Search bioRxiv⌕ Search
Home/Archive/bioRxiv

bioRxiv · 10.1101/2023.03.02.530745

Adaptive dynamics of memory-1 strategies in the repeated donation game

LaPorte, P.·Hilbe, C.·Nowak, M.
View original source ↗Export metadata ↓

Abstract

Social interactions often take the form of a social dilemma: collectively, individuals fare best if everybody cooperates, yet each single individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows individuals to adopt reciprocal strategies which incentivize cooperation. The most basic model to study reciprocity is the repeated donation game, a variant of the repeated prisoners dilemma. Two players interact over many rounds, in which they repeatedly decide whether to cooperate or to defect. To make their decisions, they need a strategy that tells them what to do depending on the history of previous play. Memory-1 strategies depend on the previous round only. Even though memory-1 strategies are among the most elementary strategies of reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, most previous work relies on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-1 strategies has an invariant three-dimensional subspace, generated by the memory-1 counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-1 strategies and a full characterization for memory-1 counting strategies. Author summaryDirect reciprocity is a mechanism for evolution of cooperation based on the repeated interaction of the same players. In the most basic setting, we consider a game between two players and in each round they choose between cooperation and defection. Hence, there are four possible outcomes: (i) both cooperate; (ii) I cooperate, you defect; (ii) I defect, you cooperate; (iv) both defect. A memory-1 strategy for playing this game is characterized by four quantities which specify the probabilities to cooperate in the next round depending on the outcome of the current round. We study evolutionary dynamics in the space of all memory-1 strategies. We assume that mutant strategies are generated in close proximity to the existing strategies, and therefore we can use the framework of adaptive dynamics, which is deterministic.

Explore related subjects

evolutionary biologyPreprint

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

LaPorte, P., Hilbe, C., Nowak, M.. 2023-03-03. Adaptive dynamics of memory-1 strategies in the repeated donation game. https://doi.org/10.1101/2023.03.02.530745

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related preprints

bioRxiv2026

RELAX does not reproduce its own estimates at default settings, and its output does not show it

Selection-intensity estimates from RELAX are reported as a point value of K with a likelihood-ratio P. We report that, at default settings and on data of ordinary size, the program does not reproduce its own fits. Of 27 enzyme entries refitted under two optimiser configurations, none reproduced its log-likelihood to within 0.01 units; the median change was 103 units, the largest over 3,400, and four verdicts reversed. Eighty null orthologues reproduced none. A byte-identical command returned a distinct likelihood on every repetition, single-threaded, across three releases, and on alignments simulated under the fitted model, where 3.3 per cent of replicates reproduced. The documented random-number seed never reaches the generator when assigned on the command line, yet reads back as the value supplied. PAML localises the cause: its two-ratio model, without site classes, reproduced its log-likelihood for all 288 genes; its site-class models agreed for 27 to 67 per cent. The instability follows the mixture over sites, not the program. The output does not show it: 46 of 410 fits ended with a negative likelihood-ratio statistic, impossible under convergence, and 123 of 410 report a K re-estimated under a domain restriction rather than the unconstrained maximum. Of 234 published studies using RELAX, none reported a seed. Seeding while holding the thread count at one reproduced sixty of sixty runs on twenty genes under two releases; the seed alone reproduced none of five, and no documentation states the second condition. We recommend that fits be repeated and their dispersion published.

evolutionary biology↗
bioRxiv2026

Sequential accumulation of adaptive alleles forms an inversion supergene in deer mice

Supergenes are clusters of co-inherited loci that affect multiple or complex phenotypes. Despite the growing number of chromosomal inversions identified as supergenes in natural populations, their molecular basis and evolutionary history often remain obscure. Here, we identified two candidate genes, Slc45a2 and Npr3, within a 41-Mb inversion supergene in the deer mouse (Peromyscus maniculatus) that respectively drive darker coats and longer tails - two traits associated with forest adaptation. Mice homozygous for the inversion (inv/inv) exhibit elevated Slc45a2 expression in melanocytes relative to the congenic standard genotype (std/std), disrupting pheomelanin production. In parallel, downregulation of Npr3 in inv/inv mouse growth plates prolongs postnatal growth of caudal vertebrae, resulting in tail elongation. Population-level analyses further implicate that this supergene arose through the subsequent accumulation of the Npr3 allele within the inversion, rather than by capturing all beneficial mutations at its origin.

evolutionary biology↗
bioRxiv2026

Toxin structure shapes palatability in a chemically defended butterfly

The toxicity of chemical defences is well studied, but the potential contribution of compound structure to predator deterrence remains largely unexplored. Whether predation acts more strongly on toxicity or unpalatability remains largely untested, partly because few systems allow toxin structure to vary independently of quantity. Heliconius sara larvae provide such a system: those reared on Passiflora auriculata sequester cyclopentenyl cyanogenic glucosides (CGs), while those reared on P. biflora biosynthesise comparable quantities of aliphatic CGs. Using two invertebrate predators, Camponotus floridanus ants and Hierodula membranacea mantids, we tested whether this structural difference affects palatability independent of toxicity. Mantids rejected larvae with cyclopentenyl CGs more often than larvae with aliphatic CGs, despite no detectable difference in total CG content. This pattern was mirrored in extract-based assays with ants, independently of cyanide release: extracts with cyclopentenyl CGs remained deterrent, while extracts with aliphatic CGs did not differ in deterrence from water. Live larvae, by contrast, elicited similar responses from ants regardless of CG structure. These results show that variation in toxin structure can strongly affect palatability, with some compounds conferring greater protection than others. This demonstrates the importance of chemical structural diversity in the evolution of chemical defences.

evolutionary biology↗