Prey on Normal Human Weakness by Dominating at Yahtzee

Yahtzee- A devilish game of randomness and chance, two things people are just generally terrible at evaluating.

Yahtzee is a game played with five six-sided dice and a scorepad. The object of the game is to score a higher number of points than your opponent. Points are earned according to values assigned to different arrangements of dice values, as seen on the Yahtzee score card:

I know this deviates from the usual form of illustrations on this blog, but for god’s sake, you try to draw this thing repeatedly. Not happening.

At the beginning of each turn, a player rolls all five dice. The player may then choose to re-roll any number of dice from the results in an attempt to achieve a better result. After this roll, the player may choose one more time to re-roll any number of dice. After this third roll, the player must choose a category by which to score the roll. If the roll doesn’t qualify for the category in question, the score for the roll is recorded as zero in that category.

Initial Strategy- Setting the Rules

Yahtzee, being a dice game, has a naturally high level of variance due to the random nature of the game, that is, luck plays a large part in the final result.

Assuming for the moment that your understanding of dice-statistics and general strategy for the game are more advanced than that of your opponents (and it may very well be when you finish reading this article), your aim in setting the rules for the game is reducing the variance, or reducing the role of luck in the game.

As you can see from the scorecard, by far the highest-scoring box on the card is the Yahtzee bonus, which scores 100 points for each Yahtzee beyond the first. Since an average game of Yahtzee played with optimal strategy is (only click if you are interested in reading a ton of mathematical Yahtzee analysis, score average on page 24) about 255, scoring 100 extra points in a game is an advantage that is basically impossible to overcome without repeating the feat yourself.

Consequently, it is to your advantage, as a skilled player, to make the suggestion that Yahtzee Bonuses not be used in the game. You  don’t need the bonus to win, your higher level of understanding of the game provides you with an advantage. The lesser-skilled player, however, can more easily luck you out of the game if they can score bonuses.

Nooooooooooooooooooooooooooo!

Luckily, this is a commonly agreed-to rule. Unluckily, since reducing the variance of a game frequently reduces the fun level to the unskilled player (variance in games even plays a large part in gambling addiction), players looking to play for fun may object to this rule because it makes the game less fun for them. Luckily, again, these for-fun games should be the very games in which winning a game of Yahtzee matters the least.

Beginning Strategy – The Top Section

As you may notice, the top section of the scorecard is set up to score sets of each number. There is one part of the top section that plays a huge role in Yahtzee strategy:

Booyah! Do you see the free points?

This 35 point bonus for scoring 63 (an average of three dice of each number) for the top section is a huge portion of the total score of the game, and equal to more than an extra roll for the game. Put another way, if you have 14 rolls (1 virtual roll, for scoring the top bonus) and your opponent has only 13, you will win a large number of those games.

Concentrating on completing the top section has other advantages as well. Consider a final roll for the first turn of:

So close!

Many players would play this roll on four-of-a-kind, since it is the highest possible total for that square:

Eh?

While that does, in fact, maximize the score of the 4-of-a-kind category, it would be a huge mistake.

 

No. Don’t.

By instead playing the roll as four sixes, you open up your options by a huge amount:

That’s the ticket!

While this only scores 24 points, by logging an additional 6 in the top section (four instead of the necessary 3) you would allow yourself the ability to roll one fewer 5 or 4, two fewer 3s, or NO 2s or 1s if necessary and still complete the top section and receive the bonus. This is an example that frequently comes up in games and is an example of how beginning players give up a large advantage to knowledgeable players.

This strategy is applicable to every category in the top except the 1s category, as you can’t roll enough 1s to allow you to roll less of any other number.

A Yahtzee (5 of a kind) is generally an exception to this rule. You roll 5 of one number, you score it as a Yahtzee. The two exceptions to this exception would be a case in which you need 5 fives or 5 sixes to complete a bonus on top, since 5 x 5 = 25 + 35 = 60  and 5 x 6 = 36 + 35 = 71 are both greater than 50 for a Yahtzee.

This knowledge will also provide you with a greater advantage in the middle game, which is heavily centered in:

Middle Game Strategy – Managing Chance

The ‘chance’ category is a very important and very misunderstood category on the Yahtzee scorecard. Don’t think of ‘chance’ as a place to score as much as a pure mulligan, or do-over. Sometimes (often) when you are rolling for a particular category, say, full house or large straight, you won’t be able to get the required combination. This is when you use your chance.

By optimizing your play for the top category, you can often gain an ‘additional chance’ by allowing yourself the possibility of taking less than 3 of any given number in the top category. In many cases, taking zero points on ones or twos is even justified in order to maximize your chance to get a particular category filled or, more generally, maximize your score. If you can maximize the flexibility you have to use any necessary square as a chance, I would advise that you do so.

Advanced Strategy – Basic Dice Statistics

We will cover only extremely basic dice statistics here. Many more thorough treatments are available online, for example this one and this one.

Since there are six sides on a Yahtzee die, the chance of rolling any given number on a roll of one die is 1/6.

From this premise, if you have one more roll at a Yahtzee with four sixes and one four, by rolling the four you have a 1/6 chance of completing your Yahtzee. With two rolls to go and the same dice, your chances double to 1/3 (2/6).

Rolling two dice complicates matters, naturally, and the most common case in practice is rolling two dice with three of a kind kept. To complete a full house, your chances are 1/6, since any number on one die must only be duplicated on the other die to complete it. Among these possibilities for completing a full house is a 1/36 chance of completing a Yahtzee on the next roll (1/6 x 1/6, since each die must result in the same number as the current three of a kind).

It is worth knowing, as well, that a second Yahtzee rolled in a game without Yahtzee bonuses may be scored as a full house.

In order to use these statistical concepts in play, you would simply run the numbers for two options, say, the likelihood of completing a large straight versus the likelihood of completing a four-of-a-kind and choose the more likely option.

When Basic Strategy Breaks Down – Desperation Mode

Sometimes in the course of play, you will find yourself behind by such a large amount that only rolling a Yahtzee will allow you to catch up. If you find yourself in this position, make sure that you are correct in your analysis and then roll for a Yahtzee no matter what shows up on the dice. Losing by a little by by playing the ‘correct’ way is no excuse for not trying for the win, no matter how unlikely. It is very important to make sure that you don’t go into desperation mode without knowing that you must, however, since rolling for a Yahtzee and only a Yahtzee can often result in a very poor result.
Conclusion

Simply following the strategy of completing the top section first will provide a huge advantage against the majority of the Yahtzee players you will ever face. Making sure that you don’t misuse ‘chance’ will go a long way toward shoring up the remaining matches, and an understanding of statistics will take you to a whole different level of play once you have mastered the prior two skill sets. Regardless of your skills, there will be a certain number of games of Yahtzee that you will lose. Don’t be a dick about it, and for the sake of all that is holy don’t start spouting off statistical likelihoods of particular outcomes of the dice on any particular roll in a game played for fun. This is a fast way to lose friends, unless your friends happen to be particularly interested in statistics or number theory. Chances are they aren’t.

 

Thank you very much for reading this short introduction to Yahtzee strategy. If you liked it, hated it, or were indifferent to its existence, please leave a comment. If there is a particular game you’d like to see covered in the future, make sure to mention that in the comments below as well. Thanks again and we’ll see you next Monday (hopefully!)

 

 

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