Let's talk about cheatin' punishment. We've all had the situation where This'll Hurt in the Mornin was played, and either the cheatin' player got the same rank, or maybe even higher! Or maybe you Bottom Dealt an opponent a 4 of a kind. It feels like you've been cheated when this happens. When it happens to me and I benefit, I feel dirty if I end up winning the game. But just how likely is that to happen? Are those feelings justified? In a recent conversation with @inverted about various forms of cheatin' punishment, I wondered just how potentially bad This'll Hurt and Bottom Dealin can be, and decided to run the full odds. I'd like to share my results with you guys now, along with some additional thoughts on other cheatin' punishments along the way.
-disclaimer and credit-
I love probability in general, but I am far from an expert. I used an online hypergeometric calculator for these odds, along with some manual calculations, and I would love for someone else to check all of my numbers to make sure I'm right. I sent an example of one situation to @AdmiralGT, who was kind enough to double check and verify my work was correct, so big thanks to him for making sure I was on the right track! The rest of my math uses the same type of setup as the one example I sent to AdmiralGT, so in theory everything should be accurate, but again I'd love this to be double checked in it's entirety.
Also, these are very generalized odds, and of course it will depend upon what has been played, aced, discarded, etc. This is just meant to be a very rough calculator on the odds of various situations. These numbers are also rounded, and when multiple rounded numbers are used for further calculations, they can be further impacted. Safe to say, however, that these should all be accurate with 1-2%, which is just fine for our purposes!
There is a whole lot of various cheatin' punishment in this game, from actions to goods to various traits on all types of cards, but the two I want to focus on are This'll Hurt in the Mornin' and Bottom Dealin. If you've been playing this game awhile, you probably know that if you are playing against a very tightly structured deck, This'll Hurt will not be as good. The looser your opponents structure, the better your odds are of them seeing an off value in one of those 2 cards. Likewise if you, yourself, are playing a very tight structured deck, Bottom Dealin will not be as effective. If your structure is particularly loose there is a good chance your opponent will be dealt a low rank hand. Most other cheatin' punishment is more straight forward, with a number of ranks added or subtracted, or various other concrete in-game effects. These 2 cheatin' resolutions, however, are the most unique in their randomized effects with drawing multiple cards.
So let's get right into it: exactly just how much will this hurt in the mornin? For our first example, we are facing a deck with the tightest structure possible - 16x3, with only 4 dudes in the starting posse, and 2 regular jokers, totaling 50 cards in the deck/discard after starting dudes. Their structure is 2s, 3s, and 4's, and they just revealed a cheatin full house: 2,2,3,3,3. We hit them with This'll Hurt, taking away 2 of their 3's, and here are the odds:
*They draw all 2's or 3's (ie: zero 4's), making at least a full house again, or possibly higher: 40%
Wow, close to half the time they will be either unaffected or even gain a rank when they are hit with This'll Hurt! Those are awful odds, but realistically that sort of tightly structured deck is rare so let's try those odds again with a more average 15/15/10 structured deck: 15x2's, 15x3's, and 10x4's, with 5 starting dudes, 2 regular jokers, and 7 off value cards. Let's do 2 examples in this situation, one in which their full house includes 4's (the lesser value) and one without.
With a cheatin' draw hand of 3,3,3,4,4 we will take away 2 of the 3's with This'll Hurt:
*Odds they make at least a full house or higher again: 24.4%
Yikes, even with a more average structured deck with 7 off value cards using a hand with one of their lesser values, they still have one-in-four chance to either not be affected or get even higher on their full house.
With a cheatin' draw hand of 2,2,2,3,3 we will take away 2 of the 2's with This'll Hurt:
*Odds they make at least a full house (or higher) again: 37.1%
Ouch, those odds are not great for the player of This'll Hurt, which makes sense because in this example it's very similar to the 16x3 sample since we have both of our strong values in the hand.
The odds using a cheatin' 4 of a kind are almost identical, with the caveat that our odds calculate the chance of a full house or higher, so in those examples if they pull a full house they get dropped by 1 rank. I don't want to calculate the odds of cheatin full house vs cheatin 4 of a kind, since they are so similar. There's just a slightly added chance they'll be dropped by 1 rank with a cheatin 4 of a kind compared to a cheatin full house, otherwise the odds are nearly the same.
What about cheatin' 5 of a kind? Let's use the same 2 decks above, one super-tight 16x3 values and the other 15/15/10. With the 16x3, here are the odds with a hand of 2,2,2,2,2 (doesn't matter which value we choose):
*They pull 2 MORE 2's, staying at 5 of a kind: 7.9%
*They pull one 2, making 4 of a kind: 42%
*They pull 2 3's, or 2 4's, making a full house: 28.6%
Which, if you calculate that (warning: you cannot simply add all those percentages up for a final total, there is an equation that needs to be used) makes a 62% they will end up with a rank 7 or better, and therefore a 38% chance of being knocked down to a three of a kind (the lowest they can go).
Ok, that might look shocking at first, but is actually a lot more respectable. Odds are very much against them retaining a 5 of a kind, and are pretty good that they will at least be dropped to a rank 7 or 8. That's a 2 or 3 rank drop on average, and more than a third of the time will be a 6 rank drop. This seems to be where This'll Hurt shines the most, so keep that in mind the next time you are in this situation against an extremely tightly structured deck. If they pull a cheatin full house or 4 of a kind, you really need to weigh the odds. If you are currently ahead of them in rank, even if only by 1, you might want to save that cheatin res for another shootout if you expect one. Or, if you are behind in ranks (or have 2 This'll Hurts in your hand), it is probably much more worth the risk to try and save casualties on your side.
What about cheatin 5 of a kind with the 15x2's/15x3's/10x4's deck? One hand of 2,2,2,2,2:
*They pull 2 MORE 2's, staying at 5 of a kind: 6.7%
*They pull 1 more 2, making 4 of a kind: 40%
*They pull 2 3's or 2 4's, making a full house: 19.5%
Totaled/calculated makes for a 54.9% chance at a full house or higher, and a 45.1% chance of getting knocked down to a 3 of a kind.
Much more respectable. Against this type of deck they will see the odds of a drop of 2 ranks at 40%, a drop of 3 ranks is 19.5%, and a drop of 6 ranks is 45.1%. Still, that's a nearly 50-50 shot that they will stay at a rank 7 or higher, but as we've seen here if you see a cheatin' 5 of a kind, you will most likely want to play those This'll Hurts in nearly every situation. If they are pulling 5 of a kinds all day long, and you happen to get rank 11, ok - maybe save it if you only have one in your hand - but otherwise slap it down and see where fate takes you. It's really only the cheatin full house and 4 of a kinds you have to weigh more heavily.
Also, and perhaps most importantly, if it's a really critical shootout and winning by an extra few ranks could be the game decider, then it's very likely worth it to play This'll Hurt even if its less than 50% chance of real success. If you had a cheatin' res that said "one-in-three chance you will instantly win the game", you'd probably play it!
Quick odds for the 15/15/10 deck drawing a cheatin 5 of a kind with their lesser value: 4,4,4,4,4
*2 MORE 4's staying at 5 of a kind: 2.8%
*1 more 4, getting a 4 of a kind: 29.9%
*2 2's (13.7) or 2 3's(), making a full house: 25.5%
Total odds of a full house or higher is 49.2%, with 50.8% chance of being knocked down to 3 of a kind.
You might have expected worse odds with this build, but even the lesser value 5 of a kind has a 50-50 chance of staying a rank 7 or higher.
Overall it's clear that you will somewhat regularly see less-than-desired effects from This'll Hurt, but remember two things. First, dropping them by 1-3 ranks is still respectable for a free cheatin' res. And secondly, that free cheatin' res has a solid chance to drop them by 4-6 ranks as well, which can be a huge swing in your favor. There is always an element of chance in Doomtown, and sometimes unfortunately you will be on the short end of that stick. That is simply part of the game, though: you have to be able to weigh the risks with the rewards. Sometimes the bold decision brings you to a sweeping victory, and sometimes it leads to a crushing defeat. Over time and multiple games, however, you will be rewarded for making the right choices.
Remember this also doesn't take into account This'll Hurts other effects. There is an added bonus of them having to pay you GR, or their cards get aced, if your hand is legal. This can be especially devastating if used on certain key cards in the opponents deck, or might be a nice shift in GR.
This case is a more straight forward in guessing how it will end up, since we simply discard their draw hand, and give them the top 5 from our deck. It also doubles as a nice odds estimator for how your lowball hands will look. In this situation we will see what the difference is when we use the 16x3 deck compared to the 15/15/10 deck. I will only do the odds for full house, 4 of a kind, and 5 of a kind. Straights, flushes, and straight flushes will typically be very low in their odds or simply not possible for these types of structured decks, and generally not worth consideration for the purposes of this article.
Again, this assumes a fresh deck with no draw hand, no play hand, and nothing on the board (with This'll Hurt, at least we were able to calculate using an existing draw hand...). This works fine if we assume values are evenly distributed in those 3 areas, and obviously tilts the odds the more a certain value is removed from the deck.
If you are playing the 16x3 deck, the odds you'll give your opponent a:
*5 of a kind: 0.6%
*4 of a kind: 8.5%
*full house: 34.4%
For a total of 40.3% odds of giving them a rank 7 or higher.
Those are some pretty bad odds, and is a good reason you probably should look for a different cheatin' res other than Bottom Dealin in your tightly structured deck.
Heck, let's keep going because I'm curious:
*3 of a kind: 38.2%
*2 pair: 72.5%
(and it's impossible to get lower than 2 pair)
Combine those two, for a total of 83% odds they'll get a 3 of a kind or less. 2 pair is clearly the most common individual chance, but the cumulative of the odds say just under half the time they'll get a rank 7 or better. Not good, and don't expect to be winning many lowballs.
With the 15/15/10 deck, your odds of giving the opponent a:
*5 of a kind: 0.3%
*4 of a kind: 5.2%
*full house: 18%
For a cumulative of 22.5% giving them a rank 7 or higher. Continuing, because I haven't done nearly enough math by this point:
*3 of a kind: 28.5%
*2 pair: 43.2%
*1 pair: Ok, my brain shut down on this one, this is too complicated and time consuming to calculate! Extrapolating from the above data, I would (very roughly) estimate the combination of 1 pair and high card around 25-30%.
Ok, these odds seem much more favorable, but you'll still be stung around once in 5 times as you give them a rank 7 or higher. Still, those are good enough odds that you could realistically never see that happen in a standard, smaller tournament of 3-4 rounds. Or, then again, you could see it a couple times in one (poor luck) game.
If you've ever played Bottom Dealin in a straight flush structured deck, you know it is absolutely and reliably brutal! In my Putting the Pieces Together deck using a SF structure, it was extremely common to give my cheatin opponent a rank 1 or 2. There are a few common types of SF structures: Clubs only (common with some starting Judge and Desolation Row decks, and also my 108 WD PTPT deck... RIP), clubs + one value numbering around 14-16 in the deck to fall back on for full houses/4oaKs/5oaKs, and finally clubs + 2 values of around 14-16 each. We might assume the pure clubs structure is where Bottom Dealin shines the most, but let's compare the three and find out.
First we have a SF structure with 15x2's, 16x3, and 24 clubs, values 2 through 7 at x4 each. 2 regular jokers are in each example, and in fact all deck examples in this article. In looking through dtdb.co this seems to be the most popular SF structure. Chances of us bottom dealin a:
*5 of a kind: 0.4%
*4 of a kind: 3.4%
*full house: 8.9%
*flush (or SF): 2.2%
***side calculation, because I was curious: if you have 2 stud rating (seeing 7 cards), your chance of seeing a flush in a shootout is only 19.2%. Ouch! If you are trying for a consistent straight flush, or even flush, you'll have to do a lot of card pruning to get a reliable chance, either by acing cards or putting them in play.
I don't have the time or patience to calculate straight flush, so those results are included in the flush percentage. A rank 6 or above might be considered 'bad', so the totality of the above odds calculate at 16.1%. That is a decently low chance to give them a rank 6 or higher, so this structure is clearly superior to either of the 3-value structures above as far as Bottom Dealin is concerned (not surprisingly).
Next we have a SF structure with 15xA's, 30 clubs, and 5 off value cards (example taken directly from this extremely strong deck by @Khudzlin http://dtdb.co/en/decklist/2279/flush-desires). This one starts 6 cards (hooray for Xiaodan), so it's only 1 card off a typical example from a non-108 outfit. Chance of a:
*5 of a kind: 0.2%
*4 of a kind: 2.8%
*full house: (Only one strong value in the deck, so this is very close to zero)
*flush (or SF): 8.3%
***side calculation - chance for a flush/straight flush with 2 stud (drawing 7 cards): 46.9% Still not great. What about 4 stud (9 cards) plus one draw, which is always what Bai Yang is along with xiaodan, assuming no bullet increase or reduction: 89.9% That's more like it!
Calculated total probability of a rank 6 or higher: 11% Bottom Dealin is doing some serious work in this deck, and it's no wonder a rank 1-2 is not uncommon when being punished by this deck. For me, the hardest decision in a SF deck with 2's, is how many Muggings do I really need? Because I always want as many Bottom Dealin's as possible, but if shotguns, soul blasts, and holsters are real problems then you may want to throw a couple in and add in some (admittedly lesser) cheatin punishment in another value. Coachwhip often fits the bill, in that case.
And lastly, we have a more 'pure clubs' deck with 36 clubs, and the other 11 are off value, although likely built with supporting straight/straight flush in mind (Comin Up Roses is a likely inclusion). Here we only have to the the flush/straight flush chances, as any full house, 4oak, or 5oak's are so low they may as well be zero for rounding and our purposes. Straights are more possible, but still extremely low, and not a terrible result for a bottom dealin, especially if you hit a cheatin 5oak (5 ranks lower) .
*Chance of bottom dealin a flush/SF: 19.8%
I wasn't sure what to choose for the number of clubs here, as these sorts of decks are somewhat rare. The 36 clubs + 11 off value I thought was a middle ground, but let's take a couple of different examples from dtdb. First we have the extreme 'all in' clubs from @Jayjester, using 40 clubs and 7 off value cards. http://dtdb.co/en/decklist/2294/all-in-judgement
Chance of Bottom Dealin a flush/SF: 34.5%
Wow, that's high, and probably a good reason that Bottom Dealin is not in this deck. The next example is the other side of the coin, with my PTPT 108 WD deck I mentioned earlier (http://dtdb.co/en/decklist/2276/putting-108-flushes-together-1st-octgn-league-1st-wi-sheriff). It has only 28 clubs and 18 off value cards all meant to support the straight flush structure using Comin Up Roses. The chance for a straight is the highest in this deck, but it is still so low it's negligible, and it also has the lowest chance of the off values giving a full house, 4oak, or 5oak (nearly impossible).
Chance of Bottom Dealin a flush/SF: 5.2%
Wow, no wonder I always saw rank 1 (maybe 2 tops) whenever using this card! I honestly didn't even need the Worldly Desires ability to win lowball, but it was in there primarily to avoid cheating. That is why I've posted on it's deck list that I really wanted to have more Bottom Dealin's in the deck, but it was so weak to shotguns, etc, that I had to include them even though this was easily the most brutal cheatin punishment you'll ever see. I had plenty of other cheatin' resolutions to make up for it, but I was always extra confident against cheating when holding that card. Rest in peace, deck - you were easily my most fun, most unique, most winning, and greatest creation of a deck I'll ever make.
So of all 5 of these straight flush decks, my PTPT deck clearly had the best odds for Bottom Dealin. After that, though, it was a bit surprising to me that the next best deck was Khudzlins 30 clubs, 15 aces deck at 11% giving away a rank 6 or higher. I would have thought the 'pure clubs' with 36 clubs would be next, but it makes sense that their odds of a top decked flush is a lot higher. The classic 15/16 + 24 club structure was very respectable with only a 16.1% chance to give the opponent rank 6 or higher, and beats the odds of using This'll Hurt as your cheatin' punishment against the 2 types of decks we illustrated.
Summary (sorry for such a long article!)
I had no idea how complicated this math would be when I first set upon this, but I hope this is useful or at least interesting knowledge. This game is incredibly deep, and there are a whole host of other factors to consider other than simple odds. As one quick example, if your opponent reveals a cheatin full house or 4 of a kind, you may let it pass for a whole host of reasons, such as not caring about a tied rank casualty and wanting to feign that you don't have a cheatin' res in hand so you can really surprise-punish them in a consecutive round (you can really lay it on, with a comment like "bah, I ain't got nothin! pass"). Despite all of this, I feel having these odds is just one more bit of knowledge you can tuck away, and it will make that moment when you must decide what to do even that much more informed.