In my very first game of Can’t Stop I went first and got my markers established on the 6, 7, and 8 columns. I then proceeded to keep rolling until I had maxed out 7 and was one away on both 6 and 8. At that point I stopped my turn. All of my opponents crapped out and I won the game on my second turn advancing both the 6 and 8 one spot.

This seems like a very bad outcome for the game. My opponents sure weren’t happy about it. So it got me thinking, what is the probability of this happening? Is it enough to break the game?

I’m not good enough with probabilities to calculate this by hand so I wrote a computer program to simulate 10,000 games of Can’t Stop (someone can check my work by hand if they’re good enough).

The first question I asked was given your markers are on 6, 7, and 8, what is the probability that you will roll at least one 6, 7, or 8 and thus be able to continue your turn? The answer turns out to be 92% of the time you will roll safely.

If it takes you on average 17 rolls to max out one of the numbers (which I know it does) then your chances of making it once you start out with one 6, 7, and 8 are 0.92^16 = 0.26 or 26%

Now the question becomes, what are your chances of getting your three markers established on 6, 7, and 8? This required a much more sophisticated program, one that basically played out the entire turn. It gave a negative result if it didn’t roll such that it established the 6, 7, and 8 columns, or if it established them but didn’t push one to the top. It gave a positive result if one of the columns reached its maximum.

The result was that 10% of the turns maxed out one of the columns. Therefore, in about 10% of Can’t Stop games you would expect the first player to essentially win on his first turn if he followed this strategy (really he would win on the second turn probably, but for all intents and purposes the game would be over on the first turn since this strategy wouldn’t be available to someone else with one of the columns maxed out).

Is this a game breaking result? It’s awfully close. If you don’t establish your markers on 6, 7, and 8 then it’s moot (these are objectively the best columns to try this strategy with, all other columns will yield an inferior result). But if you do get that magic start then you are 26% to win the game from there. And you have about a 1 in 3 chance of getting that optimum start. You do the math.

We immediately played a second game and it went much more normally. I ended up winning that game too by maxing out the 7 column, 5 column, and the 12 column, but it took several turns.

A “normal” game of Can’t Stop is quite fun, but I’m afraid the “running the center” strategy might be a bit too strong. Any game that can essentially end before a player even gets their turn is a bit problematic for me.

Not at all. You just happened to have a very good run in your first game. I've never see anyone win on Turn 2.

I would also add three points.

1. Can't stop doesn't actually let you choose those numbers. Your first and second rolls on a turn will determine what numbers are possible to choose from.

2. In the many games I have played we do have fierce competition for those columns. This can result in many 'busts' as people push too far.

3. Can't Stop is too light a game for this kind of analysis. I recommend enjoying it for what it is - light fun that should be more about laughs than heavy strategy.

Thanks for the work though. It's nice to see the numbers.

That is a pretty big given. The large odds you mention are offset by the smaller odds of actually achieving this. In my plays of the game, as much as I would love to get 6,7,8 to start, the dice do not cooperate that often to actually let me have them.

If you haven't already, might want to try out James Cobb's Roll or Don't implementation of the game. Against 3 AI opponents, you can play a large number of games in short order to see how often this strategy can in fact be implemented.

there's a bunch of rolls that crap out even with 6, 7, and 8.

4 with the other 3 5's or 6's
all 1's
all 2's
all 5's
all 6's
any combo of 1's with 2's or 3's or 4's
any combo of 5's with 4's or 6's

It may not have been clear in what I wrote but I did take this into account in the second step. The first step was to assume you had 6, 7, and 8 already, and from there it's about 26% to win. Then I simulated 10,000 games from scratch. In 10% of them the player established columns 6, 7, and 8, and pushed one of them to the max. This implies that about 1 time in 3 you'll be able to get 6, 7, 8 to start.

Ok Richard, I started 100 games on that software and on 30 of them I was able to establish my markers on 6,7,8. That's about 1 in 3 like my simulation said.

I meant to keep track of how many I pushed to the top but unfortunately I misunderstood when I reached the top and kept rolling, which resulted in me busting out every time and not tracking how many times I maxed out. But upon realizing how it worked I know I reached the top several times. It seems to confirm to me that my numbers are correct.

OK, didn't quite catch that from your original post.

Depriving your opponents of one of the center columns is a strong advantage (it makes their future turns harder, and you're one number closer to winning). However, I'm not sure it's 'too strong.' Firstly, even given the first player has a 10% chance of getting one such column on the first turn (that's not exactly dependable), the next player may do something similar right back at you. When going first, I play a bit more conservative and get a decent foothold in 2 or more of the center columns (that can be done fairly reliably). Then other players may take greater risks (and get burned) rather than getting shut out of those numbers.

If you've got guts to claim a number on Turn 1 and the luck to do it, you deserve to win.

Are you saying that with 6/7/8, one can roll more before stopping than another combination? If so, I've played hundreds of games of Roll or Don't and I've not seen that. I've considered a broader range of numbers closer to optimaly, say 5/7/9, to cover dice that roll a bit more towards the extremes rather than clumping in the middle.

You got me curious.. I hacked together a little Monte Carlo with some goofy (i.e. broken, incomplete) AI on this and came up with just about your numbers:

22.3% got all three numbers (6, 7, & 8) on the first turn
59.5% got two of the three numbers on the first turn
17.6% got one ...
00.6% got none ...

Not that this has anything to do with anything.. but I was curious.

To answer the original question.. No. It's not. I believe that running up the center is no different than running up any other column because each column is proportionally sized. It "feels" better because you can maybe run 2 or 3 or 4, etc. spaces up the 7 column before busting; however, if you look at your gain as a fraction of the column height you'll see it's no different than running up the 2 or 12 columns.

Yes.

I estimate the probabily of rolling a 6, 7, or 8 at approximately 0.92; whereas a 5, 7, or 9 is approximately 0.86.

But the 6/8 tracks are longer than the 5/9 tracks... dun dun dunnnn.

Hi!

Your analysis is missing one important factor. How often does the first player win when he busts on the first turn trying to establish one of the columns. And is his total win percentage better than when using a more conservative strategy of trying to establish a good presence on one or more columns?

Jeff

Yes, this is what I'm saying.

Given that your markers are on 6,7,8 you are 92% to roll safely.

Given that your markers are on 5,7,9 you are 85% to roll safely.

Now the question is does the shorter tracks for the 5 and 9 versus the 6 and 8 make up for the decrease in roll percentage?

No, the tracks are shorter but not short enough. Compare the 5 and the 6 tracks. Out of 36 rolls you'd expect to roll four 5s and five 6s. The 6 track is 11 spaces long, therefore, to be proportional you would expect the 5 track to be 8.8 spaces long. But it's 10 9 spaces long. Over one space longer than it should be. You're better off rolling up the 6,7,8 (if you can) than the 5,7,9 (edit to add: but only barely).

edit: fixed some math.
edit2: fixed some math again.
edit3: dammit. I can't count the number of spaces on the board.

But I don't think they are proportionally sized. See the example for the 5 and 6 tracks above. Also consider the 2 and 7 tracks. Out of 36 rolls you'd expect to get one 2 and six 7s. The 7 track is 12 13 spaces long therefore, you'd expect the 2 track to be 2.2 spaces long but it's 3 spaces long.

All of the side tracks are a bit longer than they proportionally should be (edited to add: but only barely). That combined with the greater chance of rolling safely in the center means you should stay to the center as much as possible.

I played several hundred games against the computer last night and I don't think "running up the center" is too strong because there is enough variation in the dice to keep this from happening all that often. It's just going to result in a disappointing game for three of the players every once in a while, that's all.

edit because I can't count.

You're right. But if the first player busts he essentially becomes the fourth player. I do believe (but don't have any numbers) that the fourth player is at a disadvantage with respect to the first player. So if the first player busts he is hurting his chances somewhat, but it's probably not a huge amount.

It looks like you're doing your math as if the player only rolls 2d6; however, the players in this game roll 4d6 and can then choose to pair them up in anyway they like.

The probability of being able pair up the dice in order to get any specific number on a single roll is approximately this:

p(2) = 0.1293
p(3) = 0.22975
p(4) = 0.3534
p(5) = 0.4458
p(6) = 0.56055
p(7) = 0.64695
p(8) = 0.5604
p(9) = 0.45025
p(10) = 0.35795
p(11) = 0.23355
p(12) = 0.1314

Looks oddly linear.. strangely, it's also very similar to the layout of the board!

edit: misspelling, added quote for context, misc. cleanup

No, I'm definitely doing 4d6 when coming up with the probabilities. 92% for safely rolling a 6,7,8 and 85% for safely rolling a 5,7,9. It looks like you got the same result so this is probably right.



Once I finally counted the spaces on the board right I got a linear result also. My problem was not being able to count.

That's right.. but when you said, "Out of 36 rolls you'd expect to get one 2 and six 7s." you weren't. Out of 36 4d6 rolls I'd expect to be able to make a two 4.7 times and a seven 23 times.

Anyway, now you see the (near) linearity. I believe that the columns do have a very slight imbalance; however, it's damn close. Too close to matter in the span of a handful of games. To get it any closer the board would have to grow in size.

Sid Sackson was a very smart guy.

Ok, so I guess there's a small difference. Hopefully it's canceled in the long run by players competing for those rows.

Maybe use the variant that has players bid for who goes first by placing a markers on the board to show where opponents may begin if they go in later turns.

I'd thought about trying to make a bigger version of the game using a real STOP sign. I even searched eBay a few years ago, but they were pricier than I'd imagined.

So what would the spaces on the board look like if wanted to smooth out the numbers? Or just to extend the game? (It usually seems too short to me, without enough chance to catch the leader.) Or what if we used 6 dice and made 3 pairs? Or if we used 4 d8?...4 d10?..4 d12?

For me, this is the key point. You have approximately 1/4 chance of achieving this feat of 17 lucky rolls. Assuming that there are 4 players, that is your chance of winning as well. I think the game is fair enough.

Yes you are right, sorry about that.



There is still an imbalance though. I agree that the columns are set up to be (relatively) balanced per the probabilities you showed above of rolling one number on a 4d6. But what makes it unbalanced is that you still have to count the second pair of dice, and these dice are more likely to help you if your columns are towards the center.

For example, from running my simulation this is what I have found (I spotted a small bug since I first posted. It doesn't make a huge difference but the numbers below are slightly different than above, these should be more accurate).


Columns % to roll safe % to push 1 col to top Avg # of rolls
6,7,8 92 9 15
5,7,9 85 3 13.5
2,3,12 43 0.04 3.7

From this you can see that the center is still much better. It takes you more rolls in a row to complete a column but you're still much more likely to do it because you are so much more likely to roll safely.

The difference, once again, between this result and your table showing the chance of rolling a given number on a 4d6 is that this also counts the second number, and that number is more likely to benefit a center column.

Yes, you are right. And just to be clear with all these numbers flying around the 26% comes from when you are given a start on 6,7,8. When you start from nothing and have to establish 6,7,8 the number is 9%.

To go for 3 columns, you had to make choices on the rolls. Can you provide some details on the AI you used along with your simulation?


I haven't had a chance to think about this, but my feeling is that this thinking is flawed.

Yes, the strategy for the AI was to establish the 3 selected columns as quickly as possible. So, for example, if it was trying to establish the 6,7,8 columns and it rolled a 3, 3, 4, 4 it would choose to use the 6,8 instead of the double 7 because that established 2 columns instead of 1.

If, the next roll were a 2, 3, 5, 6 it would choose to establish the 7 column instead of using the double 8.

Once all three columns were established it would choose the pair that pushed the highest column, if it had a choice.



Maybe it is. Think about it and get back to me. This was my off the cuff explanation. If the board were perfectly balanced you should have the same chance of pushing the 2, 3, 12 columns starting from scratch as the 6, 7, 8 columns but I don't think you do, so there is some effect going on.

Btw, this is all an academic exercise for me. It doesn't bother me one bit if the columns are unbalanced. In fact I prefer it that way as it lends some structure to your decision making. I think it would be more boring if every column were mathematically identical to every other one. My original question was if pushing up the center in one turn is too strong. It's about a 10% chance of doing that which is probably not too strong, but is going to happen frequently enough.

If the game were longer then the unbalanced columns would be a problem because I think they lead to a runaway leader problem. Once the center columns get locked out players will have a much harder time catching up to whoever locked them out. In a short game stuff like this is easily forgivable though.