Reinforcement Density in Shaping: Picking Your W Without Guessing

Picking W in the percentile schedule is a decision a Board Certified Behavior Analyst (BCBA) makes before the formula ever runs. The formula tells you when to reinforce. W tells you how often. Those are two different jobs. The percentile schedule equation, K = M + 1(1 - W), gives you the threshold a response has to beat. But you, the clinician, pick W based on what the client can afford to lose. That choice shapes the whole program. Pick a high W and the learner barely touches extinction. Pick a low W and the learner runs into extinction often, which can actually speed things up. Neither one is right by default. The right pick depends on the response class, the risk, and the goal. This page walks through how to make that call.

What W means inside the percentile schedule#

W is the density of reinforcement. In plain words, it is the share of responses that meet criterion and get reinforced. A W of 0.9 means about nine out of ten qualifying responses get the reinforcer. A W of 0.1 means about one out of ten do. The formula uses W to set K, the value the next response has to beat. A bigger W lowers K. A smaller W raises K. So W does two things at once. It sets how often the learner contacts reinforcement, and it sets how strict the bar is on each new try.

That is why W matters more than M. M is how far back you look to set the bar. If you look at the last five responses or the last ten, K shifts a little, but not a lot. W shifts the whole feel of the program. The talk is direct on this:

Increasing the W has the biggest effect. So increasing the reinforcement density says larger W more reinforcement, meaning the learner is less likely to contact extinction and more likely to experience what's akin to errorless learning. From the talk — Matt Harrington

So when you sit down to write a shaping program, the first real decision is W. Not the steps. Not the criterion. W.

The high-W decision: when errorless makes sense#

A high W, somewhere in the 0.7 to 0.9 range, pulls the program toward errorless learning. The learner gets reinforced often. The steps stay small. Progress looks steady but slow on a graph. Each session moves the K up a little, and the learner rarely misses the bar.

Pick a high W when the cost of an extinction burst is too high to risk. Self-injurious behavior is the clearest case. So is aggression that can hurt staff, family, or peers. So is any response class where one bad session sets the program back weeks. In these cases, the small slow steps are a feature, not a bug. The learner stays in contact with reinforcement, the response class stays calm, and the staff can run the program without fear.

If the risk of this intervention not working and the risk of an extinction burst is head banging on concrete, we need to be very cautious with that response class. So you would want to do a high reinforcer density, which means you would create your intervention with very small shaping steps. From the talk — Matt Harrington

The trade is speed. A high-W program will not move fast. The acquisition graph climbs gently. If a parent or supervisor asks why progress looks slow, the answer is built in: the program was designed to keep the learner safe, and slow is the price of that.

The low-W decision: when extinction-induced variability helps#

A low W, somewhere in the 0.1 to 0.3 range, pulls the program the other way. The learner often hits the bar and gets nothing. That is extinction. And extinction has a side effect that most clinicians forget: it shakes the response loose. The learner tries new things. Some of those new things are bigger jumps than shaping would ever ask for. The graph jumps around more, but the average slope is steeper.

What about for quick acquisition? Then we can decrease the reinforcement density. A smaller W means less reinforcement, which means the learner is going to experience extinction more often, which is not necessarily a bad thing because when you experience extinction, you also have extinction-induced variability, which means that you're more likely to make big jumps in your learning. From the talk — Matt Harrington

Pick a low W when extinction is safe in this response class. Social skills, tolerance for a non-preferred task, length of independent play, length of focused work time. None of these put the learner at risk if a session goes sideways. If the worst case of an extinction burst is some whining or a quick refusal, low W is on the table.

The con is exactly what the pro is. The learner will hit extinction more often. If the team is not ready for that, they will pull back on the program and quietly bump W up themselves. So low W only works when staff understand the plan and can hold the line.

Matching W to response-class risk#

The clean way to make this call is to ask one question: what is the worst case if this learner hits extinction in this response class? If the worst case is dangerous, high W. If the worst case is uncomfortable but safe, low W. If it sits in the middle, start in the middle, around 0.5, and adjust based on data.

A rough decision table:

• Self-injury, aggression, elopement near traffic: W in the 0.8 to 0.9 range.
• Severe tantrum, property destruction in a clinic room: W in the 0.6 to 0.8 range.
• Non-compliance, mild tantrum, work refusal: W in the 0.4 to 0.6 range.
• Social skill, tolerance task, focused work time: W in the 0.2 to 0.4 range.

This is not a rulebook. It is a starting place. You will move W based on what the data show in the first few sessions. If a low-W program produces a burst the team cannot handle, raise W. If a high-W program is moving so slowly it stops being a program, lower W.

How W choice changes the shape of your acquisition graph#

A high-W program produces a graph that climbs in small, steady steps. The line is smooth. There are very few dips. Sessions look the same as the session before. This is what errorless learning looks like on paper. It is calming for the team and easy to explain to parents.

A low-W program produces a graph that jumps. Some sessions, the learner makes a huge jump. Other sessions, the learner drops back. The line wobbles, but the overall slope is steeper than the high-W graph. The big jumps come from extinction-induced variability. The drops come from sessions where the new attempts did not work and the learner reverted.

If you show both graphs to a supervisor without context, the high-W graph looks safer and the low-W graph looks faster. Both can be the right answer. The graph is just a side effect of the W choice. So before you defend a graph, defend the W. That is the real decision.

Writing W into your BIP so the RBT can run it#

A W value is useless if it lives only in your head. It has to show up in the Behavior Intervention Plan (BIP) in language the Registered Behavior Technician (RBT) running the session can follow. That means three things.

First, write the rule, not the math. The RBT does not need to see K = M + 1(1 - W). The RBT needs to see something like: reinforce any response that beats the best of the last five tries, and reinforce roughly seven out of ten of those. The math is your job. The rule is theirs.

Second, write the look-back window. M is how many past responses define the bar. Five is common. Ten is fine. Pick one and write it down. If you change M mid-program, write the new number on a dated note in the BIP so the next clinician can see why.

Third, write the W and why you picked it. One sentence is enough. Something like: W is set at 0.8 because the response class includes head-banging and the cost of an extinction burst is high. This sentence is what protects you in an audit and what trains the next BCBA on the case.

So we naturally do this. The difference here is that instead of just doing it based on instinct, you can do it objectively by reinforcing that W value. What that means is that we can better train our staff to follow through with the intervention over and over again and not have as many fidelity errors. From the talk — Matt Harrington

When the W is written down, staff fidelity goes up. RBTs run the program the way it was designed. Supervisors can spot drift. And when a new behavior analyst takes the case, the plan reads like a plan, not a vibe.

Frequently asked questions#

What reinforcement density should I use for a head-banging response class?

Start at a W of 0.8 or higher. Head-banging is the textbook case for high density. The cost of an extinction burst is direct physical harm, so the program has to keep the learner inside reinforcement. Small shaping steps, a tight criterion bar, and a high W. Move W down only after the response class has been calm for several weeks and the team agrees the risk has dropped.

Can I change W mid-program if acquisition stalls?

Yes, and you should. A stalled program is a sign that W is wrong for the current phase. If the learner is sitting at the same K for too many sessions with steady reinforcement, the W is too high and the bar is not moving. Drop W by 0.1 or 0.2 and watch the next three sessions. If the learner is missing the bar most of the time and the team is fighting bursts, W is too low. Raise it. Document each change with a date and a reason.

How do I document my W choice in a BIP for audit?

Write one sentence that names W, names the response class, and names the risk that drove the call. Add a second sentence on M, the look-back window. Add a third sentence on how the RBT will know whether to reinforce a given response, in plain words. Three sentences in the BIP procedure section is enough. An auditor wants to see that the choice was deliberate, not random.

Watch the full breakdown#

The full one-hour talk walks through the percentile schedule equation, contingency strength, and how both tie back to real BIP decisions. If you are sizing W for a current case, the talk gives you the working examples you need.