Multiplayer decision making, a cost-effectiveness analysis, [Advanced] Options Rating

[caffeine]

8/25/13 Edit: I have revised and expanded on this work. You can visit the new article here.

Quick edit: 12/21/10, Since writing this, I've thought of a better method. I'll probably revise this article some time in the future. I actually want take into account new variables to make for a much more complete (albeit complex) system, but I may save that for a whole new article altogether. Also, the combo tables are outdated

What I'm measuring, when I say "cost-effectiveness," is how efficiently an attack makes use of the tetrominoes I've invested into it. I find this value by weighing how many tetrominoes an attack requires with how many it removes from my stack with how many tetrominoes it adds to my opponent's stack. Technically speaking, the only tetrominoes that can kill me are the ones (or parts of ones) that end up on columns 4, 5, 6, and 7. But for the purpose of this article, I'm going to assume I stack perfectly even.

The object of multiplayer Tetris is to top out my opponent before my opponent tops me out. Therefore the more my opponent's stack and the less my own, the better I'm off. Dividing an attack's outcome by its cost will result in that attack's overall cost-effectiveness. An attack's cost is how many tetrominoes I must build up at first. An attack's outcome is how many tetrominoes are removed from my stack, including garbage, plus the ones added to my opponent's stack. Simply, I'm dividing the good (more to my opponent's stack and less of my own) by the bad (the time, represented by tetrominoes, it takes to set up).

Two and a half tetrominoes make up a line, and 2.5*(9/10), or 2.25, makes up a garbage row. Moreover, a garbage row costs 1/4th a tetromino to clear. I use a tetromino as the reference point, but really I could also use a line (2.5 tetrominoes) or a block (1/4 tetrominoes or 1/10 lines).

Sometimes a player can use previous garbage lines to attack back, so "Double with one garbage line" means a double that includes in it one garbage line from an opponent's previous attack.

Combos begin adding garbage on the third consecutive clear (it goes: clear a line, then another (Combo 1), then a third (Combo 2) sends one garbage row). Combo 2 and 3 send one garbage row, Combo 4 and 5 send two garbage rows, Combo 6 and 7 send three, and Combo 8 or more sends four. So a 2 Combo requires three lines to add one garbage row to my opponent. I calculate the next combos by adding on all of the combo's history (I assume all combos are singles). So a 3 combo really requires four lines and sends two garbage rows. I haven't included comboing garbage calculations since there are so many, but you don't need to guess how brutally effective they can be.

T-Spins cost more than I account for because they require more time in the form of soft drops. They also require at least one tetromino that won't count until the next attack.

A Double costs five tetrominoes and has an outcome of removing five tetrominoes from my field and adding one garbage row (2.25 tetrominoes) to my opponent . (5 + 2.25) / 5 = 1.45 . In another example, a T-Spin Double containing one garbage row costs 2.75 tetrominoes (2.5 for one line plus .25 of a tetromino to clear the garbage row) and has an outcome of removing five tetrominoes from my field and adding nine tetrominoes (2.25*4) to my opponent. (5 + 9) / 2.75 = 5.09 . In my last example, a 5 Combo costs fifteen tetrominoes (six lines worth) and has sent a running tally of six rows of garbage (two total from 2 Combo and 3 Combo, four total from 4 Combo and 5 Combo). (15 + (6 * 2.25)) / 15 = 1.9 .

Cost-effectiveness, Cost, Outcome, Type of attack

01.00, 02.50, 02.50, Single
10.00, 00.25, 02.50, Single as one garbage row

01.45, 05.00, 07.25, Double
02.64, 02.75, 07.25, Double containing one garbage row
14.50, 00.50, 07.25, Double as two garbage rows

01.60, 07.50, 12.00, Triple
02.29, 05.25, 12.00, Triple containing one garbage row
04.00, 03.00, 12.00, Triple containing two garbage rows
16.00, 00.75, 12.00, Triple as three garbage rows

01.90, 10.00, 19.00, Tetris
02.45, 07.75, 19.00, Tetris containing one garbage row
03.45, 05.50, 19.00, Tetris containing two garbage rows
06.33, 03.00, 19.00, Tetris containing three garbage rows
19.00, 01.00, 19.00, Tetris as four garbage rows

01.90, 02.50, 04.75, T-Spin Mini
19.00, 00.25, 04.75, T-Spin Mini containing one garbage row

02.80, 02.50, 07.00, T-Spin Single
28.00, 00.25, 07.00, T-Spin Single containing one garbage row

02.80, 05.00, 14.00, T-Spin Double
05.09, 02.75, 14.00, T-Spin Double containing one garbage row

02.80, 07.50, 21.00, T-Spin Triple
04.00, 05.25, 21.00, T-Spin Triple containing one garbage row

01.30, 07.50, 09.75, 2 Combo
01.45, 10.00, 14.50, 3 Combo
01.72, 12.50, 21.50, 4 Combo
01.90, 15.00, 28.50, 5 Combo
02.16, 17.50, 37.75, 6 Combo
02.35, 20.00, 47.00, 7 Combo
02.60, 22.50, 58.50, 8 Combo
02.80, 25.00, 70.00, 9 Combo
02.96, 27.50, 81.50, 10 Combo
03.10, 30.00, 93.00, 11 Combo
03.22, 32.50, 104.50, 12 Combo

In descending order (* denotes a tie):

1. T-Spin Single containing one garbage row
*2. T-Spin Mini containing one garbage row
*2. Tetris as four garbage rows
4. Triple as three garbage rows
5. Double as two garbage rows
6. Single as one garbage row
7. Tetris containing three garbage rows
8. T-Spin Double containing one garbage row
*9. Triple containing two garbage rows
*9. T-Spin Triple containing one garbage row
11. Tetris containing two garbage rows
12. 12 Combo
13. 11 Combo
14. 10 Combo
*15. 9 Combo
*15. T-Spin Single
*15. T-Spin Double
*15. T-Spin Triple
19. Double containing one garbage row
20. 8 Combo
21. Tetris containing one garbage row
22. 7 Combo
23. Triple containing one garbage row
24. 6 Combo
*25. 5 Combo
*25. T-Spin Mini
*25. Tetris
28. 4 Combo
29. Triple
*30. 3 Combo
*30. Double
32. 2 Combo
33. Single

Findings:

1. Clearing with garbage is almost always better than clearing without garbage, and usually by a huge margin.
2. If you have more than one garbage row lined up, clear them all at once rather than separately.
3. There's no use in stacking on top of a garbage row to make a bigger attack out of it unless you T-Spin Single or Mini with it (although not T-Spin double).
4. At the start of a game, comboing is only worth the risk when you make it past 9 Combo. Otherwise, you should use a T-Spin Single, Double, or Triple (not Mini).

[Corrosive]

That is a theory I never really thought about before, nice dude. I'll probably have to read it a couple more times to grasp it all.

[Kitaru]

Ahh, this makes a whole lot of sense. Thank you for writing this.

[Digital]

Looks great caff. Would you also consider adding a small garbage countering section? Just thought it may be worth mentioning as it appears in recent guideline games.

[Kevcel]

This is a great article, Caf. I've always been curious about clearing lines with or without garbage;
this theory about cost-effectiveness is very interesting and can be quite helpful to me.

ps: I think you've made a little mistake:

''Let's look at the attacks that contain only one garbage row.

1, Single as one garbage row, 0 - -1
1, Triple with one garbage row, 2 - (2 + -1)
1, Triple with one garbage row, 2 - (2 + -1)
2, Tetris with one garbage row, 4 - (3 + -1) ''

You've mentioned ''Triple'' twice.

Interesting is: Single, Double and Triple with one garbage row all have the same cost-effectiveness of 1.

[caffeine]

Thank you, Kevcel. I've corrected that typo. Digital, I've thought about applying it to T-Spins and Combos (Combos are really tricky), but not garbage canceling. Good food for thought, thanks.

BTW, I'm in love with "FASTREPLY."

[Pineapple]

A special exception may be when your opponent is about to top out, and you need a little extra attacking power (which is no more cost-effective, but in this case better in timing).

There's probably an entire strategy article that could be written just about "going for the kill". Knowing when to do this can often make the difference between losing and winning...

[jujube]

4. You may as well just clear that single garbage row unless you plan on making a Tetris out of it. A special exception may be when your opponent is about to top out, and you need a little extra attacking power (which is no more cost-effective, but in this case better in timing).

that's really interesting. i wonder about a situation where a player clears (let's say) 3 garbage singles, and could easily make 1 more garbage single which would leave their stack more or less screwed, so instead opts to stack up for a double or triple, removing most or all of the junk on top of the garbage, allowing them to downstack with a few garbage singles again.

[clincher]

Interesting I hope people keep writing articles they help me a lot
Thx for this Caffeine
PS: I was used to fast reply before TC I found it weird that TC didnt have that

[Zircean]

Interesting read, caff. However, in games like Blockbox, the garbage rarely lines up so it's not easy to clear more than one garbage line at once. In this case, does building up lines over garbage holes become more cost-effective? Otherwise, you have no garbage output, and there's no way for you to top out your opponent. O.o

[jujube]

that's true Zircean, it doesn't really work with Blockbox, unless you're playing an unrated game with different garbage.

[caffeine]

I remember there was a glitch where multiplayer games weren't actually random (in random, two in a row line up one in ten, three line up one in a hundred, and four line up one in a thousand). Last I talked with him he was planning on fixing the glitch. If you're saying garbage doesn't line up, then I'm guessing he forgot about it.

[clincher]

Well, I play a lot there and i havent seen 2 garbage holes in a row

[jujube]

yeah, it seems like the rated mode garbage in BB has 100% hole change.

[meow]

consecutive holes aren't that bad if the hole placement is random

10% chance for 2
1% chance for 3
0.1% for 4...