If you thought I was going to stop at one post about beer pong, then you were sorely mistaken. I wanted to address a few things related to my previous post, and since I'll be all graduated soon, time was running out to use the software. Thus, I spent some time during this gorgeous day where I didn't have school in a nearly-empty computer cluster running simulations. Yup, I'm awesome.
So maybe some of my previous results were exaggerated. My bad.
My original intention for this post was to add some graphs to show what I was describing in the original post. Specifically, I wanted to show the distribution of results for the GB vs. AA match that had very different outcomes depending upon which team started. As I did this, I noticed that about 4% of the games simulated ended in a tie. It turns out that the 25 simulated turns in my spreadsheet weren't enough for some of the games, and this skewed my results towards the team that went first. The stat I was using to determine who won averaged a cell that should have only been returning 1 or 2 depending on which team won. Since some games were ending without a victor, this cell would return a 0, and skew the results. Thus, all the numbers from my original post were wrong (oops). Here are the correct results for select scenarios:
GG vs. GG: The team that goes first wins 50.25% of the time.
GB vs. AA: GB wins 37.1% of the time when it goes first and 35.56% of the time when it goes second. The expected cup differentials for GB turned out exactly the same for both scenarios (-1.12).
GA vs. AA: GA wins 66.15% of the time when it goes first and 66.26% of the time when it goes second. Cup differential was once again the same for both (+1.43).
As you can see, the base scenario is still basically even while the effects of the others are greatly lessened if not completely eliminated. It appears that GB should still elect to go first in that scenario, but it doesn't actually increase win probability by 15%. I was pretty surpised by the initial results, so this correction makes sense to me.
33% made is a strong player? That seems low.
I selected 33% since that was roughly my skill level during practice and no one at the tournament severely outplayed me. Thus, I theorized that that was a good estimate for a typical strong player. Yes, I realize that people who play beer pong for a living probably have higher true talent levels than that, but I was mostly concerned with the majority of players who aren't as good.
That being said, a friend brought up an interesting point. If all four players in a game could make every shot, then the team that goes first would win every time (ignorning rebuttals and overtime). Thus, he theorized that as talent levels go up, there is an added benefit to going first. Here is what the sim said:
All four players hit 40% of shots: The team that goes first wins 50.09% of the time
All four players hit 50% of shots: The team that goes first wins 49.19% of the time
All four players hit 80% of shots: The team that goes first wins 45.83% of the time
All four players hit 95% of shots: The team that goes first wins 55.78% of the time
It appears that higher talent levels remain even for quite a while. At 80%, there is actually a slight dip for the team that goes first, but then it looks to satisfy our hypothesis as we get closer to 100%. Regardless of this, it looks like the difference is never too extreme, and of course you're unlikely to encounter a true talent 80% anywhere (Yes, I'm sure we've all had games where we've hit 80% of our shots, but good luck averaging that rate).
Doesn't beer pong get harder as it goes along?
The other thing my friend proposed was that it's harder to hit cups later in the game. While I normally wouldn't want to make this assumption without some empirical data on whether this is true, it's definitely worth simming. Since we don't have any idea of what this effect is, I did a quick and dirty estimate where each cup removed reduces the odds of hitting a cup by 5% (Thus, you're only 55% as likely to hit a one cup rack as you are a full rack). Since the initial true talent level now only represents one's likelihood of hitting a full rack, I upped the numbers a bit. Here are the results for a couple of tests:
All four players with initial true talent of 40%: The team that went first won 49.18% of the time.
All four players with initial true talent of 50%: The team that went first won 49.28% of the time.
It looks like we still get pretty even results, although these simulations show a slight advantage to the team that defers. This could suggest the building advantage for the second team that I described in the original post could come more into play here. Still, the effect is so small that it probably won't make a practical difference.
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