Nepotism masks evidence for reciprocity in cooperation networks

15 Nepotism and reciprocity are not mutually exclusive explanations for cooperation; helping 16 decisions can depend on both kinship cues and past reciprocal help. The importance of these two 17 factors can therefore be difficult to disentangle with observational data. We developed a 18 resampling procedure for inferring the statistical power to detect observational evidence of 19 nepotism and reciprocity, and applied this procedure to simulated and real datasets. We simulated 20 datasets resulting from perfect reciprocity, where the probability and duration of helping events 21 from individual A to B equaled B to A. We also simulated varying degrees of simultaneous 22 nepotism. We then assessed how nepotism and sampling effort influenced the probability of 23 detecting evidence of reciprocity. We applied the same analysis to empirical data on food sharing 24 in vampire bats and allogrooming in mandrills and Japanese macaques. Nepotism consistently 25 masked evidence for reciprocity. With perfect reciprocity and imperfect nepotism, nepotism was 26 more likely to be detected and overestimated. We explain the causes and consequences. To 27 compare the relative importance of genetic and social ties, researchers should measure the relative 28 reliability of both estimates. We provide R scripts to allow others to assess the reliability of 29 kinship and reciprocal help estimates in their own datasets. 30 3


Introduction
Definitions of reciprocity (also called reciprocation, reciprocal altruism, reciprocal cooperation, 59 contingent cooperation, and direct reciprocity) have varied between authors and sub-disciplines 60 (Carter 2014). Although the original concept was quite broad (Trivers 1971), more narrow 61 subsequent definitions of reciprocity restricted its general importance to humans (Carter 2014).

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For example, the term has been used to describe: a broad category of enforced mutual benefit 63 (analogous to kin selection), a correlation between cooperation given and received across dyads 64 or over time (analogous to kin-biased association), a conditional helping behavior that causes this 65 correlation (analogous to nepotism), and a specific psychological mechanism that might cause 66 this conditional behavior (analogous to phenotype matching) (Carter 2014). For our purposes here, 67 we define reciprocity broadly as help given that is influenced by rates of help received (i.e. Semantics aside, the importance of reciprocity is also contentious because it is difficult to 84 test especially in the presence of nepotism. Moreover, kinship bias is considered sufficient 85 evidence for nepotism, while symmetry, even in the absence of nepotism, is not sufficient 86 evidence of reciprocity (Carter 2014). A demonstration of reciprocity requires experimentally 87 manipulating helping rates and then measuring a change in reciprocal helping (Dolivo and

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In observational studies, nepotism will lead to collinearity between reciprocal help and 100 kinship as predictors of helping rates. Due to inherent differences in sampling effort, kinship 101 estimates will generally be more precise than estimates of helping rates (see discussion for 102 details). As a consequence, the more precise estimate of the correlation between kinship and 103 helping will often be over-estimated relative to the less precise estimate of the correlation 104 between help given and received, and nepotism should tend to mask evidence of reciprocity in 105 correlational datasets. To assess this idea, we developed a resampling procedure for inferring 106 power to detect both kinship bias and symmetry in mixed-kinship groups. To simulate perfect 107 reciprocity, we simulated data of helping events resulting from pairs of individuals that based 108 their decisions to help on an unobserved history of past reciprocal help that is perfectly symmetrical. We then systematically changed two variables: (1) the degree of nepotism and (2) 110 the number of observed helping events. Finally, we used permutation and bootstrapping to assess 111 how these two factors interactively influenced the probability of detecting evidence for 112 reciprocity.

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To demonstrate the application of our approach to empirical data, we applied the same

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Regurgitated food sharing in vampire bats has been a classic example of the possible co-124 occurrence of reciprocity and nepotism (Wilkinson 1984;Wilkinson 1988

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For instance, at the first step, we randomly sampled 20 observations with replacement 1000 150 times. To analyze the simulated data (described below), we created a different dataset of size N 151 observations by sampling from the input probability distributions 1000 times (i.e. Monte Carlo 152 simulation) rather than bootstrapping a single dataset 1000 times.

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For each observed dataset, we extracted the observed coefficients of kinship and 154 reciprocal help from a multiple regression quadratic assignment procedure permutation test with 155 double semi-partialling (MRQAP-DSP, (Dekker et al. 2007)). We defined the response variable 156 'help' for individual A to B as the total of duration of help from A to B, divided by the total to the empirical allogrooming and food sharing durations because they were lognormal. We ztransformed all variables to obtain standardized beta coefficients, so that an observed coefficient 162 of X for kinship indicates that a one standard deviation increase in kinship predicts an increase of 163 X standard deviations in help.

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To calculate p-values for the observed coefficients, we used a network-level permutations 165 (Farine 2017) randomizing each input variable independently using the standard approach from 166 the MRQAP-DSP function in the R package 'asnipe' (Farine 2013). We used this procedure to

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We simulated 500 observations of help among 20 individuals. To simulate perfect 180 reciprocity, we generated a network of reciprocal helping history, and set this to be perfectly 181 symmetric across dyads, such that the helping history from A to B was always equal to helping 182 history of B to A. We then created a dataset of 500 observed helping events by randomly 183 sampling one individual as an actor and selecting a remaining individual as a recipient with a 184 probability that was proportional to the helping history. The duration of help was equal to this 185 symmetrical helping history. All observed helping was therefore determined by the symmetrical helping history, which is not directly observable but that probabilistically informs the observed 187 data.

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To simulate nepotism as an additional behavior, we made the history of past reciprocal 189 help correlate with kinship to varying degrees. To do this, we constructed helping history network  (Table S1), so the observed correlations between observed help and kinship (i.e. kinship 202 bias) will also increase. Finally, we added a step to ensure that all individuals were observed 203 helping at least one other individual.

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In sum, these simulations generate an observed set of helping events where individuals 205 base their helping decisions entirely on an unobserved foundation of past reciprocal help.

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However, this reciprocity co-existed across a spectrum of nepotism from 0% nepotism, where 207 helping rates are symmetrical and kinship played no role, to 100% nepotism, where helping was 208 symmetrical but only occurred among kin, and the relative causal roles of reciprocal help and 209 kinship are therefore unknown. Of course, there are many possible causes of symmetrical helping 210 besides helping others based on reciprocal help. However, the point of this simulation is simply to ask: If perfect reciprocity did exist among individuals that were also somewhat nepotistic, how 212 likely are we to overestimate the evidence for kinship bias relative to reciprocal helping?

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Real datasets

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We also applied this resampling procedure as a power analysis for three real datasets. The

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Past analyses of these same data found that food sharing was better predicted by reciprocal

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To give another example, if we observe only one sharing event for each dyad those data 333 could be sufficient to detect and estimate an existing kinship bias (albeit the estimate might be 334 poor), whereas we cannot use the same data to detect or estimate an existing symmetry in helping 335 rates. If a cooperative relationship exists, then kinship estimates will generally be more precise 336 than estimates of helping rates. Because the more precise estimate will be over-estimated relative 337 to the less precise estimate, nepotism will tend to mask evidence of reciprocity.

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This asymmetry in precision is compounded in larger study populations. The number of 339 possible dyadic helping rates is almost the square of the number of individuals, n*(n-1).

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Observing enough dyadic helping interactions to accurately estimate a cooperation network

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Third, for any given relationship, the relative importance of reciprocity and nepotism can 415 change over time. For example, the help invested by a mother in her two daughters when they are 416 young might be 100% nepotistic and 0% reciprocal, with equal helping allocated to each 417 daughter. However, when her daughters become adults, the mother's investment might also be influenced by each daughter's reciprocal investment in her, and she may have a stronger 419 relationship with one daughter over another.

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Fourth, reciprocity can be more difficult to detect than nepotism because cooperative

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We used network permutations which account for the network structure and hold the total 520 help given and received by each individual constant. Alternate versions of our analyses suggested that the multiple regression quadratic assignment procedure we used here (Dekker et al. 2007) is 522 better at reducing collinearity, and hence the masking of symmetry, compared to permutation 523 tests applied to standard multiple linear regression coefficients. However, network permutations 524 do not account for biased sampling, so the helping rates (network edge weights) must take into 525 account the relative opportunity for individuals to help each other. We accomplished this by 526 defining edge weights as the proportion of help received from individual X divided by the total 527 help received from all other individuals that could have otherwise come from individual X 528 because X was present at the time. Another possibility is to define edges as the help from X over 529 the opportunity for X to help. If helping events are scored as yes/no events, then an even more 530 rigorous approach is to use pre-network permutations (Farine 2017), where the helping acts in the 531 dataset are permuted across individuals present at the time, rather than permuting the helping 532 rates in the network. Pre-network permutations allow for precise control over the null hypothesis 533 by swapping within time periods or locations, and also control for biased sampling; however, they 534 are most appropriate when the helping events are binary (0/1) and hence interchangeable.

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In conclusion, because nepotism masks evidence of reciprocity in cooperation networks, 536 it is useful to assess the reliability of symmetry and kinship bias as a function of sampling effort.

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We provide R code (Carter et al. 2018) to produce plots that allow one to assess the relative 538 power for detecting evidence of nepotism and reciprocity in simulated datasets or in a given