Our model of cooperation on a grid had been rewarding selfishness

Microbes that release costly products into their surroundings are exposed to cheaters that take the benefit and skip the cost. A standard result in social evolution says that when relatives stay near each other, producers can persist anyway. We tried to reproduce that on a simple grid model before using the idea in our own writing, and in the second of three preregistered rounds the model seemed to succeed, with a threshold that matched a rule from the theory literature. It had a flaw: each producer received a share of its own goods. The shuffled control, which should have collapsed, kept producers at a mean share of 0.37 (95th percentile 0.52), and with a 3 by 3 goods box a producer recovers a ninth of its own output, so producing paid for itself whenever the benefit-to-cost ratio passed 9.c1 The third round paid goods to others only. With goods paid to others only, local producers ended at a share of 0.03 at a benefit-to-cost ratio of 100, 0.19 at 200 and 0.48 at 400, and never reached the preregistered bar of 0.5.c2 Under this update rule, space alone gave producers no foothold at any benefit-to-cost ratio up to 50, where their share was 0.00.

Many bacteria release a public goodDefinition: public good social evolution and economics A product that costs its maker something and benefits others nearby, such as an enzyme or iron-scavenging molecule a microbe releases into its surroundings. Excludes products the maker keeps for itself. full definition, a product that costs its maker something and helps others nearby. Iron-scavenging siderophores are a studied case. In a population where everyone meets everyone, a cheaterDefinition: producer and cheater social evolution A producer pays a cost to make the public good. A cheater makes none and still gains from the goods of producers near it. Excludes any intention on either side; the words name two fixed strategies. full definition that makes none and takes the benefit should win. Kin selection gives one way out: if producers mostly help relatives, which are also producers, production can spread.

Space is the usual route to relatedness in microbes, because daughter cells stay near their parents. Nadell, Drescher and Foster [1] review how the spatial arrangement of clones and species in biofilms shapes cooperation and competition. Space also has a cost. Taylor [2] showed with an inclusive fitness model that in a patch-structured population the benefit of helping relatives and the cost of local competitionDefinition: local competition social evolution, after Taylor and after Griffin and colleagues Competition for space or resources among neighbours, which, when neighbours are relatives, can cancel the benefit that helping relatives brings. Excludes competition spread over the whole population. full definition among them can cancel exactly, the same result Wilson, Pollock and Dugatkin [3] reached by simulation. Griffin, West and Buckling [4] tested it in the bacterium Pseudomonas aeruginosa by setting relatedness and the scale of competition independently, and found that cooperation rose with relatedness and fell when competition was more local.

On graphs, Ohtsuki, Hauert, Lieberman and Nowak [5] found that selection favours cooperation when the benefit-to-cost ratio exceeds the average number of neighbours. We used that rule in our second round to forecast a threshold, and the forecast came true for a reason the rule had nothing to do with.

The study ran in three rounds on 31 October 2026, each under its own preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition, hash-locked before that round ran and written after reading the round before. The decision they inform was stated in the third: whether our own writing may say that space protects cooperators on the strength of our own model, or must only cite published work for it.

The SHA-256 of each preregistration is in the reproduction section.

No external data were used. The model and all its numbers are our own and invented for the check. It describes no organism. Every number in this paper is read from the three rounds' result files.

A torus gridDefinition: torus grid simulation A square grid whose opposite edges are joined, so every cell has the same number of neighbours and there is no border. Excludes grids with edges, where border cells have fewer neighbours. full definition of 100 by 100 cells holds producers and cheaters, starting half and half at random. Let $z$ be 1 for a producer and 0 for a cheater. A cell's fitness is

floored at a small positive value, where $\varphi$ is the share of producers in the cell's goods boxDefinition: goods box our usage for this study The square of cells over which one producer's goods are spread, 3 by 3 at range 1 and 5 by 5 at range 2. Excludes the 3 by 3 window from which parents are drawn, which stays fixed. full definition. Each generation is a synchronous updateDefinition: synchronous update simulation Every cell of the grid is replaced at once in each generation, from the fitnesses of the previous generation. Excludes asynchronous updating, where one cell at a time is replaced. full definition by fitness-proportional replacementDefinition: fitness-proportional replacement evolutionary dynamics The update rule in which a cell's next occupant is copied from a parent drawn with chance in proportion to the parent's fitness, here among the 3 by 3 window around the cell including the cell itself. Excludes other update rules, such as death followed by birth from neighbours only, or imitation of the best neighbour. full definition: every cell's next occupant is copied from a parent drawn from the 3 by 3 window around the cell, the cell included, with chance in proportion to fitness.

In rounds one and two the producer share counts the cell itself:

where $n$ is the number of producers in the box, the cell included, and $N$ the number of cells in the box, 9 for a 3 by 3 box. In round three, goods reach others only:

The local arm uses the rules above. The mixed armDefinition: mixed arm our usage for this study The version of the model in which every cell gains from the producer share of the whole population and parents are drawn from the whole population, so neighbours mean nothing. Excludes any spatial structure. full definition sets $\varphi$ to the producer share of the whole grid and draws parents from the whole grid. The shuffled armDefinition: shuffled arm our usage for this study The version of the model in which all positions are shuffled at random before each generation and the local rules then apply, so neighbours are strangers. Excludes the clustering of relatives that the local arm allows. full definition shuffles all positions before each generation and then applies the local rules, so neighbours are strangers.

Under eq. 2 a producer is one of the $N$ cells its own goods reach, so it receives $b/N$ of its own output. Producing then pays the producer directly whenever

that is, whenever $b/c$ exceeds $N$, which is 9 for a 3 by 3 box. That is a private return. It needs no relatives nearby, so it survives in the shuffled arm too. In round three it did so from a ratio of 20 (a share of 0.37) and fully from 30; at ratios of 12 and 16 the shuffled producers ended at 0.00 while local producers ended at 1.00, so in that band clustering still mattered.

The final producer shareDefinition: final producer share our usage for this study The share of producers in the grid at the end of a run, averaged over the replicate runs. A run stops early when one type takes the whole grid. Excludes the share of runs in which producers survived. full definition is the share of producers at the end of a run, averaged over replicate runs; a run stops early when one type fills the grid. A ratio counts as persistence when that average reaches 0.5. Round one ran 1,500 generations and 8 replicates; round two 1,200 generations and 6 replicates; round three 1,000 generations and 5 replicates.

The shuffled arm is the controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition on which the study turns. If space is what keeps producers alive, removing space must make them collapse. Rounds two and three ran it 20 times at the most favourable setting.

Round two added a known answer for the mixed arm. Producers have fitness $1+\mathrm{bp}-c$ and cheaters $1+\mathrm{bp}$, where $p$ is the producer share, so the mean-field recursionDefinition: mean-field recursion population genetics An exact formula for next generation's producer share in the mixed arm, used here as a known answer the simulation must match. Excludes the local and shuffled arms, which have no such formula. full definition is

The simulated mixed arm, on a 300 by 300 grid averaged over 8 runs, had to match it within 0.03 over the first 20 generations. It did, with a maximum error of 0.0019.

Round one's known answer for the mixed arm was a closed-form slope for the early decline of the producer share on the log-odds scale. It failed, with an error of 0.117 against a bar of 0.05. The closed form was our mistake: it held the producer share fixed at one half while the share fell. Round two replaced it with eq. 5.

At $b$ of 2 and $c$ of 0.3, a ratio of 6.7, local producers collapsed in every run, against our forecast of a final share of 0.6. The mixed and shuffled arms collapsed too. The cost sweep at $b$ of 2 kept producers at $c$ of 0.1 and 0.2, ratios of 20 and 10, and lost them from 0.3 up. Read with eq. 4, that pattern already fits the flaw.

Figure 1 shows the threshold. Local producers first reached a final share of 0.5 at a ratio of 9 for every value of $b$: 0.52 at $b$ of 1, 0.66 at 2 and 0.88 at 4. We had forecast 7 to 12 from the rule that cooperation wins when the ratio exceeds the number of neighbours, 8 here, and counted it a pass.

The control did not hold. The shuffled control, which should have collapsed, kept producers at a mean share of 0.37 (95th percentile 0.52), and with a 3 by 3 goods box a producer recovers a ninth of its own output, so producing paid for itself whenever the benefit-to-cost ratio passed 9.c1 The control ran at a ratio of 20 with 20 draws; its 5th percentile was 0.26. The same round's goods-range sweep at a ratio of 20 kept producers at range 1 and lost them at range 2, where the box holds 25 cells and the private return $b/25$ falls below $c$. The threshold was the size of the goods box.

Table 1 and Figure 2 give the side-by-side run at $b$ of 2. With self-benefit put back, local producers reached a share of 0.70 at a ratio of 9 and 1.0 from 12 up, and shuffled producers reached 1.0 from a ratio of 30 up.c3 The artefact reproduced.

With goods paid to others only, local producers ended at a share of 0.03 at a benefit-to-cost ratio of 100, 0.19 at 200 and 0.48 at 400, and never reached the preregistered bar of 0.5.c2 Mixed producers ended at 0.04 or less at every ratio, and shuffled producers with goods to others only ended at zero at every ratio.c4

At a ratio of 400, widening the goods box lowered the others-only local share from 0.47 at range 1 to 0.18 at range 2, 0.21 at range 3 and 0.15 at range 5. These come from a separate set of 5 runs; the range 1 value of 0.47 is that set's counterpart of the 0.48 in Table 1.

The round-three shuffled control, with goods to others only at a ratio of 400, ended at zero in all 20 runs. The local arm's 0.48 is above every one of them. The preregistered verdict that compares the two was recorded as untestable, because the control's draws had no spread for a 95th percentile to mean anything.

Table 2 sets the forecasts against the outcomes.

The second round looked like a textbook reproduction, and the shuffled control is what showed it was not one. Scrambling positions every generation removes relatives from a producer's neighbourhood, so any persistence that survives scrambling cannot come from kin. Producers kept a mean share of 0.37 in the scrambled grid. Reading that number led to eq. 4: the threshold of 9 was the size of the goods box, and the match with the neighbour count of 8 in the rule of Ohtsuki and colleagues [5] was a coincidence. Their rule concerns a game played between pairs of neighbours on a graph. Ours shares a public good over a box, a different model, so the match was never a test of their rule.

With the private return removed, local producers did not persist anywhere up to a ratio of 400, and widening the goods box made them worse off. This fits the result of Taylor [2] and of Griffin and colleagues [4]: when relatives compete for the same space, the competition can cancel the benefit of living near them. In our model the parent window and the goods box overlap, so a producer's goods feed the same neighbours that compete for its offspring's cells. The model does not show that space never protects cooperators. It shows that this model, with this update rule, gives them no protection at benefit-to-cost ratios up to 50.

The claim that space protects cooperators in microbes therefore stays a citation of published work, such as the review of Nadell and colleagues [1] and the experiment of Griffin and colleagues [4]. It is not a result of ours.