The tragedy of the commons is an old story.
In 1968, Garrett Hardin described a scene like this: an open pasture that anyone can graze, with every herder adding one more cow of their own. The benefit of that extra cow accrues entirely to the herder, while the cost of pasture degradation is shared by everyone. So every herder chooses to add one more, and the pasture is eventually destroyed.
The game-theoretic translation is simple: defection is the dominant strategy. Whatever others do, defection yields a higher payoff than cooperation. Replicator dynamics tells you that defectors will eventually swallow the whole population.
But this model has a hidden premise: the grass is an inanimate thing. The payoff matrix is fixed.
You step on it, and it does not feel pain. You eat it all up, and it does not change the rules of the next round.
A 2016 PNAS paper by Weitz and colleagues did one thing: they let the grass come alive.
Letting the Grass Come Alive
The paper proposes a framework called "coevolutionary game theory" β the joint evolution of games and environments. The core change is just one: the payoff matrix is no longer a fixed constant, but a function of the environment. When the environment is good (replete), defection dominates. When the environment is poor (depleted), cooperation dominates.
The intuition is easy to grasp. When resources are abundant, free-riding is the smartest move β others do the work, you reap the benefits. When resources are exhausted, no one can free-ride β refuse to cooperate and everyone dies together. In the paper's words:
mutual cooperation is a Nash equilibrium when n = 0 and mutual defection is a Nash equilibrium when n = 1.
$n$ is the environmental state, $n=0$ stands for depleted, $n=1$ for replete. The payoff matrix $A(n)$ interpolates linearly between these two extremes, with the cooperation and defection Nash equilibria each holding one end.
But the key point is not "the environment determines strategy." The key point is the other direction: strategies, in turn, change the environment.
The Closed Loop
Cooperators improve the environment β for instance, bacteria secrete public enzymes to break down nutrients, vegetation anchors soil and water. Defectors degrade the environment β they consume without producing. A closed loop emerges:
More cooperators $\to$ the environment improves $\to$ defectors gain the upper hand $\to$ the environment deteriorates $\to$ cooperators regain the upper hand $\to$
The system starts to breathe.
Inhale, exhale. Prosper, wither.
This is the part of the paper that fascinates me most. The mathematical model does not predict a stable equilibrium; it predicts an unceasing oscillation. The cooperator proportion $x$ and the environmental state $n$ trace out closed periodic orbits in phase space β more precisely, a heteroclinic cycle. The system keeps jumping between four boundary points:
$$(x=1, n=1) \to (x=0, n=1) \to (x=0, n=0) \to (x=1, n=0) \to (x=1, n=1)$$(cooperate, replete) $\to$ (defect, replete) $\to$ (defect, depleted) $\to$ (cooperate, depleted) $\to$ (cooperate, replete).
Weitz called this phenomenon the "oscillating tragedy of the commons." Why is it still a tragedy? Because the system can never settle into an optimal state. It is doomed to swing back and forth between abundance and scarcity, with cooperation and defection rising and falling in turn, neither able to hold steady. In Hardin's words, it is "the inevitableness of destiny."
The Conditions for Escape
But the paper also leaves an exit.
Whether the oscillation collapses to an interior fixed point depends on the payoff structure in the depleted state. Weitz and colleagues analysed every possible payoff ordering at $n=0$ and summarised it in an elegant phase diagram (Fig. 5 of the paper, seven regions, seven fates).
The core condition can be expressed by a single inequality:
$$\frac{P_1 - S_1}{T_1 - R_1} > \frac{S_0 - P_0}{R_0 - T_0}$$
In plain language: when the payoff of "lending a hand" for a cooperator in a poor environment is large enough β that is, when choosing cooperation while everyone else is defecting yields a high enough payoff β the system can settle stably at an intermediate environmental state. The heteroclinic cycle disappears, and the tragedy is averted.
The more people willing to extend a hand during the troughs, the less likely the system is to collapse over and over again.
This conclusion, I have to say, has a flavour of moral exhortation derived from mathematical reasoning. (Perhaps I am reading too much into it.)
More Than the Commons
The paper's discussion section lists a wealth of feedback-evolving game examples. Microbes secrete siderophores to grab iron ions β the secretors are cooperators, and when the environment is iron-poor, cooperation dominates; once iron is abundant, the free-riders arrive. Vaccination is another one β when an outbreak hits, everyone wants to be inoculated, but once the coverage rate goes up, no one wants to get the shot any more (Bauch & Earn, 2004). Water resource management follows the same pattern β in wet years, no one thinks of saving water, but by the time a drought comes, it is too late to start conserving.
All these systems share a common feature: individual rational behaviour alters the environment, and the environment, in turn, redefines what counts as "rational."
The classical iterated Prisoner's Dilemma relies on "memory" β you remember what the other player did last time, and pay them back next round. Tit-for-tat. Weitz's framework needs no individual memory at all. The environment itself is the memory. The grass remembers that you stepped on it.
Instead, a feedback-evolving game changes with time as a direct result of the accumulated actions of the populations.
The accumulated actions of individuals constitute an environment that "remembers."
Reading this section, the two models from my previous post kept coming back to me. Model 1 is instantaneous selection β looking at this step's payoff. Model 2 is historical selection β looking at the cumulative historical payoff. I said at the time that we do not know which one natural selection actually is.
Weitz's model offers a third possibility: it is not about "which time window you look at," but about "who defines the payoff itself."
Fitness does not only depend on how many people cooperate β that is the entire content of traditional replicator dynamics. Fitness also depends on the current environmental state, and the environmental state is itself the historical integral of cooperator proportions. A strategy shifts once, the stage wobbles. The stage wobbles, and the next step changes with it. This is not a simple opposition between "instantaneous versus historical"; it is a strategy and an environment dancing a never-ending duet.
The language of statistical mechanics may be more accurate: this is a non-equilibrium steady state. The system is never in equilibrium, always flowing. The grass grows, the cattle graze, payoffs shift, strategies chase. There is no single "right" strategy that lets you solve the problem once and for all.
The Morality of Grass
Hardin's original point was that the tragedy of the commons has no technical solution, and can only be addressed through "mutual coercion, mutually agreed upon."
But if the environment itself changes in response to your choices, then the real question is no longer "how to coerce cooperation," but "how to maintain resilience in the face of fluctuation." You do not need to cooperate forever β Weitz's model tells you that even permanent cooperation cannot hold the system steady, because defectors will always take advantage of you at your most successful moments. What you need is this: when the troughs come, extend a few more hands and pull the system back a little. As long as the pull is strong enough, the heteroclinic cycle will collapse into a stable interior fixed point.
It is not about eliminating defectors. It is about making sure that, when the defectors arrive, the grass is still long enough.
The grass knows how to breathe.
The only thing one can do is to avoid trampling its roots.
References
Weitz, A. (2016). An oscillating tragedy of the commons in replicator dynamics with game-environment feedback. PNAS.