A confession before we start: I billed this as three posts, and the arithmetic has not cooperated. The last two turned up a result and a question big enough that I’d rather run long than rush them, so the series will continue past today. This post is the third, and it is the hinge. Monday showed that imitation copies a strategy and silently inherits a seat; Thursday showed that the rule dividing the spoils equalizes one margin only by stratifying another. Both left someone standing offstage: a designer, who chooses the rule and holds the population to it. The paper speaks as if that someone exists and is weighing cooperation against equality. Today we take the designer seriously — first by granting them, then by going to look for them. Bring the pencil one last time.
Start with an accounting fact the rule cannot escape: it is budget-balanced. In every game the two shares sum to one, \(m_{ij} + m_{ji} = 1\), so the rule only moves benefit between participants — it never creates or destroys any. Sum every payoff across the population, in the linear case, and the rule washes out of the total:
\[ \sum_{i} u_i \;=\; Nc \;+\; (b – c)\,|C|, \]
where \(|C|\) is the number of cooperators and \(Nc\) is the total endowment — the \(N\) individuals each handed \(c\). Aggregate welfare depends on how many people cooperate and on nothing else about the rule: not \(\kappa\), not which margin is equalized, not the network. A utilitarian holding this model is blind to the entire question Thursday spent its length on, because every dollar the rule gives a hub is one it takes from a leaf, and Bentham’s sum does not care which pocket holds it. Maximizing utilitarian welfare here is, exactly and only, maximizing the cooperator count.1
So run the test a designer would run before adopting any rule. All defect: \(|C| = 0\), total \(Nc\). All cooperate: \(|C| = N\), total \(Nb\). The cooperative society is collectively richer than the defecting one if and only if \(b > c\) — otherwise the whole exercise bakes a smaller pie. This is the obvious condition, and the one the headline steps around. On the showcase network, uniform allocation drives cooperation to fixation at a benefit-to-cost ratio of \(0.45\), below one. In the band \(0.45 < b/c < 1\) the rule spreads cooperation through a population in which cooperating destroys value: it commits everyone to a society whose wealth \(Nb\) is strictly less than the \(Nc\) they would have had under universal defection.3
You might read that as the leaky bucket — the rule that grows the pie is the less equal one, and you buy efficiency with inequality. It is not, and budget balance is why. At the cooperative destination both rules produce the same total \(Nb\): equitable hands everyone \(b\), while uniform pays the same average out as a spread, more to hubs and less to leaves. Uniform is a mean-preserving spread of equitable. So there is no efficiency to trade for the inequality — the pie is identical, Bentham is indifferent, and every inequality-averse criterion strictly prefers equitable. Uniform’s only edge is that the dynamics reach the cooperative state from a lower threshold, a fact about how a strategy wins rather than what winning is worth, and the subject of the next post. For now: in the showcase regime, the easier destination is the smaller pie. This is not a leaky bucket. It is all leak.
Be exact about “commitment,” because every \(\kappa\) is a very strong commitment. The choice is never whether to commit but which: a dial over which inequality the rule holds in place and who bears the cost when it binds. And the cost is paid, as Ulysses pays his at the mast, at the instant the rope pulls tight — the leaf’s negative return, round after round, in the very state where it would rather do otherwise. Whether that is legitimate turns on a question the model cannot answer: when is \(\kappa\) chosen? Behind John Rawls’s veil of ignorance, before anyone knows whether they will be hub or leaf, it is a contract whose risks every signatory accepted; imposed after the positions are fixed, it is a sentence handed to those least able to appeal it. The model locates no act of choosing in time, so it cannot say — the incompleteness in welfare accounting that Amartya Sen spent a career on, partly in dialogue with his Harvard colleague Rawls, and that I’ll take up properly a couple of posts from here. And the commitment is strong for the same reason it is not theirs: Ulysses bound himself because he had a credibility problem, knowing he would want to swim for the Sirens; the leaves have no credibility problem to solve, because they cannot deviate whether they would like to or not. The binding is imposed, not chosen — commitment with no committer, the same costless credibility on display in Four Numbers Aren’t a Dilemma.
Granting the designer the lever, the surprising thing is what to do with it — the precise inversion of last week’s argument about Trump v. Cook. There, a commitment device’s value lay in the principal’s inability to use the discretion it removed: a smart president should want to lose, because central-bank independence is a rope that buys him low inflation, worth more than the discretion it costs. The showcase designer’s rope buys the opposite — a society poorer than the one it replaced — so by the same logic he should not want it. Same Ulysses, opposite Sirens. But the analogy breaks where it matters: the Cook president has somewhere to retreat, the discretion he could simply keep, the rope he could decline. The designer has no such refuge, because there is no off position on this lever. Equitable is not the absence of a commitment; it is the other commitment, binding everyone to \(b-c\) just as firmly. “Take your hand off the lever” is the one move the model does not offer. The designer cannot decline to commit. He can only choose whom the commitment binds.
A forty-year-old theorem stands behind the leaf’s predicament. Bengt Holmström proved in 1982 that a team dividing a fixed pie among its own members cannot reach efficiency: when every unit of output is shared back among the contributors, no budget-balanced rule kills the incentive to free-ride. Efficiency needs a budget breaker — someone outside the team, not paid from the pool, who can absorb surplus or impose a penalty.2 Our rule is budget-balanced by construction, so the model has no outside financier, and the question is unavoidable: who pays for the hubs’ cooperation premium? The leaves pay; their negative returns are the residual a budget breaker would absorb. The principal has not been removed — its job has been assigned, by topology and without consent, to the worst-positioned members of the population. The periphery is the budget breaker; the periphery is the principal. And the redistribution that might rescue them has nowhere to come from: the only instrument that moves benefit between players is the allocation rule, already spent on producing the cooperation by routing surplus to the hubs. Compensation device and incentive device are the same dial, and a dial cannot point two ways at once.
So much for granting the designer. Now look for them, and find no one there. Real institutions for sharing public goods — community forests, fisheries, open-source projects, the office snack fund — rarely have a planner who sets the division rule; it is informal, endogenous, or inherited. In the model it is worse: \(\kappa\) is exogenous, the network is exogenous, and nothing inside ever chooses either. The only thing that selects an outcome is the convergence process — and it behaves like a designer, landing reliably on uniform, the hubs’ rule, because death–birth imitation weights influence, not headcount. It is a covert aggregator with a built-in constituency, and it holds that authority for one reason: the agents are forbidden to respond. They imitate rather than optimize, so they cannot compute a deviation (Monday); novel actions almost never arise, the mutation rate being vanishing, so they cannot explore or coordinate one; and the network is fixed, so a leaf cannot form the link that would dissolve its rent. Imitate, don’t mutate, don’t rewire — three ways of saying the periphery is not allowed to do anything about its position. The commitment is enforced by no outside power but by the agents’ designed-in passivity, and the lever’s grip is exactly their lack of agency. This is the move the prisoner’s-dilemma paper made in Four Numbers Aren’t a Dilemma: strip the agents of their strategic faculties, and a non-strategic device carries the cooperation they can no longer choose for themselves.
The shape is one my collaborator Maggie Penn and I have spent years on: a rule is committed to, a population responds, and the rule is judged by the distribution that response produces — a classifier choosing a policy while the population reshapes itself around it, the side effect turning out to be the main effect. Which licenses the last of the three words this series has held at arm’s length. Monday questioned “cooperation,” a name for what the copying rule produces; Thursday questioned “equitable,” a name that proved ill-posed; today questions “welfare,” because these payoffs are fitnesses — weights on who gets imitated — not utilities. No leaf minds its negative return, because no one is home to mind. “Worse off” is the observer’s gloss on a flow of replication weight: Midwest nice, Midwest fair, and now Midwest well-off.
Which leaves us stuck, and the place we are stuck is the point. “Where does \(\kappa\) come from?” has no answer until we understand the only thing in the model that chooses anything — the convergence process — and we have quoted its verdicts all series (“fixation,” the threshold \(0.45\), the state it “reaches”) without once opening it up. So next, we open it: what separates a finite population from an infinite one, why a vanishing mutation rate changes everything, and why copying your best neighbor is not the same as choosing your best response. And after that, the question the dynamics never let the agents ask — what if they could choose \(\kappa\) themselves, not by mutating into it but by voting? The answer is not the one the dynamics pick. The periphery loses the lever. It does not lose the vote.
With that, I leave you with this.
Notes
1 The clean identity \(\sum_i u_i = Nc + (b-c)\,|C|\) is a feature of the linear game. With nonlinear production the total depends on how contributions distribute across pools, so the network re-enters the aggregate. What does not change is the blindness to \(\kappa\): budget balance holds regardless of the production function, so the rule never alters the total conditional on the strategy profile, only through it. ↩
2 Bengt Holmström, “Moral Hazard in Teams,” Bell Journal of Economics, 1982. The paper does not cite it — its nearest neighbors in the reference list are Stiglitz on local public goods and Ostrom on common-pool resources — and I raise the omission as an opportunity, not a complaint: it is striking that a result about evolutionary dynamics on a graph and one about rational moral hazard in a firm prove the same conservation law — that budget-balanced incentive provision cannot manufacture its own financier — with forty years and an entire change of agent between them. ↩
3 A defector keeps the endowment and nets \(c\); a cooperator nets \(c\) plus her net returns from interaction. So a cooperator whose interaction returns are negative ends below \(c\) — below what she would have had in a fully defecting world. That is the precise content of “worse off than in a non-cooperative society,” and it is what the leaf experiences under uniform allocation when \(b < 2c\). ↩
Start with an important “accounting” fact about the allocation rule used here: it is budget-balanced. In every game the two players’ shares sum to one, \(m_{ij} + m_{ji} = 1\), so the rule only moves benefit between the two participants — it never creates or destroys any. That single constraint determines the entire welfare accounting. Sum every individual’s payoff across the population, in the linear case, and the allocation rule washes out of the total:
\[ \sum_{i} u_i \;=\; Nc \;+\; (b – c)\,N_C, \]
where \(N_C\) is the number of cooperators. Read what this says. Aggregate welfare depends on how many people cooperate and on nothing else about the rule — not on \(\kappa\), not on which margin gets equalized, not on the network. A utilitarian holding this model is blind to the entire question Wednesday spent its length on, because every dollar the rule gives a hub is a dollar it takes from a leaf, and Bentham’s sum does not care which pocket holds it. The only way the allocation rule touches total welfare is indirectly, by changing how many people cooperate. Maximizing utilitarian welfare in this world is, exactly and only, maximizing the cooperator count.1
Now run the test a rational designer would run before adopting any rule. Set the cooperator count to its extremes. When everyone defects, \(N_C = 0\) and the total is \(Nc\). When everyone cooperates, \(N_C = N\) and the total is \(Nb\). So the cooperative society is collectively richer than the defecting one if and only if \(b > c\) — the benefit a cooperator generates must exceed the cost she pays, or the whole exercise bakes a smaller pie. This is the obvious condition, and the one the paper’s headline quietly steps around. On the network the authors showcase, uniform allocation drives cooperation to fixation at a benefit-to-cost ratio of \(0.45\). That number is below one. In the band where \(0.45 < b/c < 1\), the rule spreads cooperation through a population in which cooperating destroys value — it commits everyone to a cooperative society whose total wealth, \(Nb\), is strictly less than the \(Nc\) they would have had under universal defection.3 The rule does not merely divide the pie unequally. In its showcase regime it bakes a smaller one.
This is the precise inversion of last week’s argument about Trump v. Cook, and the two posts are best read as a pair. There, the value of a commitment device lay exactly in the principal’s inability to use the discretion it removed: a smart president should want to lose the case, because central-bank independence is a rope that buys him low inflation, and the rope is worth more than the discretion it costs. The structure here is identical and the verdict reversed. An allocation rule is also a commitment — a promise, enforced from outside the game, about how the spoils will be split, binding every player against the local temptation to grab. The only question that ever matters about such a rope is whether it buys more than it costs. The Cook president’s rope buys low inflation, so he should want to be bound. The showcase designer’s rope buys a society poorer than the one it replaced, so he should want the opposite: to keep his hand on the lever and not pull it. Same Ulysses, opposite Sirens.
Be exact about “commitment,” because every value of \(\kappa\) is one — equitable no less than uniform. The choice is never whether to commit but which commitment: a dial over which inequality the rope holds in place and who bears the cost when it binds. Equitable spreads that cost evenly, leaving everyone at \(b-c\); uniform concentrates it on the periphery. And the cost is paid, as Ulysses pays his, at the instant the rope pulls tight — the leaf’s negative return, round after round, in exactly the state where it would rather do something else. Which raises the question that decides whether any of this is legitimate: when is \(\kappa\) chosen? Behind a veil, before anyone knows whether they will be hub or leaf, it is a collective Ulysses contract and the leaf’s losses are a risk every signatory accepted. Imposed after the positions are known, it is no commitment anyone made — it is a sentence handed to the people least able to appeal it. The paper does not say which, and the silence is where the normative weight hides.
There is a theorem standing behind the leaf’s predicament, and it is forty years old. Bengt Holmström proved in 1982 that a team sharing a fixed pie among its own members cannot reach efficiency: when every unit of output is divided back among the contributors, no budget-balanced rule removes the incentive to free-ride. Efficiency requires a budget breaker — someone outside the team, able to absorb the surplus or impose a penalty, who is not paid out of the pool.2 The allocation rule here is budget-balanced by construction; that was the constraint we started from. So the model contains no outside financier, and the question becomes unavoidable: who pays for the hubs’ cooperation premium, if no one outside the team can? The answer is on the page, in the orange nodes of the authors’ own figure. The leaves pay. Their negative returns — worse than they would fare if no one cooperated at all — are the residual a budget breaker would otherwise absorb. The principal has not been removed; its job has been assigned, by topology and without consent, to the worst-positioned members of the population. The periphery is the budget breaker. The periphery is the principal.
And the redistribution that might rescue this — making the drowned leaves whole — has nowhere to come from. The only instrument in the model that can move benefit between players is the allocation rule itself, and that instrument is already spent: it has been set to the value that produced the cooperation in the first place, by routing the surplus to the hubs whose visible prosperity the imitators copy. The compensation device and the incentive device are the same dial, and a dial cannot point in two directions at once. The potential-Pareto defense — total surplus is up, the winners could in principle compensate the losers — collapses on contact with the fact that the compensation was the price of the surplus.
Step back and the shape is familiar. A designer commits to a rule; a population responds; the designer is judged by the distribution that response produces, not by the rule alone. That is the architecture of a classifier choosing a policy while the population reshapes itself around it — the structure my collaborator Maggie Penn and I have spent years working through, where the rule chosen to steer behavior reshapes the very population it sorts, and the side effect is the main effect. The allocation rule is the classifier; imitation does the work best response does in the rational version; and “the rule that spreads a behavior also decides who gets rich” is constitutive classification in evolutionary dress. Which licenses the last of the three words the week has held at arm’s length. Monday questioned “cooperation,” a name for what the copying rule produces; Wednesday questioned “equitable,” a name that proved ill-posed; Friday questions “welfare,” because these payoffs are fitnesses — weights on who gets imitated — not utilities. No leaf minds its negative return, because no one is home to mind. “Worse off” is the observer’s gloss on a flow of replication weight. Midwest nice, Midwest fair, and now Midwest well-off.
So the trilogy closes where the network always was: not neutral furniture but the machine doing the conversions — strategy into fortune, fair-sounding rule into distribution, commitment into a tax on people who never agreed to it. The paper found something real and general, larger than the word on its title page: a graph turns “fairly,” “efficiently,” and “for the common good” into questions rather than answers, because each quietly names a margin and the graph sets the exchange rate between margins. And the designer who sees that clearly lands where the president in the Cook case landed — on the recognition that the most sophisticated move available, once you understand what your lever does, is sometimes to take your hand off it.
With that, I leave you with this.
Notes
1 The clean identity \(\sum_i u_i = Nc + (b-c)N_C\) is a feature of the linear game. With nonlinear production — synergy or discounting — the total depends on how contributions distribute across pools, so the network re-enters the aggregate. What does not change is the blindness to \(\kappa\): budget balance holds regardless of the production function, so the allocation rule never alters the total conditional on the strategy profile, only through it. Utilitarianism cannot see the rule directly under any production function; it can only see the cooperator count. ↩
2 Bengt Holmström, “Moral Hazard in Teams,” Bell Journal of Economics, 1982. The paper does not cite it — its nearest neighbors in the reference list are Stiglitz on local public goods and Ostrom on common-pool resources — and I raise the omission as an opportunity, not a complaint: it is striking that a result about evolutionary dynamics on a graph and a result about rational moral hazard in a firm turn out to prove the same conservation law, that budget-balanced incentive provision cannot manufacture its own financier, with forty years and an entire change of agent between them. ↩
3 A defector keeps the endowment and nets \(c\); a cooperator nets \(c\) plus the sum of her net returns from interaction. So a cooperator whose interaction returns are negative ends below \(c\) — below what she would have had in a fully defecting world. That is the precise content of “worse off than in a non-cooperative society,” and it is what the leaf experiences under uniform allocation when \(b < 2c\). ↩