’ve got the attention span of an autistic kid at chucky cheese, so I didn’t read all of it. So I’ll just say good job anyways.. lol. Where’s the reply, Foster?
August 7, 2007 at 5:26 am #5886I’ve already got the simplistic version done, which is 1on1 pong, and the results are interesting: I see a small drop with the inclusion of a single rollback, and then the win pct does come back for unlimited rollbacks. I will try and extend this to regular pong, and let you know how it turns out…
For 1on1 pong you only need one stat: Austin’s win percentage is 100%.
August 7, 2007 at 7:19 am #5887Austin –
A couple of things…
For the start of the game, when a team only has one shot – I like how you took the better of two players.I noticed that the end of the game is a little funny. I saw one game where the team apparently kept shooting at the last cup (0 remaining… get balls back, hit, 1 remaining, hit, 2 remaining) It might make things screwy when it comes to OT.
Well done though. I obviously enjoy seeing the playbyplay when the results are printed out.

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