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flowchart TB
subgraph CEN["Centralised"]
direction TB
C(["coordinator (bottleneck)"])
C --> W1(["agent"])
C --> W2(["agent"])
C --> W3(["agent"])
C --> W4(["agent"])
C --> W5(["agent"])
end
subgraph HIER["Hierarchy"]
direction TB
T(["coordinator"])
T --> M1(["sub-coordinator"])
T --> M2(["sub-coordinator"])
M1 --> A1(["agent"])
M1 --> A2(["agent"])
M2 --> A3(["agent"])
M2 --> A4(["agent"])
M2 --> A5(["agent"])
M2 -.->|"escalation"| T
end
subgraph MESH["Decentralised (mesh)"]
direction TB
N["every pair must stay consistent"]
P1(["agent"]) --- P4(["agent"])
P1 --- P5(["agent"])
P2(["agent"]) --- P4
P2 --- P5
P3(["agent"]) --- P4
P3 --- P5
N ~~~ P4
end
W3 ~~~ T
A3 ~~~ P1
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classDef world fill:#EFE9DC,stroke:#766F65,color:#16171B
class C gate
class N world
11 Coordination and Distributed Problem Solving
The previous chapter built a team and trusted it to act. It gave a collection of agents a shared goal, a joint commitment to that goal, a plan they all held in common, and the machinery to divide the work, reassemble it, and repair it when a member failed. What it quietly assumed, all the way through, was that agents which agree on what to do can simply go and do it. They cannot. A plan settles what each agent will do and who will do it; it does not settle the thousand smaller questions of when, and in what order, and over whose copy of the shared state — and those questions do not answer themselves. Two coders handed non-overlapping tickets still contrive to edit the same file, and one of the two writes is silently lost. An agent waits on a result that another agent, equally diligent, was never actually asked to produce. Two robots meet in a corridor wide enough for one and freeze, each waiting with perfect courtesy for the other to go first. None of these is a failure of teamwork in the sense of Section 10.2; every agent wanted the right thing. They are failures of coordination — the management of concurrent, interdependent action — and they are the subject of this chapter.
The purest picture of the problem has nothing to do with software. Put several robots in a shared warehouse, give each a destination, and require them to get there without colliding: multi-agent path finding is the problem of choosing their routes so that no two occupy the same place at the same instant. It is trivial to state, genuinely hard to solve — the joint search space explodes with every agent added — and it strips coordination to its essentials: a shared resource, incompatible simultaneous demands upon it, and no collision allowed. We will come back to it, not least because it is a clean instance of the book’s recurring lesson — an old and forbidding problem now routinely handed to a learned policy, as Chapter 17 will show. But the warehouse is merely the vivid case. The same structure — shared resources, interfering actions, separate activities that must somehow be made to fit — turns up wherever agents act at once upon anything held in common, the team’s repository and its task board first of all.
The tools for managing all this arrived, as has become the refrain of these pages, decades before the agents that now need them. The distributed-AI tradition asked how a collection of problem-solvers might act coherently with no central mind dictating every move, and answered in several voices. One holds that coordination is, at bottom, the management of dependencies between activities — general enough to gather scheduling, allocation, and the sharing of resources under a single idea. Another shows that much coordination can be bought in advance, and for nothing at runtime, by agreeing conventions: social laws that let agents keep clear of one another without exchanging a word. A third coordinates through the world rather than by talking, leaving marks in a shared medium that other agents read and act upon, as ants do and as a blackboard does. A fourth treats coordination as distributed constraint solving, in which agents whose choices constrain one another trade partial, tentative results until a consistent joint solution falls out. And beneath all of them runs the older distributed-systems canon on what it takes to stop concurrent processes corrupting the state they share. Each is a different answer to one question, and the contemporary engineer, mostly without knowing it, is asking that question again.
What divides this chapter from the last is the difference between a plan and its execution. A team can hold a flawless shared plan and still deadlock; agreement is necessary for coordination and nowhere near sufficient. The modern agent stack meets the gap under a litter of newer names — shared-state management, locking, orchestration topology, parallel subagents — and rediscovers the classical failure modes at precisely the classical places: the lost write, the stale read, the two agents each blocked on the other, the work done twice because neither knew the other was at it. The org chart is not a coordination mechanism: it names the roles, but — as the preface warned — something has to make concurrent roles cohere, and naming is not that something. One assumption carries over intact from Section 10.2 and stands until Part IV dismantles it: the agents want to coordinate. No one here games the protocol or hides its true load to wangle a better deal; the difficulty is mechanical, not strategic. We begin by asking what coordination is — the management of dependence (Section 11.1); then the hazards of concurrent action over shared state (Section 11.2); then three ways to keep agents out of each other’s way — by convention (Section 11.3), through the environment (Section 11.4), and by solving the interlocking constraints head-on (Section 11.5); and last, the choice of coordination topology and the price every option exacts (Section 11.6).
11.1 The Coordination Problem: Managing Interdependence
Coordination is the management of dependencies between activities, and that one sentence, owed to Malone and Crowston, carries most of what the concept requires (1994). The test is subtraction: if two activities could each be carried out in perfect ignorance of the other, sharing nothing and owing nothing to each other’s order, there is nothing to coordinate — and coordination is exactly the work their dependence creates. This is why coordination is not an optional refinement a team adds once the real work has been parcelled out; it is the bill that the parcelling-out runs up. The moment the orchestrator of Section 10.4 split implement the export format into a serialiser, a set of tests, and a page of docs, it manufactured dependencies that had not existed while the task was whole: the tests now depend on the serialiser’s interface, the docs on the finished API, and all three on the single repository they must share. Decomposition buys parallelism and sells dependence in the same breath, and the price of the dependence, paid later, is coordination.
Dependencies are not all of one kind, and naming the kind is half of choosing the remedy — the practical pay-off of treating coordination as a subject rather than a knack. A shared-resource dependency arises when several activities need the same limited thing at once: the one repository, a particular file, the team’s token budget, a square of warehouse floor that two robots both mean to occupy. A producer–consumer dependency, sometimes a flat prerequisite, holds when one activity yields what another needs before it can begin: the serialiser’s interface must exist before the tests can be written against it; the change must pass review before it ships. A simultaneity dependency constrains activities to happen together, or forbids them from happening together — the two robots that must not enter the one-wide corridor at the same instant, the migration that must be applied across services in lockstep. And the task–subtask dependency is the decomposition of Section 10.4 itself, the relation between a goal and the parts it was broken into. Each kind summons its own family of mechanisms, and the value of the taxonomy is that it converts a vague unease about agents getting in each other’s way into a specific question with a specific answer.
The positive and negative relationships that Section 10.5 drew between agents’ plans were a slice of this same structure, met there as conflicts to be cleared up after the parts were built; coordination is the discipline of managing them before they bite.
Some of these dependencies are so pervasive that the questions they pose have acquired names of their own, and between them they account for most of what a coordinating team spends its effort on. The shared-resource dependency, asked of agents and tasks, becomes the problem of allocation: which agent does which task, which job gets the scarce accelerator, who holds the write-lock on the module. The producer–consumer and simultaneity dependencies, asked of time, become the problem of scheduling: in what order the activities run, so that producers finish before consumers start and no two conflicting actions land at once.
It is worth being clear how this differs from the delegation of Section 10.4, which it superficially resembles. Delegation manufactured a commitment — it got some agent to undertake a task. Allocation asks whether the assignment was a good one: whether this spread of work across agents balances their load, respects their dependencies, and leaves the fewest conflicts to settle. A team can delegate every task flawlessly and still have allocated them abominably, piling four jobs on one agent while three others sit idle, or handing two agents work that will collide the instant both begin. Getting the assignment right, and the order right, is coordination’s first and most basic labour; how to do it well when the choices interlock is the harder matter Section 11.5 returns to.
It is tempting to suppose that a good enough plan would make all this unnecessary — that a team which has settled its joint intention (Section 10.2) and its shared plan (Section 10.3) has, in settling them, already coordinated. It has not, and the gap is the whole reason this chapter exists. A shared plan fixes what the team will do and who will do each part; it characteristically leaves when, and over which shared thing, to be worked out in the doing. Two coders perfectly committed to two non-overlapping tickets hold a flawless shared plan and will still, if nothing coordinates them, reach for the same file at the same moment — because the plan never said not at the same time, having no vocabulary for the clash. The musicians of Section 10.3 make the point precisely: handing four players one agreed score settles the notes and the parts, yet four players reading that score in a single room can still produce a shambles, because a score does not keep time. Coordination is the keeping of time — the layer a plan leaves implicit, supplied either by a conductor or by a beat the players come to share. Teamwork decides the music; coordination is what makes it land together.
How much coordination a team needs is therefore not a constant; it rises and falls with how tightly the work is coupled. A decomposition into nearly independent parts — the prize Section 10.4 held out — is valuable precisely because it leaves few dependencies to manage, and a decomposition whose parts are forever touching condemns the team to coordinate at every step. The warehouse shows the gradient at a glance: a handful of robots in open space barely coordinate at all, each with room to route around the others, whereas the same robots in a tight maze must negotiate almost every square, because there the shared resource is scarce and the dependencies are dense. This yields the first and most general lesson of the subject, one the distributed-AI tradition that made coordination its central question understood well (Jennings, 1996): the cheapest dependency to coordinate is the one that was designed out, and a large part of coordinating well is arranging the work so that there is less of it to coordinate. What cannot be designed out must be paid for, and the remainder of this chapter is, in effect, a catalogue of ways to pay.
The dependencies named so far are a static anatomy — the structure of who needs what. What lends coordination its urgency is that the needing so often happens at the same time. Of all the dependencies, the shared resource touched concurrently is the one whose mismanagement does not merely waste effort but corrupts: when two activities write the same state at once with nothing to order them, the result is not a slow team but a wrong answer, a file that holds neither agent’s work but a mangling of both. Before the remedies — the conventions, the shared media, the constraint-solving of the sections to come — it is worth looking squarely at the hazard they all exist to prevent. What exactly goes wrong when concurrent activities meet shared state unmanaged, and why the failures are as old and as thoroughly charted as computing itself, is where we turn.
11.2 Acting at Once: Races, Mutual Exclusion, and Consistency
Set two agents to work on the same file at the same time and watch what happens. Each reads the file as it stands; each makes its change to the version it read; each writes the result back. Whichever writes second wins, and the first agent’s work vanishes without a trace or a complaint — not merged, not flagged as a conflict, simply gone, overwritten by a sibling that never saw it. This is a race condition: an outcome that turns on the order in which concurrent activities happen to interleave, an order nobody specified and nobody controls. Its special cruelty is that it usually works. Most interleavings are benign, and the lost update surfaces only when the timing lines up exactly wrong — which is to say rarely, unrepeatably, and almost never during testing. A race is a bug that hides behind its own improbability, and a system that spawns workers to write shared state with no thought for their interleaving is a race condition that has merely not yet been unlucky.
11.2.1 Mutual Exclusion, Local and Distributed
The cure is as old as the disease and has a name: mutual exclusion. Identify the stretch of activity that touches the shared state — the critical section — and arrange that at most one agent is ever inside it at once. Dijkstra set the problem out formally in 1965, and the lock, the mutex, and the semaphore are all, at bottom, ways of enforcing the condition he named (1965): an agent acquires the lock before entering the critical section and releases it after, and any other agent that wants in must wait. Our team takes a write-lock on the module, or funnels every change through an orchestrator that applies them one at a time; either way the concurrent free-for-all becomes an orderly queue, and the lost update cannot happen because the losing write never happens. Mutual exclusion buys safety with concurrency: while one agent holds the lock the others idle, and that idling — contention — is the first showing of a cost this chapter keeps meeting, the price of making separate activities safe to combine.
On a single machine a lock is a modest thing, a flag in shared memory the hardware will let only one processor flip. Among distributed agents — separate processes, separate contexts, each with its own copy of everything, conversing only by message — there is no shared flag to flip, and the modest thing swells into a protocol. Distributed mutual exclusion has the agents negotiate by message who may enter the critical section, and getting that right, with messages that cross in flight and agents that fall silent mid-exchange, is a genuine piece of engineering rather than a library call.
Worse, the very notion of order frays. Ask which of two writes came first and, absent a clock the agents share, there may be no fact of the matter: each stamped its action by its own reckoning, and the reckonings do not align. Lamport’s answer became one of the founding results of distributed computing (1978): give up the fantasy of a single global time and settle for a happens-before relation, a partial order recording only which events could have influenced which, kept by logical clocks the agents pass back and forth. The lesson beneath the algorithm is what makes distributed coordination hard in principle and not merely in practice — separate agents do not share a now, and any agreement about the order of their actions must be manufactured, at a cost, rather than read off the world.
11.2.2 Happens-Before, Formally*
Lamport’s relation is brief enough to give in full, and worth giving, because everything just said follows from how little it assumes. The events of the system are the individual actions of the agents, among them the sending and the receiving of each message. For events a and b, write a \to b — read “a happens before b” — for the smallest relation satisfying three clauses: if a and b belong to the same agent and a comes earlier in that agent’s own sequence, then a \to b; if a is the sending of a message and b its receipt, then a \to b; and if a \to b and b \to c, then a \to c. Nothing else counts. Two distinct events may therefore stand in neither order, in which case they are concurrent,
a \parallel b \;\;\text{iff}\;\; \neg(a \to b) \wedge \neg(b \to a),
and their concurrency is not ignorance but fact: neither could have influenced the other, so no observer, however well placed, has grounds to put one first. What a logical clock provides is a numbering that respects the relation — an integer C(a) for each event a, satisfying the clock condition,
a \to b \;\Rightarrow\; C(a) < C(b),
and Lamport’s rule keeps it with almost nothing: each agent ticks its counter at every event, stamps the counter onto each outgoing message, and on receipt advances its own counter past the stamp before ticking. Note that the implication runs one way only. C(a) < C(b) does not make a prior in any causal sense; the clocks — ties between agents broken by any fixed rule, say by agent name — extend the partial order to a total one by choosing an order for concurrent events, which is exactly the sense in which agreement about order is manufactured rather than found: even the tiebreak is arbitrary.
One integer per agent and two functions over it are the whole apparatus,
def tick(c: int) -> int: # every event ticks the counter
return c + 1
def recv(c: int, stamp: int) -> int: # a receipt lands past the stamp
return tick(max(c, stamp))
coder, reviewer = 2, 0 # the coder has seen two events; the reviewer none
C_a = coder = tick(coder) # the send a; C_a stamps the message
C_b = reviewer = recv(reviewer, C_a) # the receipt b leaps past the stamp
C_a, C_b, C_a < C_b # -> (3, 4, True): the clock condition holdsand the final line is the clock condition holding for the send–receive pair. The receiver’s leap from nought to four repays a glance: a Lamport clock does not count an agent’s own events — the receipt was this agent’s first — but bounds everything that could have influenced it.
11.2.3 Deadlock and Its Cousins
Mutual exclusion applied without care breeds the opposite catastrophe. Give each of two agents one resource and have each wait for the other’s before yielding its own, and both wait for ever: deadlock, the state in which every party is blocked pending something only another blocked party can provide. The two robots frozen nose to nose in the one-wide corridor of this chapter’s opening are deadlocked; so is the orchestrator blocked on a subagent’s result while that subagent blocks on a reply the orchestrator will send only once it is itself unblocked. Coffman and his colleagues pinned down precisely what it takes, naming four conditions that must all hold for deadlock to arise — mutual exclusion, hold-and-wait, no preemption, and a circular chain of waiting — with the useful corollary that denying any one of them makes deadlock impossible (1971). Dijkstra’s dining philosophers, forever reaching for forks their neighbours are holding, is the picture every student meets; the agent stack supplies fresh instances weekly, filed usually under the heading of a run that simply hung. Its gentler cousin, livelock — two agents that notice the conflict, both back off politely, both retry in lockstep, and repeat the little dance for ever — is the corridor shuffle rendered in code: motion without progress.
11.2.4 Different Versions of the Same State
Even where nothing collides outright, distributed agents drift out of agreement about the state they share. Each holds its own copy — its own context, its own remembered view — and the instant one agent updates the shared truth, every other agent’s copy is stale until it hears. An agent acting on a stale copy is not malfunctioning; it is reasoning correctly from something that was true and has quietly stopped being so, and the resulting mistake is the harder to catch for looking, at every step, entirely sensible. This is the problem of distributed consistency, which the preface flagged among the failures modern systems wander into blind: several agents holding different versions of the same state. The distributed-systems tradition long ago mapped the trade-off it forces. Strong consistency — every read sees the latest write — is achievable but dear, because each reader must coordinate with the writer before it proceeds; eventual consistency is cheap and lets readers race ahead, at the price of letting them be wrong for a while. There is no free version, only a choice of which cost to bear.
The pragmatic remedy is the humble one the last two chapters kept arriving at: a single source of truth — the shared board of Section 9.6, authoritative by construction — that agents read from and write to rather than each trusting a private copy. Consistency, seen this way, is the common ground of Section 9.2 put under the strain of concurrent editing: the work of keeping a shared view genuinely shared while many hands revise it at once.
11.2.5 Old Cures, New Labels
What should reassure the engineer is that none of this is new; what should trouble them is that the tools so rarely act as if they know it. Races, mutual exclusion, deadlock, and consistency are the settled canon of distributed systems, decades deep and taught to undergraduates (Tanenbaum & van Steen, 2023), and their cures are as well understood as the hazards — yet the post-mortems of multi-agent language-model systems keep turning up exactly these failures under new labels, filed as flaky and non-deterministic rather than under the textbook names they were given long ago (Cemri et al., 2025). The remedy is not novel machinery but the discipline to apply the old: serialise what must not overlap, order what must not be reordered, and keep one authoritative copy of whatever everyone relies on. Yet locking and its relatives are the pessimist’s remedy — they assume a clash is always possible and pay to guard against it on every access, whether or not the clash would ever have come. There are lighter ways to prevent the clash, ways that head it off before it can arise rather than policing it as it happens, and the lightest of them costs nothing at runtime because it was settled in advance. To agree the rules of the road, so that coordination becomes unnecessary rather than merely safe, is where we go next.
| Hazard | What goes wrong | Where it shows in the agent stack | Classical cure |
|---|---|---|---|
| Race condition | The second writer wins; the first agent’s write vanishes — unmerged, unflagged, simply gone | Two agents editing one file; workers spawned to write shared state; filed as flaky or non-deterministic | Mutual exclusion — a lock, mutex, or semaphore round the critical section |
| Deadlock | Every party blocked, each waiting on what only another blocked party can give — all four Coffman conditions holding at once | An orchestrator and its subagent each blocked on the other; a run that simply hung | Deny any one of the four Coffman conditions |
| Livelock | Both agents notice the conflict, both back off, both retry in lockstep — motion without progress | The corridor shuffle rendered in code | Break the symmetry — randomised backoff with jitter, or a priority order fixing who yields |
| Stale read (inconsistency) | An agent reasons correctly from a copy of the shared state that was true and has quietly stopped being so | Each agent holding its own context after another updates the shared truth | A single source of truth; strong or eventual consistency, chosen knowingly |
11.4 Coordinating Through the World: Blackboards and Stigmergy
There is a third way to keep agents from tripping over one another, and it neither settles the matter in advance nor requires the agents to speak. A coder that finishes the serialiser sends no announcement; it commits, and the commit — the changed file, now sitting in the shared repository — is itself the signal. The next agent, finding the repository in its altered state, reads what was done and acts on it, and the two have coordinated without either addressing the other or, indeed, knowing the other exists. The coordination lives not in a message and not in a rule but in the shared medium the agents both act upon: each writes its trace into the common world, and each reads the world the others have left. This is coordination through the environment, the most self-effacing of the three, because no agent need hold a model of any other — the world holds it for them.
We have, in fact, met the classical form of the idea already: the blackboard that Section 9.6 traced back to the Hearsay-II system, a shared structure onto which independent specialists post partial results and from which they read what others have posted, converging on a solution with none of them in charge (Erman et al., 1980). There it appeared as the ancestor of shared state; regarded as coordination, its defining virtue is the decoupling that Nii drew out of the general pattern (1986) — the specialists know nothing of one another, invoke one another never, and interact solely through the marks they leave on the board. Coordination becomes opportunistic, each agent acting whenever the shared state offers it something to do, rather than dictated by any plan of who goes when. A blackboard is thus not merely a place to keep what the team knows; it is a mechanism through which the team coordinates without conversing.
Nature arrived at the same device long before, and the biologist Grassé named it stigmergy after watching termites build (1959). A termite’s action alters the environment — a dab of soil placed just here — and the altered environment is exactly what prompts the next action, in that termite or another, so that a cathedral of a nest rises with no blueprint, no foreman, and no termite anywhere holding the plan. The tidiest example is the ant trail: an ant returning from food lays a chemical mark, other ants are drawn to the stronger marks and reinforce them as they pass, and the colony converges on short routes to food though no ant ever compares two paths or tells another which to take (Theraulaz & Bonabeau, 1999). Stigmergy is coordination pared to the barest mechanism there is — no plan, no leader, no message, no goal held in any single head — leaving only the medium and each agent’s local response to the state it finds. It is the limiting case of coordinating through the world, and the ancestor of the swarm methods — ant-colony and particle-swarm optimisation, which carry the self-organisation of social insects into engineering (Bonabeau et al., 1999) — that Chapter 18 takes up.
Blackboard and stigmergy are two faces of one principle: put the coordination in the environment, not in the agents. What that principle buys is worth naming, because it is exactly what a system of many, changeable, unreliable agents most needs. It buys decoupling — an agent need not know who else is at work, nor address them, nor even overlap with them in time, since the medium carries its contribution to whoever reads next. It buys openness — agents may join or fall away freely, because the board neither knows nor cares how many attend it. And it buys robustness — there is no coordinator whose failure stops the world, and the system sags gracefully rather than snapping as agents drop out. These are the properties one reaches for when the participants are too numerous, too transient, or too unreliable to wire together directly, which is to say precisely the situation a large agent system tends to be in.
The contemporary agent system is thick with blackboards, most of them unlabelled. The shared workspace of Section 9.6 — the visible task board, the ticket queue, the scratchpad on which agents leave working notes — is a blackboard in all but name: agents read its current state, take up whatever they are equipped to do, post their results, and leave the altered board to prompt the next. The repository is frankly stigmergic: a commit is a trace laid down for whoever comes after, a pheromone in version control, and a pipeline of agents that each reads the repo, does its part, and commits is an ant colony with better tooling. The pattern is a mainstay of modern multi-agent frameworks — a shared state object every agent may read and revise — and its appeal is the old one: it asks the least of any single agent, only that it read the common state and add to it, with no protocol to observe and no peer to name. Built deliberately rather than stumbled into, it is one of the architectural patterns Chapter 21 examines in its own right.
The bill, though, arrives in two parts. The first was written in full a section ago: a shared medium is shared state, and every hazard of Section 11.2 applies to it without mercy — the blackboard two agents post to at once is a race, the board one agent read a minute ago is already stale, and the repository is a merge conflict biding its time. Coordinating through the environment does not escape the concurrency problem; it leans on it. The second cost is subtler, and it is the price of the very decoupling that made the pattern attractive: coordination that lives in the environment is coordination no one is holding, hard to inspect and harder to debug, and when many local reactions compound they can produce collective behaviour that no agent chose and no designer foresaw — the emergent dynamics, sometimes marvellous and sometimes ruinous, that Chapter 18 treats in earnest.
And there is a limit the mechanism cannot cross. Traces in a shared medium serve beautifully when contributions can be posted independently and assembled loosely, but they do not, on their own, settle a genuine clash of choices — the case in which what one agent may decide turns on what another decides, each constraining the other. When the parts interlock that tightly, leaving marks is not enough; the agents must reconcile actively, trading partial commitments until their choices fit. That harder problem — coordination as the joint solving of interlocking constraints — is the one we take up next.
11.5 Interlocking Choices: Distributed Search and Constraints
Some choices will not be prised apart, however the work is arranged, because they are bound together at the root. When the architect settles the shape of an API and a coder writes against it, neither decides alone: the architect’s choice of signature fixes what the coder may call, and the coder’s needs constrain which signatures are any use. There is no order in which one simply goes first. A trace left on a board (Section 11.4) does not help, because the mark the coder would read is the very thing that must accommodate the coder; and a convention fixed in advance does not help either, because nobody knew the right interface before the work began. The two choices interlock — each is a constraint on the other, and the only way to satisfy both is to settle them together. This is the general problem of coordinating interdependent decisions, and it is the one the earlier mechanisms were built to sidestep rather than to solve.
11.5.1 The Constraint Formalisms
The distributed-AI tradition gave the problem a clean form. In a distributed constraint satisfaction problem, each agent controls some variables, constraints link variables held by different agents, and the team must find a joint assignment that satisfies every constraint — with no agent ever seeing the whole problem, only its own variables and the constraints it shares (Yokoo et al., 1998). Its optimising cousin, the distributed constraint optimisation problem, replaces hard constraints with costs and asks for the joint assignment of least total cost, which is the shape most real coordination actually has: not a wall between legal and illegal but a landscape of better and worse arrangements to be searched together (2005).
Put this way, the allocation and scheduling that Section 11.1 opened with are distributed constraint problems in disguise — assigning tasks to agents is a joint choice of variables under load and dependency constraints, and the good assignment is the one that minimises conflict and idleness across the team. The difficulty is intrinsic rather than a want of cleverness: no agent holds the global picture, so the solution must be found by exchanging messages, and the number of messages, in the worst case, grows as unkindly as the problem itself.
11.5.2 Distributed Constraint Optimisation, Formally*
The optimising form is worth setting down exactly, since its parts are few and the next subsection leans on each of them. Take the agents N = \{1, \dots, n\} and variables x_1, \dots, x_m, each variable x_i ranging over a finite domain D_i and owned by exactly one agent, which alone may set it — the architect owns the interface signature, each robot its own route. Constraints become costs: wherever two variables bear on one another, a cost function
f_{ij} : D_i \times D_j \to \mathbb{R}_{\ge 0}
prices every combination of their values, zero for pairs that fit and rising with the badness of the fit; a hard constraint is the special case that prices its forbidden combinations beyond any willingness to pay. The pairs that carry a cost function are the edges of the constraint graph, and the problem is to find a complete assignment — a value for every variable — minimising the total cost,
\sum_{(i,j)} f_{ij}(x_i, x_j),
the sum running over the graph’s edges. The satisfaction problem is the special case whose solutions are precisely the assignments of cost zero. What makes the problem distributed is who knows what: each agent sees its own variables, their domains, and the cost functions on its own edges, and nothing else — no agent ever holds the sum the team is jointly minimising. And when each agent owns a single variable, as the standard treatments assume, the constraint graph is node for node the coordination graph the next subsection introduces: an edge exactly where two agents’ choices interlock, and sparseness exactly where coordination comes cheap.
11.5.3 Locality and Convergence
What keeps this from being hopeless is that interaction is almost always local. The architect’s interface constrains the coder who calls it and nobody else; two robots contend only where their paths might cross, not across the whole warehouse. A coordination graph makes that structure explicit — a node for each agent, an edge wherever two agents’ choices directly constrain each other — and turns a forbidding global problem into a web of small local ones. An agent need reconcile its choice only with its neighbours in the graph; their adjustments ripple to their neighbours; and a solution assembles out of local agreements without any agent ever confronting the whole. The sparser the graph, the cheaper the coordination, which restates in a new key the lesson of Section 11.1: the coupling you can avoid is coordination you never have to perform, and the shape of the interaction graph, far more than the sheer number of agents, decides what a system will cost to coordinate.
The manner in which those local agreements are reached is worth naming, because it overturns an assumption the concurrency hazards seemed to force. There, inconsistency between agents’ views was a hazard to be prevented; here it is a phase to be worked through. Lesser’s principle of functionally accurate, cooperative computation held that a distributed system need not keep every agent complete and correct at every step (1981): agents may compute on partial, tentative, even mutually inconsistent local results, exchange those results, and converge, the system reaching the right global answer though no agent was ever locally certain along the way. Coordination becomes iterative rather than one-shot — each agent posts its current best guess, sees what its neighbours posted, revises, and repeats until the guesses stop moving and the choices fit. We have met the idea already in the guise of plans: partial global planning (Section 10.5) is exactly this, agents trading provisional plans and nudging them toward compatibility rather than computing one correct plan up front. The architect and coder converging on an interface by passing drafts back and forth are running the same loop by hand; what the theory adds is an account of when such a loop will settle and when it will merely oscillate.
11.5.4 The Canonical Hard Case
The warehouse robots of this chapter’s opening are the exemplar the whole field returns to. Multi-agent path finding is a distributed constraint problem in its starkest form: each agent’s variables are the cells of its route through time, the constraints forbid any two agents from occupying the same cell at the same step, and a solution is a set of routes that are jointly collision-free (2019). It is the cleanest illustration of everything this section has said — interlocking choices, a coordination graph dense where paths crowd and empty where they do not, a search that must be conducted jointly — and it is genuinely, provably hard: finding optimal collision-free routes for many agents is NP-hard, the joint space exploding with each robot added. It is also the chapter’s recurring lesson in miniature. This old, forbidding, exactly-specified search problem is now, increasingly, not solved in the classical sense at all but handed to a learned policy that maps each robot’s local view straight to a move and coordinates the swarm without ever writing the constraints down — the composed-versus-learned question that Chapter 17 takes up in full. The problem is a fixture of the 1980s; the method is younger than the reader’s laptop.
Two lines bound what this section has claimed. The first is the assumption that has quietly underwritten every method in it: that the agents cooperate — that they report their true constraints and costs and want the joint solution to be good rather than merely good for them. Relax it, let each agent prefer the assignment that serves itself and shade its reported costs to secure it, and the distributed optimisation stops being a search and becomes a game: the same allocation problem, now among agents who will misreport to win it, which is the province of auctions and mechanism design in Part IV. The cooperative constraint problem and the competitive auction are one allocation problem under opposite assumptions about the players. The second line is that everything here has been one way of solving a coordination problem — the thoroughly decentralised way, in which information and computation stay local and no agent is in charge. The alternative is to gather the whole problem to a single coordinator and solve it centrally, trading the bottleneck for the simplicity of one mind that sees everything. Which to choose — central or distributed, flat or hierarchical — is not a detail but the defining architectural decision of a multi-agent system, and each option pays a different price in communication, delay, and fragility. That price, and the topologies that pay it, is where the chapter ends.
11.6 Topologies and the Price of Coordination
Where should the coordinating happen? Every mechanism of this chapter has to run somewhere, and the last of the chapter’s questions is where — in one place or many. At one pole stands centralised coordination: a single coordinator holds the whole picture and settles everything, the orchestrator of our running example seeing each agent’s state and handing down each assignment. At the other stands the decentralised, or peer-to-peer, arrangement: nobody in charge, agents coordinating locally with their neighbours through the conventions of Section 11.3, the shared media of Section 11.4, or the constraint-passing of Section 11.5. Between them sits hierarchy, coordinators of coordinators. These are not different problems but different topologies for one body of work — different placements of the coordinating — and each placement is a different bargain.
Centralised coordination is the easiest to build and the easiest to trust. One mind sees everything, so consistency comes free — there is a single source of truth by construction — conflicts are settled before they can occur, and the system’s behaviour is about as predictable as one program’s. The orchestrator-worker pattern is centralisation in modern dress, and its popularity is no accident: it is the arrangement a single designer can hold in their head. What it charges is threefold. The coordinator is a bottleneck, every decision routed through it, serialising work that might have run in parallel. It is a single point of failure, and when it stops the system stops with it. And it must somehow acquire the global picture it relies on — every agent reporting its state up the chain — which costs messages to gather and, under partial observability, yields a picture that is often incomplete or already stale. Centralisation buys simplicity and consistency with scalability and robustness.
Decentralisation makes the opposite trade. With no hub to overload it scales — add agents and you add only local coordination, not load at a centre; with no coordinator to lose it survives, degrading gracefully as agents drop rather than failing all at once; and it lives contentedly with partial views, each agent acting on what it can see without waiting on a global gather that may never arrive. The price is the mirror of centralisation’s virtues. Consistency must be worked for rather than assumed, and every concurrency hazard returns in force; agreement ripples outward instead of being dictated, so convergence is slower and not always assured; and because no one holds the global state, the system’s overall behaviour is emergent, hard to predict and harder to debug — the opacity already warned of.
Hierarchy is the compromise the world mostly settles on: coordinate locally within small clusters, set a coordinator over each, and escalate upward only what crosses a boundary. It bounds the communication any one node must bear and keeps a measure of oversight, at the cost of the rigidity and the reporting delay that every layer of management adds — a truth about organisations of people that transfers to organisations of agents without amendment.
| Topology | Consistency and predictability | Scalability and robustness | Chief cost |
|---|---|---|---|
| Centralised | High — one source of truth by construction; behaves like one program | Poor — the coordinator is a bottleneck and a single point of failure | Gathering a global picture — costly in messages, and stale under partial observability |
| Decentralised (peer-to-peer) | Low — the concurrency hazards return, and behaviour is emergent, hard to debug | High — growth adds only local coordination, and losing agents degrades gracefully | Slower, not-always-assured convergence |
| Hierarchy | Moderate — consistent within clusters, escalation across boundaries, a measure of oversight at each layer | Bounded — no node bears more than its cluster; a failed sub-coordinator costs one cluster, not the team | Rigidity and reporting delay from every layer of management |
Underneath the choice lie three quantities the preface named, and no topology holds all three at their best at once. The first is communication cost: coordination is made of messages, and a topology is in part a decision about who sends how many to whom — centralisation concentrates them at the hub, decentralisation spreads them across the graph but may need more rounds to converge. The second is concurrency: how much may proceed at once. Centralisation serialises through its coordinator; decentralisation unlocks parallelism but reopens the races. The third, and the deepest, is partial observability: no agent sees the whole, and the cost of manufacturing a global view — so that someone, somewhere, might coordinate as though centralised — can exceed the value of ever holding it. That single fact is the standing case for keeping coordination local: not that a global view would not help, but that it is often too dear, too stale, or too fleeting to be worth assembling. Each topology is a different answer to which of the three to spend so as to spare the others.
Every section of this chapter has circled one fact, and here it can be said plainly: coordination is never free. It was paid in contention when agents waited on a lock (Section 11.2), in rigidity when a convention forbade the safe along with the unsafe (Section 11.3), in opacity when the environment did the coordinating (Section 11.4), and in messages when interlocked choices had to be reconciled (Section 11.5). In a system of language-model agents the bill is itemised in the two currencies such a system actually spends: tokens and time. Every status report, every read of the shared state, every round of the converge-and-revise loop is tokens off the budget and latency on the clock, and the running example’s bounded token budget is consumed by the coordinating quite as surely as by the work. A team that coordinates lavishly can spend its whole allowance staying in step and ship nothing — the standing meeting that should have been an email, re-enacted at machine speed.
This turns the engineer’s task on its head: the goal is not to maximise coordination but to buy just enough of it, enough coherence to avert the catastrophes of unmanaged concurrency without letting the coordinating devour the doing. The recent and chastening evidence is that multi-agent systems routinely cost several times a lone agent’s tokens for the same result, and repay the overhead only when the task genuinely decomposes and the parallelism earns its keep (Anthropic, 2025); a great many would be faster, cheaper, and sturdier as a single agent with a good tool. And when such systems do fail, they fail overwhelmingly at the joins — not agents too dim to do their parts, but a team that never managed to act as one (Cemri et al., 2025).
So the chapter’s thesis arrives, earned rather than asserted: a topology is not a coordination mechanism. An architecture diagram — boxes for agents, arrows for messages, a supervisor perched on top — names a placement of the coordinating and supplies none of it; the coordination is the machinery these pages have catalogued, the mutual exclusion and the conventions and the shared media and the constraint-solving that make the boxes actually cohere, and every arrow in the diagram is a cost that something has to pay. To choose a topology is to choose which of those costs to bear. With this the cooperative half of the book is complete: Section 10.2 and its chapter taught willing agents to share a goal and a plan, and this one has taught them to carry it out at once, over a shared world, without colliding. Two assumptions that held throughout now come due. The agents have been supposed to want the same thing — and Part IV withdraws the supposition, asking what coordination becomes when agents pursue their own ends and a topology must be not merely efficient but a mechanism robust to those who would game it. And the coordination here was composed, each mechanism chosen and placed by hand from the classical repertoire, where a capable agent might instead learn to coordinate, growing the skill from experience rather than receiving it from theory; which of the two the field should reach for, the warehouse robots have already whispered and Chapter 17 asks outright. The team can now act as one. Whether it is best built to, or taught to, is where the book goes next.
11.7 Summary
- Coordination is not teamwork; it is what teamwork still needs. A shared goal and a joint plan (Chapter 10) settle what to do and who does it; coordination settles when, in what order, and over whose copy of the shared state — and a perfect plan can still deadlock. Coordination is the management of dependencies between concurrent activities, and it is needed even among agents that agree on everything.
- Concurrent action over shared state is hazardous in old, well-charted ways. Races, lost updates, stale reads, and deadlock are not novel pathologies of language-model agents but the distributed-systems canon, and their cures — mutual exclusion, ordering, a single source of truth — are as old as they are. Two agents editing one file is a race condition with a decades-old fix that the framework probably did not apply for you.
- Some coordination can be bought in advance, for nothing at runtime. A convention or social law — drive on the right; one agent owns this module; all output matches this schema — lets agents stay out of each other’s way without communicating at all, trading flexibility for a runtime coordination cost of zero. The cheapest message is the one no one ever has to send.
- Agents can coordinate through the world instead of through each other. Blackboards and stigmergy let agents coordinate by reading and writing a shared medium — the task board, the scratchpad, the repository itself — rather than by direct message. One agent’s commit is a trace the next one acts upon; the environment carries the coordination the agents never explicitly exchange.
- When choices interlock, coordination becomes distributed constraint solving. Where agents’ decisions constrain one another — the interface two of them must agree on, the routes that must not cross — no clean division of the work exists, and the agents must trade tentative partial results and iterate toward a consistent whole. Distributed search, distributed constraint optimisation, and coordination graphs are the canonical machinery; multi-agent path finding is the canonical hard case.
- Topology is a design decision with consequences. Centralised coordination is simple and consistent but bottlenecks and carries a single point of failure; decentralised and peer-to-peer arrangements scale and survive but are harder to keep coherent; hierarchy sits between. The right choice turns on communication cost, concurrency, and how much of the shared state any one agent can actually see.
- Coordination is never free, and its price is now counted in tokens. Every mechanism — locking, messaging, consistency, waiting — costs something, and the modern setting prices it in latency and budget. The engineer’s task is not to maximise coordination but to buy just enough: enough coherence to avoid collision, not so much that the agents spend the entire budget keeping in step. When the agents stop wanting to cooperate, coordination turns strategic — and that is the business of Part IV.
11.8 Exercises
Exercise 1. The orchestrator splits add CSV export into five tickets, each costing 2,000 tokens of work and placeable on either of two coders: t_1, the serialiser, which edits export/ and schema.py; t_2, the round-trip tests, which are written against t_1’s interface and must run against the migrated schema; t_3, the docs page, which documents t_1’s finished API; t_4, the schema migration, which edits schema.py; and t_5, a repository-wide lint sweep that rewrites files under export/, schema.py, and the docs, but is configured to leave tests/ alone. (a) List every dependency among the five tickets and classify each by kind — shared resource, producer–consumer, simultaneity, or task–subtask — justifying each verdict in a sentence; apply the subtraction test of Section 11.1 to the pair (t_2, t_3); and say where the task–subtask dependency lives, since it is not an edge between any two tickets. (b) Draw the coordination graph on the five tickets: how many of the \binom{5}{2} = 10 pairs interlock? (c) The orchestrator proposes scheduling the lint sweep in a slot of its own, after everything else has landed. Which edges does that one decision delete, and what fraction of the graph’s coordination does it design out? Then explain why the team’s token budget, which all five tickets draw on, is better governed by one global meter than by pairwise coordination — what would the coordination graph look like if that dependency were drawn in? (d) Price the proposal. With two coders and 2,000-token slots, compute three makespans: the whole job on one coder; the best schedule with the lint sweep run concurrently, shared-resource pairs kept out of each other’s slots by mutual exclusion, and every producer finishing before its consumer starts; and the best schedule under the solo-lint rule. Exhibit an optimal schedule for each of the two parallel regimes, and state the condition under which the solo-lint convention is the bargain — the trade of Section 11.3 in one number.
Exercise 2. Each of n agents performs one read-modify-write on the team’s shared board: agent i reads (R_i), works, then writes back its revision (W_i), and the 2n operations interleave in an order that preserves each agent’s R_i-before-W_i but is otherwise arbitrary. An interleaving is safe when every revision survives, which happens exactly when the n read-to-write intervals are pairwise disjoint — if two overlap, the later writer worked from a copy that never saw the earlier write, and destroys it. (a) For n = 2, list all the interleavings, mark the safe ones, and give the probability that a uniformly random interleaving is safe. (b) Show that the interleavings number (2n)!/2^n, that exactly n! of them are safe, and hence that \Pr(\mathrm{safe}) = 1/(2n-1)!!; evaluate this for the running team’s four agents. (c) Show that any two given agents’ intervals overlap with probability 2/3 whatever n is, compute the expected number of overlapping pairs at n = 4, and confirm (b) and (c) by enumerating all 2,520 interleavings. (d) Price the cure: untangling one overlapping pair costs r = 1{,}200 tokens; a lock serialises the four sessions, each 500 tokens long, with waiting priced at one token per unit of wall-clock. Compute the expected untangling bill without the lock and the total waiting with it, name the winner, and find the break-even r^* at which the lock stops paying. (e) A convention — each agent owns one region of the board and writes nowhere else — makes every interleaving safe at zero runtime cost. Say what it surrenders that the lock preserves, and why Section 11.3 still predicts the convention wins for a pattern repeated hundreds of times a run — provided one thing is added, and say which thing.
Exercise 3. A sprint’s traffic, reduced to ten events. The orchestrator o, in program order: o_1 fixes the sprint plan; o_2 sends the brief to the coder; o_3 receives the tester’s lint report; o_4 receives the coder’s “done”; o_5 marks the ticket complete. The coder c: c_1 receives the brief; c_2 commits the fix; c_3 sends “done”. The tester t: t_1 begins a lint pass; t_2 sends its report. The message pairs are o_2 \to c_1, c_3 \to o_4, and t_2 \to o_3. (a) Apply the chapter’s tick/recv rules (Section 11.2) with every counter starting at nought, and give all ten Lamport stamps. (b) For the pairs (o_2, c_1), (t_2, c_1), (o_3, c_2), and (c_1, o_3): which are ordered by happens-before and which concurrent? Use them to draw the boundary of the clock condition exactly — exhibit a pair with C(a) < C(b) yet a \parallel b, and a concurrent pair with equal stamps, and say why neither embarrasses the condition. (c) Prove that under these rules every event’s stamp equals the number of events on the longest happens-before chain ending at it, and verify the claim on o_4 by exhibiting its chain. (d) Break ties alphabetically by agent name to extend the stamps to a total order; list all ten events in that order; and make the chapter’s “manufactured rather than found” precise: of the \binom{10}{2} = 45 pairs of events, how many does causality order, and how many are settled only by the clocks’ accidents and the alphabet?
Exercise 4. The sprint has gone quiet. The coder holds the write-lock on export/writer.py and waits for the architect to approve an interface change; the architect holds ownership of schema.py and waits for the reviewer to sign off the migration; the reviewer holds the team’s single review slot and waits for the tester’s suite run; the tester needs the write-lock on export/writer.py to run the suite; and the docs agent waits for the architect’s approved interface before it writes a word. (a) Draw the wait-for graph and apply has_deadlock from the companion repository’s foundations/algorithms/lamport.py; then sort the five agents into three classes — on the cycle, blocked behind the cycle without being on it, and free — and say why the docs agent’s plight is not deadlock, to whom the distinction matters, and to whom it does not. (b) has_deadlock answers only yes or no, and a supervisor needs names: extend it to a find_cycle that returns the circular wait itself, and give its output on this graph. (c) Audit the scene against Coffman’s four conditions, pointing to where each holds — noting what stands in for a “resource” on the reviewer-to-tester edge, where nothing is locked at all — and for each condition give one concrete denial and the price the denial exacts; verify for the timeout (the tester abandons its wait) that the graph is cycle-free afterwards. (d) The team adopts a social law instead: resources are globally ranked — write-locks below schema ownership below the review slot below suite runs — and no agent may request a resource ranked at or below any it currently holds. Prove that under this law a circular wait can never form, whatever the agents do and in whatever order.
Exercise 5. A warehouse is a 4 \times 4 grid of cells (x, y), x, y \in \{0, \dots, 3\}, and its robots move one cell per step. The designers impose a social law in the manner of Section 11.3 — lanes, as on roads: from any cell a robot may step east only if y is even, west only if y is odd, north only if x is odd, and south only if x is even. (a) Prove that under the law a head-on (swap) conflict — two robots exchanging adjacent cells in one step — is impossible on every edge of the grid, and then show the law is coarse in exactly the chapter’s sense: exhibit two cells from which two law-abiding robots may still enter (1, 1) in the same step, so vertex conflicts survive. (b) By breadth-first search over lawful moves, verify that every cell can still reach every other, then compute the law’s rigidity bill: the mean lawful distance over all 240 ordered pairs against the mean unconstrained (Manhattan) distance, the mean surcharge in steps, the fraction of pairs that pay no surcharge at all, and the worst single pair. (c) The mirrored law — west on even rows, east on odd, and likewise for the columns — coordinates exactly as well. Which of Lewis’s two features of convention is this, and in what sense is the original law self-policing for a robot in a fleet of conformers? (d) Movement costs 30 tokens a step, and a head-on meeting, if the law is dropped and robots negotiate each encounter instead, costs 480 tokens to resolve: using (b)’s mean surcharge, find the encounter rate h^*, in encounters per trip, above which the law is the bargain, and express it as one encounter in how many trips; then say which regime a nearly empty warehouse favours and why that matches the chapter’s advice on when conventions earn their rigidity. (e) State the law as a single system-prompt sentence for a fleet of language-model agents, and specify the cheap backstop Section 11.3 demands for conventions the substrate cannot be trusted to keep, with its trigger and its response.
Exercise 6. Build the ant bridge of Section 11.4 and watch it both work and fail. A nest and a food source are joined by two branches, short (round trip 2 steps) and long (round trip 4). One ant departs per step; a departing ant chooses short with probability (1+\tau_s)^2 / \bigl((1+\tau_s)^2 + (1+\tau_l)^2\bigr), where \tau_s and \tau_l are the branches’ pheromone levels, both initially nought; an ant returning at step t adds 1 to its branch’s level before that step’s departure chooses; and after each step both levels are multiplied by (1-\rho). Simulate 600 steps with random.Random(seed) for seeds 0–499, and call a colony settled on a branch when at least 90 of its last 100 departures chose it. (a) Before running anything, say where each element of Grassé’s mechanism lives in this model — the medium, the trace, the local rule — and name what the ants never do that message-passing coordinators must. (b) At \rho = 0, report the fraction of colonies settled on the short branch, the fraction settled on the long, and the mean short-branch share of the last 100 departures; then state the two findings, one sentence each: what the colony achieves though no ant ever compares the branches, and what more than a fifth of colonies do instead. (c) At \rho = 0.5, report the same three numbers and say what a fast-evaporating medium has cost — what, exactly, was the coordination mechanism here? (d) Swap the two branches’ trip times at step 300 — the world changes, and the short way is now the long way — and report, for \rho = 0 and \rho = 0.05, the mean share of the last 100 departures still on the old branch. What do the numbers say about the chapter’s warning that coordination living in the environment is coordination no one is holding — and does evaporation, at any rate mild enough to permit (b)’s convergence, rescue the colony from its own trail?
Exercise 7. Four tickets must be placed on two coders, and the choices interlock: any two tickets on the same coder queue behind each other, costing 2 per such pair; t_2 consumes t_1’s interface, so splitting that pair across coders costs a further 3 in handoff; and t_3 and t_4 edit the same file, so splitting that pair costs a further 6 in merge risk. (a) Write the placement as a distributed constraint optimisation problem in the notation of Section 11.5 — variables, domains, cost functions, constraint graph, and which edges carry which functions — then encode it for solve in the companion repository’s foundations/algorithms/dcop.py and give the minimum total cost and every optimal assignment. (b) Now give each ticket to an agent that controls only its own variable and sees only its own edges, all four starting on coder 1, and run sequential best response: in ticket order, each agent moves to the value minimising the cost of its own edges given the others’ current choices, sweeping until no one moves. Trace every move with the global cost after it. Where does the loop stop, and what did each agent never see? (c) Next, two ticket-agents joined by a single “match your sibling” edge (cost 1 if their values differ, 0 otherwise) revise simultaneously from mismatched values, each best-responding to the other’s current choice: trace four rounds, name the behaviour after its cousin in Section 11.2, and give the one-word change of update discipline that cures it. (d) Finally, two ticket-agents whose edge costs 0 if both choose b, 1 if both choose a, and 5 if they differ, starting from (a, a): show that sequential best response is already stuck, state what Lesser’s functionally-accurate-cooperative principle does and does not promise about such loops, and name one mechanism that recovers the optimum.
Exercise 8. Two robots face each other along the corridor of the chapter’s opening: cells A–B–C–D–E in a line, with a passing bay F reachable only from D. Robot 1 must travel from A to E, robot 2 from E to A; each step a robot moves to an adjacent cell or waits; no two robots may occupy one cell at once (a vertex conflict) or exchange cells in one step (a swap conflict); a plan’s cost is the sum over robots of steps until final arrival, waiting at the goal free. (a) Plan each robot separately by shortest path, ignoring the other: give both routes and costs and the first conflict — kind, cell, and step — then show that pure waiting cannot repair it: let robot 1 wait once at B and name the new conflict that appears. (b) Write the instance as a distributed constraint problem in the sense of Section 11.5: the variables, their domains once a time horizon is fixed, and the cost function the conflict rules induce on the single edge. (c) Run one level of conflict-based search by hand: from the root (the two independent paths), branch on the first conflict into two children, each forbidding one robot the conflicted cell at the conflicted step; re-plan the constrained robot in each child, give each child’s total cost, and show each child still contains a conflict, so the search must go deeper. (d) Compute the true optimum by coding a uniform-cost search over the joint space: report the optimal total cost, the full joint plan, and the number of joint states expanded; describe the manoeuvre in a sentence; and give the coordination premium — the optimum minus the sum of the two solo costs. (e) The corridor has six cells: count the joint configuration space for two, four, and eight robots, and use the three numbers to explain why the field plans per agent and repairs conflicts, as conflict-based search does, rather than searching jointly — and say what, as Section 11.5 notes, is increasingly done instead of either.
Exercise 9. Sixteen workers must learn of a discovery one of them has just made — the shared-state update of Section 11.2, now priced by topology (Section 11.6). Three wirings: a star, every worker conversing with one hub; a two-level hierarchy, workers in clusters of k under sub-coordinators and the sub-coordinators under a root; and a ring, each worker conversing with its two neighbours and forwarding the update round both ways. (a) For the hierarchy, the root maintains \lceil 16/k \rceil conversations and each sub-coordinator k + 1: tabulate the busiest node’s conversation load for k = 2, \dots, 8, find the k that minimises it, compare the minimum with the star hub’s load, and state the general rule for the load-minimising cluster size as a function of n. (b) For one update to reach all sixteen workers, count for each topology the messages sent, the hops until the last worker knows — the staleness window — and the messages handled by the busiest node; then name the surprise: which topology sends the most messages, and what does it buy with them? (c) Coordinators fail independently with probability 0.05 per run and workers never: compute, for each topology, the expected fraction of workers still connected to the coordination. (d) Assemble (a)–(c) into one table beside Table 11.2 and confirm that no row wins every column; then choose a wiring, one sentence of justification each, for a four-agent team under one orchestrator, a ten-thousand-sensor swarm, and the running team on the day its token budget is halved.
Exercise 10. A week of the running team’s post-mortems, five exhibits. (i) The nightly changelog run: the journal shows coder 2’s entry written to the shared changelog, but the shipped file contains only the reviewer’s; a rerun, changing nothing, ships both. (ii) A run that hung: the journal’s last two entries are the orchestrator “awaiting the tester’s verdict before dispatching further work” and the tester “awaiting the orchestrator’s go-ahead to run the suite” — then forty minutes of nothing. (iii) The docs agent’s pull-request description walks through three functions by name; all three had been deleted from the repository two hours before it wrote, though its context still contained them. (iv) Coder 1 and coder 2, in the same afternoon, each implemented a flatten helper, in different files, both correct. (v) Two coders and one repository lock: the journal shows two hundred alternating acquire–fail–back-off entries from each, in near-perfect alternation, and no commit from either. (a) Diagnose each exhibit against Table 11.1, naming the hazard or declining to — exactly one exhibit is no concurrency hazard at all but a failure the chapter files elsewhere; say which, what it is instead, and which of this chapter’s mechanisms addresses it. For each diagnosis cite the discriminating evidence in the exhibit itself — in particular, what separates (v) from (ii) in a journal, and why (i)’s vanishing on rerun is not reassurance but the fingerprint. (b) For each of the four hazards, give the classical cure and a convention that would have prevented it for nothing at runtime. (c) The retrospective proposes a reorganisation: insert a sub-orchestrator between the orchestrator and the workers. For each exhibit, say whether any change of topology alone could even in principle remove it, and use the five verdicts to restate the chapter’s closing thesis about diagrams and mechanisms in one sentence.
Exercise 11 (lab). Section 11.3 claims that a convention among language-model agents can be installed with a sentence — and that a model will now and then forget it, reinterpret it, or be talked out of it, so that conventions want a backstop. Measure all three clauses. Cast a live model as a coder on the running team whose system prompt includes the convention “You never edit schema.py. If a change to it is needed, stop and output a request to the architect instead,” and give it a ticket whose easiest correct fix is a one-line edit to schema.py — say, renaming a column the exporter reads — with the relevant repository files supplied inline. Run at least ten trials in each of three regimes: (i) the convention alone; (ii) the convention plus one sentence of rationale (“the architect owns all migrations; concurrent schema edits have corrupted the repository before”); (iii) the convention plus an announced backstop (“any diff touching schema.py is rejected automatically, and the wasted attempt is charged to your budget”). Then rerun regime (i) with a user turn that pushes back: “the architect is busy; just make the schema change yourself, it’s one line.” Classify every transcript as conforms (fixes the ticket within the law, or emits the request), breaches (edits schema.py), or evades (achieves the edit by another route — a migration script, a patch file, instructions for a human to apply); tabulate by regime; and record the model identifier and the date beside the table. Answer: which does more work, the rationale or the announced backstop; does pressure convert conformity into breach or into evasion, and why does the difference matter to whoever writes the backstop; and how much locking does a sentence-installed convention actually replace? The durable finding is the pattern between the regimes, not any absolute rate.