Phase transitions and symmetry breaking of cooperation on lattices
Bibliographic record
Abstract
The donation game is an instance of a social dilemma with a single parameter given by the cost-to-benefit ratio of cooperation, r. In spatial settings, limited local interactions and clustering are capable of supporting cooperation by reducing exploitation from defectors. Traditionally, the interaction and competition neighborhoods are identical. Here we discuss intriguing differences in the dynamics that arise when separating the neighborhoods. On the square lattice, disjoint interaction and competition neighborhoods are easily realized by considering nearest-neighbor interactions and second-nearest-neighbor competition. Incidentally, the number of first and second neighbors is the same. More importantly, this separates the population into two competing subpopulations, with interactions solely between subpopulations but competition within subpopulations. For negative cost-to-benefit ratios, r, the donation game turns into a harmony game, and defection becomes an act of spite. In the traditional setup, the extinction of cooperators under harsh conditions, large r, and that of spiteful defectors, r<0, exhibits critical phase transitions with characteristics of directed percolation. In contrast, with two subpopulations spiteful behavior cannot persist, while the extinction of cooperators exhibits the same characteristics. Most intriguingly, however, for smaller r spontaneous symmetry breaking in the levels of cooperation between the two subpopulations is observed. The symmetry breaking resembles the sublattice ordering occurring in the antiferromagnetic Ising model. Within the twofold-degenerate phases, decreasing the cost-to-benefit ratio induces extremely large fluctuations (bursts) in the frequencies of cooperation. These bursts eventually drive the system into one of the absorbing states: occasionally homogeneous defection in both sublattices, but usually only in one and homogeneous cooperation in the other, achieving perfect asymmetry.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".