Introduction

For decades, humans have been intrigued by the remarkable ability of animals to navigate their environment [1–4]. Fascinating examples include the annual pole-to-pole migration of Arctic terns [5, 6] and sooty shearwaters [7], that achieve flight distances spanning thousands of kilometers or the precise navigation of anadromous salmon, swimming from the open ocean to return to their natal spawning grounds [8, 9]. To achieve these marvelous feats, animals utilize a variety of environmental information to guide their journey at different length scales – for example, polarized light [10] or geomagnetic cues [11, 12] can roughly guide animals across long length scales, while olfactory cues [13] and visual landmarks [14] are more useful at shorter length scales.

Animals living in groups can further reduce directional uncertainty through social interactions. One way this can be achieved is by the direct social transfer of information. In such cases, the group leader (or a few members of the group, who are often older or more experienced) guide the rest of the group in their preferred direction via a leader-follower relationship [15–21]. For example, dominant female meerkats have higher influence in leading group movements, especially movement turn and speed, due to their experience and frontmost position in the group [22]. Through these leadership dynamics, expertise can propagate through the group, allowing less knowledgeable individuals to exhibit improved navigational performance.

An alternative socially-mediated mechanism that can improve migration efficiency is the ‘wisdom of crowds’, where large groups achieve optimum decision by averaging individual opinions [21]. This mechanism benefits from the law of large numbers, whereby individual errors cancel each other out with increasing group size, resulting in greater collective accuracy [23, 24] (assuming that the distribution of opinions is roughly centered around the optimal value [25]). In contrast to the leadership mechanism described above, here the group outcome is emergent. In other words, the collective decision may be highly accurate even if there are no knowledgeable individuals in the group at all. Furthermore, this mechanism has low cognitive requirements, since it does not require group members to recognize differences in knowledge or experience. In both humans and non-human animals, across different behavioral contexts, the wisdom of crowds has been shown to improve outcomes as group sizes increase [26–31].

However, the two mechanisms described above are not mutually exclusive. Weighted average strategies [32, 33], where more experienced, knowledgeable or confident individuals hold greater influence in the collective decision, but other individuals still have some decision weight, may capture the benefits of both existing knowledge and emergent solutions. While sociality can theoretically support a multitude of distinct and non-mutually-exclusive strategies for improving navigation efficiency, the extent to which animals use each strategy in practice remains unclear. In particular, how the optimal strategy depends on the type of decision that animals face, the cognitive ability of the animals, and other details of the species’ life history, remains poorly understood.

Learning through time

Many animals usually make repeated trips between two locations in a given period of time. At small spatial scales, these journeys may be between known locations within their home range [18], or at larger spatial scales, they may be seasonal migrations that occur annually [34]. A solitary individual may improve its performance across bouts by sampling multiple routes and preferentially remembering better ones. However, the individual faces a classic exploration-exploitation trade-off [35] between spending time exploring new routes (which may be associated with unknown and possibly severe risks) or utilizing previously known routes that are less costly. Furthermore, an individual can attempt a trip only a finite number of times, ultimately limited by its lifespan, which sets an upper bound on the ability of a single individual to improve its learned route over time.

One way in which animals can overcome the finite time horizon set by an individual lifespan is by genetically encoding migratory information — such as when and where to migrate — so that it can be inherited by future generations [36, 37]. However, genetically encoding information can be slow and susceptible to environmental variations [38–40]. Alternatively, traveling in a group can allow less-informed individuals to incorporate information from others into their own learned route, thereby permitting information to pass across individuals (and time). When such social learning is coupled with the ability to recognize beneficial solutions and prune out costly mistakes, improved solutions can be preferentially spread horizontally and/or vertically. Such

‘ratchet’-like improvements is referred to as cumulative cultural evolution (CCE), in recognition of its functional similarity to genetic evolution (i.e., variation, heritability, and selection). CCE has been observed in humans and some non-human animals [40–44], including primates [45–47], birds [48–51] and cetaceans [52–54]. However, CCE is usually inferred as the underlying process when improvements ‘accumulate’ over time, even though the social learning mechanism(s) that lead to these improvements have not been directly observed (see [55] for core criteria required for CCE and [48, 56] for examples of such cases). Yet, as previously discussed, alternative mechanisms such as the wisdom of crowds could in principle also cause an increase in task performance, without the need for other components associated with CCE [55].

In this study, we build upon one such example, by Sasaki and Biro [49], in which homing pigeons (Columba livia) were shown to improve their navigational performance over time but the mechanism by which they do so remains a mystery. Homing pigeons are a highly social species that have been extensively studied for their ability to develop idiosyncratic routes and recall them across repeated flights, making them an ideal model organism for studying navigational strategies [16, 17, 49, 57–59]. In their experiment, Sasaki and Biro created ‘chains’ of birds, similar to a game of telephone, and allowed them to fly back home repeatedly from a release site 8.4 km away. In the experimental condition, each chain was composed of five ‘generations’, with each generation consisting of 12 flights home. In each generation, an experienced bird, having developed knowledge about the homing task from the previous generation, was paired with a naïve bird that lacked this information. At the end of the generation, the experienced bird was removed and a new naïve bird was paired with the remaining (now-experienced) bird (see Methods for details). As control conditions, solo and fixed pairs of birds were flown for the same total amount of flights as in the experimental condition (60 flights). To quantify the performance of the birds, the route efficiency, calculated as the ratio of the beeline distance (straight line distance between the release and home sites) to the actual distance traveled by the pair, was measured (a value of 1 indicates a beeline flight, while a value of 0 indicates an infinitely long flight).

The study found that the experimental chains of birds significantly outperformed both the solo and fixed pair controls by the end of the fifth generation. The researchers proposed CCE as the mechanism driving this result, where chains of birds improve their routes by exploring different options and selectively choosing better ones. Valentini et al. examined the trade-offs between exploration and exploitation of new routes by the naïve and experienced birds [60], showing that both birds had an equally likely chance of initiating collective explorations. Despite these insights, a detailed mechanistic understanding of how route improvements emerge in pigeon chains is still lacking. For instance, it remains unclear what (social and/or environmental) information birds are sensing and using, and what kinds of computations they are performing on that information, to produce the observed increase in route efficiency across generations.

Here, we hypothesized several plausible learning mechanisms that bird pairs might employ during homing navigation. We simulated each of these mechanisms using artificial chains of birds composed of trajectories from the solo control dataset and applied a weighted averaging procedure (which depended on the mechanism under study) to mimic the outcome of paired flights in a generation, analogous to the actual experimental design. Our learning strategies can be organized into three classes based on the kind of information required to execute the strategy. We quantitatively compared the predictions generated by each strategy with the experimental data to infer which strateg(ies) real birds are most likely to be using. By doing so, we can assess what kind of information and cognition pigeons may be using during paired homing flights. In particular, we ask whether the cognitive requirements of CCE are necessary for the observed improvement in navigation ability.

The navigational model

To simulate the experimental condition, we created artificial chains of bird pairs — similar to the experimental design — using bird trajectories from the solo control condition. We used the 12th flight of each solo control bird, since birds typically established stable idiosyncratic routes by this point, with minimal improvements in subsequent releases. In our model, we assumed that when two birds are paired together, a shared route is formed from their individual routes, which is the outcome of the learning process that the two birds engage in over the course of 12 releases. We model this learning process using a weighted averaging procedure for each pair (see Methods for details). The naïve bird in that pair replaces its idiosyncratic route with this new averaged trajectory and becomes the ‘experienced’ bird in the next generation, where it is paired with a new naïve bird drawn from the solo control condition to continue the sequence. This iterative process — sampling a bird, generating the averaged route, assigning it as the ‘experienced’ bird for the next generation, and pairing it with another bird— was repeated for five generations, matching the actual experimental chains. The weighting parameter w_E used in our model quantifies the influence (i.e., leadership) of the ‘experienced’ bird. As a comparison, for the experimental birds, we measured leadership based on the relative front-back positions of the birds during flights (see Methods for details). We compared the route efficiencies and social weights of the experienced bird resulting from each simulated strategy with the experimental data to identify which candidate strateg(ies) are compatible with the empirical data.

We simulated seven strategies, which can be categorized into three types based on the kind of information required to execute the strategy. The first type (T₁) represents the simplest process, where birds have no knowledge of their partner’s level of experience or performance (Fig. 1A). This includes only the averaging strategy, where both birds weigh their individual routes equally in each generation, regardless of their respective navigational abilities.

Figure 1.

The second type (T₂) assumes that birds are capable of recognizing the more experienced individual within the pair and adjust their social weighting to maximize performance using this knowledge (Fig. 1B). Indeed, empirical evidence suggests that individuals can rely less on social information as they gain more experience [61]. Within type (T₂), the experienced bird weighting strategy assumes that the experienced bird maintains a constant weight w_E across all generations, where w_E is chosen such that route efficiency, on average, is maximized for the fifth generation. Alternatively, the maximize generation strategy allows the experienced bird to adjust its weight w_E based on the generation the bird is in, thereby maximizing the route efficiency for each generation.

The third type (T₃), introduces the highest level of cognitive complexity, requiring birds to actively evaluate their individual or paired performance (Fig. 1C). The strategies in T₃ align with the mechanistic criteria required for CCE, where individuals can recognize and select for better-performing routes. Within this type, the first strategy, better bird weighting, assigns the better bird (i.e., the bird in the pair that has a shorter individual route, regardless of its experience level) a weight w_B that maximizes efficiency in the fifth generation. Here, information from the worse-performing bird is still incorporated in the weighting process (with a weight 1−w_B) before passing to later generations. A more extreme version of this strategy is the all-or-nothing strategy, wherein the better bird takes full despotic control (i.e., w_B = 1), propagating only the best-performing routes across generations, with no contribution from the worse bird. The remaining two strategies focus on adjusting ‘local’ aspects of a paired flight. In the maximize flight strategy, birds aim for finding the best achievable route as a pair. For a particular pair of birds, this is a unique pair of weights w and 1 − w that maximizes the route efficiency of the resulting trajectory. Finally, the maximize time steps strategy assumes that birds can judge their joint performance moment-to-moment and make adjustments to their heading at each time step during a flight. In this strategy, the bird with a preferred route closer to the beeline at a given moment, on average, exerts greater influence over the paired flight, pulling the pair closer to the correct homing direction.

Collectively, these strategies range from the minimal cognitive requirement (T₁) to recognizing one’s own experience level (T₂) to evaluating the relative quality of the preferred routes of the two birds (T₃). We systematically tested the seven strategies by generating all possible combinations of chains from the solo control trajectories and measured the resulting distribution of route efficiencies and social weights used by the experienced bird for each generation. Using a mixed modeling approach, we fitted two different models, one for the responses of route efficiency and one for the social influence of the experienced bird (see Methods for details) and compared the results to the experimental data.

Results

Similar route improvements emerge from simple and complex strategies

Across all strategies, we observed an improvement in route efficiency over generations (Fig. 2). This suggests that a wide range of decision-making mechanisms can lead to navigational improvements, not just the ones compatible with the definition of CCE. Interestingly, we found that the experimental birds performed relatively poorly compared to the majority of our hypothesized strategies (Fig. 2B).

Figure 2.

After adjusting for generation effects, we found that three strategies significantly outperformed the experimental birds in efficiency gains (Fig. 2A and SI Appendix, Table S1). These strategies all belonged to type T₃ (where birds are able to evaluate route performance) and comprised the better bird weighting (OR = 1.37, 95% CI = [1.15, 1.63], p < 0.001), the maximize flight (OR = 1.55, 95% CI = [1.30, 1.86], p < 0.001), and the maximize time steps (OR = 1.26, 95% CI = [1.06, 1.51], p < 0.01) strategies, with the maximize flight strategy (where a pair of birds uses the specific set of social weights that optimizes their joint route) producing the highest route efficiency among all strategies tested. These observed effects suggest that the ability to evaluate the performance of routes (type T₃) can result in greater improvements compared to strategies that do not measure route efficiency (types T₁ and T₂).

Furthermore, we found that both strategies within type T₂ (where birds are able to detect which is the more experienced one), the experienced bird weighting (OR = 1.13, 95% CI = [0.94, 1.34], p = 0.34) and the maximize generation (OR = 1.15, 95% CI = [0.96, 1.37], p = 0.19) strategies, showed some effect but had non-significant efficiency gains compared to the experimental data.

The remaining two strategies — averaging (OR = 1.05, 95% CI = [0.88, 1.25], p = 0.93) (type T₁) and all-or-nothing (OR = 0.99, 95% CI = [1.15, 1.63], p = 1.00) (type T₃) — aligned closely with the experimental data with little to no difference in improvements compared to the experimental birds. Interestingly, the all-or-nothing strategy, which passes the best routes without integrating any information from the worse-performing bird (i.e., a greedy strategy), performed almost identically (poorly) to the simplest strategy of averaging individual routes. This is despite the fact that the all-or-nothing strategy has the ability to identify which of the two birds knows the shorter route, while the averaging strategy does not.

Examining how the performance of these strategies varied across generations provides further insights into the dynamics of route improvements (Fig. 2B and SI Appendix, Table S3). Several strategies showed significant positive effects, but only from generation 3 onwards. These included the better bird weighting (Gen 3 : OR = 1.66, 95% CI = [1.16, 2.39], p < 0.01; Gen 5: OR = 1.57, 95% CI = [1.05, 2.37], p = 0.02), the maximize flight (Gen 3 : OR = 1.89, 95% CI = [1.32, 2.72], p < 0.001; Gen 4: OR = 1.51, 95% CI = [1.00, 2.27], p = 0.046; Gen 5: OR = 1.96, 95% CI = [1.30, 2.94], p < 0.001), and the maximize time steps (Gen 3 : OR = 1.54, 95% CI = [1.07, 2.22], p = 0.01; Gen 5: OR = 1.49, 95% CI = [0.99, 2.24], p = 0.06) strategies. However, the increase in effect in generation 3 specifically can be attributed to the dip in efficiency of the experimental birds during this generation, inflating the contrasts with other strategies.

Overall, these observations helped us narrow down our search for candidate mechanisms, excluding three strategies from type T₃ which showed significant deviation from the experimental data. However, the strategies in types T₁ and T₂, as well as the all-or-nothing strategy in type T₃, remain statistically indistinguishable from the experimental data, when comparing their route efficiencies.

Experimental birds rely on simple route averaging for navigation

Next, we examined how the social weights of the birds varied across different strategies in comparison to the experimental birds (Fig. 3). Although several of our hypothesized strategies assigned social weights to the better/worse bird (type T₃), rather than the experienced/naïve bird (types T₁ and T₂), for all strategies we calculated the mean social weight of the experienced/naïve bird to facilitate comparison with the experimental data. Furthermore, because the social weights of a pair of birds sum to 1, we report just the social weight of the experienced bird in this analysis.

Figure 3.

Certain strategies — averaging, experienced bird weighting, and maximize generation — had, by definition, constant social weights assigned to the experienced bird. Specifically, the averaging strategy assigned a weight of 0.5, meaning that both birds contributed equally to homing decisions. For the experienced bird weighting strategy, the social weights were optimized to achieve maximum efficiency in the last generation, which was estimated to be 0.64 for the experienced bird. For the maximize generation strategy, the weights were optimized for each generation, with the experienced bird assuming weights 0.5 (generation 2), 0.67 (generation 3), 0.75 (generation 4), and 0.80 (generation 5) (note that this appears to follow a proportional relation 1/M for M generations, which we will return to later). Given their fixed weights, we excluded these strategies from our mixed model and manually contrasted their effects with the experimental birds.

After adjusting for generation effects, we found that the leadership values of the experienced bird in the experiment aligned with only one of the strategies: averaging (Mean Difference = 0.003, 95% CI = [-0.08, 0.09], p = 0.99) with almost no difference between the two (Fig. 3A and SI Appendix, Table S2). In other words, the experienced bird and the naïve bird shared, on average, equal social weight across all generations. All other strategies predicted a significantly higher influence of the experienced bird compared to that observed in the experimental data (experienced bird weighting (Mean Difference = 0.14, 95% CI = [0.05, 0.23], p < 0.001), maximize generation (Mean Difference = 0.18, 95% CI = [0.09, 0.27], p < 0.001), better bird weighting (Mean Difference = 0.17, 95% CI = [0.06, 0.29], p < 0.001), all-or-nothing (Mean Difference = 0.23, 95% CI = [0.11, 0.34], p < 0.001), maximize flight (Mean Difference = 0.23, 95% CI = [0.11, 0.34], p < 0.001), and maximize time steps (Mean Difference = 0.12, 95% CI = [0.00, 0.23], p = 0.04)). Combined with our previous conclusions, only one of our hypothesized strategies quantitatively matched both the route efficiency of the real birds (for all generations) as well as the social weights of the real birds (for all generations): the averaging strategy.

We observed that the average influence of the experienced bird monotonically increased across generations, except for the averaging and experienced bird weighting strategies (Fig. 3B and SI Appendix, Table S4). By the end of generation 5, the strategies that showed the strongest positive deviations from the experimental birds included the better bird weighting (Mean Difference = 0.30, 95% CI = [0.08, 0.52], p < 0.01), all-or-nothing (Mean Difference = 0.40, 95% CI = [0.18, 0.62], p < 0.001), maximize generation (Mean Difference = 0.34, 95% CI = [0.12, 0.56], p < 0.001), maximize flight (Mean Difference = 0.39, 95% CI = [0.17, 0.61], p < 0.001), and maximize time steps (Mean Difference = 0.26, 95% CI = [0.04, 0.48], p = 0.01).

Interestingly, we found that strategies that had similar average social weights can lead to very different route efficiency outcomes. For instance, the all-or-nothing and maximize flight strategies had almost identical trends for the social weights of the experienced bird. However, these strategies had substantially different route efficiency outcomes, with the all-or-nothing strategy being the worst performing and the maximize flight strategy achieving the highest efficiency gain. Conversely, strategies that exhibited similar route efficiencies could be driven by very different average social weights. For example, the all-or-nothing and averaging strategies produced similar route efficiency outcomes, yet the former had the highest influence of the experienced bird, while the latter had the lowest influence. These observations demonstrate that the average social weights are not predictive of performance, nor is performance predictive of social weights, at least in their aggregate.

Chaining with replacement experimental design leads to reduced effective group sizes

Although homing pigeons appear to be simply averaging their flights when flown together, the structure of the ‘chaining with replacement’ experimental design does not result in the five birds in a chain equally contributing to the final learned flight. Instead, the chain structure produces a ‘geometric forgetting’ of past information, with the last bird contributing 1/2 of the final flight path, the penultimate bird contributing a 1/4, the third bird contributing an 1/8, etc. Indeed, we predict that the first bird contributes only 1/32 of the final flight path. These unequal contributions of each bird to the final flight trajectory is in contrast to the classic ‘wisdom of crowds’, for which collective performance is maximized when all individuals are weighed equally (assuming that individuals are unable to perceive differences in competence).

We asked to what extent the chain structure, and correspondingly the unequal weighting of birds, decreases the collective performance of the group. To answer this, we defined a measure that we call ‘effective group size’, which is the size of a democratic (i.e., equally weighted) group that exhibits equivalent performance to the focal group under study (which is composed of a certain number of individuals, each with a certain social weight). To estimate this, we first mapped the size of democratic flocks to a corresponding route efficiency (Fig. 4, inset) by simulating the outcome of democratic flocks through averaging the routes of independent control flights and measuring the route efficiency of the resulting collective flights (see Methods for details). We then re-mapped the route efficiencies resulting from our hypothesized strategies, as well as the experimental data (i.e, Fig. 2B), to an equivalent effective group size.

Figure 4.

Most of the hypothesized strategies exhibit an effective group size smaller than five by the end of the fifth generation (i.e., fall below the 1:1 line shown as a dashed orange line in Fig. 4). This demonstrates that the chain structure tends to be an inefficient way to combine information, since for these strategies, the five birds that comprise a chain show a diminished performance compared to a hypothetical flock of five birds flying together. This is even true for all of the strategies of type T₂, where there is knowledge of who is the more experienced bird (knowledge that our hypothetical flock of five birds does not possess). By contrast, two of the strategies of type T₃ exhibit an effective group size greater than five. This is possible because birds using this strategy have direct knowledge of the quality of a route, information that is unavailable to our simulated flocks of five birds.

Our results suggest that the experimental design limits the propagation of information from the past to the future. For a chain of five birds, the average effective group size from the experimental dataset barely exceeds two birds (2.12) (Fig. 4). This suggests that in each generation, contributions from prior individuals are hardly retained and chains of experimental birds fly only marginally better than pairs of birds. In contrast, more complex navigational mechanisms, facilitated by, for example, perceiving route quality, could surpass this limit and even reach effective group sizes exceeding five individuals. Hence, continuous replacement of birds within a chain leads to a substantial loss of information, and, unless compensated by a more cognitively complex mechanism, constrains the chain’s ability to improve the efficiency of its route over time.

Discussion

In this study, we investigated potential navigational strategies, which varied in the level of cognitive complexity required to execute them, to infer the mechanism of collective route improvements in homing pigeons. Our work builds upon a 2017 experiment which demonstrated that pairs of experienced and naïve pigeons could iteratively improve homing routes over generations. Specifically, our aims were three-fold: 1) provide insights into the mechanism(s) pigeons use to improve route efficiency, 2) investigate how cognitive complexity, as well as experimental design (turnover of birds), relate to performance and 3) investigate if the identified mechanism(s) fall under the criteria required for CCE. To achieve this, we developed seven plausible mechanisms, broadly categorized into three types (T₁, T₂, and T₃), with increasing cognitive complexity. We created a simple navigational model by artificially pairing solo bird trajectories, similar to the experimental design, and tested our strategies to identify candidate strateg(ies) that are quantitatively compatible with the empirical data. All of our strategies resulted in route improvements, regardless of their underlying complexity. However, when combined with the results from the social weight analysis, the experimental data overall aligned best with the simplest strategy: averaging individual routes.

The route efficiency achieved generally increased with the cognitive complexity of the strategy — strategies in type T₃ tended to perform the best. However, this was not always the case. Specifically, the all-or-nothing strategy (of type T₃), where only the best route was passed to the next generation, performed no better than the simple averaging strategy (of type T₁). This can be explained by recalling that there are two sources of information: leadership from experts and emergent wisdom of crowds. The all-or-nothing and the averaging strategies, respectively, represent these two ‘pure’ strategies. By contrast, all of the other strategies represent mixed strategies, whereby individuals with better information have greater social weight, but individuals with worse information still contribute to the collective decision. The mixed strategies all outperformed the pure strategies, demonstrating their ability to exploit both sources of information.

Furthermore, the strategies of type T₃ are compatible with the definition of CCE [42, 55] but showed a quantitative mismatch with the experimental data. Therefore, an improvement in route efficiency alone is not sufficient evidence for cultural transmission, as the experimental birds did not exhibit evidence of some of the criteria of CCE (evaluation of performance, for instance). This is because of the possibility of the wisdom of crowds improving routes ‘for free’ without the need for individuals to specifically recognize and maintain improvements. In other words, the wisdom of crowds functions like a statistical ratchet, rather than a deterministic one. Moreover, as evidenced from our social weighting analysis, we did not find direct evidence of social learning in context of imitative copying of prior routes (see [62] for similar results). Birds in this strategy, on average, equally influenced each other’s routes disregarding any differences in experience. This observation raises broader questions about which forms of social learning mechanisms truly align with the requirements for CCE [63, 64].

We note that the wisdom of crowds is beneficial in some but not all task types [65–67]. In the context of the homing task, the pigeons apparently have a general (noisy) tendency to find their way home. Because of this, averaging the flights of multiple pigeons is an effective way of canceling out noise, resulting in an increasingly efficient route (on average). Hence, averaging routes can lead birds to stumble upon the best routes by chance. By contrast, in human cultural evolution, complex innovations are unlikely to arise de novo by averaging existing behaviors [44, 68]. In these types of tasks, selection on novel behaviors at the tails of the distribution — not the average — is necessary to ‘evolve’ towards improved solutions. Future work should investigate the range of tasks that group-living animals typically face, and categorize them into ones that can benefit from the wisdom of crowds and ones that require CCE [69].

This study also reveals that collective strategies must be understood within the context in which it is implemented. Specifically, here the experimental design caused birds to fly in pairs in a particular chain structure. Combined with the averaging strategy, this resulted in a geometric dilution of influence over generations. This progressive loss of information due to repeated turnover of birds reflects an inefficient version of the ‘wisdom of crowd’, where instead of equally incorporating the knowledge of all past individuals, information from early generations is disproportionately lost. This is further supported by our effective group size analysis, which showed that, despite consisting of five birds, experimental chains of birds exhibited an effective group size only slightly better than two individuals. We expect the behavior of animals to be adaptive in the conditions in which they evolved. The chain structure of this experiment may approximate the dynamics of fission-fusion groups, and additionally forced group sizes to be just two birds. While urban pigeon flocks are highly fluid, they are typically of sizes much larger than two [70]. The averaging strategy that the experimental birds exhibited may have performed relatively poorly within this chain structure, but it should exhibit higher performance in larger groups, where there is more scope for the wisdom of crowds. A simple strategy such as the averaging strategy may also be beneficial in large temporary groups, where recognizing experienced or more knowledgeable individuals may be difficult.

It is also important to consider what the pigeons themselves are trying to optimize [69]. While our analysis focused on route efficiency, pigeons may not necessarily prioritize minimizing travel distance. Indeed, the difference in travel distance between the best-performing and worst-performing strategies is only a few hundred meters. Previous studies have shown that pigeons rely on specific landmarks for navigation, suggesting that familiar, slightly longer routes may be preferable to unfamiliar, shorter ones [71–73]. In this context, birds may be prioritizing maximizing their probability of successfully returning home rather than minimizing flight distance per se (which would require exploring alternative routes, with an associated risk of getting lost). The averaging strategy is the most likely outcome for birds maximizing the probability of returning home.

Another assumption of our model is that the birds execute the strategies perfectly. In other words, the birds are able to accurately identify who is the more experienced bird or the bird with the shorter route, and can apply the optimal social weight for that strategy. Real animals, if they did implement these strategies, would have some error rate (either in differentiating the birds or applying the optimal social weight), resulting in a lower measured performance. If the error rate is high enough, the performance could even be worse than that exhibited by the averaging strategy. Therefore, the averaging strategy could be a less risky one, guaranteeing (statistically) some measure of improvement.

In conclusion, our study opens avenues for future research into collective decision making across time [74]. Collective performance is the emergent outcome of the social structure, task type, and social strategy. Understanding how these interact with each other could improve our predictions about what collective strategies we should expect to observe in nature given uncertain shifts in timings of events in the future [75]. For this navigation task, simple averaging is sufficient to quantitatively explain the experimental results, but other tasks may be less amenable to the wisdom of crowds. It is crucial to understand collective strategies within the context in which they evolved, including typical group sizes, turnover rate, error rate, and how many times a task is repeated. While more cognitively complex strategies can result in greater performance, they may not be necessary, and, indeed, mixed strategies that draw on both individual and collective sources of intelligence may result in the highest levels of performance.

Methods

Data pre-processing and experimental procedure

We used GPS trajectory data from a previous study by Valentini et al. [60], which was based on a 2017 study conducted by Sasaki and Biro in homing pigeons (Columba livia) [49]. The study consisted of three conditions: experimental, solo control, and fixed control. Each condition consisted of birds that were allowed to fly back home from a site 8.4 km away, either individually or as pairs. The experimental trials were structured around sets of 12 bird flights, termed a ‘generation’. In generation 1 of the experimental group, a single bird flew 12 times before being paired with a new individual (the naïve bird) in generation 2. This pattern continued in subsequent generations, wherein the prior naïve bird (now experienced) was paired with a new bird, creating a transmission chain. Each chain consisted of five generations, or 60 total flights across the entire chain. The birds in the control conditions completed an equal number of flights (60 releases), with solo control birds flying individually and the fixed control birds flying in constant pairings. In total, each condition had 10 independent replicate chains.

For our analysis, we focused on the final flights (release 12) of each generation for the experimental chains for comparison with our model predictions. Following Valentini et al., we only considered flights where the two birds were less than 250m from each other at all times, as well as excluding flights of birds that never reached their home destination (see Materials and Methods in [60]). After accounting for missing flights in the experimental group, we were left with ten flights each for generations 1 to 3, eight flights for generation 4, and seven for generation 5. Each of the trajectories were down-sampled to a frequency of 1 Hz (from 5 Hz) using the R package trajr [76]. The trajectory points were converted from their original geographic coordinate system (GPS) to the Universal Transverse Mercator (UTM) system for all of our distance calculations. The joint route flown by a pair of birds was calculated as the center of mass of the two birds at each time point. The route efficiency was then obtained as the ratio of the beeline distance to the actual distance traveled by the birds as a pair. To avoid biases due to circling behavior of the pigeons near release and home sites, we excluded trajectory points within a radius of 800m and 200m from the release and home site, respectively. Our beeline distance was adjusted accordingly to account for these zones.

Weighted averaging

To simulate the experimental chains of birds, we first assumed that each naïve bird has an initial ‘preferred’ route that it would follow if flown alone. We compiled a set of these preferred routes by taking the 12th release for each of the solo control birds, to match our procedure of analyzing the 12th release of each generation in the experimental condition. Moreover, the route efficiency of the solo control birds reached their maximum, on average, by this flight, and therefore represents the idiosyncratic learned preference of that bird.

We then used a weighted averaging method to simulate the consensus flight t hat would emerge when two birds are flown together. Each run in our simulation proceeds as follows: In generation 1, we randomly chose a trajectory from our set, which becomes the ‘experienced’ bird in generation 2. For generation 2, our experienced bird is paired up with a naïve bird, which is another trajectory sampled with replacement from the representative set. We then computed an averaged route from these paired flights using the following steps: First, we synchronized the number of data points across the two trajectories to match the size of the longer trajectory. This is done by interpolating missing coordinates from two consecutive time points in the shorter trajectory. We used the TrajResampleTime() function available from the R package trajr to perform our resampling [76]. Next, the x and y coordinates of the paired trajectories were combined using the following weighted averaging equations:

where w ∈ [0, 1] and t ∈ {t₁, t₂, …, t_L} corresponds to the time steps along a trajectory of length L. Here, w is the weighting parameter, which quantifies the social influence of the experienced bird (E) relative to the newly sampled naïve (N) bird. Therefore, a weighting of w = 0 indicates that the experienced bird has no influence on the averaged route, while w = 1 implies full control of the experienced bird on route formation. To account for all possible weighting contributions, w was incremented in steps of 0.01 during simulations of each strategy. We assume that the previously naïve bird adopts this averaged route as its preferred route by the end of the generation, and this averaged route is then propagated as the experienced bird’s preferred route for generation 3. A new naïve bird is drawn from our dataset, and its preferred route is again averaged with the experienced bird’s. This process of sampling, pairing, averaging, and propagation continued iteratively using all possible combinations of birds in each generation, until generation 5. By systematically varying the weighting parameter w, we investigated seven plausible strategies of navigation and calculated the resulting route efficiencies, similar to the method described for the experimental group.

Leadership calculation

Previous studies have shown that pigeons follow a front-back leadership, wherein the bird at the front of the flock assumes a greater leadership role compared to birds at the back [16, 17]. We calculated a leadership metric for the final release in each generation of the experimental data to compare with our simulation results. To achieve this, we used the method outlined in Valentini et al. (see Appendix 1, Methods [60]) that is based on calculating the relative position of the birds from each other d_EN, at each time step and projecting it onto the direction of motion of the bird pair in the subsequent time step. We used the sign of d_EN to determine which bird among the pair was leading at a given time step. The distance d_EN is positive if the experienced bird is in front of the pair and negative if the naïve bird is leading the pair. For our analysis, we used the experienced bird as our reference bird in the pair. We calculated the average proportion of time spent by the experienced bird in the front for each flight and used this for comparing with the simulated social weightings.

Statistical analysis

We utilized a mixed-effects modeling approach for comparing our simulation results with the experimental data. We created two different models to address our two response variables: route efficiency and social weight of the experienced bird across strategies. Since the original experiment provided data from only nine birds in the solo control condition, we were limited by the number of representative trajectories for our simulations. As a result, the same bird trajectories were reused in our simulated chains (i.e., we sampled with replacement), introducing an unwanted dependency in our analysis. This non-independence was evident across generations and simulated chains, effectively reducing the variance within each sample relative to the true variance. To account for this non-independence, we incorporated bird ID (1-9, corresponding to each solo trajectory) and its interaction with each generation as our random effect in our model. For the experimental data, we conditionally excluded the random effect term due to independent bird IDs in the experimental chains. Thus, the modeled bird-specific deviations in the random effect term γ_Gen:bird are:

where the random effect variance is dataset specific (experimental vs simulated).

To predict route efficiencies, we combined the experimental an d simulated da tasets an d fitted a beta regression model with a logit link using the glmmTMB() function from the R package glmmTMB (see SI Appendix for model details and performance checks) [77]. We used generation, strategy, and their interactions as fixed predictors for our models. Given this, our model equations for the expected route efficiency µ_ij for generation i and strategy j are:

Similarly, for the response variable social weight of the experienced bird, we fitted a linear mixed effect model using the lmer() function from the R package lme4 (see SI A ppendix for model details and performance checks) [78]. We used an identical fixed and random effect structure as the route efficiency model specified abo ve. However, since the estimates for the averaging, better weighting, and maximize generation strategies were constant values, they were removed from the mixed model and contrasted manually with the experimental data. The model equation for the expected social weight ω_ij for generation i and strategy j is given by:

From both our models, we computed estimated marginal means (EMMs) adjusting for the random effects, u sing t he emmeans package i n R [79]. Post hoc analysis tests comparing t he simulated and experimental data was also performed using the emmeans package. To validate our model predictions and results, we performed a bootstrapping procedure (see SI Appendix, Tables S5 and S6). We generated bootstrap samples for both the experimental and simulated strategies for each generation, using a sample size matching the number of experimental samples in the respective generation, and conducted 10,000 runs. Our model predictions were in accordance with the bootstrap results within reasonable margin of error. We adjusted the calculated p-values and confidence intervals f or multiple comparisons within each generation using the Dunnett’s method against our experimental control.

Effective group size analysis

We conducted an analysis to map the predicted route efficiencies from our model with those achievable by democratic flocks of varying group sizes (i.e., they make decisions by averaging the preferred routes of all of the individuals in the group). This approach enabled us to better understand the cumulative effects of different mechanisms within the ‘chaining with replacement’ experimental design used in the study. To achieve this, we first generated groups of different sizes by artificially pairing birds from the final flights of the experimental and solo control conditions in generation 1 (19 trajectories total). Using a weighted averaging approach, similar to our previous analysis (but with equal weights), we calculated an overall mean route from the individual trajectories. This mean route served as the measure of the flock’s average route efficiency. We then calculated mean route efficiencies for each group size and observed a non-linear relationship between group size and route efficiency. To characterize this trend, we applied non-linear least squares regression (NLS) in R [80], using various functional forms with group size as the predictor and route efficiency as the response variable. Among these evaluated models, the best fit was obtained using a hyperbolic function (see SI Appendix, Fig. S1 for further details regarding model comparisons and goodness of fit). Finally, we used the model coefficients to back-transform our simulated efficiencies and derive corresponding effective group size values.

Data availability

The dataset and code used for this study can be made available upon request.

Acknowledgements

We would like to thank Almond Stöcker, Cooper Kimball-Rhines and Jacob Adamczyk for their valuable feedback on our analysis. S.C.B, F.A.F and A.B.K acknowledge support from the U.S. National Science Foundation (BRC-BIO DBI-2233416) and a generous anonymous gift fund.

Additional files

Supplementary Material

Additional information

Funding

National Science Foundation (BRC-BIO DBI-2233416)