Stacking is a variance decision, not a points decision

Here is the claim, stated plainly so it can be argued with: stacking a quarterback with his receiver does not add a single expected point to your lineup. The expected value of a sum is the sum of the expected values, always, regardless of how the two are related. If your projections say the quarterback scores 20 and the receiver scores 14, then a lineup containing both projects for 34 whether they are teammates, strangers, or the same person twice.

What changes is the spread around that 34. That is the whole trade, and once you state it that way, most stacking questions become answerable: you are not asking "does this help my lineup score more?" You are asking "do I want a wider distribution, and how much am I paying for it?"

The arithmetic, in one line

For two random scores, the variance of their sum is:

Var(A + B) = Var(A) + Var(B) + 2 * rho * SD(A) * SD(B)

The first two terms are what you get from any two players. The third term — the correlation term — is the only thing stacking changes.

Take a worked hypothetical. Suppose both players have a standard deviation of 8 fantasy points, and suppose the pair's mean is 34. Correlation rho is the dial:

Every row has the same mean of 34. The 8-point standard deviations and the correlations are invented for the example; the arithmetic is not, and you can check any row with a calculator.

To see why the spread is worth paying for, put a threshold on it. Treat the pair as roughly normal — a convenience, not a claim, since real fantasy scores are right-skewed — and ask for the chance the pair clears 55, twenty-one points above its mean. At a standard deviation of 11.31 that is about 3%. At 14.31 it is about 7%. Roughly double the chance of the outcome that wins a tournament, bought without touching the projection.

The same arithmetic tells you the cost, which people skip. The low tail widens by the same mechanism. The chance the pair comes in under 24 rises from about 19% to about 24%. Stacking is not a way to get a ceiling for free. It is a way to trade floor for ceiling at a stated rate.

That is also the argument against stacking in a cash game, and it comes with a condition worth stating rather than assuming. Widening the spread moves probability out of the middle and into both tails — which is exactly what the two paragraphs above measured, one tail each. Whether that helps you depends entirely on which side of your mean the threshold sits. A cash line below your lineup's projected mean is cleared by the middle of the distribution, so thinning the middle strictly lowers your chance of clearing it. That case is the "under 24" row of the calculation, not the "over 55" one: nothing about cash rewards the right tail, and the left tail is where the entry dies.

That condition is the ordinary case, because it is the case a maximized-median build is constructed to produce: you are trying to project above the line, and if you have, the wider distribution is a pure cost. But it can flip, and it is worth recognizing when it has. If your lineup projects below the line you have to clear, the arithmetic reverses and variance becomes the only thing that can get you there — the tightest build is then the one most reliably short. Read that as a diagnosis rather than a strategy. A cash lineup that needs variance to reach the line is a cash lineup with a problem upstream of its correlation structure.

Where the correlation actually comes from

You do not need a study to know why a quarterback and his receiver are correlated. The scoring rules do it. On DraftKings, a passing touchdown pays the quarterback four points and pays exactly one receiver six. A 40-yard completion pays the passer 1.6 points of passing yardage and the catcher 4.0 of receiving yardage, before the reception point. These are not two events that tend to co-occur; they are one event, entered on two rows of your lineup.

The same reasoning tells you where correlation is weaker or negative, which is the more useful half:

• Quarterback and his second receiver. Same mechanism, thinner. He catches fewer of the touchdowns, so the shared events are rarer.
• Two receivers on the same team. At a fixed team total there is a fixed pile of targets, and every one that goes to your first receiver does not go to your second. They correlate through the team's overall scoring and anti-correlate through target share, and the net can land near zero or below.
• Quarterback and his running back. A rushing touchdown by the back pays the back and pays the quarterback nothing. Goal-line carries are, mechanically, passing touchdowns that did not happen.
• A running back and the defense he is playing against. Straightforwardly opposed. Your defense is paid when that offense fails; the back is paid when it succeeds. This is the clearest negative correlation available on a Classic roster, and it is why GameScript's default stack rules refuse to put a running back in the same lineup as the defense he faces.

Notice that the last one is a default on. Negative correlation inside a roster is the one correlation choice that is close to unambiguously bad in tournaments, because it shrinks the spread without improving the mean — the top row of the table, applied by accident.

What a bring-back actually buys

A bring-back — a skill player from the opposing team, added to a stack — is usually described as a hedge. That description is wrong, and the wrongness matters.

A naked quarterback-and-receiver stack is a bet on one team scoring. Its failure modes are not symmetric. The obvious one is the game where nobody scores. The less obvious and more common one is the game your team wins comfortably: the offense goes conservative in the second half, the quarterback throws 14 times after halftime, and your ceiling evaporates precisely because the bet worked.

A bring-back removes that branch. The game states in which your stack posts its ceiling are the ones where the other team keeps scoring and forces your offense to keep throwing. A bring-back is positively correlated with your stack's best outcomes, not with its worst ones. It is not insurance; it is a second bet on the same event, which is the game total rather than the team total.

So the honest description is: a bring-back converts a bet on one team's scoring into a bet on the game being both high-scoring and competitive. That is a narrower event than "this team scores," and narrower events are what tournaments pay for. It also means the bring-back is the wrong tool for a game the market expects to be lopsided — a heavy favorite's implied blowout does not need insurance, it needs a different game.

The roster budget

Correlation is not free, and the constraint is roster slots.

A Classic entry is nine players spanning at least two games and at least two teams, and GameScript defaults to a maximum of four players from any one team. A quarterback plus two pass catchers plus a bring-back commits four of nine slots to a single game. That is nearly half the roster expressing one opinion.

This is why the stack rules are constraints rather than preferences. The defaults are: at least one same-team pass catcher with the quarterback, no required bring-back, at most four players per team, and no running back opposite your own defense. Turning min_pass_catchers up to 2 and min_opposing_skill_players up to 1 is a real narrowing of the feasible set, and if you also lock three players and set a high salary floor, the answer may be that no legal roster exists. That is information about your preferences, not a failure of the solve.

When a naked running back is fine

Given all of the above, the case for an unstacked running back is stronger than the stacking orthodoxy usually admits.

A workhorse back's floor is carries. Carry volume rises when his team leads and the game is being run out, which is the same game state that suppresses pass attempts. His production is therefore close to independent of the passing games in your lineup — and in a roster that has already spent four slots on one game, an independent slot is exactly the thing you are short of.

Three situations where the naked back is the right answer:

• Your roster is already concentrated. If four slots are in one game, the marginal value of a fifth correlated slot is lower than the marginal value of a second, unrelated outcome. Two independent bets that each need to hit is a worse structure than one concentrated bet plus a player whose Sunday does not depend on it.
• The usage is script-independent. A back who plays all three downs and catches passes scores in a 31-28 game and in a 16-13 game. That is a different asset from a two-down back whose whole case is a lead.
• It is a cash lineup. Independence lowers variance, and for a build that already projects above the line, lowering variance around the same mean is the objective. In cash, the naked back is not a compromise; he is the point.

The failure mode to avoid is not the naked back. It is the uncorrelated tournament roster — nine good players from nine different situations, each one defensible, collectively a bet that nine independent things go right at once. That roster has a fine median and a ceiling far below what the contest requires.

The one-sentence version

Correlation moves probability from the middle of your distribution into both tails. Tournaments pay for one tail and cash games are killed by the other, so the question is never "should I stack?" but "which contest is this, and how many of my nine slots am I willing to spend on a single game script?"

GameScript expresses these as explicit rules on a build — minimum pass catchers, minimum opposing skill players, maximum players per team — and reports how many of the finished lineups satisfied each one, so the shape you asked for and the shape you got are two things you can compare rather than assume.