"Best" column combination

[thg0724]

If 6-8-10 and 6-8-9 had the same failure rate, 6-8-10 would be better because 10 is two steps shorter.

Is 6-7-11 (90.0%) better than 6-7-10 (90.3%)? You give up 0.3% but 11 is two steps closer.

[jerjerjaii]

Best combination:
Pay bga to be premium player and get rigged moves

[BarnardsStar]

First turn of the game, if the strategy is to roll until you either complete a column or lose all progress, what is three column combination most likely to produce a completed column?

I was thinking about this recently and decided to actually do the Monte Carlo simulations needed for this. But first, strategy… if you have three columns and you are partway up them, and you get a choice between column A and column B, how do you choose? For the purposes of this simulation I had it favor the column that would be the greatest percentage done after the move. For example, if you have 2,7,8 and you are one space up on 2 and 5 spaces up on 7, and you roll (1,1,6,6) do you choose the 2 or the double 7? I said two, since that would put you ⅔ the way up 2 but only a little more than halfway up 7.

So… if you are really gonna press your luck until you complete a column, the triple you most want to have is … <drumroll> 2-7-8! Or its counterpart 6-7-12. With one of these you have about 33.6% chance of completion.

At the other end of the spectrum, the worst triples to try to press your luck on are 3-4-5 and 9-10-12, with only a 3.5% completion rate. This is noticeably worse than the second worst combo, 3-4-11/3-10-11, at 5.07%. Where does 2-3-4 rank? It’s the third-worst, at 5.45%.

The other combos that would give you a greater than 25% completion rate are (in descending order of likelihood) 2-6-8, 4-6-8, 6-7-8, 5-7-8, 2-6-7, 4-7-8 and 3-7-8. (And their counterparts as well, of course.)

One interesting observation, to me at least: 2-6-8 has about a 30.6% and 4-6-8 has about a 29.5% chance of press-your-luck completion… but 3-6-8 is comparatively terrible, with only about a 16.5% chance.

[Ceaseless]

I was thinking about this recently and decided to actually do the Monte Carlo simulations needed for this. But first, strategy… if you have three columns and you are partway up them, and you get a choice between column A and column B, how do you choose? For the purposes of this simulation I had it favor the column that would be the greatest percentage done after the move. For example, if you have 2,7,8 and you are one space up on 2 and 5 spaces up on 7, and you roll (1,1,6,6) do you choose the 2 or the double 7? I said two, since that would put you ⅔ the way up 2 but only a little more than halfway up 7.

So if you were 8 up on the 7 and 1 up on the 2 and you rolled for a single 7 and a single 2, you'd pick the 7? Just to give an example that eliminates things like the percentage increase or percentage completion.

[BarnardsStar]

I was thinking about this recently and decided to actually do the Monte Carlo simulations needed for this. But first, strategy… if you have three columns and you are partway up them, and you get a choice between column A and column B, how do you choose? For the purposes of this simulation I had it favor the column that would be the greatest percentage done after the move. For example, if you have 2,7,8 and you are one space up on 2 and 5 spaces up on 7, and you roll (1,1,6,6) do you choose the 2 or the double 7? I said two, since that would put you ⅔ the way up 2 but only a little more than halfway up 7.

So if you were 8 up on the 7 and 1 up on the 2 and you rolled for a single 7 and a single 2, you'd pick the 7? Just to give an example that eliminates things like the percentage increase or percentage completion.

You make a good point. I probably would not do that… I’ll have to rerun it with different decision algorithms…

[Meeplelowda]

I was thinking about this recently and decided to actually do the Monte Carlo simulations needed for this.

You might be interested in the article I linked to in this Can't Stop thread.

Ed.: Unfortunately the link to the article is dead, but I did quote the abstract in that thread.

[BarnardsStar]

Take two: choose the move that will leave you closest to the top (in terms of number of steps) of that column. So if you are on step 1 of col 2 and step 8 of col 7, a 2 will put you one space from the top but a 7 will put you 4 from the top, so choose the 2. The other hand if you’re on step 2 of col 2, and step 12 of col 7, then a seven lets you win, and a two does not… So choose the seven.

This change … doesn’t actually change all that much. The actual percentages are a little higher across the board… but the relative rankings of the combos are largely unchanged. The best combo is still 2-7-8 at 34.8%, followed by 2-6-8 at 31.6% and 4-6-8 at 29.9%. Rounding out the >25% cohort are 6-7-8, 2-6-7, 5-7-8, 4-7-8 and 3-7-8. The dregs are still 3-4-5 at about 3.7% completion rate. And 3-6-8 (16.6%) is still much worse than 2-6-8 or 4-6-8.

[BarnardsStar]

I was thinking about this recently and decided to actually do the Monte Carlo simulations needed for this.

You might be interested in the article I linked to in this Can't Stop thread.

Ed.: Unfortunately the link to the article is dead, but I did quote the abstract in that thread.

I found the paper again: https://cdn.aaai.org/ocs/123/123-2430-1-PB.pdf

It’s a very interesting read.

[thg0724]

Take two: choose the move that will leave you closest to the top (in terms of number of steps) of that column. So if you are on step 1 of col 2 and step 8 of col 7, a 2 will put you one space from the top but a 7 will put you 4 from the top, so choose the 2. The other hand if you’re on step 2 of col 2, and step 12 of col 7, then a seven lets you win, and a two does not… So choose the seven.

This change … doesn’t actually change all that much. The actual percentages are a little higher across the board… but the relative rankings of the combos are largely unchanged. The best combo is still 2-7-8 at 34.8%, followed by 2-6-8 at 31.6% and 4-6-8 at 29.9%. Rounding out the >25% cohort are 6-7-8, 2-6-7, 5-7-8, 4-7-8 and 3-7-8. The dregs are still 3-4-5 at about 3.7% completion rate. And 3-6-8 (16.6%) is still much worse than 2-6-8 or 4-6-8.

Thanks for this. I am not sure the choice of column to advance on is quite right. I think of the choice is between a move that leaves you three from the top of column 7 or 2 from the top of column 11, you'd probably rather column 7. It must be better odds to get 3 sevens than 2 elevens. I doubt, however, that it will change the ranking of triples, just end up with minor changes to the percentages. (It's probably not a big deal to create a table for choice of single advances on either of two columns, more difficult when you are comparing advancing on two middle columns vs advancing on one outer-ish column.)

[BarnardsStar]

Thanks for this. I am not sure the choice of column to advance on is quite right. I think of the choice is between a move that leaves you three from the top of column 7 or 2 from the top of column 11, you'd probably rather column 7. It must be better odds to get 3 sevens than 2 elevens.

Hmm, I think I’d actually probably take the 11 in that case. I get this, actually, from my own odds above… the generalization is that the best triples have a really short column and then two of the middles. I’d need to break it down to be sure, but it seems to me like they are the best because you end up completing the short column, while the two very likely columns act as safety nets… take the 2 if you can, but if you can’t you’ll probably have a 7 or an 8 to fall back on.

But also, in an actual game, you’re not really pressing your luck until you complete a column, and you are trying to complete 3 columns. So there I’d definitely take the 11 because, while I can probably easily complete 7, 11 is gonna be harder so I should take it when I can.

I wish we had statistics on which columns got completed. That would be useful.