When does doing it once beat doing it n times? · AI capability management

The problem

The second invoice

Two teams in the same company buy the same lesson twice. This lesson asks when they should buy it once.

Picture a company that makes and services industrial pumps. Two of its teams have started using AI to read documents. Warranty Claims reads contracts and scanned annexes to decide whether a repair is covered. Field Repair Dispatch reads manuals and service logs to decide which crew to send. Both need the same hard thing: a system that reads messy scanned pages reliably.

Each team can do the work alone. It picks a vendor model, builds a set of test documents, writes the instructions, finds out where the model fails, fixes what it can and trains its people. Or the company can do the common part once, in a shared route, and let each team adapt the result to its own documents.

This is not only a question about training models. Most companies never train one. They introduce, adopt and use AI: they choose a service, test it on their own work, teach people to use it and keep it working when the vendor changes it. What is scarce is the same in every case: specialist time, compute, or vendor budget. The question is how to spend it so the company learns each lesson once, not once per team.

Throughout this research, the shared route means doing the common work once for several teams, and the local route means each team doing it alone. An AI capability management function (ACM, for short) is whoever runs the shared route: one person, a small network or a team.

The intuition

One payment against many

Sharing has an obvious attraction. The hard part is paid for once. Whoever does it sees every team's documents, so they learn faster. There is one set of tests, one vendor contract and one place to look when something breaks. And the more teams need the same thing, the bigger the saving: the shared route's fixed cost is spread thinner with every team that joins.

Sharing also has costs that the local route never pays. The shared work still has to be adapted to each team: Warranty's contracts are not Dispatch's manuals. Someone has to agree on priorities, which takes meetings. Teams wait in a queue instead of starting today. And a mistake in shared work reaches every team at once.

The local route is the mirror image. It fits the team's own work exactly, needs no meetings and has no queue. But it pays the full cost every time, and each team learns the same painful lesson on its own. When the second team discovers the failure the first team already paid to find, the company receives a second invoice for the same lesson.

So the answer could go either way, and slogans will not settle it. "Centralize for scale" and "keep it close to the business" are both true sometimes. To know which, we have to count. This first lesson is pure counting: one decision-maker, no strategy, no uncertainty. Later lessons add what the counting hides. How much to invest is a question of optimization. What teams choose to say when they compete for scarce help is a question of game theory. Those lessons need the tools. This one only needs a careful list.

The model

Add up both routes

Fix one planning horizon, say a year, and one unit of cost. If each of $n$ teams does the work alone at a cost of $d$ , the company pays $n$ times $d$ . If the company shares, it pays the one-off cost $f$ once, the adaptation cost $a$ for each team, and then the three costs of sharing: coordination $q$ , waiting $w$ and the expected loss from failures $l$ .

Definition

\begin{aligned}\text{local route}&=n\,d\\\text{shared route}&=f+n\,a+q+w+l\end{aligned}

In words: doing it alone costs the same for every team; sharing costs one fixed amount, one adaptation per team, and the costs of coordinating, waiting and shared failure.

Subtract one from the other and rearrange. The shared route is cheaper exactly when this holds:

Checked in Lean · sharing_threshold

n\,(d-a)\;>\;f+q+w+l

In words: what each team saves by adapting instead of doing it alone, times the number of teams, must beat what sharing costs once.

Read it from left to right. The bracket $d - a$ is the saving per team: the full cost of doing it alone minus the cost of adapting the shared work. The $n$ in front says that saving repeats for every team that joins. The right-hand side is the price of sharing, paid once however many teams there are. Everything a shared route does well or badly must show up in one of these seven letters.

Take the bracket one letter at a time, with the two document teams. $d$ is what a team pays to do the work alone, from scratch. For Warranty Claims that means building its own set of scanned contracts to test on, writing the instructions, finding the exclusions the reader misses and fixing what it can. Say $d$ = 40. $a$ is what the same team still pays when the shared route has done the common part: reworking the instructions for its own contract wording, tuning on its own annexes, and running its own acceptance test before it trusts the result. Say $a$ = 10. The difference, $d - a$ = 30, is what Warranty saves by starting from the shared result instead of from nothing. Field Repair Dispatch saves its own 30 on its own manuals.

That saving is per team, so it repeats: with $n$ teams it is $n(d - a)$ . The costs of sharing are not per team. The shared work, the meetings, the queue and the failures are paid once, whether two teams use the result or ten. The whole inequality is a saving that multiplies against costs that do not.

What each letter stands for

$n$: Number of teams with the same needteams
$d$: Cost for one team to do the work aloneillustrative units per team
$a$: Cost to adapt the shared work to one teamillustrative units per team
$f$: One-off cost of the shared work: building it and keeping it upillustrative units
$q$: Cost of coordinating: meetings, reviews, agreeing prioritiesillustrative units
$w$: Cost of waiting for the shared routeillustrative units
$l$: Expected loss from failures on the shared routeillustrative units

Two rules keep the sum honest. Count every cost once: if waiting is already priced inside coordination, it cannot also appear in $w$ . And give both routes the same horizon and the same unit. "Illustrative units" means exactly that: the numbers below are chosen to show the shape, not measured at any company.

Start with the two document teams. Doing it alone costs each 40, so 80 in total. Sharing costs 60 once, 10 per team to adapt, 5 to coordinate and 15 in waiting: 100. Two teams are not enough. Now slide the number of teams to three, as if Customer Support joined. Doing it alone costs 120; sharing costs 110. The lines have crossed. The break-even point is 80 ÷ 30, about 2.7 teams.

Then push the cost of waiting up. Nothing changes in the per-team saving, yet the break-even point slides right, and at some point no realistic number of teams rescues the shared route. Finally, raise the adaptation cost $a$ until it reaches $d$ . The lines become parallel or diverge. When adapting shared work costs as much as starting from scratch, no number of teams helps.

What Lean checks

Lean, a program that checks every step of a proof, accepts a claim only when every step follows from the definitions. It does not take our word for anything. Here the definitions are the two sums above, written in whole numbers, and Lean checks three facts about them.

Checked in Lean

With costs added up this way, the shared route is cheaper exactly when the overhead $f + q + w + l$ is smaller than $n$ times $(d - a)$ .

sharing_threshold in CraftCosts.lean

Show the Lean statement

theorem sharing_threshold (n d a f q w l : Int) :
    shared n a f q w l < separate n d ↔
    f + q + w + l < n * (d - a)

Checked in Lean

The two routes cost the same exactly when the overhead equals $n$ times $(d - a)$ : that is the break-even point drawn in the figure.

cost_tie in CraftCosts.lean

Show the Lean statement

theorem cost_tie (n d a f q w l : Int) :
    shared n a f q w l = separate n d ↔
    f + q + w + l = n * (d - a)

Checked in Lean

If the overhead is larger than $n$ times $(d - a)$ , the local route is strictly cheaper.

overhead_reverses_saving in CraftCosts.lean

Show the Lean statement

theorem overhead_reverses_saving (n d a f q w l : Int)
    (h : n * (d - a) < f + q + w + l) :
    separate n d < shared n a f q w l

Checked in Lean

Two concrete cases: four teams that would each pay 50 alone pay 80 in total by sharing; raise waiting from 5 to 200 and sharing costs 275, more than the 200 of working alone.

shared_saving_example · waiting_reversal_example in CraftCosts.lean

Show the Lean statement

theorem shared_saving_example : shared 4 10 30 5 5 0 < separate 4 50
theorem waiting_reversal_example : separate 4 50 < shared 4 10 30 5 200 0

Be clear about how much this is. These are rearrangements of a sum; a careful student could check them on paper in a minute. Lean's value here is not depth but certainty: the floor every later lesson stands on has no arithmetic slip in it. The source file is 39 lines long.

What Lean does not check matters more. It does not check that costs simply add up, with nothing interacting. It does not check that these seven costs are the right ones or that none was counted twice. It does not know any real value of $d$ or $a$ . It says nothing about which route a real company should choose. The equation tells you what to count; it cannot count for you.

What it means

Scale is the main reason to share

The saving per team is multiplied by $n$ ; the overhead is not. A shared route that loses with two teams can win with five. This is the one advantage of a shared route that needs no further argument, and it is why the question "how many teams have this same need?" comes before any question about organization charts.

Fit can cancel scale

If teams' needs differ so much that adapting the shared work costs nearly as much as doing it alone, $d - a$ shrinks towards zero and so does the saving, however many teams join. Lesson 2 shows where that gap comes from: shared expertise widens it, mismatch narrows it.

Queues and meetings are real costs

The second Lean example is the one to remember. Nothing about the work changed except waiting, and a clear saving became a clear loss. A shared route that makes teams wait is not free just because nobody sends an invoice for the wait.

A shared failure is shared

The term $l$ is the expected cost of something going wrong on the shared route. When one shared model misreads a clause, it misreads it for every team that uses it. Here we held $l$ at zero; a correlated failure across teams is a reason to raise it, not to ignore it.

The name earns nothing

Nothing in the equation rewards a route for being called "ACM", "centre of excellence" or "platform". This is deliberate, and every later lesson keeps it: each model gives a capable local team the same information, people and rules as the shared route, so a shared route wins only through something specific it does that a local team cannot or will not do, never through its label. That is a rule we chose for building the models, not something we discovered.

When the shared route loses

It loses when few teams share the need, when their needs differ, when the shared route makes them wait or meet too much, or when one shared failure can hurt them all. If the shared route is optional, free to try and knows at least what the team knows, the team can always ignore it, so its existence cannot leave the team worse off. Losses come through the doors this equation names: cost and compulsion, and, in later lessons, incentives.

The equation treats $d - a$ , the saving per team, as a number you already know. It is not. It depends on two forces that pull in opposite directions. A shared route that has seen many teams' documents makes fewer slips. But a method built for many jobs fits each one a little less well. Lesson 2 puts a number on that trade: better expertise, worse fit.

The sharing equation

The second invoice

One payment against many

Add up both routes

What Lean checks

What it means

Scale is the main reason to share

Fit can cancel scale

Queues and meetings are real costs

A shared failure is shared

The name earns nothing

When the shared route loses

Next: where the saving per team comes from