When you face a decision you don’t know how to make, you look sideways. You find someone who is doing well — the colleague who made tenure, the neighbor whose kids turned out kind, the friend who somehow retired comfortably a decade early — and you copy what they did. But the rule has a catch, easiest to feel at the limit. Not everybody can be Taylor Swift. You can study her songwriting note for note and you will not become her, because most of what you are looking at is not the songwriting — it is the stadium, the catalog, the place she occupies in a few hundred million people’s attention, none of which the behavior created and none of which comes home with you when you copy it.
Both the rule and its catch have exact statements. Start with the rule: when a population copies the successful, the practice has a name and a law of motion. Write \(x_s\) for the share of the population using strategy \(s\) and \(f_s\) for that strategy’s fitness; the share grows in proportion to how far the strategy beats the population average,
\[ \dot{x}_s = x_s\,\bigl(f_s – \bar{f}\bigr). \]
This is the replicator equation, the engine underneath most of evolutionary game theory and underneath the paper I want to spend this week on, and it deserves more respect than its simplicity invites. The individual rule that aggregates into it — sample someone, and if they did better than you, switch toward their strategy with probability proportional to how much better — is not a lazy heuristic we tolerate in agents too dim to optimize. Karl Schlag proved it is the best rule available to anyone who can do nothing but observe others’ outcomes.1 Imitation is not the absence of optimization. For a creature that can only watch, it is optimization.
Now look at what the notation assumes. Writing fitness as \(f_s\) — indexed by the strategy alone — declares that payoff is a property of what you do. That declaration is false on a network, and the falseness is the whole subject. The payoff to an individual is not determined by her strategy; it is determined by her strategy and by where she sits. Decompose it:3
\[ f_i = \underbrace{\alpha_{s_i}}_{\text{strategy}} + \underbrace{\beta_i}_{\text{position}} + \varepsilon_i. \]
The first term is what the replicator equation thinks it is tracking — the value of the action, the part that travels with you when you adopt it. The second is the value of the seat: the audience, the connections, the place in everyone else’s attention that the behavior did not create and cannot transfer. In a well-mixed population, where everyone faces the same crowd, \(\beta_i\) is the same for all and drops out, and the notation \(f_s\) is honest. On a network it is not. Different positions have different \(\beta\), and that single fact is where the week lives.
Here is the consequence. An imitator who copies her most successful neighbor and credits his strategy is estimating \(\alpha\) while ignoring \(\beta\) — a regression of behavior on payoff with the position term left out. The omission is not random: the friendship paradox guarantees that the neighbors she samples are disproportionately the high-degree hubs, the large-\(\beta\) individuals whose success is least about their strategy. The omitted variable is correlated with the regressor, so the estimate of \(\alpha\) is biased upward by exactly the part of the payoff that came from the seat — the part she cannot take home. “Not everybody can be Taylor Swift” is the opening paragraph in symbols: \(\beta_{\text{Swift}}\) is enormous, \(\beta_{\text{you}}\) is not. The quiet version is the one that catches you — the café that thrived on its corner, not its espresso; the analyst who looked brilliant because the market rose underneath him. The more striking the success, the larger the share you should charge to \(\beta\), because large \(\beta\) is what makes success striking.
Static omitted-variable bias would be bad enough, but the replicator equation does not estimate once and stop — it acts on its estimate and then re-estimates on the result. Every round, copying is drawn from the same position-correlated sample, and nothing in the equation ever subtracts \(\beta\) back out, so the bias does not average away. It compounds. The strategy that sweeps the population is the one whose carriers did best, and on a heterogeneous network the carriers who did best are the high-\(\beta\) hubs, so a strategy can fix not because the action is worth adopting but because it was sampled on good seats. The end state is a consensus that is individually wrong for nearly everyone who reached it: each low-\(\beta\) agent now playing the winning strategy earns \(\alpha + \beta_{\text{low}}\), which may be worse than the alternative she abandoned, while the signal that recruited her was \(\alpha + \beta_{\text{high}}\) from someone she cannot resemble. It is omitted-variable bias in a feedback loop — a regression that hires on its own fitted values and then refits on the people it hired.4 The replicator equation is innocent of all this; it faithfully climbs mean realized fitness, exactly as advertised. The error, if there is one, is ours — for reading a population-fitness maximizer as advice to an individual. The wedge between the two is \(\beta\).
This also fixes the boundary of Schlag’s theorem. Proportional imitation is the best rule when the sampled payoff is informative about what the strategy would be worth to you — that is, when \(\beta\) is exchangeable between the person you copy and yourself. In a well-mixed world that holds by construction, and imitation is optimal; on a heterogeneous network it fails by construction, and imitation is biased in a direction the agent cannot detect from inside. The field’s foundational learning rule is optimal in the world without seats and systematically wrong in the world with them — and the interesting models all live in the second.
Which brings us to the paper, published June 3 in Nature by Anzhi Sheng, Qi Su, Alex McAvoy, Long Wang, and Joshua Plotkin — the last of whom regular readers met in Four Numbers Aren’t a Dilemma. They build exactly the second world: a population on a social network, each agent playing a contribution game with its neighbors and updating by the replicator rule above. On top of it they hand a designer one lever — a rule for dividing the goods that contribution produces — and ask which setting of the lever spreads contribution most widely. The headline, which we will dismantle over the week, is that the rule which spreads contribution best is the rule that concentrates the proceeds in the high-\(\beta\) hubs, leaving the low-\(\beta\) periphery sometimes worse off than if no one had contributed at all. In the language above: the designer is choosing the \(\beta\) distribution, and the choice that maximizes the spread of the strategy is the choice that maximizes the inequality of the seats.
Notice the word the paper uses for the strategy: cooperation. It is \(\alpha\)-talk — a name for a property of the action — applied to an outcome the paper itself will show is mostly \(\beta\). And it imports a warmth the model does not contain. Nobody here weighs the common good; each agent copies whoever is prospering, and “cooperation” is just the label on the contribution that, in clusters, makes its users prosperous enough to imitate. Anyone who has lived in the upper Midwest knows the gap between that and virtue: the niceness is real, reliable, and not a window into anyone’s heart — an equilibrium sustained by a convention nobody chose. The model’s cooperation is Midwest nice, which is the right object to study and the wrong word to trust.
Questions about cooperation always become questions about institutions, for a reason the folk theorem made unavoidable: once people interact repeatedly, nearly any pattern of behavior can be propped up as an equilibrium, so “cooperation is an equilibrium” explains nothing, and what needs explaining is which equilibrium gets selected.2 The answer is always commitment — what binds a population to one arrangement against the local temptation of another. The division rule in this paper is a commitment of that kind, a promise enforced from outside the game about who collects. And as last week’s post on Trump v. Cook argued, the value of a commitment device lies in your inability to use the discretion it removes — which leaves the question of who, in a society of imitators, enforces the rule, and what they pay to be bound by it.
So the week studies \(\beta\). Today: a position term exists, and the population’s optimal learning rule is blind to it by construction. Wednesday: we compute \(\beta\) — it has a closed form, and it takes about three lines — and find out what “the rule that equalizes” equalizes. Friday: we ask who chose the rule that sets the \(\beta\) distribution, and whether a rational designer, like the principal in the Cook post, should want the lever at all.
With that, I leave you with this.
Notes
1 Schlag asked which rule a boundedly informed agent should use when all she can do is observe one other individual’s action and realized payoff, and proved that switching toward the more successful party with probability proportional to the payoff gap maximizes expected improvement — and that a population running it reproduces the replicator dynamics. The result rests on an assumption made visible in the decomposition above: that the observed payoff is informative about the action’s value to the observer, i.e., that the position term is exchangeable between sampler and sampled. The rest of this post is what happens when it isn’t. (Karl Schlag, “Why Imitate, and If So, How?”, Journal of Economic Theory, 1998.) ↩
2 Called the “folk theorem” because it circulated among game theorists well before anyone wrote down a careful proof. Its formal versions are associated with, among others, Aumann and Shapley, Rubinstein, and Fudenberg and Maskin. What matters here is not the proof but the embarrassment it created: a result so permissive that the field’s signature claim — good behavior can be an equilibrium — became true of nearly everything, and therefore informative about nothing. Selection, not possibility, is the live question, and selection runs on commitment. ↩
3 The decomposition \(f_i = \alpha_{s_i} + \beta_i + \varepsilon_i\) is written additively here for legibility; the paper’s payoffs carry a genuine strategy-by-position interaction, which is where Wednesday’s arithmetic does its work. Nothing in today’s argument depends on additivity — only on \(\beta\) varying across positions and being correlated, through the sampling process, with the strategies an imitator observes. ↩
4 “Optimal” needs a referent, and three plausible ones diverge here. Against the highest-\(\alpha\) strategy — the best transferable action — the dynamics can fail, fixing a low-\(\alpha\) strategy that rode high \(\beta\). Against the imitator’s own realized payoff, they can also fail: the leaf adopts what shone on the hub and collects the leaf’s version of it. Against what the replicator equation is actually built to do — climb mean realized fitness across the population — they succeed exactly. The dynamics are not broken; they maximize a population aggregate that no single individual’s welfare is guaranteed to track. The failure is a category error, not a bug, and the whole week is an argument about which referent the word “cooperation” is quietly borrowing. ↩