On Monday I split the payoff an agent earns into a part that travels with the strategy and a part that stays with the seat, \(f_i = \alpha_{s_i} + \beta_i + \varepsilon_i\), and promised that the positional term \(\beta\) has a closed form that takes about three lines to derive. Today we derive it. The exercise is worth doing slowly, because it turns the paper’s headline — that the rule which best spreads cooperation is also the rule that worsens inequality — from a surprising empirical finding into something closer to a definition, and because it exposes a question hiding inside an innocent word. The paper compares two ways of dividing what cooperation produces, and calls one of them equitable. By the end of this post the right response to that word will be a question: equitable on which margin?
Here is the machine, stripped to what we need. A population sits on a network. Each round every individual receives an endowment \(c\) and chooses one strategy to play with all of their neighbors: cooperate or defect. A cooperator spreads her endowment evenly across the games she plays, so an individual with \(k_i\) neighbors contributes \(p_{ij}c\) to each game, where \(p_{ij} = 1/k_i\). A defector contributes nothing and keeps the \(c\). In any game, the two players’ contributions form a public pool, and the pool produces a benefit larger than itself: in the linear case, a pool of size \(c\) yields a benefit \(b\).1 That benefit is then divided between the two players by an allocation rule. Two rules present themselves, and both look fair. Under equitable allocation, your share of a pool is proportional to what you could contribute to it. Under uniform allocation, the two players split the benefit evenly, fifty–fifty, regardless of who put in what. We will compute what each rule does in the state where everyone cooperates — the state the dynamics are driving toward — so that the only question left is how the spoils land.
Take equitable allocation first. With both players contributing, the pool on edge \((i,j)\) has size \((p_{ij}+p_{ji})c\), and in the linear case produces benefit \(B_{ij} = b\,(p_{ij}+p_{ji})\). The equitable share is \(m^{(e)}_{ij} = p_{ij}/(p_{ij}+p_{ji})\). So individual \(i\)’s net return from this one game is her share of the benefit minus her contribution,
\[ m^{(e)}_{ij}\,B_{ij} – p_{ij}c \;=\; p_{ij}b – p_{ij}c \;=\; p_{ij}(b-c), \]
and summing over all of her neighbors, with \(\sum_j p_{ij} = 1\),
\[ \sum_{j \in N(i)} p_{ij}(b-c) \;=\; b – c. \]
Every individual nets exactly \(b-c\), on every network, regardless of degree. The distribution of payoffs is perfectly flat; its Gini coefficient is zero. In the language of Monday, “\(\beta_i = 0\) for everyone!”
Before celebrating equitable allocation as the fair rule, look at why that happened, because it is not a discovery about networks — it is an identity. The equitable share \(m^{(e)}_{ij} = p_{ij}/(p_{ij}+p_{ji})\) is built as the exact inverse of the contribution rule \(p_{ij} = 1/k_i\). The cost convention spread your endowment by degree; the equitable rule then hands the benefit back in the same proportion, undoing the positional asymmetry term for term.2 “Equitable allocation produces equality” is true in the way “the inverse function inverts” is true. It tells you the rule was designed to flatten the payoff, not that the network is benign. All of the real content lives in the other rule, the one that does not undo the cost convention.
So take uniform allocation, where each player simply gets half of every pool, \(m^{(u)}_{ij} = 1/2\). Individual \(i\)’s net return from a single game is
\[ \frac{1}{2} B_{ij} – \frac{c}{k_i} \;=\; \frac{b}{2}\left(\frac{1}{k_i} + \frac{1}{k_j}\right) – \frac{c}{k_i}, \]
and summing over her \(k_i\) neighbors — the \(b/2k_i\) terms add to \(b/2\), the cost terms add to \(c\), and the cross terms survive —
\[ u_i – c \;=\; \underbrace{\frac{b}{2} – c}_{\text{common to all}} \;+\; \underbrace{\frac{b}{2}\sum_{j \in N(i)} \frac{1}{k_j}}_{\beta_i}. \]
There is the closed form, three lines as promised. The first piece is a constant every individual receives identically. The second piece is \(\beta\), the positional term — and it is worth staring at, because of what it does not contain. Your own degree has cancelled. It is nowhere in \(\beta\). The quantity that determines your fortune under uniform allocation is not how many neighbors you have but the sum of the inverses of your neighbors’ degrees. You are rich when your neighbors are poor in connections — when you are a hub surrounded by leaves, people for whom you are a large fraction of their entire social world. \(\beta\) does not count your connections; it counts your dependents, weighted by how dependent they are. The hubs that looked so successful on Monday, the ones an imitator was busy copying, are collecting precisely this: not a return on a better strategy, but a tax on their neighbors’ lack of alternatives.
One check makes the mechanism unmistakable. On a regular network, where every individual has the same degree \(k\), each neighbor contributes \(1/k\) to the sum and there are \(k\) of them, so \(\beta = b/2\) for everyone and the payoff is flat — uniform allocation produces exactly the same equality as equitable allocation, Gini zero under both. Heterogeneity in degree is not a complicating detail in this result; it is the entire engine. Strip it out and the two rules become indistinguishable.
Now set the two results side by side, because together they dismantle the word. Equitable allocation makes every person equal: everyone nets \(b-c\). Uniform allocation makes every game equal: each pool is split exactly in half. Both rules equalize something, and the something is different, and there is no rule that equalizes both — because individuals sit in different numbers of games, so equality per interaction and equality per person cannot hold at once on a heterogeneous network. “Equalize” was never a complete instruction. It always carried a hidden argument: equal across what? A network is a machine for converting equality on one margin into inequality on the other, and the degree distribution is the exchange rate between them. That is what a graph does to an adverb. It makes “fairly” ambiguous in a way no amount of good intention can resolve, because the ambiguity is in the arithmetic and not in the ethics. The rule that is perfectly even in every single interaction is the rule that is grossly uneven across people, and the rule that levels people does so by being deliberately uneven in each interaction. You cannot have both, and choosing a rule is choosing which inequality to keep.
The cost of the choice is not abstract. Consider a leaf attached to a single hub. Its one neighbor has high degree, so \(1/k_j\) is nearly zero, so its \(\beta\) is nearly zero, and its net return collapses to the common constant \(b/2 – c\). That is negative whenever \(b < 2c\) — the leaf, while faithfully cooperating, ends up worse off than if no one in the population had cooperated at all, since a lone defector at least keeps its endowment \(c\). This is not a marginal case. On the network the authors showcase, cooperation under uniform allocation takes hold at a benefit-to-cost ratio of \(0.45\), and at that threshold ninety-six percent of the population is underwater — contributing, and losing, so that a handful of hubs can collect. The inequality here is not a story about cooperators versus free-riders, because in the all-cooperate state there are no free-riders; everyone contributes. It is pure positional rent under unanimous cooperation. The strategy converges everywhere. Only the payoffs refuse to.
So the two fair-sounding rules were never describing fairness. Each was equalizing one margin while quietly stratifying the other, and the gap between them is filled entirely by the degree distribution — by where people sit, not by what they do. Which leaves the question the arithmetic cannot answer on its own. Someone selects the rule. Someone enforces it from outside the game, holding the whole population to a division of spoils that, on most networks, some of them would never consent to. That is a commitment in the sense of last week’s post — a promise enforced against the temptation to renege. On Friday we ask who holds that lever, who pays for the commitment when it binds, and whether a clear-eyed designer, like the president in the Cook case, might be wiser to refuse it.
With that, I leave you with this.
Notes
1 Linear here means the benefit produced is proportional to the pool: a pool of size \(s\) yields \(f(s) = (b/c)\,s\), so a full endowment \(c\) yields \(b\). The paper also allows nonlinear production — synergy or discounting — which changes the magnitudes but not the qualitative split between the two rules; the conflict between cooperation and equality survives across the nonlinear range. The linear case is the one you can do on a napkin, which is why it is the one here. ↩
2 That the equitable rule exactly cancels the cost convention is worth one more beat: it depends on cooperators spreading a fixed endowment by degree. Had the model instead charged a fixed cost per game, the contribution proportions would differ, the inverse that defines “equitable” would differ, and the rule that flattens payoffs would be a different rule with a different name. The choice of cost convention is upstream of everything here, and it is itself a modeling decision the word “equitable” quietly inherits rather than earns. ↩