I Solved Yahtzee*

[00:00:01]What happens when you roll a dieice?

[00:00:03]Okay, silly question. We all know that there are six things that can happen.

[00:00:06]The numbers 1 through six, all equally likely. Slightly better question. What happens when you roll five dice? As you increase the number of dice, the total goes from 6 to 21 [music] to 56 to 126 and finally to 252 total [music] outcomes, although some are much more likely than others. So, what happens when you play a game of yatsi? Each turn, you roll the dice three times for 756 total [music] situations you can find yourself in in a given turn.

[00:00:34]Combine that with over 8,000 combinations of boxes on your card that you can fill out. Each of which can [music] have as many as billions of possible score combinations, and you get over 385 billion ways to fill out a [music] Yatsi scorecard. Multiply these numbers together, throwing out the cases where the game is already over, and you get 259 trillion possible positions you might find yourself in in a game of gatsi. And that's not even thinking about what your opponents are doing. But what if I told you that despite all of that, a computer can play this game perfectly? And you and I can come pretty close. To figure out how we can play Yatsi like a computer, we'll have to think like a computer. Starting from the end of the game and working our way back, we'll have to think about not only the best way to roll a small straight, but also the best ways to think ahead and keep our options open. We'll see the surprising ways in which some boxes are a lot more valuable than others. We'll learn the one thing you should never do on your first turn. And by the end, we'll know what it takes to beat the entire human race in a winner takes all game of Yatsi. So, without further ado, let's get started. First, a refresher on the rules of Yatsi. You have five dice which you have to roll in various combinations to fill in 13 spots on the scorecard. Each box has to be filled in exactly once during the game. So every game of Yatsi lasts 13 rounds. Each round you roll all five dice. Once you do that, then you can choose some dice to keep and roll the rest. Then you can reroll a second time again choosing whichever dice you want to keep and roll. Once you finish three rolls, you have to fill in one of the empty boxes on the scorecard with whatever you have.

[00:02:09]You could fill in a box for something you don't have on your dice, but then you have to put a zero. The 13 boxes on the card work as follows. The top six boxes are for ones through sixes. You can get as many of a particular number as you can, and then you add up the dice of just that number to get points for that box. There's also a bonus for the top section. If you get at least 63 points in the numbered boxes, then you get a 35 point bonus added to your score. The bottom section has a bunch of specialized combinations. There's three of a kind and four of a kind where you need well three or four of a kind. If you get one of these, you get points equal to the sum of all of the dice.

[00:02:45]Next is full house. Like a full house in poker, you need three of one number and two of another. This gives you 25 points. Then we have small straight.

[00:02:54]Four numbers in a row worth 30 points.

[00:02:55]And large straight, five numbers in a row worth 40 points. Finally, there's the big one, the yatsi. If you get five of the same number, you get 50 points.

[00:03:04]There's also a chance box at the bottom.

[00:03:06]You can fill this in with the sum of your dice regardless of what's on them.

[00:03:11]In addition to being the most valuable turn in the game, the yatsi is also special for another reason. It's the only thing you're allowed to get multiple times. If you get a yatsi after the yatsi box is already filled in, then you get to fill in a different box according to some weird rules we won't get into. If you fill in the full house, small straight or large straight using these rules, you get the full points, even though your five of a kind wouldn't normally qualify. If you filled in the yatsi box with a 50 and not a zero, then you get a 100 point bonus for every additional yatsi that you roll. Now, my first video was about Battleship. Yatsi is a very different game, but with a few key similarities. For one thing, both are basically one-player games disguised as multiplayer games. You spend most of your time working on your own, and then you compare your performance to your opponent at the end. The other is that when I told people that I was working on a strategy video, people were surprised to learn that the game had much strategy at all. How are you supposed to make decisions or plans when your results depend so much on some random dice roll?

[00:04:05]It turns out there's a very involved strategy to this game and it starts with some calculations. If you're just here to crush your family members at Yatsi, feel free to skip ahead to the timestamp down here. But I think the math is cool and this is my video. So the rest of you are stuck in here with me for the next few minutes. To understand yatsi, we first have to understand dice. When you roll five dice, the number of outcomes is 6 * 6 * 6 * 6 * 6 or 7,776.

[00:04:34]But the number of different outcomes is a lot less, only 252 since we only care about how many of each number show up and not which die gives which number.

[00:04:43]That doesn't mean that each of the outcomes is equally likely. For cases with all dice the same or just a yatsi, there's only one way for that outcome to occur. For cases with all dice different, including both large straits, there are a lot more ways, 120 to be exact. This means that you are 40 times more likely to get a large straight on your first roll than you are to get a yatsi, even though there are six ways to get a yatsi, but only two ways to get a large straight. Other combinations of dice can have different frequency.

[00:05:12]Rollles definitely change the math around which of these outcomes are the most likely, but the game strategy is going to depend a lot on which of these outcomes can be achieved in the most ways.

[00:05:23]Now, first a few caveats on that perfect play thing that I mentioned. I don't mean that we can get the best possible score every time. We still have to rely on the dice after all. I also don't mean that we're optimizing for beating certain opponents, although we'll talk about how to do that later. Instead, we're using a computer science technique called dynamic programming to make every possible decision in a way that maximizes our average points over the course of the game. If you're worried that you accidentally clicked on a video trying to teach computers to play Yatsi, don't worry. This is just what we need to do to figure out good strategies for humans to use. So, here's the plan. We start at the end of the game and work backwards. After your last roll on your last turn, you just fill in whichever box is open and you get some points for it. So, what do we do on the second roll of our last turn? Well, once we choose which dice to roll, we can figure out the chances of every possible outcome after the roll. We take these probabilities, multiply by the points we get in each case, and add the results together to get the average number of points we'll get if we reroll those specific dice. We can do this for every possible group of dice to roll. And whichever gives us the biggest number is the group that we should reoll. And if you're not an expert at calculating probabilities in your head, then again, that's mostly what the computer is for.

[00:06:35]Once we've done this for every possible position we could be in before the second roll, we can go back a step to the first roll. For every possible roll of the dice, we see how many points we get, but instead of using the final points, we use the average points that we just calculated from our second reroll. We again choose the dice to roll to maximize the average points, and we keep track of what that average is. We can then average over all fossil first rolls to get the expected score before the beginning of the last turn. From there, we can keep going back to the previous turn. Now, we might have a decision to make at the end of the turn for which box to fill in, but we can make that decision the same way as before. We get the average points for each possible decision. In this case, that's the points we get for this turn plus the average points starting next turn and make the decision with the highest average. We then do the same thing for the second roll of turn 12, then the first roll of turn 12, then go back to turn 11 and keep going. At every step, we get the average points until the end of the game for every possible position that we can be in so that we're ready to do the calculations when we have to calculate the previous step.

[00:07:39]Once we get back to the beginning of the game, we'll have calculated a full strategy.

[00:07:44]Now, you may remember from the beginning of this video that the number of possible positions in a game of yatsi is 259 trillion. If you're not familiar with programming, that number probably sounds impossibly large. If you're a bit familiar with programming, you know that giant numbers are much more manageable.

[00:07:58]If you're more familiar with programming, you know that 259 trillion is manageable if you happen to have access to a supercomput. I do not have access to a supercomput. I am an aspiring YouTuber who tried to buy a desktop when people were hoarding RAM like it was toilet paper in 2020. But it turns out that if we're careful, we can simplify the problem a lot because a lot of these positions might as well be the same.

[00:08:19]Take a look at these two scorecards. In both positions, we have the same boxes filled in. We also have 33 points total and nine points in the top section, meaning we need another 54 points to get the top section bonus. For our purposes, these might as well be the same scorecard. Nothing about our score or our strategy changes between the two.

[00:08:37]So, it turns out that we can get all the information we care about for our yatsi card from just a few pieces of information. Which boxes are filled in, our score in the top section for figuring out the bonus, our score in the bottom section, and the number of yatsis we've gotten so far. This reduces the number of positions substantially all the way down to 80 billion. Now we're in the realm of numbers computers can deal with eventually. I was able to generate all the positions, but actually calculating anything took days at a time. So I had to simplify further.

[00:09:06]Thankfully, it turns out that if we're only trying to maximize points, there's a lot more information we can throw out.

[00:09:12]Because we're calculating our average scores by thinking into the future, we only need to think about the parts of our card that affect our ability to get future points.

[00:09:20]We still care about which boxes are filled in, but we can throw out a lot of other information. We don't need the number of yatsis. We just need to know whether we are eligible for future 100 point bonuses. We only care about the top section score if it's less than 63.

[00:09:34]Once we get the bonus, we don't care if we have 63 points or 93. The only thing that affects our future points is which boxes are still open. We don't need the bottom section score at all. Once we know which boxes are open, we have all the information we need. Once we make these simplifications, our number of positions goes from 80 billion all the way down to 405 million. And suddenly, we can get everything we need to figure out how to play the best possible game of yatsi. That starts with our average number of points per game if we play perfectly, just a bit under 255.

[00:10:07]Now that we've talked through where a strategy is coming from, we can get into the strategy itself. We're going to take a similar approach to the computer.

[00:10:12]Start at the end and work our way back.

[00:10:15]It's much easier to think about what to do at the end of the game when there are fewer things we're trying to do at once.

[00:10:19]Then once we've talked through the later stages of the game, it will be much easier to make sense of why we're making certain decisions earlier on. Fair warning, there's a lot here. We could probably study everything in this video and memorize a bunch of cases, but I'm mostly hoping that people will take away a better feel for the game and the decision-m skills that go with it. With that in mind, let's start at the end. On the last turn of the game, we'll be going all out trying to fill in one specific box. Let's figure out what we need to do for each one. Starting with the easiest cases, we're going to ignore bonuses that you get for extra yatsis and just focus on getting the thing in each box. If your last box is in the top section, the strategy is straightforward. Keep every die with the number that you need. Reroll everything else. After three rolls, you'll get about two of your given number on average. Here's how often you'll get each number of your given value. If you need two, you're in decent shape. If you need three or more, you'll need some luck. The strategy for getting a yatsi is pretty similar. Just keep whichever number you have the most of and reroll the rest. If there's a tie, just choose which value to keep. Notice that this might change from your first roll to your second. Unfortunately, your chances of actually getting a yatsi are less than 5%. So hopefully you're not counting on this to win. Now, you'd think that the next simplest cases are three and four of a kind, but they're actually more complicated for weird reasons. We'll get to them later.

[00:11:33]Full house is pretty straightforward.

[00:11:35]You should keep pretty much any groups of dice that you have. If you get two pairs, three of a kind, or a single pair, you should keep those dice and reroll the rest. If you get four of a kind, you should keep three of them and reroll the other two. The chance of successfully filling in the full house box on your last turn is about 1 and three.

[00:11:51]Getting a bit more complicated is the large straight. There are two ways to get a large straight. 1 through 5 or 2 through six. Or to put it another way, you need 2 3 4 5 and either one or six.

[00:12:02]The strategy looks like this. If you roll a 2, three, four, or five, then keep one of each. For one or six, it's a bit more complicated. You should never keep more than one, but if it allows you to keep at least four dice, then you should keep a one or a six. If you would be keeping three or fewer dice, then you're better off re-rolling all of your ones and sixes because the increase in combinations helps more than keeping the extra die. Doing this strategy will get you the large straight box about a quarter of the time.

[00:12:27]The small straight adds more complexity.

[00:12:30]There are now three ways to get it. 1 through 4, 2 through 5, and 3 through six. Plus, there's an extra die that doesn't get used, adding to the number of combinations. You still always need a three and a four, but the other dice are more complicated. You should reroll everything if you don't have a three or four. If you do have threes or fours, you should keep one of each, as well as one each of your twos and fives if you have any. You should keep a one if you have a one and two, but no five, and you should keep a six if you have a five and six, but no two. Your chances of filling the small straight box if you do this are pretty good. That's why small freight is one of the better boxes to leave around until the end. The next one is fun from a math perspective, chance.

[00:13:06]Since we only care about getting the highest total possible, we can treat each die individually. How do we decide which ones to reoll? In the spirit of doing things backwards, let's do the second reroll first. If you roll a dieice, the average value that you get is 3 and 1/2. This helps us to think about what to roll on our second roll.

[00:13:22]For each die above 3 and 1/2, it would get lower on average if we rerolled it.

[00:13:26]For each die below three and a half, it would get higher on average. So, we should reroll the ones, twos, and threes and keep the fours, fives, and sixes.

[00:13:33]Averaging all these out gives 4.25 points on average. Why do we care about this number? Well, we need it to figure out what to do on our first roll. After one roll, we know that any die we roll will have two chances to give us a higher number. This means that any die we roll will now give us that 4.25 on average instead of just three and a half. This means that we now want to roll the four as well as the 1 2 3 and keep the five and the six. So, some complicated math gives us a simple strategy. Keep fives and sixes on your first roll and four fives and sixes on your second. Doing it this way gives you an average of four and 2/3 per die or 23 and a3 total for chance. That brings up our most complicated cases, the three and four of a kind. At first, they seem like they should be simple. Just try for as many of a number as possible. And if your only goal is to put something in those boxes, then it is that easy. We do basically the same thing as if we were trying for a yatsi. But if we want to play perfectly, these boxes are a little weird. The score for these boxes is the sum of the dice. Because of that, they combine the all or nothing nature of full house yatsi in the straits with the score maximizing of chance. To get this right, we have to combine the two approaches in sometimes counterintuitive ways. Starting with the four of a kind, the last rule is mostly what you expect.

[00:14:45]Keep whatever you have the most of. If you already have a four of a kind, then you reroll your last die if it's three or less to try to get more points. The first roll, however, is a bit strange.

[00:14:54]If you have four of a kind, you should keep it and roll the other die if it's a five. But otherwise, it depends a lot on which numbers you have the most of.

[00:15:01]Because our points come from the sum of the dice, we care about the chance of success, but also the value of those dice if we succeed. Because of that, we sometimes want fewer larger dice over more smaller ones. If we have, for example, three 1's and two sixes, our chances of success will be highest from keeping the ones. However, we will get significantly fewer points on average if we get four 1's compared to four sixes.

[00:15:24]Because of this, the average number of points that we get overall is actually higher if we keep the sixes. In fact, there are several cases where it's best to choose fewer of a larger number instead of more of a smaller one. Even though we're not quite trying to maximize her chances, we end up with a 28% chance of getting four of a kind with an average score of about six points. Three of a kind is similar. On the second reroll, just keep whatever you have the most of. If you already have three of a kind, we roll the other two dice if they're less than four. On the first roll, we again usually want to keep the number we have the most of, but there are a lot of exceptions. Using this strategy gives us a 71% chance of getting three of a kind and an average of 15 points. Now, that was a lot for just the last turn of the game, but hopefully it helps with a few things.

[00:16:06]First, these strategies can work on any turn, not just the last one. Earlier on, we often care more about keeping our options open and having a backup plan if our strategy doesn't work out. But once you decide what box you're going for, the strategies work the same way at any time of the game. Second, this hopefully gives a sense of which boxes are easy or hard, which helps us to think about what we should focus on earlier in the game.

[00:16:26]That brings us to the middle game, which focuses on which boxes to prioritize as our options start to narrow. We talked about how the average points using the strategy is 255, but that doesn't tell the whole story. [music] Here's a histogram of how often every score is achieved using our strategy. This plot has a lot more peaks and valleys than we might expect. And it turns out [music] that understanding these peaks and valleys is key to understanding the strategy here. Where exactly do these come from? Well, the two on the right are more straightforward. This one comes from getting a 100 point yatsi bonus, and this one comes from getting two.

[00:16:56]There are a lot more bumps like these from getting more and more yatsis, but the rest are too small to see on this plot. But what about the rest of these bumps? [music] Why do we get all these little peaks with the pair of big peaks in the middle? That comes down to what I'm going to call the big bonuses and little bonuses in this game. The top section boxes and chance on their own are unlikely to make or break your game.

[00:17:17]The difference between a good and a bad score in those boxes is usually just a few points. But there are eight bonuses in this game that are sort of all or nothing, meaning that getting one or not can really swing the game. These bonuses are actually so important that I'm going to create a new point system just to talk about them. The biggest is the 100 point yatsi bonus. This is so massive that it belongs in its own category worth four bonus points in our system.

[00:17:39]If you get one of these and your opponents don't, you're very likely to win. But they're also rare, so it's not usually worth it to try for these. But the rest can be divided into big bonuses and small bonuses. [music] The big ones worth two points each are the top section bonus, large straight, and yatsi. And the small ones worth one point each are three of a kind, four of a [music] kind, bullhouse, and small straight. Why did I divide them up like this? That'll make more sense when we go back [music] to our graph. Before we move on, let's take a look at our chances of getting each of these with our perfect strategy along with the average number of points that we get from each one. Yatsis, as expected, are pretty hard, while the straits are pretty easy. In fact, we should be getting the small straight almost every game. These are so easy that the average points from the small straight is actually higher than the yatsi and the top bonus. Three and four of a kind give us surprisingly few points, even though we saw before that they're not that hard. We'll see why that is in a bit.

[00:18:33]So, why did I divide up the bonuses this way? Even though the small freight is worth more on average than almost anything else in the game, it's because of what happens if you miss one of these bonuses. Missing a big bonus is about twice as bad as missing a small one.

[00:18:46]Let's go back to the [music] plot. We already talked about those bumps on the right, but what about the others? These peaks [music] near the right come from the games where we don't get an extra 100 point yatsi, but we do get all the big bonuses. In particular, this last big peak here comes from us getting every single one of our regular bonuses.

[00:19:02]The peak next to it comes from getting everything except for one of the small bonuses, usually four of a kind. The big peaks are mostly for missing the yatsi.

[00:19:10]This one comes [music] from missing two bonus points, either one big bonus or two small ones. And this one comes from missing three bonus points, [music] one big bonus and one small one or three small ones. Then these peaks come from other combinations. This is from four bonus points. This is for missing five [music] bonus points. Then the peaks mix together after that. This tells us some very important things about our strategy. The yatsi, large straight, and top bonus are really valuable. Yatsi is hard and mostly out of our control. Even if we spent the entire game just trying to get a yatsi, we'd fail most of the time. This makes the large straight and top bonus extremely important to our success in the game. Sometimes meaning that it's worth it to zero out one of the smaller bonuses if it means keeping our chances alive for the top bonus.

[00:19:53]That brings up an interesting question.

[00:19:54]How do we decide which boxes to prioritize as we go throughout the game?

[00:19:58]Which ones do we sacrifice when roles don't go our way? Ideally, we'll do a good job of keeping interruptions open so that we don't have to do this very often, but even if we play perfectly, we sometimes can't avoid it. Is it a lowcoring box like ones or maybe a hard one to get like yatsi? Just like an aspiring YouTuber who's realizing no one's going to watch a 40-minute video on yatsi. Sometimes figuring out what to cut is the key to keeping our hopes alive. So, using our model from before, we can start to answer this question. If we're going to have to zero out a box, we want it to be the one that probably wouldn't be giving us at any points anyway. Here's a plot that might help.

[00:20:32]This is a plot of the round number versus average points for each box given that the box hadn't been filled by that round. To start with an example, here's a small straight over the course of the game. We're almost guaranteed to get the small straight at some point. So, it's worth almost 29 12 points out of a possible 30 at the beginning of the game. But the longer it goes unfilled, the bigger the chances that we won't be able to fill the box. If we get to the last turn and are still waiting on a small straight, our odds are much lower.

[00:20:59]So, we only get about 18 points on average. Now, we can fill in this chart with the rest of our bonuses. The yatsi line also includes points from the 100 point bonuses since we'd also be giving those up if we zeroed out the yatsi.

[00:21:11]With slight exceptions, this chart tells us what is safest to zero out. Whatever appears lowest on this chart for the turn that we're on is generally the best to zero out.

[00:21:20]And rather than memorizing this full chart, we can notice a few patterns. The safest bet to zero out changes halfway through the game. It's four of a kind for the first half, but yatsi for the second half since yatsi is worth more, but also harder to get. Similar things also happen at the top of the chart.

[00:21:36]Large straight starts out at the top, but it's a weird case. There's a decent chance it will come up on its own, but it's actually hard to get if you're still trying. small straight ends up at the top as the last thing to get rid of because you still have a pretty good chance of getting the 30 points even if it sticks around until the last turn.

[00:21:53]But the big takeaways here are that you should feel safest zeroing out four of a kind or yatsi and the last ones to try to keep around at the end of the game are the small straight followed by three of a kind and large straight. The other big thing to focus on during the middle of the game is the top bonus. It's one of our three big bonuses and probably the one we have the most control over with about a 2/3 chance of succeeding.

[00:22:14]Since it takes six turns to complete, it should probably be one of the main focuses of the game. 63 points may seem like a strange number to choose for the bonus, but there's a good reason for it.

[00:22:23]It's exactly what you get if you add together three of each number. This gives a pretty clear approach to the strategy for the top bonus. Just try to get three of everything. Fours are great. Twos and below are quite bad. We want to aim for three, but if you get four of a higher number, it can give you the freedom to get less of a lower number or even to zero out the ones box if you need to later in the game. It's often helpful to track how far ahead or behind you are compared to getting three of everything on the boxes you filled in so far. Here's a table that might help.

[00:22:53]On average, we get 24 points from the top bonus. If on the first turn, we fill in three of any number, our odds basically don't change. If we fill in a four, our odds go way up, especially if we're filling in four of a larger number. But filling in twos or less can ruin our chances very quickly. Filling in two ones won't do much, but two of a larger number can really hurt your chances. Putting a 12 in the six box immediately plummets your average score by over 17 points. So, you're not only losing six points from that box, you're effectively losing another 17 points worth of bonus. And zeroing out one of the top boxes can also have a huge impact on your chances. Ones and twos are harmful, but threes will make things much harder, and anything higher is basically giving up on the bonus. This gives some useful information. Zeroing these out early in the game can be very harmful, making it better to zero out four of a kind or yatsi instead. Later in the game, if you already know whether you'll get the bonus, then zeroing out the ones or twos can often be the safest play. Now that we have a better feel for the game, what is important, what we should focus on, and what we can leave out, we can get back to the beginning.

[00:23:56]We'll take a look at the turn that's hardest to think about but easiest to plan for the first one. Let's use what we know and our computer model to map out our first turn. What are our best outcomes for turn one? The best turn, not surprisingly, is a yatsi. But what about the second best? What would you think? Is it a big bonus like a large straight or maybe something we're less likely to get like four of a kind? Or maybe something that will help us to get the top bonus? It turns out it's the last one. The second best outcome for the first turn is four sixes, putting a 24 in the six box. After that come four fives, large straight, four fours, and four threes. Now, if you've ever studied a game like chess, you know that opening theory can quickly turn into loads of memorization. So, while I'll throw the details into my blog post linked in the description, I'm going to try to just focus on the general principles here.

[00:24:42]Getting four of a number is great, and three of a number is good enough. Full houses are really good, too, but you'll probably get them by accident when you're trying to go for three of a number. So are straights, but they're usually not the best things to try for unless you get a small straight by accident on one of your earlier rolls.

[00:24:58]Getting two or fewer of a number is really bad. You should never score two or fewer four, fives, or sixes. And you should never score one of anything but one. There's one box you should never fill in on your first turn, four of a kind. That's not because it's a bad outcome, but because you're always better off putting it in the top section instead. Three of a kind is also extremely rare and only in cases with three sixes and other numbers that are pretty high. In case you're curious, the worst possible turn one is 2 3 4 4 6.

[00:25:27]All of these numbers are too bad to write in the top box, and you just miss out on a small straight. The right move here is a 19point chance, which wastes your chance on a score that is not particularly good right off the bat.

[00:25:39]Now that we've talked through every part of the game, we can think more clearly about how to handle every box on our scorecard. We've already talked through the top section, so we'll focus on the rest. Starting at the bottom, chance is like a free space. While you'd prefer not to put a low number in this box, its real value comes from the fact that you can use it to throw away a bad turn. Try to mostly use it to avoid a bad score elsewhere. And even then, be careful.

[00:26:02]You can afford to take a lot more risks when chance is open than when it's already been used.

[00:26:07]Yatsi is the best box on the board, but it's rare. If you get it, great, but don't push your luck. Don't go for a Yatsi unless you have somewhere else you can go if you fail. You should get both straits most games. There's an 82% chance you roll a small straight on your first roll at some point during the game. So, it's usually best to just look out for one whenever you do roll. The large straight is easy to get randomly, but harder if you have to try for it. If you roll a small straight and still have both straights open, or if you roll four out of five and still have chance open, then it's often worth it to go for the large straight. Otherwise, only do it if you're prepared to zero something out.

[00:26:43]Full house also usually comes up on its own. Take it when you get it, but you can usually count on seeing one while you try for other things. Three and four of a kind are tricky since anytime you get one, it's usually best to fill in a top box instead. You should usually only go for these if you get a bunch of a number that you've already used up top.

[00:27:01]Four of a kind is also much harder than three of a kind with no extra reward. So anytime you have to choose which of these boxes to fill in or zero out, always choose four of a kind. You can probably get the three of a kind later.

[00:27:13]So hopefully that should give you a sense of how to play through the game.

[00:27:16]The dice rolling principles from the endame should help a lot on the earlier turns and the prioritization from the beginning should continue to help in later turns. Now there's one really important part of this game that we haven't talked about at all. You probably have an opponent. Now, there are a few challenges in trying to put multiplayer strategy into this video.

[00:27:33]One is that my computer was barely able to do all the computations with one player, and every additional player increases the difficulty exponentially.

[00:27:41]The other is that I don't know anything about who you're playing against. Trying to beat an optimal computer and trying to beat a human are usually very different. It's also hard to see everyone's scorecards during a game, making it harder to adjust to your opponents. Anyway, we're not going to be able to play perfectly anymore, but we can talk through some general principles that might help. The main approach, as in most games, is to play aggressively when you're behind and conservatively when you're ahead. But this raises two questions. What does it mean to play aggressively? And for that matter, what does it mean to be ahead? Just looking at people's points so far doesn't really tell you who's winning. Putting a 12 in the three box isn't a lot of points, but it's a good turn. Putting a 12 and four of a kind is pretty bad. Putting a 12 in the six box is really bad. So, how do we figure out who's ahead when there are 13 boxes that could be filled in in any combination with many different numbers?

[00:28:27]To figure this out, we're going to bring back our simplified scoring system. Four points for yatsi bonuses, two points for our big bonuses, and one point for our small ones. If two players are using our point maximizing strategy, and one beats the other by a single point on our simplified scale, there's a 97% chance that they won the game, too. This can help us evaluate our chances and figure out our strategy. To figure out how we're doing in a game, we can just look at how many bonus points each player has, as well as how easy or hard it's going to be to get the rest of the bonuses. If yatsi is empty, it will usually be a zero. If small straight is empty, it will usually still get points.

[00:29:04]For the top bonus, see if each player is ahead or behind the pace of three dice per box filled in so far. This can help figure out where you stand. So, how do we change our strategy? Well, if you're ahead, focus on sure points. Make sure to keep around easy categories like small straight and try to get good top section scores to get the top bonus. If this means zeroing out the yatsi or other hard to get boxes, that's probably okay. If you're behind, try to go big.

[00:29:30]Keep the yatsi on the board, even if that means you have to sacrifice less valuable boxes like ones or four of a kind. You'll probably lose points by doing this, but it will give you more opportunities to win the game. So, how does this change with more players?

[00:29:43]Well, that depends on what you want. If you're trying to do as well as possible, then the strategy probably doesn't change much. Just get as many points as you can, and you'll probably do better.

[00:29:51]It's probably better to be a bit more aggressive when you're behind and a bit more conservative when you're doing well, but otherwise, not much changes.

[00:29:58]But for a lot of us, that's not how we play. We want to win, and anything less will lead to hurt feelings and a flip table. So, if we want to crush our families at this game, how well do we need to do? Well, here's the plot from earlier showing how often someone gets every score if they're trying to maximize their average points. The average score here, as we mentioned, is 255, and the median is 248. So, if you're playing against one really good player, that's the number of points you'll probably need to be more likely than not to win. But what about more players? If two players are both using the strategy, we can look at the frequencies of the maximum of the two scores, which looks like this. The median goes up to 268, which means that if you're playing a three-player game against two good opponents, you can't afford to miss more than one big bonus or two small ones. [music] And even that might not be enough. With three opponents, the median is now 279, [music] meaning that you probably can't afford to miss more than a single small bonus in a four-player game against good opponents. This means that you probably need a yatsi to win, so try not to zero it out, even if it loses you points on average. With five opponents, that median goes up to 307, meaning that if you don't get every single bonus on the board or multiple yatsis, you probably won't win. This requires some luck, so you have to try as hard as you can to not miss a single bonus, even if it means taking some risks in the top section.

[00:31:14]Note that we've been assuming that our opponents are using our point maximizing strategy. In reality, most human players won't be this good. So, if you're playing against worse opponents, you can approximate that by assuming that you have fewer, better opponents instead.

[00:31:26]Here's where our analysis starts to get fun, though. Even though a score sheet only has room for six players, the box just says two plus players, so I want to keep going. With 10 opponents, you'll probably need multiple Yatsis just to have a good chance at winning. So, it's probably best to just focus on Yatsis above all else. There are 80 score sheets in a typical Yatsi box. [music] If you handed them all out and played a game with them, you'd probably need three Yatsis to win. Hasbro, the company that makes Yatsi, has about 5,000 employees. If they had a whole company picnic and decided to play Yatsi, the winner would probably have four yachts.

[00:31:59]Hasbro is based in the great state of Rhode Island, which has about a million people. If they held a mandatory statewide yatsi game, it would probably take six yatsis to win. And of course, the world has about 8.3 billion people.

[00:32:11]If every single one became an expert at yatsi and we all played a game, we would probably see someone get nine yatsis. If we were somehow able to get a larger game than that going and we were able to get 51 quintilion participants dedicated to optimal play, we would be more likely than not to see a perfect game of Yatsi, a 1575.

[00:32:31]Yatsi may be a game of chance, but it's also very much a game of skill. Beyond just thinking about which dice to roll, it also forces you to think about advanced planning and option value. If you don't want to be stuck with boxes that you can't use at the end of the game, sometimes the best thing you can do is to make luck work in your favor.

[00:32:50]Just a few quick things before I go.

[00:32:52]One, thank you all so much for watching.

[00:32:54]I was so amazed by the reaction that I got to my first video. I love talking to you all in the comments about Battleship, and I hope to be able to do the same on this video about Yatsi. Um, I also have a blog post linked in the description and in the comments where I talk about some more of the process and about some of the strategy and things that were a bit more detailed that I couldn't get into uh during the rest of the video. Thank you all so much.