The tension in Yahtzee game strategy arises from the need to make decisions about which objective to pursue after seeing the outcome of the 1st and 2nd rolls in any given round. There are so many possible situations in Yahtzee, that I don't think anyone has ever calculated optimum strategy for the entire game. There are lots of websites with general strategy principles/rules but some of the strategy decisions when you have only a few rolls left still seem to be relatively unanalyzed.
So, I am starting this thread because this is the best damn forum to take some bites out of the Yahtzee strategy problem. And, if nothing else, we can pose some interesting problems for the math geeks on this forum to analyze.
To get us started, here is some information I've calculated on what occurs on the initial roll of 5 dice.
Initial Roll Category | Permutations | Probability |
---|---|---|
Yahtzee (5oak) | 6 | 0.0008 |
4 of a kind | 150 | 0.0193 |
Full House | 300 | 0.0386 |
3 of a kind + 2 Sing. | 1200 | 0.1543 |
2 Pair + 1 Sing. | 1800 | 0.2315 |
1 Pair +3 Sing. | 3600 | 0.4630 |
No Pair | 720 | 0.0926 |
TOTAL | 7776 | 1 |
| | |
No Pair|Long Straight | 240 | 0.0926 |
No Pair|Short Straight | 240 | 0.0926 |
| | |
66 Pair|Short Straight | 60 | 0.0077 |
55 Pair|Short Straight | 120 | 0.0154 |
44 Pair|Short Straight | 180 | 0.0231 |
33 Pair|Short Straight | 180 | 0.0231 |
22 Pair|Short Straight | 120 | 0.0154 |
11 Pair|Short Straight | 60 | 0.0077 |
Total Short Straight | 960 | 0.1235 |
And if you are chasing a single top category such as 6's, here are the probabilities I've calculated for the first roll.
First Roll | Permutations | Probability |
---|---|---|
Five 6s | 1 | 0.0001 |
Four 6s | 25 | 0.0032 |
Three 6s | 250 | 0.0322 |
Two 6s | 1250 | 0.1608 |
One 6 | 3125 | 0.4019 |
Zero 6s | 3125 | 0.4019 |
TOTAL | 7776 | 1 |
Here are some problems to get us started:
Problem #1: There is only one round left, and you need to fill the FULL HOUSE line. What is the probability of being successful?
Problem #2 There is only one round left, and you need to fill the LONG STRAIGHT line with either 65432 or 54321. Your initial roll, unluckily enough, is a Yahtzee: 11111. Do you hold one dice and roll four dice? Or do you reroll all five dice? What is the strategy for the third roll? What is the expected probability of making a LONG STRAIGHT with this initial roll?
Problem #3 There are three rounds left. You need to fill "THREES", "SIXES" and "FULL HOUSE". You need 27 points between the "THREES" and "SIXES" to make the 35 point bonus for the top categories. The FULL HOUSE category is 25 points.
You first roll is: 66333. What should you do?
Including the 35 point bonus as appropriate, what is the EV in points for the last three rounds if you decide to :
- keep all five dice as a FULL HOUSE?
- keep the two 6s and reroll hoping to roll more 6s
- keep the three 3s and reroll hoping to roll more 3s
(I never said these would be simple problems.)
Any general comments on the Yahtzee Strategy problem are welcome as well!
Quote: DRichDo you have or have you read Olaf's Yahtzee book? He used to be a slot machine designer and BJ strategy guy.
No, I have never heard of it until this moment. I imagine that he has probably worked out all the expectations for most of the situational decisions that I am proposing.
I enjoy working on the mathematics of game strategies. Many of us have worked out the probabilities of many blackjack decisions even though books and the WOO site are available. The same is true for many casino games -many of us do the math ourselves. I was proposing that we do that for Yahtzee; if no one else is interested I'll just go ahead an do it myself unless I lose interest.
There's an interesting read at https://en.wikipedia.org/wiki/Yahtzee which describes some of the decisions. It looks a tough game to analyse.Quote: gordonm888...Any general comments on the Yahtzee Strategy problem are welcome as well!
Quote: charliepatrickThere's an interesting read at https://en.wikipedia.org/wiki/Yahtzee which describes some of the decisions. It looks a tough game to analyse.
Thanks Charlie. I also found this viewchart presentation from the Netherlands Optimal YAHTZEE strategy . Its slow to get going, but the second half of it has some interesting analysis.
LAST TURN Category | Expected Value | Prob. of Zero Pts, % |
---|---|---|
Aces | 2.11 | 6.49 |
Twos | 4.21 | 6.49 |
Threes | 6.32 | 6.49 |
Fours | 8.43 | 6.49 |
Fives | 10.53 | 6.49 |
Sixes | 12.64 | 6.49 |
| | |
3 of a kind | 15.19 | 28.76 |
4 of a kind | 5.61 | 72.26 |
Full House | 9.15 | 63.39 |
Short Straight | 18.48 | 38.40 |
Long Straight | 10.61 | 73.47 |
Yahtzee | 2.3 | 95.40 |
Chance | 23.33 | 0.00 |
This is from the link I referenced in my previous post.
I can see the recursive approach, similar to how to do BJ. Also I noticed that One thru Sixes are always going to be X thru 6X, since it's just a matter of how many of "what you want" you get multiplied by the value.Quote: gordonm888Here is some statistical information...when its your last turn...
So I guess at N options to go you look at the {throw,# rolls left} and which dice to keep. Then like BJ look at where you're at and where you might land up and work out the EVs (or expected score). I suspect once you get going (programming wise) it's a matter of cranking a very big handle.
I feel a challenge looming!
Quote: charliepatrickI can see the recursive approach, similar to how to do BJ. Also I noticed that One thru Sixes are always going to be X thru 6X, since it's just a matter of how many of "what you want" you get multiplied by the value.
So I guess at N options to go you look at the {throw,# rolls left} and which dice to keep. Then like BJ look at where you're at and where you might land up and work out the EVs (or expected score). I suspect once you get going (programming wise) it's a matter of cranking a very big handle.
I feel a challenge looming!
Parts of the problem involving the three rolls in a given round can be addressed by Markhov chains e.g., pursuing 6's, i.e. whenever you have one 6 you will roll four dice and the outcome probabilities for one, two, three, four, five 6's are always the same; when you start a roll with 2 sixes you will roll 4 dice and there will be another set of transiotion probabilities for two, three, four and five dice, etc.
The problem comes when you are rolling for Yahtzee or 4oak and you get say, two 6s on the first roll, but the second roll is three 4's, then you switch to fours. The rolls then switch to looking for 4s.
I agree with the lower numbers on the assumption that on the last round (of three rolls) a yahtzee can provide a winner in any of the sections (except "chance"). I don't quite get the top numbers (currently getting near 2.08 not 2.106481), assuming an incorrect yahtzee is 0. I'll have another look at my code tomorrow as I possibly have an error when the player should re-roll everything which can't happen in the bottom half.Quote: gordonm888Here is some statistical information...
Aces | 2.079 564 |
Twos | 4.159 128 |
Threes | 6.238 692 |
Fours | 8.318 256 |
Fives | 10.397 819 |
Sixes | 12.477 383 |
3 of a kind | 15.194 661 |
4 of a kind | 5.611 263 |
Full House | 9.153 620 |
Short Straight | 18.477 282 |
Long Straight | 10.612 741 |
Yahtzee | 2.301 432 |
Chance | 23.330 717 |
Aces | 2.106 481 |
Twos | 4.212 963 |
Threes | 6.319 444 |
Fours | 8.425 926 |
Fives | 10.532 407 |
Sixes | 12.638 889 |
3 of a kind | 15.194 661 |
4 of a kind | 5.611 263 |
Full House | 9.153 620 |
Short Straight | 18.480 750 |
Long Straight | 10.612 742 |
Yahtzee | 2.301 432 |
Chance | 23.333 333 |
I'd probably fill in 4 for twos.
Quote: WizardLet's say it's your first turn and after the third roll you have #$%*, like 1-2-2-4-6. Which field should you fill in?
I'd probably fill in 4 for twos.
Personally, I'd fill in the 1. If you fill in the two 2s, you have to get a fourth 3, 4, 5 or 6 to get the upper bonus. If you fill in the 1, you have the additional benefit of being able to get four 2s to get the bonus.
Quote: rsactuaryPersonally, I'd fill in the 1. If you fill in the two 2s, you have to get a fourth 3, 4, 5 or 6 to get the upper bonus. If you fill in the 1, you have the additional benefit of being able to get four 2s to get the bonus.
Good point. It just feels so feeble but write in a "1," but I suspect you're right.
http://gunpowder.cs.loyola.edu/~jglenn/research/optimal_yahtzee.pdf
and...
http://www-set.win.tue.nl/~wstomv/misc/yahtzee/slides-2up.pdf
Here is an optimal strategy analyzer which suggests playing the two 2s instead of the one 1. Seems counter-intuitive to me. I guess the advantage of having the ability to get four 2s on a future roll is more than offset by the probability of not getting at least two 2s again?
http://www-set.win.tue.nl/~wstomv/misc/yahtzee/osyp.php
1. First turn, first roll: 1 1 6 6 6
a. Keep: 6 6 6 _____265.12 +/- 61
b. Keep all and score 25 in Full House: 253.91 +/- 57
I suspect the ranking of these options would be flipped if this roll occurred late in the game, because Full House is so hard to get points for when you target it, rather than get it accidently. Although it probably depends upon what is needed on top to get a total of 63 or higher.
2. First turn, second roll: 1 1 3 4 6
a. Keep 3 4: 245.17 +/- 57
b. Keep 1 1: 245.14 +/- 57
c. Keep 4: 244.96
d. Keep 3: 244.74
e. Keep none: 244.55
f. Keep 6: 244.52
The fact that “f: Keep 6” scores the lowest seems to show the importance of targeting the SIXES when your first roll has at least two 6s. Scoring at least 18 with a shot at scoring 24 in SIXES is one of the keys to getting a high score.
3. First turn, third roll: 6 6 6 6 1
a. Score 24 in Sixes: 268.23 +/- 53
b. Score 25 in Four of a Kind: 260.54 +/- 54
I suspect the difference between these two options would be smaller (by 4) if it were 6 6 6 6 5.
cool game
Have it on my switch, part of a 50 game package of games
My strategy is to try to hit the straight and take ones or 2's or 3's when I dont because those dont score much anyway
Overall, seems whoever can hit a straight 1st will win.
You hit it early, you will score high
You hit it late, you will score low
Quote: WizardAt the end of your first turn you have 6-6-6-6-5. Do you fill in 24 for sixes or 29 for a four of a kind?
I'm going for 24 for sixes in a heartbeat as it practically ensures you're going to get the 35 point bonus.
Quote: rsactuaryI'm going for 24 for sixes in a heartbeat as it practically ensures you're going to get the 35 point bonus.
I'm not saying you're wrong, but often I have to put in a zero for a four of a kind. If you play the 4K, that's 29 points and you still have a lot of hope at the 35 bonus points. I think I achieve the bonus about 2/3 of the time.
Quote: WizardI'm not saying you're wrong, but often I have to put in a zero for a four of a kind. If you play the 4K, that's 29 points and you still have a lot of hope at the 35 bonus points. I think I achieve the bonus about 2/3 of the time.
I ran this through the optimal strategy link above and the top three results were as follows (assuming the first roll of a game):
Score 24 in Sixes 268.23±53
Score 29 in Four of a Kind 264.54±54
Score 29 in Three of a Kind 254.96±56 -- obviously no one is going to do this.
Quote: rsactuaryI ran this through the optimal strategy link above and the top three results were as follows (assuming the first roll of a game):
Score 24 in Sixes 268.23±53
Score 29 in Four of a Kind 264.54±54
Score 29 in Three of a Kind 254.96±56 -- obviously no one is going to do this.
Thanks. Good to know.
Quote: rsactuaryI ran this through the optimal strategy link above and the top three results were as follows (assuming the first roll of a game):
Score 24 in Sixes 268.23±53
Score 29 in Four of a Kind 264.54±54
Score 29 in Three of a Kind 254.96±56 -- obviously no one is going to do this.
But I do think that the answer is different if it is late in the game and you roll 66665 and you still have SIXES and FOUR OF A KIND open and four or five rounds to go - especially if you only need 18 points in SIXES to get the 35 point bonus.
Quote: terapinedIve been playing lately
cool game
Have it on my switch, part of a 50 game package of games
My strategy is to try to hit the straight and take ones or 2's or 3's when I dont because those dont score much anyway
Overall, seems whoever can hit a straight 1st will win.
You hit it early, you will score high
You hit it late, you will score low
Concerning straights, a few scenarios...
What would you do if...
1) It's your first turn and you roll a 2-2-4-5-6?
2) It's your first turn and you roll a 2-3-4-6-6?
3) You scored a full house on your first turn and the first roll of your second turn is 2-2-3-4-4?
Quote: GialmereConcerning straights, a few scenarios...
What would you do if...
1) It's your first turn and you roll a 2-2-4-5-6?
2) It's your first turn and you roll a 2-3-4-6-6?
3) You scored a full house on your first turn and the first roll of your second turn is 2-2-3-4-4?
I have no idea if this is right, but my hunch would be to...
1. Keep the 2-2. Second choice would be to keep the 2-4-5 and go for either straight.
2. Keep the 6-6 for sure.
3. Keep the 4-4
Anyone know where we can play Yahtzee against each other? There is a Facebook game that plays it, but it doesn't follow the proper rule on more than one Yahtzee.
Quote: GialmereConcerning straights, a few scenarios...
What would you do if...
1) It's your first turn and you roll a 2-2-4-5-6?
2) It's your first turn and you roll a 2-3-4-6-6?
3) You scored a full house on your first turn and the first roll of your second turn is 2-2-3-4-4?
According to the optimizer:
1) Keep 2 2 _ _ _ 250.56±59
Keep _ _ _ 5 _ 249.94±58
Keep _ _ 4 5 _ 249.89±58
Keep _ _ 4 _ _ 249.88±58
Keep 2 _ 4 5 _ 249.87±57
2) Keep _ _ _ 6 6 253.94±59
Keep _ _ 4 6 6 251.74±58 <--- interesting...
Keep _ 3 _ 6 6 251.71±58
Keep 2 _ _ 6 6 251.64±58
Keep 2 3 4 _ _ 250.42±58
3) Keep _ _ _ 4 4 251.54±56 (assumes full house is not available)
Keep 2 _ 3 4 _ 249.99±55
Keep _ _ 3 4 4 249.86±55
Keep 2 2 _ _ _ 249.67±56
Quote: WizardI have no idea if this is right, but my hunch would be to...
1. Keep the 2-2. Second choice would be to keep the 2-4-5 and go for either straight.
2. Keep the 6-6 for sure.
3. Keep the 4-4
Anyone know where we can play Yahtzee against each other? There is a Facebook game that plays it, but it doesn't follow the proper rule on more than one Yahtzee.
Wizard,
Use the multiplayer option here:
https://cardgames.io/yahtzee/
Dog Hand
I'm just wondering if there is any first turn, first roll scenario where going for the (non pat) straight becomes optimal.
Quote: GialmereWhat if, instead of a 2-3-4-6-6, you roll a 1-1-3-4-5 to begin your first turn.
I'm just wondering if there is any first turn, first roll scenario where going for the (non pat) straight becomes optimal.
I'm going for the 3-4-5 every time, in this situation.
Quote: DogHandWizard,
Use the multiplayer option here:
https://cardgames.io/yahtzee/
Dog Hand
Thanks! Who is up for a game? I'll keep an eye on this thread for a spontaneous challenge.
Quote: rsactuaryI'm going for the 3-4-5 every time, in this situation.
How about a 1-1-4-5-6?
Quote: GialmereConcerning straights, a few scenarios...
What would you do if...
1) It's your first turn and you roll a 2-2-4-5-6?
2) It's your first turn and you roll a 2-3-4-6-6?
3) You scored a full house on your first turn and the first roll of your second turn is 2-2-3-4-4?
Chuckle
I was playing quite a bit on my switch due to this thread
But
I just finished the Queens Gambit, great netflix series.
I've been playing a lot of chess on my switch now
I'm a pawn to King 4 opening person my whole life
Been playing a lot on my switch using Queens gambit opening and really loving my position.
The series is fascinating as it uses real historical games picked by Gary Kasporov. I've watched analysis of those games on youtube.
Quote: terapinedI just finished the Queens Gambit,...
I think I'll split off your Queen's Gambit post to it's own thread, as I thought the show was outstanding too. However, feel free to make your own original post, if you wish.
Back to Yahtzee, what would be the optimal strategy and mean score if the only thing left on the card was Chance?
Quote: rsactuaryAccording to the optimizer:
1) Keep 2 2 _ _ _ 250.56±59
Keep _ _ _ 5 _ 249.94±58
Keep _ _ 4 5 _ 249.89±58
Keep _ _ 4 _ _ 249.88±58
Keep 2 _ 4 5 _ 249.87±57
2) Keep _ _ _ 6 6 253.94±59
Keep _ _ 4 6 6 251.74±58 <--- interesting...
Keep _ 3 _ 6 6 251.71±58
Keep 2 _ _ 6 6 251.64±58
Keep 2 3 4 _ _ 250.42±58
3) Keep _ _ _ 4 4 251.54±56 (assumes full house is not available)
Keep 2 _ 3 4 _ 249.99±55
Keep _ _ 3 4 4 249.86±55
Keep 2 2 _ _ _ 249.67±56
When the margins are 1 to 2 tenths of a point and the standard deviations change from ±55 to ±59, it s a bit hard to feel conviction about the results. Maybe the results are just not being presented in a very good way.
Quote: gordonm888When the margins are 1 to 2 tenths of a point and the standard deviations change from ±55 to ±59, it s a bit hard to feel conviction about the results. Maybe the results are just not being presented in a very good way.
I think what it really means is that in the first few rolls, the order you do things doesn't really matter. But.... if you're the type to pay attention to micro VP edges.....
There are a few different strategies, depending on what you want to accomplish. You can play to get the highest score, you can play to optimize the highest score for the game that you're playing, you can play to beat the other player, or you can play to avoid losing.
Playing for the highest score? Every roll try to get a yahtzee. You'll wind up with a lot of games in the 100s, and every now and then you'll hit five hundred or more.
Play to optimize your score? That's the strategy most mentioned in this topic.
Play to beat the other player? If it's the last few rolls and you need two Yotz, you gotta do what you gotta do.
Play to avoid losing? This is a variation of the optimum strategy, I can best describe it as thinking "Where am I going to put this roll if I miss it?" In this strategy, 3oak, ss, 1s and 2s, and Chance are all safety spots, places to put rolls that missed. If you roll 3 1s, and already have them filled, you can shoot for Yotz, knowing it might only be 10 in the 3oak. But it's not 0 in the Yotz. If you missed your 4s, you can zero your 1s. You still need four 4s, but that's down the road.
I play at Board Game Arena, https://en.boardgamearena.com/gamepanel?game=yatzy
It is free. You can play with specific opponents, or random, anywhere from 2 to 5 players. There are players from all over the world. There is a nice chat feature for use during the games. It helps to learn how to say "Good Luck!" and "Good Game!" in French, Spanish, German, and Italian. Also chess, checkers, backgammon, hearts, spades, etc.
Wiz, I'll play you, but I can't right now I'm at work.
New high score today!
Quote: DRichWizard, have you read Olaf Vancura's book on Yahtzee?
I have it, but only skimmed it. As I recall, it seemed more academic than strategic.
Quote: WizardI've been playing quite a bit of Yahtzee online. I still haven't seen any WoV members in the player pool.
New high score today!
My daughter laid a 502 on me today. What stunk was I had a 313.
Quote: WizardDon't you hate it when people quit on you?!! In my first six turns I had two Yahtzees and a large straight. Then my opponent quit. No closure to the game. No opportunity to beat my high score of 498. I think whoever runs this game should give you the opportunity to finish it out alone.
I used to play bridge online on POGO.com. Everyone had a rating, and your rating would go up or down depending on whether you won or lost, and it took into account the rating of your partner and opponents. After a while my rating reached near the top. But then you would play the occasional opponent who would not quit or resign or finish, but would stall. So I would be stuck mid game unless I wanted to resign. You would make note of the scoundrel and never play with him/her again, but I eventually got frustrated enough that I don’t play there anymore.
However, there are very few of the classic games there. They are more focused on newer games.
Quote: terapinedWe should arrange a time ahead of time to all get on at once and play each other
I agree. How about we try 7:00 PM PST and whoever shows up shows up shows up?
Here is just one bit of advice. Let's say it's late in the game and you have to zero in a category due to a lousy hand. Here is the order you should do it in, from first to last.
1. Yahtzee
2. Four of a kind
3. Full house
4. Large straight
5. Three of a kind
6. Small straight
This was helpful. I previously would have zeroed in a large straight before a full house, but otherwise I think I had the order correct.