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Study site and contextAmbatovy is a very large nickel, cobalt and ammonium sulphate mine in central-eastern Madagascar owned by a consortium of international mining companies50. It represents the largest ever foreign investment in the country24 (US$8 billion by 201650) and a substantial source of fiscal income49. In 2018, the company contributed ~US$50 million in taxes, tariffs, royalties and other payments49 and employed >9,000 people (93% of whom were Malagasy)51. Commercial production began in January 201424 (Supplementary Fig. 1). As key components in batteries, supplies of nickel and cobalt are critical to the green energy transition and demand for these metals is predicted to increase notably in future52.The mining concession covers an area of 7,700 ha located in the eastern rainforests of Madagascar (Fig. 1) which have very high levels of biodiversity and endemism53,54. After avoidance and minimization measures were applied (Supplementary Methods) the mine was predicted to clear or substantially degrade 2,064 ha of high-quality natural forest at the mine footprint and upper pipeline24. Any impacts on plantations or secondary habitat are not included in this estimate. Losses at the impact site were not discounted in relation to a background rate of decline, meaning that the company took responsibility for the full area of forest lost25. Independent verification by our team (by measuring the size of the mine footprint on Google Earth) confirms the extent of forest loss at the mine footprint (Supplementary Fig. 2). Clearance of the footprint accounts for most of the forest loss associated with the mine as losses associated with the pipeline are small54.Ambatovy aims to generate biodiversity gains to offset the mine-induced losses by slowing deforestation driven by shifting agriculture elsewhere26. To this end the company designated four sites, totalling 28,740 ha, to be protected as biodiversity offsets; Ankerana, Corridor Forestier Analamay-Mantadia (CFAM), the Conservation Zone and Torotorofotsy54 (Fig. 1). The offsets are considered like-for-like30 and were selected on the basis of similarity to the impact site in terms of forest structure and type, geology, climate and altitude24. The large combined area of the offsets relative to the impacted area was designed to allow flexibility, account for uncertainty and incorporate as many of the affected biodiversity components as possible24. Ankerana is the flagship offset, selected on the basis of its size, connectivity to the Corridor Ankeniheny-Zahamena (CAZ) forest corridor and the presence of ultramafic outcrops thought to support the same rare type of azonal forest lost at the mine site54. Extensive surveys conducted within Ankerana to establish biological similarity concluded the offset to be of higher conservation significance than the forests of the mine site due to the presence of rare lowland tropical forest24.The Conservation Zone is directly managed by the company, given its location within the concession area, while the other offsets are managed in partnership with local and international NGOs24,25. Ambatovy funds the management of Ankerana by Conservation International and local NGO partners (although before 2015 Ankerana was directly managed by Ambatovy via a Memorandum of Understanding with Conservation International24), supports BirdLife partner Asity with the management of Torotorofotsy and a number of local NGOs including Voary Voakajy25 are involved in CFAM26. The company is also working to secure formal, legal protection for CFAM26 as part of a proposed Torotorofotsy–CFAM complex new protected area (although progress on this has stalled).Overview of methodsTo estimate the impact of the offsets on deforestation and determine whether this has prevented enough deforestation to offset forest loss at the mine site, we combined several complementary methods for robust impact evaluation. First, we used statistical matching to match a sample of pixels from each biodiversity offset to pixels from the wider forested landscape with similar exposure to drivers of deforestation. Then we used a site-based difference-in-differences regression for each matched offset–control sample and a fixed-effects panel regression on the pooled data, to estimate the effect of protection. We systematically explored how arbitrary modelling choices (including the statistical distance measure used in matching, caliper size, ratio of control to treated units, matching with or without replacement and which, if any, additional covariates were included) affected our inference, exploring the robustness of our results to 116 alternative model specifications.MatchingThe former province of Toamasina was selected as the geographic area from which control pixels were sampled as it encompasses forests of the same type as the concession area with varying degrees of intactness and accessibility. The four biodiversity offsets are located within this province (Fig. 1).The unit of analysis is a 30 × 30 m2 pixel that was forested in the baseline year 200045,55. It is important that the scale of analysis aligns with the scale at which the drivers of deforestation (in this case, small-scale shifting agriculture) operate56. The median agricultural plot size (from 564 measured plots) in the study region is ~36 × 36 m2 (ref. 57). We took a subsample of pixels to reduce computational effort while maintaining the capacity for robust statistical inference58,59. We used a grid-based sampling strategy ensuring a minimum distance between sample units to reduce spatial autocorrelation60 and equal coverage of the study area58. A 150 × 150 m2 resolution grid, aligned to the other 30-m resolution data layers (Fig. 1c), was overlaid on the province and the 30 × 30 m2 pixel at the centre of each grid square was extracted to produce a subsample of pixels that are 120 m away from their nearest neighbour. The 120 m is larger than the minimum distance between units used in another matching study in Madagascar (68 m; ref. 59) but smaller than that used in other studies (200 m; ref. 61) and so strikes an appropriate balance between the avoidance of spatial autocorrelation and maximizing the possible sample cells.Protected areas in the study area managed by Madagascar National Parks were excluded from our control sample as they are actively managed and therefore do not represent counterfactual outcomes for the biodiversity offsets in the absence of protection (Fig. 1). However, control pixels were sampled from within the CAZ new protected area as legal protection was only granted in 2015 and resources for management are limited and thinly spread62. Additionally, Ankerana and parts of CFAM overlap with the CAZ and would have experienced the same management, and likely trajectory, as the rest of the CAZ, had they not been designated biodiversity offsets. Areas within 10 km of an offset boundary were excluded from the control sample to reduce the chance of leakage (where pressures are displaced rather than avoided) biasing results17,29. The 10 km was selected as it is a commonly used buffer zone within the literature17,58.To test for leakage effects, we used Veronoi polygons to partition the buffer area for CFAM, the Conservation Zone and Torotorofotsy (which overlap) into three individual buffer areas according to the nearest offset centroid and took a subsample of pixels from each (Fig. 1). Areas that overlapped with the established protected areas of Mantadia National Park and Analamazotra Special Reserve were excluded from the buffer zones.The outcome variable is the annual deforestation rate sourced from the Global Forest Change (GFC) dataset34. Following Vieilledent et al.45 these data were restricted to only include pixels classed as forest in a forest cover map of Madagascar for the year 200045,55, reducing the probability of false positives (whereby tree loss is identified in pixels that were not forested). The resulting tree loss raster was snapped to the forest cover 2000 layer to align cells, resulting in a maximum spatial error of 15 m. The GFC product34 has been shown to perform reasonably well at detecting deforestation in humid tropical forests63. In the north-eastern rainforests of Madagascar, Burivalova et al.39 found that GFC data performed comparably to a local classification of very high resolution satellite imagery at detecting forest clearance for shifting agriculture (although it was not effective at detecting forest degradation from selective logging). As clearance for shifting agriculture is considered the principal agent of deforestation in the study area22 and the forests of the study area are tropical humid ( >75% canopy cover), the GFC data are an appropriate tool for quantifying forest loss. Although recent evidence suggests that GFC data may have temporal biases64, this phenomenon affects our control and treated samples equally and so is unlikely to impact our results.The choice of covariates is extremely important in matching analyses. They must include, or proxy, all important factors influencing selection to treatment and the outcome of interest so that the matched control sample is sufficiently similar to the treated sample in these characteristics to constitute a plausible counterfactual, otherwise the resulting estimates may not be valid33. On the basis of the literature and a local theory of change we selected five covariates that we believe capture or proxy for the aspects of accessibility, demand and agricultural suitability that drive deforestation in the study area22,59,65,66. These are slope, elevation, distance to main road, distance to forest edge and distance to deforestation (Supplementary Methods). These five essential covariates comprise the main matching specification and form the core set used in all alternative specifications that we tested in the robustness checks. We also defined five additional variables (annual precipitation, distance to river, distance to cart track, distance to settlement and population density) and tested the effect of including these in the robustness checks. The additional covariates were so defined because they were of poorer data quality (population density and distance to settlement), correlated with an essential variable (annual precipitation and population density) or simply considered less influential (distance to river and distance to cart track; Supplementary Methods).Statistical matching was conducted in R statistics using the MatchIt package v.4.1 (ref. 67). To improve efficiency and produce closer matches we cleaned the data before matching to remove control units with values outside the calipers of the treated sample in any of the essential covariates (see Supplementary Methods for caliper definition). Following the recommendations of Schleicher et al.68 we tested several matching specifications and selected the one that maximized the trade-off between the number of treated units matched and the closeness of matches as the main specification (Supplementary Table 7). This was 1:1 nearest-neighbour matching without replacement, using Mahalanobis distance and a caliper of 1 s.d. This specification produced acceptable matches (within 1 s.d. of the Mahalanobis distance) for all treated units within all offsets. The maximum postmatching standardized difference in mean covariate values between treated and control samples was 0.05, well below the threshold of 0.25 considered to constitute an acceptable match69. This indicates that, on average, treated and control units were very well matched across all covariates.Matching was run separately for each offset. The resulting matched datasets were aggregated by treated status (offset or control) and year to produce a matrix of the count of pixels that were deforested each year (2001–2019) in the offset and the matched control sample. Converting the outcome variable to a continuous measure of deforestation avoids the problem of attrition associated with binary measures of deforestation and is better suited to the framework of the subsequent regressions70.Robustness checksStatistical matching requires various choices to be made68, many of which are essentially arbitrary. There therefore exists a range of possible alternative specifications that are all a priori valid (although some may be better suited to the data and study objectives69) but which could influence the results20,28. We tested the robustness of our results to 116 different matching model specifications (Fig. 4). First, we tested the robustness of the estimates to the use of three alternative matching distance measures (Mahalanobis, standard propensity score matching using generalized linear model regressions with a logit distribution and propensity score matching using RandomForest), three different calipers (0.25, 0.5 and 1 s.d.), different ratios of control to treated units (one, five and ten nearest neighbours) and matching with/without replacement. Holding the choice of covariates constant (using only the essential covariates), the combination of these led to the estimation of 54 different models. Second, we tested the robustness of results to the inclusion of the five additional covariates. Holding the choice of distance measure and model parameters constant, we constructed 31 models comprising all possible combinations of additional covariates with the core set of essential covariates. Finally, we explore the robustness of results for 31 randomly selected combinations of distance measure, model parameters and additional covariates. All 116 specifications are a priori valid, assuming that the covariates capture or proxy for all important factors influencing outcomes, but may fail to satisfy the parallel trends condition or produce matches for insufficient numbers of treated observations ( More

Genetic diversity and population structureThe 614 bp length of mtCO1 sequences was successfully amplified and sequenced from 75 individuals of E. tetradactylum and 89 individuals of E. rhadinum from different sites. Based on the CO1 analysis, we detected 5 and 16 haplotypes, respectively, from E. tetradactylum and E. rhadinum (Table 1). Only one haplotype was inter-specifically shared in E. tetradactylum populations, as showed in the TCS haplotype networks (Fig. 2a). A total of 77 polymorphic sites was identified in E. rhadinum but 3 polymorphic sites in E. tetradactylum. Among these sites, a total of 3 and 11 parsimoniously informative sites was detected in E. tetradactylum and E. rhadinum, respectively. In E. tetradactylum, the number of CO1 haplotypes was 2 in ZS and 3 in PA and ZJ. The haplotype diversity was also much higher in ZJ (0.211) and PA (0.197) than ZS (0.105). In E. rhadinum, CO1 haplotypes varied from 3 (JH) to 8 (ZZ). The haplotype diversity was the highest in ZZ (0.663). The populations of ZJ and ZZ showed the statistically negative Tajima’s D value, which could signify the demographic expansion. The MDA revealed similar results (Fig. S3).Table 1 Genetic polymorphisms and neutrality tests of Eleutheronema tetradactylum and Eleutheronema rhadinum inferred from CO1 and 16s rRNA.Full size tableFigure 2The unrooted TCS haplotype networks were constructed based on the haplotypes of CO1 (a) and 16s rRNA (b) of Eleutheronema tetradactylum (left) and Eleutheronema rhadinum (right). Haplotype frequency was related to the size of the circle. Different colors within the nodes refer to various sampling sites.Full size imageThe mitochondrial 16s rRNA (574 bp in length) was also successfully sequenced from 75 and 89 individuals of E. tetradactylum and E. rhadinum (Table 1), which yielded 5 and 6 haplotypes, respectively (Fig. 2b). No haplotype was interspecifically shared of 16s rRNA both in E. tetradactylum and E. rhadinum. A total of 4 and 14 polymorphic sites of E. tetradactylum and E. rhadinum were identified, respectively, of which 3 and 4 were parsimoniously informative sites. Table 1 shows that only four haplotypes with 0.200 haplotype diversity were identified in E. tetradactylum from PA. In E. rhadinum, relatively high haplotype diversity (H = 0.481) and nucleotide diversity (π = 0.00170) were found in populations SA. Overall, the populations from Thailand showed higher genetic diversity than the China population both for E. tetradactylum and E. rhadinum.The TCS network was constructed to identify the phylogenetic relationships in E. tetradactylum and E. rhadinum between China and Thailand populations, as shown in Fig. 2. In E. tetradactylum, 5 haplotypes were closely related to a small number of mutation steps, and the Hap_1 was likely the most primitive haplotype, which evolved into others. In E. rhadinum, 16 haplotypes were distributed between the two branches, including China and Thailand branches. Only the Hap_7 was shared in ZJ and SA of the Thailand branch. One (hap_1) in E. tetradactylum and two (Hap_2 and Hap_8) in E. rhadinum were used as the central radiation distribution for most haplotypes. Other haplotypes were formed by a small number of mutations of these haplotypes. As shown in the TCS network of 16s rRNA haplotypes, the Hap_1 in E. tetradactylum and Hap_4 in E. rhadinum were the most primitive haplotype, which showed central radiation distributions. Also, in E. rhadinum, the haplotypes of China and Thailand populations were divided into two branches; only Hap_2 was shared in ZJ and SA.The level of population genetic differentiation between China and Thailand populations was also evaluated (Table S3). In E. tetradactylum, the average Fixation index (Fst) between PA and the other two sites was 0.81344 in ZS (p  More

Scat sample collection and preparationAt known harbour seal haulout sites individual scat samples were collected using a standardized protocol (Fig. 1). Disposable wooden tongue depressors were used to transfer deposited scats into 500 ml single-use jars or zip-style bags lined with 126 µm nylon mesh paint strainers18. Samples were either preserved immediately in the field by adding 300 ml 95% ethanol to the collection jar, or were taken to the lab and frozen at −20 °C within 6 hours of collection19. Later, samples were thawed and filled with ethanol before being manually homogenized with a disposable wooden depressor inside the paint strainer to separate the scat matrix material from hard prey remains (e.g. bones, cephalopod beaks). The paint strainer containing prey hard parts was then removed from the jar leaving behind the ethanol preserved scat matrix for genetic analysis20. The paint strainer containing prey hard parts was refrozen for subsequent parallel morphological prey ID.Fig. 1The 52 harbour seal scat collection sites in the Salish Sea represented in this dataset.Full size imageMolecular laboratory processingScat matrix samples were subsampled (approximately 20 mg), centrifuged and dried to remove ethanol prior to DNA extraction. DNA was extracted from scat with the QIAGEN QIAamp DNA Stool Mini Kit according to the manufacturer’s protocols. For additional details on the extraction process see Deagle et al.21 and Thomas et al.20.The metabarcoding marker we used to quantify fish and cephalopod proportions was a 16S mDNA fragment (~260 bp) previously described in Deagle et al.15 for pinniped scat analysis. We used the combined Chord/Ceph primer sets: Chord_16S_F (GATCGAGAAGACCCTRTGGAGCT), Chord_16S_R (GGATTGCGCTGTTATCCCT), and Ceph_16S_F (GACGAGAAGACCCTAWTGAGCT), Ceph_16S_R (AAATTACGCTGTTATCCCT). This multiplex PCR reaction is designed to amplify both chordate and cephalopod prey species DNA. A blocking oligonucleotide was included in the all 16S PCRs to limit amplification of seal DNA22. The oligonucleotide (32 bp: ATGGAGCTTTAATTAACTAACTCAACAGAGCA-C3) matches harbour seal sequence (GenBank Accession AM181032) and was modified with a C3 spacer so it is non-extendable during PCR22.A secondary metabarcoding marker was used in a separate PCR reaction to quantity the salmon portion of seal diet, because the primary 16S marker was unable to reliably differentiate between coho and steelhead DNA sequences. This marker was a COI “minibarcode” specifically for salmonids within the standard COI barcoding region: Sal_COI_F (CTCTATTTAGTATTTGGTGCCTGAG), Sal_COI_R (GAGTCAGAAGCTTATGTTRTTTATTCG). The COI amplicons were sequenced alongside 16S such that the overall salmonid fraction of the diet was quantified by 16S, and the salmon species proportions within that fraction were quantified by COI.To take full advantage of sequencing throughput, we used a two-stage labeling scheme to identify individual samples that involved both PCR primer tags and labeled MiSeq adapter sequences. The open source software package EDITTAG was used to create 96 primer sets each with a unique 10 bp primer tag and an edit distance of 5; meaning that to mistake one sample’s sequences for another, 5 insertions, substitutions or deletions would have to occur23.All PCR amplifications were performed in 20 μl volumes using the Multiplex PCR Kit (QIAGEN). Reactions contained 10 μl (0.5 X) master mix, 0.25 μM of each primer, 2.5 μM blocking oligonucleotide and 2 μl template DNA. Thermal cycling conditions were: 95 °C for 15 min followed by 34 cycles of: 94 °C for 30 s, 57 °C for 90 s, and 72 °C for 60 s.Amplicons from 96 individually labeled samples were pooled by running all samples on 1.5% agarose gels, and the luminosity of each sample’s PCR product was quantified using Image Studio Lite (Version 3.1). To combine all samples in roughly equal proportion (normalization), we calculated the fraction of each sample’s PCR product added to the pool based on the luminosity value relative to the brightest band. After 2013, amplicon normalization was performed using SequalPrep™ Normalization Plate Kits, 96-well.Sequencing libraries were prepared from pools of 96 samples using an Illumina TruSeq DNA sample prep kit which ligated uniquely labeled adapter sequences to each pool. Libraries were then pooled and DNA sequencing was performed on Illumina MiSeq using the MiSeq Reagent Kit v2 (300 cycle) for SE 300 bp reads. Samples were sequenced on multiple different runs as part of the larger study; however, typically between 4 and 6 libraries (each a pool of 96 individually identifiable samples) were sequenced on a single MiSeq run.BioinformaticsTo assign DNA sequences to a fish or cephalopod species, we created a custom BLAST reference database of 16S sequences by an iterative process. First, using a list of the fish species of Puget Sound, we searched Genbank for the 16S sequence fragment of all fishes known to occur in the region (71 fish families 230 species)24,25. Reference sequences for each prey species were included in the database if the entire fragment was available, and preference was given to sequences of voucher specimens. When the database was first generated (November, 2012) Genbank contained 16S sequences for 192 of the 230 fish species in the region, and the remaining 38 species were mostly uncommon species unlikely to occur in seal diets. Following a similar procedure, we added to this database sequences for all of the regional cephalopods for which 16S data were available (7 squid species, 2 octopus species). A separate reference database was generated for the COI salmon marker containing Genbank sequences for the nine salmonid species known to occur regionally: Oncorhynchus gorbuscha (Pink Salmon), Oncorhynchus keta (Chum Salmon), Oncorhynchus kisutch (Coho Salmon), Oncorhynchus mykiss (Steelhead), Oncorhynchus nerka (Sockeye Salmon), Oncorhynchus tshawytscha (Chinook Salmon), Oncorhynchus clarkii (Cutthroat Trout), Salmo salar (Atlantic Salmon), Salvelinus malma (Dolly Varden)24.To determine if some species in the database cannot be distinguished from each other at 16S (i.e. have identical sequences in the reference database) a distance matrix was performed on the complete database using the DistanceMatrix function in the R package DECIPHER26. Species with identical sequences were identified as having a distance of “0.00”. In some cases, one haplotype for a species was identical to another species but other haplotypes were not. When two species’ sequences were identical, we ultimately reported both species in the prey_ID field.Sequences were automatically sorted (MiSeq post processing) by amplicon pool using the indexed TruSeqTM adapter sequences. FASTQ sequence files for each library were imported into MacQIIME (version 1.9.1-20150604) for demultiplexing and sequence assignment to species27. For a sequence to be assigned to a sample, it had to match the full forward and reverse primer sequences and match the 10 bp primer tag for that sample (allowing for up to 2 mismatches in either primers or tag sequence).Next, we clustered the DNA sequences that were assigned to scat or tissue samples with USEARCH (similarity threshold = 0.99; minimum cluster size = 3; de novo chimera detection), and entered a representative sequence from each cluster into a GenBank nucleotide BLAST search28,29. If the top matching species for any cluster was not included in the existing database (or the sequence differed indicating haplotype variation), we put the top matching entry in the reference database. We repeated this procedure with every new batch of sequence data to minimize the potential for incorrect species assignment or prey species exclusion. This process was conducted for both the 16S and COI reference databases with each new batch of samples.For all DNA sequences successfully assigned to a sample, a BLAST search was performed against our custom 16S or COI reference databases. A sequence was assigned to a species based on the best match in the database (threshold BLASTN e-value  More

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Pure coordination gameIn this section we study the Pure Coordination Game (PCG) (also known as doorway game, or driving game) in which (R=1), (S=0), (T=0), and (P=1), resulting in a symmetric payoff matrix with respect to the two strategies:$$begin{gathered} begin{array}{*{20}c} {} & {quad ; {text{A}}} &; {text{B}} \ end{array}hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & 0 \ 0 & 1 \ end{array} } right) hfill \ end{gathered}$$
(2)
There are two equivalent equilibria for both players coordinating at the strategy A or B (a third Nash equilibrium exists for players using a mix strategy of 50% A and 50% B). As the absolute values of the payoff matrix are irrelevant and the dynamics is defined by ratios between payoffs from different strategies, the payoff matrix (2) represents all games for which the relation (R=P >S=T) is fulfilled.In the PCG the dilemma of choosing between safety and benefit does not exist, because there is no distinction between risk-dominant and payoff-dominant equilibrium. Both strategies yield equal payoffs when players coordinate on them and both have the same punishment (no payoff) when players fail to coordinate. Therefore, the PCG is the simplest framework to test when coordination is possible and which factors influence it and how. It is in every player’s interest to use the same strategy as others. Two strategies, however, are present in the system at the beginning of the simulation in equal amounts. From the symmetry of the game we can expect no difference in frequency of each strategy being played, when averaged over many realisations. Still, the problem of when the system reaches full coordination in one of the strategies is not trivial. We address this question here.Figure 1Time evolution of the coordination rate (alpha) (in MC steps) in individual realisations for different values of the degree k in a random regular network of (N=1000) nodes, using (a) the replicator dynamics, (b) the best response, and (c) the unconditional imitation update rule.Full size imageFigure 2Coordination rate (alpha) and interface density (rho) vs degree k of a random regular network for (N=1000) using (a) the replicator dynamics, (b) the best response, and (c) the unconditional imitation update rule. Each green circle represents one of 500 realisations for each value of the degree k and the average value is plotted with a solid line, separately for (alpha >0.5) and (alpha le 0.5). Results are compared to the ER random network ((alpha _{ER})) with the same average degree.Full size imageFirst, we look at single trajectories as presented in Fig. 1. Some of them quickly reach (alpha =0) or 1, or stop in a frozen state without obtaining global coordination. Other trajectories take much longer and extend beyond the time scale showed in the figure. What we can already tell is that the process of reaching coordination is slower in the replicator dynamics where it usually takes more time than in the best response and unconditional imitation to reach a frozen configuration. For all update rules the qualitative effect of the connectivity is similar—for bigger degree it is more likely to obtain full coordination and it happens faster. For the UI, however, larger values of degree than for the RD and BR are required to observe coordination. For example, in the case of (k=10) or 20 the system stops in a frozen disorder when using UI, while for the RD and BR it quickly reaches a coordinated state of (alpha =0) or 1.To confirm the conclusions from observation of trajectories, we present the average outcome of the system’s evolution in the Fig. 2. The first thing to notice is that all plots are symmetrical with respect to the horizontal line of (alpha = 0.5). It indicates that the strategies are indeed equivalent as expected. In all cases there is a minimal connectivity required to obtain global coordination. For the RD and BR update rules this minimum value is (k=4), although in the case of BR the system fails to coordinate for small odd values of k due to regular character of the graph. This oscillating behaviour does not exist in Erdős–Rényi random networks. When nodes choose their strategies following the UI rule much larger values of k are required to obtain full coordination. Single realisations can result in (alpha = 0), or 1 already for (k=15). However, even for (k=60) there is still a possibility of reaching a frozen uncoordinated configuration.The important conclusion is that there is no coordination without a sufficient level of connectivity. In order to confirm that this is not a mere artefact of the random regular graphs we compare our results with those obtained for Erdős–Rényi (ER) random networks76,77 (black dashed line in Fig. 2). The level of coordination starts to increase earlier for the three update rules, but the general trend is the same. The only qualitative difference can be found in the BR. The oscillating level of coordination disappears and it doesn’t matter if the degree is odd or even. This shows that different behaviour for odd values of k is due to topological traps in random regular graphs78. Our results for the UI update rule are also consistent with previous work reporting coordination for a complete graph but failure of global coordination in sparse networks40.Figure 3Examples of frozen configuration reached under the UI update rule for small values of the average degree k in random regular networks (top row) and Erdős–Rényi networks (bottom row) with 150 nodes. Red colour indicates a player choosing the strategy A, blue colour the strategy B. Note the topological differences between random regular and ER networks when they are sparse. For (k=1) a random regular graph consists of pairs of connected nodes, while an ER network has some slightly larger components and many loose nodes. For (k=2) a random regular graph is a chain (sometimes 2–4 separate chains), while an ER network has one large component and many disconnected nodes. For (k=3) and (k=4) a random regular graph is always composed of one component, while an ER network has still a few disconnected nodes.Full size imageSince agents using the RD and BR update rule do not achieve coordination for small values of degree, one might suspect that the network is just not sufficiently connected for these values of the degree, i.e. there are separate components. This is only partially true. In Fig. 3, we can see the structures generated by random regular graph and by ER random graph algorithms. Indeed, for (k=1) and 2 the topology is trivial and a large (infinite for (k=1)) average path length23 can be the underlying feature stopping the system to reach coordination. For (k=3), however, the network is well connected with one giant component and the system still does not reach the global coordination when using RD or BR. For the UI update rule coordination arrives even for larger values of k. Looking at the strategies used by players in Fig. 3 we can see how frozen configuration without coordination can be achieved. There are various types of topological traps where nodes with different strategies are connected, but none of them is willing to change the strategy in the given update rule.We next consider the question of how the two strategies are distributed in the situations in which full coordination is not reached. Looking at the trajectories in Fig. 1 we can see that there are only few successful strategy updates in such scenario and the value of (alpha) remains close to 0.5 until arriving at a frozen state for (k=2) (also (k=7) for UI). This suggests that there is not enough time, in the sense of the number of updates, to cluster the different strategies in the network. Therefore, one might expect that they are well mixed as at the end of each simulation. However, an analysis of the density of active links in the final state of the dynamics, presented in Fig. 2, shows a slightly more complex behaviour. When the two strategies are randomly distributed (i.e. well mixed) in a network, the interface density takes the value (rho =0.5). When the two strategies are spatially clustered in the network there are only few links connecting them and therefore the interface density takes small values. Looking at the dependence of (rho) on k, we find that for the replicator dynamics the active link density starts at 0.5 for (k=1), then drops below 0.2 for (k=2) and 3 indicating good clustering between strategies, to fall to zero for (k=4) where full coordination is already obtained. When using the best response update rule the situation is quite different. For (k=1) there are no active links, (rho =0), and hardly any for (k=2). There is a slight increase of the active link density for (k=3), to drop to zero again for (k=4) due to full coordination. Because of the oscillatory level of coordination there are still active links for odd values of (kP) (otherwise we can rename the strategies and shuffle the columns and rows). What defines the outcome of a game are the greater than and smaller than relations among the payoffs. Therefore we can add/subtract any value from all payoffs, or multiply them by a factor grater than zero, without changing the game. Thus, the payoff matrix (1) can be rewritten as:$$begin{gathered} begin{array}{*{20}c} {} & {qquad {text{A}}} & {quad quad {text{B}}} \ end{array} ;; hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & {frac{{S – P}}{{R – P}}} \ {frac{{T – P}}{{R – P}}} & 0 \ end{array} } right) hfill \ end{gathered}$$
(3)
which, after substituting (S’=frac{S-P}{R-P}) and (T’=frac{T-P}{R-P}), is equivalent to the matrix: $$begin{gathered} begin{array}{*{20}c} {} &quad ;;{text{A}} &; {text{B}} \ end{array} ;quad quad quad quad quad quad begin{array}{*{20}c} {} & quad; {text{A}} & ;{text{B}} \ end{array} hfill \ begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & {S^{prime}} \ {T^{prime}} & 0 \ end{array} } right)xrightarrow[{{text{apostrophes}}}]{{{text{skipping}}}}begin{array}{*{20}c} {text{A}} \ {text{B}} \ end{array} left( {begin{array}{*{20}c} 1 & S \ T & 0 \ end{array} } right) hfill \ end{gathered}$$
(4)
From now on we omit the apostrophes and simply refer to parameters S and T. This payoff matrix can represent many games, including e.g. the prisoner’s dilemma14,46 (for (T >1) and (S More