Yahtzee - by Patrick Liscio - Ballpark Figures
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Yahtzee
Patrick Liscio<br>Jul 19, 2026
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My second video, on Yahtzee strategy, is out. You can find it here. This post covers some more details about strategy that I couldn’t fit in the video, as well as some interesting problems in Yahtzee strategy that I think are still open. I also made a post about the response to my Battleship video, and some of the changes I made for the new video, which you can find here.<br>In Battleship, a lot of my time was spent trying to salvage something from the increasingly complex placement strategy, ultimately delaying the video by quite a bit. For Yahtzee, I decided to avoid this by focusing on what worked: leaning into the single-player point maximization and relegating multiplayer approaches some small segments at the end. For what it’s worth, I do think a vast majority of the advantage in Yahtzee comes from point maximization and not from multiplayer adjustments. However, this post will mostly just be providing some more insight and data on single player cases, although there are some interesting open questions on multiplayer strategy at the end.<br>Thanks for reading Ballpark Figures! Subscribe for free to receive new posts and support my work.
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The Math
Especially compared to Battleship, the Yahtzee math was relatively straightforward. We just do some dynamic programming over all possible scorecard states that you can encounter in a game. The 342 billion state version doesn’t actually appear anywhere in my code, other than a standalone script designed specifically to calculate that number for the video.<br>My original code centered around the version in which you just ignore which boxes gave which numbers of points, focusing on boxes filled in, top and bottom section scores, and number of Yahtzees. I believe this is the most simplified version of our scorecard that still allows us to make all possible strategic decisions, with 105,285,166 non-terminal scorecard states. If you’re trying to maximize your chances of reaching a specific number or of beating a specific opponent, you might need to know all of this information.<br>However, the video requires us to do a lot of computation on these states. We need to know probabilities and expected points for each state, as well as various other calculations like “expected points for a box given that we haven’t filled that box by each turn number” for the line graph scene. In most cases, we can’t calculate these numbers for the start state without calculating them for every other state, so it became very important to shrink the state space to do these computations.<br>This is part of the reason why I focused almost the entire video on the point-maximizing strategy. If the only thing we care about is maximizing expected points, then you can ignore pieces of information that don’t directly affect future points, like the bottom section score. This brings us down to 536,320 scorecard states, which actually allows for efficient computation of any statistic I may need in just a few minuts.<br>The Endgame
The endgame is fun to think about because it essentially gets down to what most people think about when they play Yahtzee: how do you get one box in particular? The interesting ones here are the straights, 3 and 4 of a kind, and chance.<br>Chance is fun because it’s basically just a quant interview question showing up in the middle of a board game. 3 and 4 of a kind do a good job of illustrating this tradeoff between probability and raw numbers that make up an expected value, which I hope I got across in the video. 3 and 4 of a kind had a bunch of weird exceptions where you give up probability of success to maximize points. I briefly showed most of these examples in the video, but I’ll put the full tables with these exceptions at the bottom of this article (notice how eg a single 5 appears ahead of 2-2 in the 3 of a kind table, and how 1-1 doesn’t appear at all. There’s always something in your roll that you’d rather have than a pair of ones).<br>Getting the straight strategies into the video was one of the more fun challenges. It’s an interesting puzzle to try to come up with rules like “reroll everything if you have no 3s or 4s” and “only keep a 1 if you have 1 and 2 but no 5” from a table like this:
This is a table of all of the 252 possible dice rolls prior to the last reroll of a game that is only missing a small straight (and doesn’t have Yahtzee bonuses available). Notice that the expected points depend only on the dice that you choose to keep (this seems obvious in hindsight but took me a minute to realize). Here’s a simplified version, grouped by the kept dice. There are surprisingly few cases.
The way to read this chart is that you find go through the list form top to bottom until you find a subset of the dice you currently have, and then you choose those to keep.<br>One thing that adds weirdness to these tables is the strange rules around...