Lesson 10 of 10

Who pays for the backup nobody needs today

What should a company preserve before an expert leaves, and who will pay for a backup that every team values but no team needs today?

The problem

The expert who leaves on Friday

A capability that lives in one person’s head is lost when that person leaves. This lesson asks what is worth preserving before that happens, and why the teams that would gain from the backup may still not pay for it.

At the pump company, one engineer worked out why the AI reader skips exclusions printed sideways in scanned annexes, and how to check for it. Warranty Claims and Field Repair Dispatch both rely on that knowledge, though neither needs it this week. Now the engineer may move on. Perhaps to another company, perhaps only to another project; either way, the knowledge stops being on call.

Three separate decisions are hiding in that one worry. Should someone write a usable record: the diagnosis, the test documents that expose the failure, the check that catches it? Should someone be taught to handle the next case? And should the support for this kind of work stay open through a quiet spell, or close and reopen later? Each has a different cost, and they need not have the same answer.

Then there is the awkward fourth question: who pays? The record and the trained backup help both teams. Each team’s own budget pays only for what that team buys.

This is the loss term $l$ of the sharing equation in lesson 1, seen over time. A company that loses the lesson when the person leaves pays for it again, which is the second invoice in its purest form. The evidence that this matters is real, if from other fields. When eminent life scientists die unexpectedly, their collaborators’ output falls and stays lower. Studies of aircraft production and of organizational forgetting find that experience depreciates, and separate losses through people leaving from skills simply fading. None of these calibrates an AI team; they say the risk is not imaginary.

The intuition

Insurance nobody wants to buy alone

Backup is a kind of insurance. You pay now, for a loss that may not happen, to soften it if it does. That makes it easy to underfund in two different ways.

The first is a planning question with one decision-maker. Preservation is never as good as the expert: a record and a trained person together cover part of what one experienced engineer did, not all of it. So the question is whether the partial cover is worth its cost, given how likely the loss is and how likely the teaching is to work. That is optimization: list what can be bought, count the value of each, keep the best.

The second is about incentives. Suppose each team decides for itself. A record written by Warranty also helps Dispatch, but Warranty pays all of it. Each team buys protection as if only its own benefit counted, and the shared benefit leaks away. The result can be that nobody pays for a backup that is clearly worth its cost to the company. Economists call this a public good. To see whether it happens, and when, we need game theory: what each team does, given what it expects the other to do.

Here a shared function has a natural role: it can pay for what everyone values and nobody buys. But it is not the only answer. Two teams can sign an agreement to split the cost. And a shared budget has costs of its own: someone has to run it, and it may preserve things nobody needs, such as a second copy of the same documentation. The model has to let both alternatives win when they should.

The model

Preserve, then decide who pays

What to buy before the expert goes

Take two periods. The expert works in the first and certainly leaves after it. Opening the support service costs 1, running it costs 1 a period, and closing it costs 1/2. The expert produces 8. After the expert has gone, what remains can still do some of the work: a record alone produces 2, a successful apprentice alone 3, and both together 6. Even the best backup is short of the expert’s 8.

A record costs $C_{\text{record}} = 1$ to make and 1/4 a period to maintain. An apprenticeship costs $C_{\text{teach}} = 1$ and works with chance $p_{\text{learn}}$ . Both must be bought while the expert is still there; nobody can teach what nobody knows. The best plan, whatever is bought, is to run the service in the first period and close it in the second. Counting each choice:

Derived by hand · checked in Lean at a one-in-two chance that teaching works

\begin{aligned}V_{\text{none}}&=\tfrac{11}{2}, & V_{\text{record}}&=\tfrac{25}{4},\\ V_{\text{teach}}&=\tfrac92+3\,p_{\text{learn}}, & V_{\text{both}}&=\tfrac{21}{4}+4\,p_{\text{learn}}\end{aligned}

In words: preserving nothing is worth 11/2; a record alone adds a fixed amount; an apprenticeship adds more the likelier it is to work, and more still when there is a record to work from.

Buying both beats the record alone exactly when 21/4 + 4 $p_{\text{learn}}$ exceeds 25/4, that is, when teaching works more than one time in four. At one in two, the best plan buys both and is worth 29/4. Raise the cost of preservation and the picture changes: at 4 per item, nothing is worth buying.

What each letter stands for

$p_{\text{learn}}$: Chance an apprenticeship produces a usable backup (as in lessons 7 and 9)probability
$C_{\text{record}},\ C_{\text{teach}}$: Cost of making one record, or of one apprenticeship attemptillustrative units
$V_{\text{none}},\ V_{\text{record}},\ V_{\text{teach}},\ V_{\text{both}}$: Value of preserving nothing, a record only, an apprenticeship only, or bothillustrative units

Lesson 10 · Figure 1

Once teaching works one time in four, buy the apprenticeship as well as the record.

Shown at Chance the apprenticeship works: 50 %; Cost of each record or apprenticeship: 1.00 units. Turn on JavaScript to move the values.

At a 50% chance that teaching works, buy both a record and an apprenticeship: together they cover the most work once the expert has gone. The best choice changes: record only below 25%, record and apprenticeship above.

Record and apprenticeship: 29/4 ≈ 7.25
Record only: 25/4 ≈ 6.25
Apprenticeship only: 6
No preservation: 11/2 ≈ 5.5

The expert certainly leaves after the first period; backup covers part of the expert’s work. Each line is the best plan after the first choice, solved by the tested model code. Lean checks the four values at a one-in-two chance and cost 1, and that no plan beats 29/4 there. Raise the cost to see preservation stop paying.

Model result · checked in Lean

The service itself has a separate answer. Take a version where the expert stays and the work comes in bursts: needed, then quiet, then needed again. With one quiet period, keeping the service open through it is cheaper than closing and reopening. With two quiet periods, closing and reopening wins. A lull is a reason to close only when it is long enough to repay the cost of starting again.

Who pays

Now give the decision to the two teams. The expert leaves with chance $p_{\text{leave}}$ . Each team chooses, at the same time, to fund nothing, a record, an apprenticeship, or both. A record or an apprenticeship attempt costs 3/2, paid entirely by the team that funds it. Each attempt works with chance 1/2, independently. Whatever is funded helps both teams: a record is readable by anyone, and a successful apprentice can cover one backup job for each team. If the expert leaves, a team’s backup is worth 2 with a record only, 3 with a trained person only, and 6 with both.

Assumption · what one team weighs

\text{team's gain}=p_{\text{leave}}\times(\text{backup it gets if the expert leaves})-(\text{what it paid itself})

In words: a team gets the backup that both teams’ choices produce, but pays only for its own.

For the company, the best package is one record and one apprenticeship. Across both teams it is worth:

Checked in Lean · one_each_welfare, one_each_gain_threshold

V_{\text{package}}=2\,p_{\text{leave}}\Bigl[\bigl(1-p_{\text{learn}}\bigr)\,2+p_{\text{learn}}\,6\Bigr]-C_{\text{record}}-C_{\text{teach}}=8\,p_{\text{leave}}-3

In words: two teams, each getting the record’s backup or, if the apprenticeship works, the full backup, against the cost of one of each.

So the package is worth buying once the expert is more likely than 3 in 8 to leave. At an even chance it adds 1. But look at it from one team’s side. Funding a record alone, against a partner who funds nothing, gains $p_{\text{leave}} \times 2 - 3/2$ ; funding both gains $p_{\text{leave}} \times 4 - 3$ . Both are negative until $p_{\text{leave}}$ reaches 3/4. Below that, funding nothing is the better choice for each team whatever the other does. So between 3/8 and 3/4, the backup is worth buying and nobody buys it.

What each letter stands for

$p_{\text{leave}}$: Chance the expert leavesprobability
$V_{\text{package}}$: Value to the company of one record and one apprenticeship, after their costillustrative units
$C_{\text{admin}}$: Cost of running the shared route that pays for the packageillustrative units

Lesson 10 · Figure 2

Between 37.5% and 75%, the backup is worth buying and no team buys it.

Shown at Chance the expert leaves: 50 %; Cost of running a compulsory shared route: 0.00 units. Turn on JavaScript to move the values.

Between 37.5% and 75%, one record and one apprenticeship would add 1 for the company, yet each team pays the full bill and gets only its own share, so neither buys them.

Best the company could fund: 1
What the teams fund alone: 0
Value nobody pays for: 1

Two teams; each pays for what it funds and shares what it gets. Every point checks all 16 pairs of choices with the tested model code; at each break-even both lines come from straight-line payoffs. Lean checks the case where the expert leaves with chance 1/2: nothing funded is the only stable outcome, the best package adds 1, and a route cost above 1 makes the compulsory route lose.

Model result · checked in Lean

A shared route can close the gap. Pay for one record and one apprenticeship from a common budget, and the company gets the 1. But the route has its own cost, $C_{\text{admin}}$ . If the route is compulsory, the company gets $1 - C_{\text{admin}}$ ; above 1, it loses to doing nothing. If it is optional and can decline before paying, it never does worse than nothing. And the teams have another way to get the same result: a binding agreement to fund the package together and split the bill. With no administration cost, each team gains 1/2. The shared route wins only if it is cheaper to run than that agreement is to write and enforce.

What Lean checks

What the proofs cover

Lean, a program that checks every step of a proof, verifies the planning results by bounding every plan, however it reacts to whether the apprenticeship worked, and the funding game by checking every pair of choices.

Checked in Lean

With teaching working half the time and each item costing 1, the first-period choices are worth 0 (close now), 11/2, 25/4, 6 and 29/4, and no plan beats 29/4.

c1_initial_action_values · c1_every_history_policy_bounded in CapabilityContinuity.lean

Show the Lean statement

theorem c1_initial_action_values :
    (feasibleActions initial).map (actionValue c1Model c1U 1 initial) =
      [0,11/2,25/4,6,29/4]
theorem c1_every_history_policy_bounded (pi : Policy Action Outcome 2)
    (hp : Feasible c1Model 2 initial pi) : evaluate c1Model 2 initial pi ≤ 29/4

Checked in Lean

The best plan keeps the service open through one quiet period, and closes and reopens it around two.

c5_selected_standing_path · c6_selected_pop_up_path in CapabilityContinuity.lean

Show the Lean statement

theorem c5_selected_standing_path :
    c5Sigma 2 initial = .on ∧
    c5Sigma 1 ⟨true,true,false,false⟩ = .on ∧
    c5Sigma 0 ⟨true,true,false,false⟩ = .on
theorem c6_selected_pop_up_path :
    c6Sigma 3 initial = .on ∧
    c6Sigma 2 ⟨true,true,false,false⟩ = .off ∧
    c6Sigma 1 initial = .off ∧ c6Sigma 0 initial = .on

Checked in Lean

At an even chance the expert leaves, funding nothing is strictly better for each team whatever the other does, so funding nothing is the only stable outcome.

base_none_strictly_dominates · base_unique_equilibrium in PreservationInvestment.lean

Show the Lean statement

theorem base_none_strictly_dominates (rival other : Action) (h : other ≠ Action.none) :
    utility base other rival < utility base .none rival
theorem base_unique_equilibrium (a b : Action) :
    Equilibrium base a b ↔ a=.none ∧ b=.none

Checked in Lean

At an even chance, the company’s best is 1, reached exactly by the four ways of funding one record and one apprenticeship; and in general one of each beats nothing exactly when its expected backup across both teams exceeds its cost.

base_all_planner_maximizers · one_each_gain_threshold in PreservationInvestment.lean

Show the Lean statement

theorem base_all_planner_maximizers (a b : Action) :
    welfare base a b = 1 ↔ OneEach a b
theorem one_each_gain_threshold (e : Economy) :
    0 < welfare e .library .apprentice ↔
      e.recordCost+e.apprenticeCost <
      2*e.departure*((1-e.learning)*e.recordValue+e.learning*e.bothValue)

Checked in Lean

A compulsory shared route that funds the package loses to funding nothing exactly when its own cost exceeds 1; and two teams that fund the same package by agreement get exactly what the shared route gets.

base_fee_reversal · same_programme_coalition_parity in PreservationInvestment.lean

Show the Lean statement

theorem base_fee_reversal (fee : Rat) :
    sponsored {base with fee:=fee} .library .apprentice < 0 ↔ 1 < fee
theorem same_programme_coalition_parity (e : Economy) :
    coalitionTotal e = sponsored e .library .apprentice

What is not proved. The lines in $p_{\text{learn}}$ and the break-evens 3/8 and 3/4 in $p_{\text{leave}}$ are derived by hand and computed by the tested model code; Lean checks the cases at $p_{\text{learn}} = 1/2$ and $p_{\text{leave}} = 1/2$ , plus the general package formula. Teams that randomize their choices were not studied. The models have no record that goes out of date, no skill that fades, no replacement hire once the expert has gone and no doubt about whether the apprentice really learned. Nothing is measured: every chance and cost is illustrative.

What it means

Fund the insurance, not the department

Preserve while the expert is still there. Teaching needs a teacher. A record and a trained person are cheap to buy before the expert leaves and impossible after. The best plan in the example buys both in the first period and then closes the service: preservation is an act at a moment, not a permanent program.

Records and people are different kinds of cover. A record is reliable but narrow; a trained person is broader but uncertain. Below a one-in-four chance that teaching works, buy the record alone. Above it, the two together beat either. When preservation is expensive, buy neither, and accept the loss.

Close a quiet service only for a long lull. One quiet period is cheaper to wait through than to close and reopen; two are not. The same logic as keeping or releasing a person in lesson 8.

Backup is underfunded for a structural reason. Between 3/8 and 3/4, each team, acting sensibly for itself, funds nothing, though the company would gain. This is the clearest case on this site for something above the teams: a budget, or a rule, that pays for what everyone values and nobody needs today. Note what makes it work: the benefit is shared and the cost is not. That is a reason to fund the insurance, not a reason to build a department.

When the shared route loses. Every model here gives a capable group of teams the same options as a shared route; the label earns nothing. So the shared route loses when its own running cost is above the gain (above 1 in the example, it is worse than nothing); when the teams can agree among themselves at lower cost; when the expert is unlikely to leave, so there is nothing to insure; when teaching does not work, so only a record is worth buying; and when it pays for duplicate cover, which here turns a gain of 1 into a loss of 1. Above a three-in-four chance of losing the expert, a team funds backup on its own, and the case for a shared budget shrinks to the small amount the teams still leave on the table.

Honest limits. One expert, two teams, at most four periods, and backup that never goes stale. Real records age, real tools change underneath them, and real experts rarely vanish without notice. The model shows the shape of the problem, not its size.

Next: what the ten lessons add up to

This is the last lesson. Across all ten, the same pattern holds: a shared route that is optional, free and as well informed as the teams cannot make things worse for a single planner, and it loses through its cost, through compulsion, or through incentives. Where it wins, it wins through something specific: work done once for many teams, one expert serving several, a memory of failures, a fairer way to award scarce help, or insurance that no team buys alone. The research overview collects what is proved and what is not. The business article turns the lessons into decisions, and the management book into one practitioner’s way of running them.