Starting with Yeast Cells

Yeast cells secrete invertase outside the cell wall to break down sucrose, and the digested sugar is freely available to everyone. That is what makes this interesting. A cell can choose to "cheat": use the enzymes secreted by its neighbours without secreting any itself. Researchers call yeast with a functional SUC2 gene "cooperators" and yeast with SUC2 deleted "cheaters," then pit them against each other in competition.

The results are counterintuitive:

• In sparse populations (low social density), cheaters have a fitness of only 0.87 — worse than cooperators
• In dense populations (high social density), cheaters have a fitness as high as 1.19 — better than cooperators

Why? The higher the social density, the more likely cooperators are to encounter other cooperators. They all secrete enzymes together, the public pool grows, and everyone's payoff rises. But that is precisely when cheaters slip in, free-riding on the public output without paying the cost, and their payoff explodes.

At extremely high densities, cheaters can almost always find a cooperative target and steal more than they could produce on their own. Cooperators end up being dragged down.

This fits my intuition perfectly. But I wanted to run the experiment myself — not to verify the paper's conclusions, but to "see" the process with my own hands, write the equations into a grid, and watch the numbers come alive.

Model Setup

An $n \times n$ grid, with population density controlling the probability of placing an agent in each cell. Agents play one of two strategies: C (cooperate, secrete enzymes) or D (defect, do not secrete).

When two agents meet, they play the standard Prisoner's Dilemma with the payoff matrix:

R is the reward for mutual cooperation, T is the temptation to defect, S is the sucker's payoff, and P is the punishment for mutual defection. Following the classical setup: $T > R > P > S$, and $2R > T + S$ (mutual cooperation beats alternating defection).

At each step:

1. An agent looks for a neighbour within its Von Neumann neighbourhood (the four cells above, below, left, and right)
2. The two play one round of the game
3. They compare this step's payoff $\pi$
4. With probability proportional to the payoff difference, one imitates the other's strategy
5. The agent moves one grid unit in a random direction

Only the current step's payoff is considered. No history is kept, no planning is done.

Theoretical Prediction

Replicator dynamics gives the mean-field equation:

$x$ is the proportion of cooperators, and $\pi_C$ and $\pi_D$ are the expected per-step payoffs of the two strategies.

Payoffs depend on encounter probabilities. Under the mean-field approximation, a cooperator meets a cooperator with probability $x$ and a cheater with probability $1-x$; the reverse holds for cheaters:

Substituting:

With the concrete values $R=3, S=0, T=5, P=1$:

$\pi_C - \pi_D < 0$ holds for all $x$ — no matter how high $x$ is, cooperators' current payoff is always below cheaters'.

This means $\frac{dx}{dt} = x(1-x)(-1-x) < 0$. The cooperator proportion can only decline monotonically, eventually tending toward $x=0$ — total defection.

The mean-field approximation describes a "well-mixed" world: every individual meets every other individual with equal probability. My grid is not like that. Spatial structure might be what holds cooperation up.

Simulations and Results

The experiments below run on a $50 \times 50$ grid, with synchronous updates at each step and enough rounds for the system to stabilise.

Experiment 1: Hold density fixed and observe evolutionary trajectories

Set population density $\rho = 0.3$ and vary the initial cooperator fraction $x_0$.

A natural guess: the higher $x_0$, the longer cooperators hold on; low $x_0$ collapses immediately. But spatial structure may let locally dense cooperation zones survive an early siege.

Running it, the cooperator "survival window" felt narrow. After spatial structure bought a brief defence by maintaining local clusters, the D strategy eventually seeped in from the boundaries — the periphery of a cooperator cluster is always the first to be eroded.

Experiment 2: Hold the initial proportion fixed and observe the density effect

Set $x_0 = 0.5$, vary population density $\rho \in [0.02, 1.0]$, and after the run compute the mean difference between $\pi_C$ and $\pi_D$.

The y-axis is $P_C - P_D$, the cooperator payoff minus the cheater payoff. It is positive throughout — at every density, cooperators' average payoff exceeds cheaters'. This is something the pure random-mixing mean-field equation could never produce: spatial structure carves out a defensive zone for strategy C.

But the trend is unmistakable. At low density, $P_C - P_D$ is close to 0.05; at high density, it drops to roughly 0.002. A 25-fold difference.

At 2% density, the grid is sparse and agents are mostly isolated. An isolated agent has no neighbour to pair with — that step has no game, no payoff fluctuation, and holding the current strategy is a break-even outcome. Cooperators still occasionally bump into their own kind and engage in mutualism. Cheaters are equally often isolated, but isolation means nothing happens, and cheating has nothing to gain.

The higher the density, the more frequent the encounters. Every encounter is a chance for C to be exploited by D. Even when C-C mutualism is more frequent, both sides are still paying the cost to feed the public pool — which conveniently benefits the D sitting next door that contributes nothing.

Greig's yeast experiment points in the same direction: the higher the density, the more cheating pays. The difference is the angle — they looked at it from the cheater's perspective: cheating fitness > 1 at high density. My plot looks at it from the cooperator's perspective: the advantage is still there, just shrinking.

Interestingly, the critical density at which the advantage hits zero never appears in this parameter set. $P_C - P_D$ approaches zero but does not touch it. Perhaps under more extreme payoff settings (such as pushing T higher), the cooperator's advantage would be erased entirely.

(Or swapping Von Neumann for Moore — eight neighbours instead of four would expose agents to D more often. I have not had a chance to try that yet.)

Afterthoughts

Greig's paper taught me something: the "dilemma" of the Prisoner's Dilemma is not that it is unsolvable, but that its solution depends on structure. The same game rules push in opposite directions under sparse and dense social networks.

Replicator dynamics provides the baseline prediction — the direction of evolution in a well-mixed world where only the current payoff matters. But drop in the spatial structure of a real system — whether it is the neighbour radius on an agar plate or the Von Neumann neighbourhood on a grid — and the prediction changes.

That is the value of ABM: insert structure, and watch what emerges.

Experiments 1 and 2 suggest that spatial structure acts like a buffer — not reversing the direction, but slowing the speed. Cooperators are destined to lose to D, but on a grid they lose more slowly than the mean-field equation predicts.

But after writing all this, the more I thought about it, the more I felt something was missing.

Greig's experiment was run on a sucrose plate. Sucrose was the only carbon source. On a sparse plate, sugar is abundant and cells are few — the sucrose around each cell is effectively infinite. On a dense plate, cells are packed together, and the sugar runs out quickly — how much sugar you have depends on how much enzyme your neighbours secreted.

My ABM takes the payoff matrix as fixed. $T=5, R=3, P=1, S=0$ — the payoff of a single game never changes, regardless of density. But a yeast's invertase is not a game engine that runs independently of its environment. It digests sucrose. Less sugar means less digestion. Digestion depends not only on whether you secrete the enzyme, but also on how much sucrose is left.

If sucrose runs out — a cluster of cells packed together, none of them secreting the enzyme — the mutual-defection payoff may not be $P=1$, but a negative number. Nothing gets digested. The cells starve. Whereas mutual cooperation, although it still has to pay the enzyme cost, can at least split whatever sugar is left.

In other words, Greig's yeast may not always be playing the Prisoner's Dilemma.

In the sugar-rich, high-density regions, cheaters gorge themselves by exploiting cooperators — standard PD. But on the sugar-starved fringes — after a sparse cluster exhausts the local sucrose — the underlying game slides away from the Prisoner's Dilemma into something else. Mutual defection is no longer a low payoff; it is a negative payoff. The relative disadvantage of cooperation suddenly disappears.

Strategies consume resources, resources rewrite the matrix, and the matrix reshapes strategies. Once a closed loop forms, the "dilemma" of the Prisoner's Dilemma is no longer an optimisation problem with fixed boundary conditions — the boundary conditions themselves are variables.

This is still speculation. To verify it, enzyme dynamics would need to be written into the ABM — sugar concentration as a local variable, payoffs computed in real time from the sugar concentration. Not "density determines the payoff matrix," but "density indirectly alters the payoff matrix through resource consumption."

One direction.

References

Greig, D., & Travisano, M. (2004). The Prisoner's Dilemma and polymorphism in yeast SUC genes. Proceedings of the Royal Society of London. Series B: Biological Sciences, 271, S25 - S26.