Lesson 7 of 10

Teaching people, then stepping back

When is an expert’s time better spent teaching a team than doing the work, and will the support step back once the team is ready?

The problem

The expert everyone queues for

A shared route that does the hard cases forever becomes a queue. A shared route that teaches can make itself unnecessary. This lesson asks when teaching pays, and whether the support will actually step back.

Go back to the pump company. Warranty Claims and Field Repair Dispatch both use an AI reader on scanned documents. Both have hit the same failure: the reader skips a clause printed sideways in a scanned annex. One senior engineer understands why, and knows how to check for it. Every hard case goes to that one person. The queue grows.

There are two ways out. The engineer can keep doing the hard cases, which is reliable today and leaves the company exactly as dependent tomorrow. Or the engineer can stop for a while and teach someone in Dispatch to handle the familiar cases alone, keeping only the exceptions. Teaching costs the engineer’s time now, and it costs the learner’s time too. It pays later, if it works.

This is the everyday form of the question behind this whole site. Most of what an AI capability function does for a company is not building models. It helps teams introduce, adopt and use AI: choose a tool, learn its failures, check its output. Scarce specialist time is the resource being spent. In the sharing equation of lesson 1, support is how a shared route lowers $a$ , the cost of adapting shared work to each team. But support that never hands over keeps the teams waiting, which is the cost $w$ , and keeps them from learning what the centre knows.

Two separate questions hide here. First: is teaching worth the expert’s time at all? That depends on whether learning works, whether it lasts, and whether there is work for the newly able person to do. Second: once the team can manage, will the people providing support step back? That depends on what stepping back is worth to them. The first question needs a careful sum. The second needs a little game theory.

The intuition

Doing the work, or giving it away

Start with the centralized answer: the expert does every hard case. It is the safest choice for today’s work. Nobody untrained touches a document where a missed exclusion is expensive. But the company learns nothing it did not already know. The queue stays. And when the expert is ill, busy or gone, the work stops.

The opposite answer leaves each team to learn alone. It needs no expert time. It is also slow and expensive in a way that is easy to miss: every team makes the same early mistakes, on real documents, and pays for them. AI help can hide the problem for a while. A team can produce acceptable work with an assistant without learning to do the work itself. A small randomized coding study found weaker independent quiz scores right after learning with AI help, and a high-school mathematics trial found an exam penalty with unrestricted AI help that teacher-designed safeguards largely removed. Those settings are far from ours; they only warn that finished work is not proof of a skill.

Teaching sits in between. The old idea of cognitive apprenticeship describes it well: show your reasoning, let the learner practise, then fade the support. It can go wrong in three plain ways. The teaching may not work. What was learned may fade before the next hard case arrives. And there may be no second job for the learner to do, in which case the expert’s time bought nothing.

Then comes a fourth way, which is different in kind. The people who support a team are not neutral. Supporting a team can bring budget, visibility, a reason to exist. If staying involved is worth something to them, they may keep advising a team that no longer needs it. Worse, the team may see that coming and stop investing in its own learning, since being ready will not change who does the work. No single planner makes that mistake on purpose. It comes from two parties, each sensibly pursuing its own interest. That is why the second half of this lesson is a game, not a sum.

The model

Teach, test, then let go

Take the smallest case that keeps every real constraint. There are two jobs to do this period. The expert can do one, worth 8. A learner who has become able to work alone can do the second, also worth 8. A learner who is not able, but is sent out alone anyway, botches it: the botched job costs 24, the price of a missed exclusion. Whether the learner is able is hidden; only results are seen.

Mentoring makes the learner able with chance $p_{\text{learn}}$ . It costs 1 in time: half the learner’s, half the expert’s displaced work. Then a short test, done without AI help, costs a further 1/4 of the expert’s time. It passes an able learner 3 times in 4, and an unable learner 1 time in 4. The rule is simple: the learner works alone only after a pass. Last, the skill must still be there when the second job arrives; it is, with chance $p_{\text{keep}}$ . If nobody is taught, the expert does one job and the second goes undone: value 8.

Add it up. The expert’s job is always worth 8. An able learner passes and keeps the skill in a share $p_{\text{learn}} \times 3/4 \times p_{\text{keep}}$ of cases and earns 8. Every other pass sends an unable person out alone and costs 24. Subtract the 5/4 spent on mentoring and the test.

Derived by hand · value of teaching, testing, then letting go

\begin{aligned}V_{\text{teach}}&=8+\tfrac34\,p_{\text{learn}}\bigl(32\,p_{\text{keep}}-24\bigr)-\tfrac14\bigl(1-p_{\text{learn}}\bigr)24-\tfrac54\\[2pt] V_{\text{stay}}&=8\end{aligned}

In words: the expert’s job, plus what passing learners earn or lose on the second job, minus the time spent teaching and testing; against the expert working alone.

When the skill always lasts ( $p_{\text{keep}} = 1$ ) the first line collapses to 3/4 + 12 $p_{\text{learn}}$ . Teaching beats leaving the expert on the work exactly when that is more than 8:

Checked in Lean · concrete_learning_threshold

V_{\text{teach}}>V_{\text{stay}}\iff p_{\text{learn}}>\tfrac{29}{48}\approx 0.60\qquad(p_{\text{keep}}=1)

In words: teaching pays only if it works about three times in five.

And when teaching works three times in four, the skill must last almost surely:

Checked in Lean · concrete_retention_threshold

V_{\text{teach}}>V_{\text{stay}}\iff p_{\text{keep}}>\tfrac{65}{72}\approx 0.90\qquad(p_{\text{learn}}=\tfrac34)

In words: if more than one learned skill in ten fades before the job arrives, the botched jobs eat the whole gain.

What each letter stands for

$p_{\text{learn}}$: Chance teaching makes the learner able to work aloneprobability
$p_{\text{keep}}$: Chance the skill is still there when the second job arrivesprobability
$V_{\text{teach}}$: Value of teaching, testing, then letting the learner work alone after a passillustrative units
$V_{\text{stay}}$: Value of leaving the expert on the work and teaching nobodyillustrative units

Lesson 7 · Figure 1

Teaching pays only when it works often enough and the skill lasts.

Shown at Chance teaching works: 75 %; Chance the skill lasts until the job: 100 %. Turn on JavaScript to move the values.

Above 60.4%, teaching pays: the learner who passes the test can take the second job alone, and that job repays the mentor’s time.

Teach, test, let go: 39/4 ≈ 9.75
No training: 8
Best plan a careful planner picks: 39/4 (mentor first)
Chance an unready learner is sent out alone: 1/16 ≈ 6.3%

One expert, one learner, two jobs. The learner works alone only after passing a test that also passes one unable learner in four. With the skill lasting (100%), the straight line 3/4 + 12 × (chance teaching works) and its break-even 29/48 are checked in Lean; at other values the line comes from the tested model code. Lower the chance the skill lasts and watch the break-even leave the chart.

Model result · checked in Lean

Look at what the figure does when you lower $p_{\text{keep}}$ . At 90% the break-even moves almost to the right edge; at 67% it is gone. Teaching three people who forget before the next hard case is worse than teaching nobody, because the test cannot see the forgetting that happens after it.

The test has a price

Why test at all? Without the rule, the planner can mentor and then let the learner practise instead, and send the learner out without a test. In this case that earns 21/2, more than the 39/4 of the tested route. But it doubles the chance that an unable person handles a document alone, from 1 in 16 to 1 in 8. The test is not free safety. It is a trade between value and risk, and the model only prices it at 24 per botched job. A team for which a missed exclusion is much worse than that, as it is for Warranty Claims, would weigh the doubling differently. The sum does not decide that for you; it shows you what you are trading.

The support game

Now the second question. Suppose teaching works. Will the support step back? To see the incentive cleanly, the next model makes a strong simplifying assumption: everyone can see how ready the team is. Readiness has three levels, 0, 1 and 2. The model also tracks whether the team has written a usable diagnostic record of the failure, the kind of note that lets someone else recognize the sideways clause next time.

There are three rounds. In each, the support team moves first and chooses to teach alongside the product team, to advise it (possible only once the team is at level 1 and a record exists), or to leave the work to the team. The product team then chooses whether to put in learning effort, which costs it 1/2, and whether to write the record, which costs 1/4 with support and 1/2 alone. Effort with teaching moves the team from level 0 to 1; effort with advice moves it from 1 to 2. Practice and the record are separate: a team can learn without writing anything down, and a good record can help others without changing anyone’s skill.

The work is worth 6 to the company when it is taught or advised, and 0, 3 or 6 when the team works alone at level 0, 1 or 2. Teaching costs the company 2 in support time; advice costs 1. The product team cares about its own payoff: 2 when taught, 3 when advised, and 0, 1 or 5 alone at each level, minus its own effort. The support team cares about the company’s value, plus a private benefit $\theta$ for every round in which it teaches or advises.

Assumption · what each side wants

\text{support team's payoff}=\text{company value}+\theta\times(\text{rounds it teaches or advises})

In words: the support team values what the company values, plus something of its own for staying involved.

Everything turns on the last round, when the team is ready. Handing over is worth 6 to the company. Advising a ready team is worth 6 minus 1 of support time, so 5, plus $\theta$ to the support team. The gap is:

Checked in Lean · ready_state_release_gap

V_{\text{alone}}-\bigl(V_{\text{advise}}+\theta\bigr)=6-(5+\theta)=1-\theta

In words: the support team hands a ready team its work only while its private benefit from staying is below 1.

What each letter stands for

$\theta$: Private benefit the support team gets from each round it teaches or advisesillustrative units per round
$V_{\text{alone}}$: Company value of a round in which the ready team works aloneillustrative units
$V_{\text{advise}}$: Company value of a round in which the support team advises a ready teamillustrative units

With $\theta$ below 1, the stable plan is the hopeful one: teach in round 1 while the team learns and writes the record, advise in round 2 while it learns more, hand over in round 3. The company earns 55/4. With $\theta$ above 1, the support team keeps advising in round 3. The product team sees this coming. Its round-2 effort would make it ready, but being ready no longer changes who does the work, so it saves the effort. The company earns 53/4, while the support team’s own payoff goes up.

Lesson 7 · Figure 2

When staying involved pays the support team, the team stops learning.

Shown at What staying involved is worth to the support team, $\theta$ : 0.50 units per round. Turn on JavaScript to move the values.

Below 1, the support team steps back once the product team is ready. The team keeps learning, and the company gets 55/4 ≈ 13.75.

Company value: 55/4 ≈ 13.75
Product team’s own payoff: 35/4 ≈ 8.75
Support team’s own payoff: 59/4 ≈ 14.75
Last round: the team works alone in round 3

Three rounds, readiness visible to both sides. Each point solves the game exactly with the tested model code. Lean checks the stable plans at $\theta = 0$ and $\theta = 2$ and the gap $1 - \theta$ that places the switch at 1. At exactly 1 the support team is indifferent and the model picks no winner.

Model result · computed

Can a promise fix this? Suppose the support team promises only this: once the team is fully ready, it will hand over. With $\theta = 2$ , that promise is not enough. The support team can simply avoid the advising round that would make the team ready, and keep teaching instead. Over the last two rounds that earns it 13, against 25/2 for developing the team and then letting go. A rule that fixes the whole path, teach, then advise, then hand over, restores the company’s 55/4. But it costs the support team 3/2 of its own payoff compared with acting freely. It will not sign up for it voluntarily; someone has to mandate the rule, or change what the support team is rewarded for.

What Lean checks

What the proofs cover

Lean is a program that checks every step of a proof. "Checked in Lean" below means the conclusion follows from the stated assumptions; it does not mean the assumptions describe your company. The full files are linked from each result.

Checked in Lean

With the skill always lasting, teaching, testing and letting go beats leaving the expert on the work exactly when teaching works with a chance above 29/48.

concrete_learning_threshold in PeopleDevelopment.lean

Show the Lean statement

theorem concrete_learning_threshold (l : Rat) :
    8 < fixedValue {base with mentorLearning := l} (mentorAssessment true) ↔ 29/48 < l

Checked in Lean

With teaching working three times in four, the same plan wins exactly when the skill lasts with a chance above 65/72.

concrete_retention_threshold in PeopleDevelopment.lean

Show the Lean statement

theorem concrete_retention_threshold (r : Rat) :
    8 < fixedValue {base with retention := r} (mentorAssessment true) ↔ 65/72 < r

Checked in Lean

Dropping the test and practising instead earns more (21/2 against 39/4), while the chance an unable learner is sent out alone becomes 1/8.

ungated_practice_exceeds_gated_witness · ungated_bad_assignment_mass in PeopleDevelopment.lean

Show the Lean statement

theorem ungated_practice_exceeds_gated_witness : fixedValue base (mentorAssessment true) < fixedValue noGate mentorPractice
theorem ungated_bad_assignment_mass : (afterMentorPractice noGate).weights (false,false) = 1/8

Checked in Lean

In the three-round support game, for $\theta = 0$ and for $\theta = 2$ , neither side can do better by changing its own plan alone, at any round, whatever happened before.

concrete_sponsor_history_deviations · concrete_team_history_deviations in AdaptiveSupport.lean

Show the Lean statement

theorem concrete_sponsor_history_deviations (rent : Bool) (n : Nat) (hn : n ≤ 3)
    (past : List (State × Mode)) (s : State) (pi : HistoryStrategy State Mode)
    (hp : HistoryFeasible (sponsorProblem rent) pi n past s) :
    historyValue (sponsorProblem rent) pi n past s ≤
      recommendedValue (sponsorProblem rent) (sponsorSigma rent) n s
theorem concrete_team_history_deviations (rent : Bool) (n : Nat) (hn : n ≤ 3)
    (past : List ((State × Mode) × Action)) (p : State × Mode) (hv : validTeam p)
    (pi : HistoryStrategy (State × Mode) Action)
    (hp : HistoryFeasible (teamProblem rent) pi n past p) :
    historyValue (teamProblem rent) pi n past p ≤
      recommendedValue (teamProblem rent) (teamSigma rent) n p

Checked in Lean

With a ready team in the last round, handing over beats advising, for the support team, by exactly $1 - \theta$ .

ready_state_release_gap in AdaptiveSupport.lean

Show the Lean statement

theorem ready_state_release_gap (rho : Rat) :
    welfare .r1 .local .coast - (welfare .r1 .advise .coast + rho) = 1-rho

Checked in Lean

A support team that values staying involved lowers the company’s value from 55/4 to 53/4, and a promise to hand over only once the team is ready does not stop it from blocking the team’s last step.

retention_reduces_welfare · release_only_promise_does_not_prevent_obstruction in AdaptiveSupport.lean

Show the Lean statement

theorem retention_reduces_welfare : organizationValue true 3 .n0 < organizationValue false 3 .n0
theorem release_only_promise_does_not_prevent_obstruction : developThenReleaseValue < obstructValue

Checked in Lean

A rule fixing the whole path (teach, advise, hand over) costs the support team 3/2 of its own payoff, compared with acting freely.

voluntary_contract_participation_loss in AdaptiveSupport.lean

Show the Lean statement

theorem voluntary_contract_participation_loss :
    recommendedValue (sponsorProblem true) (sponsorSigma true) 3 .n0 - contractedSponsorValue 3 .n0 = 3/2

In the Lean file the support team is called the sponsor, its private benefit is called rho in the gap theorem, and in the game proofs it is fixed at 0 or 2 rather than any $\theta$ . So here is what is not proved. The figure’s sweep over $\theta$ is computed, not proved; the proofs cover the two cases and the gap $1 - \theta$ that explains the switch. The game proofs show these plans are stable; they do not show that no other stable plan exists, and at one point the game never reaches, two choices tie and the proof picks one. Uncertain learning, changing tasks and teams that learn alone were explored by computer, not proved. And nothing here measures a real learning chance, a real forgetting rate, or whether a short test detects the skill that matters.

What it means

Teach where it lasts; pay for letting go

Teaching is an investment with three conditions. It must work often enough, the skill must last until it is used, and there must be work waiting for the person who learned it. Fail any one and the expert’s time is better spent on the work. In this example, a single extra job is not enough: with only one job available, teaching loses (27/4 against 8). Choosing not to teach in those conditions is not a failure of the shared route. It is the right answer.

Assistance can beat independence. If a reliable AI tool lets an untrained person do the second job well with assistance, the best plan is to teach nobody: 12, against 45/4 for teaching. Teaching people to work alone earns its keep when the tool fails sometimes. With the tool down half the time, teaching comes back to 21/2 against 10. The model makes the point concrete: plan for the outage, not the demo.

A test before letting go is a choice about risk. It costs value, here 21/2 down to 39/4, and halves the chance of an unable person working alone. Where a single mistake is very expensive, it is likely worth it. Where mistakes are cheap and caught downstream, as when a dispatcher checks every route, perhaps not. The same model, two teams, two different right answers.

Who supports the team matters less than what the support is rewarded for. The game gives a capable local expert and a shared support team the same moves, so neither has an advantage by name; that is a design rule of every model on this site, not a finding. The result that matters is about incentives. When staying involved is worth more than 1 per round to the people providing support, they keep advising a ready team, and the team, seeing this, stops learning. A promise to let go is not enough. A rule that enforces handover works, but it takes 3/2 from the support team in this example, so it needs a mandate or a different reward. Baker, Gibbons and Murphy make a related point about delegation a boss could later overturn: authority that is worth giving must also be credible. And Garicano counts a specialist’s attention as a cost even when it helps; our model adds learning through that attention, which is our hypothesis, not his result.

When the shared route loses. It loses when teaching rarely works, when skill fades before use, when there is no work for the trained person, when a reliable tool makes independence unnecessary, and when the expert and every learner can be absent at the same time, which training cannot cover. It also loses when it is compulsory and has its own cost: forcing the same tested curriculum through a route that charges 2 to enter leaves 31/4, below the 8 of teaching nobody. Teams free to skip that route would skip it.

Honest limits. All numbers are illustrative. The support game assumes readiness is visible to everyone, which is exactly what is hard in practice; the first model keeps skill hidden but has no strategy. Joining the two, hidden skill and self-interested support, is open work.

Next: who to reserve, and for which work

Teaching changes who can do the work. It does not decide who should be booked for it, or what happens when two hard cases arrive on the same day and the one person who can do both is already busy. Lesson 8 turns to staffing: generalists or specialists, handoffs and deadlines, and when to hire before the work arrives.