Multiplayer Efficiency

This article employs a hypothetical game of Tetris, absent of T-Spins, Perfect Clears, Combos, B2B bonuses, and garbage canceling. This game also gives randomized garbage, where holes line up 10% of the time. This makes it easier to consider multiplayer strategy at its foundation.

Please read my "Multiplayer Metrics" article before this one. It explains some of the ideas and jargon found here.

As mentioned elsewhere, newer players tend to outright ignore the downstack. Even with the best attacks, the “downstack ignorer” is unable to send garbage fast enough (on a per piece basis). As long as his opponent sends and clears more total garbage, the "downstack ignorer" loses. To gain some intuition, consider the following table.

 A ‘G’ affixed to a particular line type represents the number of garbage lines that that line clear contains. The illustration below shows a “2G Triple." It consists of two rows of garbage and one row of upstack.

Line Type	Pieces	Garbage Cleared	Garbage Sent	Garbage Total	Efficiency
0G Single	2.50	0	0	0	0.00
0G Double	5.00	0	1	1	0.20
0G Triple	7.50	0	2	2	0.27
0G Tetris	10.00	0	4	4	0.40
1G Single	0.25	1	0	1	4.00
1G Double	2.75	1	1	2	0.73
1G Triple	5.25	1	2	3	0.57
1G Tetris	7.75	1	4	5	0.65
2G Double	0.50	2	1	3	6.00
2G Triple	3.00	2	2	4	1.33
2G Tetris	5.50	2	4	6	1.09
3G Triple	0.75	3	2	5	6.67
3G Tetris	3.25	3	4	7	2.15
4G Tetris	1.00	4	4	8	8.00

The least efficient line clear that includes garbage still does better than the most efficient line clear that does not include garbage.  Also of note is how the more a player stacks on top of a garbage row, the less efficient it becomes. This is true everywhere except for a 1G Tetris being more efficient than a 1G Triple.

A Tetris requires ten pieces to send four rows. One piece can clear one line of garbage. Should the garbage holes line up, a piece can clear more than one at the same time. It is tempting to jump to the conclusion that a player should then only focus on downstacking. The idea is to forego Tetrises in order to clear down garbage with as few pieces as possible. With this "freestyle downstacking" strategy, you only stack Tetrises in two cases. In the first case, four garbage rows already line up. In the second case, you have no garbage left to downstack. This might be a winning strategy, if it were not for one obstacle. Upstack tends to interfere with future garbage clearing. The following diagram shows a rare example where upstack does not interfere.

Yet in real games, such a one-to-one ratio between pieces and garbage cleared is next to impossible. When downstacking messy garbage, each piece that clears a garbage row leaves behind three filled cells. Since the field is ten cells wide, every 10/3 = 3⅓ pieces that clear a garbage row leave a Single's worth of upstack. Suppose a player clears one garbage row every two pieces. This requires him to leaves behind 1.75 pieces worth of upstack for every garbage row he clears. This inevitably leads to covered-up garbage holes.

Each filled cell covering an upcoming hole requires an additional upstacked line in order to uncover that hole. For example, a hole with two filled cells directly over it requires no less than two line clears. These upstack rows cost additional pieces before a player is able to continue downstacking.

A useful question to ask is, "just how many garbage rows can I clear within a given number of pieces?" There was a downstacking tournament to find out who could clear garbage with the fewest number of pieces. The game gave each player 18 randomly positioned garbage holes. Contestants had all the time they needed to figure out the most efficient moves. The runner-up, Anonymous, used only 38 pieces. The winner, Paul676, used only 36. Despite having no reward for sending lines, these players ended up clearing Doubles and Triples as a natural consequence of downstacking. Anonymous sent the equivalent of nine garbage rows. Paul676 sent eleven.

Player	Pieces	Garbage Cleared	Garbage Sent	Garbage Total	Efficiency
Paul676	36	18	11	29	0.81
Anonymous	38	18	9	27	0.71

Suffice to say, a long-term efficiency of over 0.7 is extremely good. The next question might then be, "should I be happy with sporadic Doubles and Triples since freestyle downstacking is so efficient?" The rest of this article deals with cases in which the answer to that question is, "probably not."

When should I stack up a 0G Tetris?

The technical answer to this question is, "any time those ten pieces could not otherwise have, in total, cleared and added more than four rows of garbage." Drawing from the aforementioned tournament, a garbage row generally requires at least two pieces to clear. (Granted, it requires two pieces if you play almost perfectly.) Hence, those ten pieces might have cleared five rows of garbage instead. Moreover, they probably would have sent at least one Double as a natural consequence of freestyle downstacking. That accounts for six total garbage events.

That is why a 0G Tetris is usually only cost effective when there already are no garbage rows on the field. Typically, there is only one other time when it is desirable to sink ten pieces into a 0G Tetris. That is when your opponent is so nearing the top that sending garbage at all costs would deliver the final blow. In that case, there is no value in clearing your own garbage since doing so squanders a chance to win. Even in such a case, a 1G Tetris requires less pieces but sends four lines just the same.

When should I clear a 4G Tetris?

A player should clear a 4G Tetris any time she has an I-piece and four lined-up rows of garbage. It is rare to do better by clearing four lined-up rows of garbage with more than one piece and without scoring the Tetris.

When should I stack up a 1G, 2G, or 3G Tetris?

There is a scenario where incorporating garbage holes into a Tetris is a no-brainer. This is when you must clear upstack before gaining access to the next hole anyway.

In such a case, clearing the first-to-come garbage hole results in cells covering up the second-to-come garbage hole.

By building up to the fourth row above the second-to-come hole, you score a Tetris. You also do this without need for any extra pieces than had you uncovered the hole through any other means.

Doing so adds more garbage sent than had you broken the Tetris down into lesser line clears. Remember to carefully place the pieces in such a way so that you do not cover the upcoming hole with more than four filled cells.

When upstack does not already cover future garbage holes, then the 1G, 2G, and 3G Tetrises become more complicated to evaluate. In the very short run, it appears more efficient to clear each garbage row with one piece rather than to stack up a Tetris on top of them. However, this only shows the efficiency of this decision over the course of one piece!  It does not say anything about whether that decision negatively affects the efficiency of future pieces.

Why might clearing a garbage row immediately with one piece cause future pieces to decrease in efficiency? Imagine a player who can clear a garbage row every three pieces. Three pieces and one garbage row later, he has enough upstack to pair with the next garbage hole.

However, he can delay liquidating the upstack. Imagine he clears the next garbage hole with a 1G Single. After six pieces, he could clear two garbage holes and have 5.5 pieces worth of upstack left. Depending on the flatness of the upstack, it may be possible to clear a 1G Triple with it.

With an I-piece and a couple of others, it is possible to upgrade to a 1G Tetris. Doing so sends an extra two garbage rows.

Grouping line clears together sends more garbage rows per piece than clearing lesser line clears separately. For example, clearing three lined-up garbage rows as a Triple sends out two garbage rows.

If you stack three pieces on top of that instead, you can then clear a Tetris with the fourth piece (an I-piece). Doing so sends four garbage rows.

The Triple results in an efficiency of (3+2)/1 = 5.00, while the 3G Tetris results in an efficiency of (3+4)/4 = 1.75. The 3G Triple is more efficient only for the first piece. What about the next three pieces? How did the decision to clear the 3G Triple right away affect the efficiency of those later pieces?

Suppose the next three rows of garbage do not line up. It is not always possible to be able to clear all three garbage rows with the next three pieces. Frequently, there is no choice but to cover the next garbage hole with upstack residue.

If just one piece does not clear a garbage row, and the other two do, then this leaves a total efficiency of (5+2)/4 = 1.75. This breaks even with the 3G Tetris. In the case of clearing one or none at all, you actually lose efficiency.

In the absence of clean garbage, the best to hope for is only needing around two pieces for each garbage hole. In real games, a minimum of at least half of all pieces do not go towards clearing garbage rows. Rarely do garbage holes position themselves in such a way that you can avoid stacking on top of them completely. The point is that upstack is inevitable. A player might as well accept it and use it to send as much garbage as possible.

The 3G Tetrises are more efficient than 2G Tetrises, which are in turn more efficient than 1G Tetrises. The more pieces needed to complete a partially garbage Tetris, the more certain you must be that it is not more efficient to instead use those pieces to clear garbage holes. A 1G Tetris requires about eight pieces of upstack. If you can clear that garbage hole with one piece, then the question becomes, "what could I have done differently with those other seven pieces?" If they could have instead gone on to clear three more garbage rows while clearing two Doubles in the process, then perhaps you were better off not stacking up to the 1G Tetris. Compare the 1G Tetris (4+1)/8 = 0.625 with the four garbage rows and two Doubles (2+4)/8 = 0.75.

An xG Tetris's cost-effectiveness diminishes in the following two cases. The first case is when they require more pieces to complete. For example, a 1G Tetris requires 4.5 more pieces than a 3G Tetris. The second case is when upcoming garbage holes are easy to clear. Awkward upcoming garbage and awkward piece combinations likely require extra upstacking to stabilize the surface. When you think you will need extra pieces of upstack, this is the time to consolidate the upstack into an xG Tetris. The crux of the issue is whether you will have leftover upstack anyway.

On the other hand, there are cases where it is unsafe to stack up a Tetris, but not so to downstack. Clearly, in these cases, you will gain more efficiency from breaking down the surface and digging into the garbage. Generally, holes on the very sides are easiest to build up to an xG Tetris. Holes in the middle disconnect the surface and make upstacking more challenging. The least appealing of holes may be the ones a single column away from the sides.

Frequently, these holes will require either an extra I-piece or some skimming in order to complete the xG Tetris. In the case of needing to skim, the extra pieces needed make the xG Tetris that much less cost-effective.

When should I stack an xG Triple or xG Double?

Similar principles to xG Tetrises apply to xG Triples and xG Doubles. Say two cells of upstack cover the second-to-come garbage hole. It may be obvious to stack up to the 1G Triple yet unclear if it is worth it to add an extra row for the 1G Tetris.

Triples and Doubles send less garbage rows per piece, but they also require less pieces. They are the middle ground between easily downstacking right away and stacking to an xG Tetris due to pessimistic garbage outlook.

The same principle applies to an xG Double. Although with a Double, building up to a Triple does not render the same payout as does going from a Triple to a Tetris. It costs three more pieces to upgrade to a Triple, but it only adds one more garbage row. Therefore, the downstack outlook for those three pieces must appear even more pessimistic in order to justify committing them to the upstack.

Converting an xG Double into an xG Tetris costs even more pieces (at least five). However, you do get a more favorable payout. The Tetris upgrade will send three more garbage rows. Compare upgrading a 1G Triple 1/2.5 = 0.4 to upgrading a 1G Tetris 3/5 = 0.6. Doing so squeezes 50% more efficiency output from the extra upstack. Usually, it is nicer to survey 0.6 efficiency over the course of five pieces opposed to 0.4 efficiency over the course of three pieces. Therefore, when asked to stack up for a 1G Triple, it may be worth your while to put in a few extra pieces toward that 1G Tetris instead.

On a final note, it is prudent to bear in mind that stacking xG Tetrises, Triples, and Doubles increase the chances of covering upcoming garbage holes. Ideally, you can avoid covering them. In practice, it does not always work out that way. In any case, it is a risk worth factoring into the equation.