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Conceived and designed the experiments: APE GGdP. Performed the experiments: APE GGdP. Analyzed the data: APE GGdP. Wrote the paper: APE GGdP.

The authors have declared that no competing interests exist.

Animals living in groups make movement decisions that depend, among other factors, on social interactions with other group members. Our present understanding of social rules in animal collectives is mainly based on empirical fits to observations, with less emphasis in obtaining first-principles approaches that allow their derivation. Here we show that patterns of collective decisions can be derived from the basic ability of animals to make probabilistic estimations in the presence of uncertainty. We build a decision-making model with two stages: Bayesian estimation and probabilistic matching. In the first stage, each animal makes a Bayesian estimation of which behavior is best to perform taking into account personal information about the environment and social information collected by observing the behaviors of other animals. In the probability matching stage, each animal chooses a behavior with a probability equal to the Bayesian-estimated probability that this behavior is the most appropriate one. This model derives very simple rules of interaction in animal collectives that depend only on two types of reliability parameters, one that each animal assigns to the other animals and another given by the quality of the non-social information. We test our model by obtaining theoretically a rich set of observed collective patterns of decisions in three-spined sticklebacks,

Animals need to act on uncertain data and with limited cognitive abilities to survive. It is well known that our sensory and sensorimotor processing uses probabilistic estimation as a means to counteract these limitations. Indeed, the way animals learn, forage or select mates is well explained by probabilistic estimation. Social animals have an interesting new opportunity since the behavior of other members of the group provides a continuous flow of indirect information about the environment. This information can be used to improve their estimations of environmental factors. Here we show that this simple idea can derive basic interaction rules that animals use for decisions in social contexts. In particular, we show that the patterns of choice of

Animals need to make decisions without certainty in which option is best. This uncertainty is due to the ambiguity of sensory data but also to limited processing capabilities, and is an intrinsic and general property of the representation that animals can build about the world. A general way to make decisions in uncertain situations is to make probabilistic estimations

An additional source of information about the environment may come from the behavior of other animals (social information)

Social animals have a continuous flow of information about the environment coming from the behaviours of other animals. It is therefore possible that social animals use it at all times, making probabilistic estimations to counteract its ambiguity. If this is the case, estimation of the environment using both non-social and social information might be a major determinant of the structure of animal collectives. In order to test this hypothesis, we have developed a Bayesian decision-making model that includes both personal and social information, that naturally weights them according to their reliability in order to get a better estimate of the environment. All members of the group can then use these improved estimations to make better decisions, and collective patterns of decisions then emerge from these individuals interacting through their perceptual systems.

We show that this model derives social rules that economically explain detailed experiments of decision-making in animal groups

We derived a model in which each individual decides from an estimation of which behavior is best to perform. These behaviors can be to go to one of several different places, to choose among some behaviors like forage, explore or run away, or any other set of options. For clarity, here we particularize to the case of choosing the best of two spatial locations,

We are seeking a model based on probabilistic estimation that can simultaneously give us insight into social decision-making and fit experimental data. For this reason we simplify the model by assuming that the focal individual does not make use of the correlations among the behaviour of others, but instead assumes their behaviours to be independent of each other. This is a strong hypothesis but allows us to derive simple explicit expressions with important insights. The section ‘Model including dependencies’ at the end of Results shows that this assumption gives a very good approximation to a more complete model that takes into account these correlations.

The assumption of independence translates in that the probability of a given set of behaviors is just the product of the probabilities of the individual behaviors. We apply it to the probabilities needed to compute

To summarize, using Eqs.

We have so far only considered the perceptual stage of decision-making, in which the deciding individual estimates the probability that each behavior is the best one. Now it must decide according to this estimation. A simple decision rule would be to go to

Finally, using Eqs. 10 and 11 we have that the probability that the deciding individual goes to

In the following sections, we particularize Eq.

We first considered the simple case of two identical equidistant sites,

(

Using Eq.

The behavior of the group is obtained by applying the decision rule in Eq.

The relevance of the symmetric case is that the model has a single parameter and a single variable, enabling a powerful comparison against experimental data. We tested the model using an existing rich data set of collective decisions in three-spined sticklebacks

(

Despite the simplicity of the behavioral rule in Eq.

Group decision-making in three-spined sticklebacks shows a single type of distribution in which probability is minimum at the center and increases monotonically towards the edges, denoted here as U-shaped distribution (or J-shaped when there is a bias to one of the two options). However, the model in Eq.

(

An interesting prediction of our model is that, for a given number of bins, the shape of the distribution of choices changes with the number of decided individuals, and the dynamics of this change depends on

An interesting modification of the experimental set-up consists in using replicas of the animals that we can modify to potentially alter their reliability estimated by the animals. We considered the particular case, motivated by experiments in

(

We finally considered the case in which sites

The stickleback dataset reported in reference

(

In the experiments in

In this section we will remove the hypothesis of independence among the behaviors of the other individuals (Eq.

(

To calculate the probability of the observed sequence of behaviors provided that

So the problem reduces to computing the individual decision probabilities

The assumption that all animals apply the same rules translates into the following. To apply an equation like Eq.

The equations above constitute the model taking into account dependencies. The new parameters of this model are

The new model also matches very well with the experimental data discussed in this paper. Results for the case of two different replicas are shown in

(

We now ask how different is the model including dependencies from the model that neglects them. To compare the two models, we plot the probability of going to

We find an interesting prediction of the new model: There are some states in which the most likely option is to choose the option chosen by

We have shown that probabilistic estimation in the presence of uncertainty can explain collective animal decisions. This approach generated a new expression for each experimental manipulation, Eq.

For the data used in this paper, previous empirical fits used more parameters

Collective animal behavior has been subject to a particularly careful quantitative analysis. Previous studies have given descriptions led by the powerful idea that complex collective behaviors can emerge from simple individual rules. In fact, some systems have been found empirically to obey rules that are mathematically similar or the same as some of the ones presented in this paper, further supporting the idea that probabilistic estimation might underlie collective decision rules in many species. For example, a function like the one in Eq.

Our model is naturally compatible with other theories that use a Bayesian formalism to study different aspects of behavior and neurobiology, thus contributing to a unified approach of information processing in animals. For example, it may be combined with the formalism of Bayesian foraging theory

Our approach also makes a number of predictions. For example, it derives the probability of choosing among

We also predict a quantitative link between estimation and collective behavior. The parameters

We can also extend the estimation model to use, instead of the location of animals, their predicted location. We would then find expressions like the ones in this paper but for the number or density of individuals estimated for a later time. Consider for example the case without non-social information, described in Eq.

The estimation rules presented in this paper refer to a single individual. To simulate the behavior of a group, we use the following algorithm: The current individual decides between

For the case of the model including dependencies, the model always starts at state 0∶0, with

We computed log likelihood as the logarithm of the probability that the histograms come from the model. We searched for the model parameters giving a higher value of log likelihood, corresponding to a better fit. This search was performed by optimizing each parameter separately (keeping the rest constant) and iterating through all parameters until convergence. In all cases convergence was rapidly achieved. We performed multiple searches for best fitting parameters starting from random initial conditions and always found convergence to the same values, suggesting there are no local maxima. Indeed, we observed that log-likelihood is smooth and with a single maximum in all the cases with 1 or 2 parameters (see

For model comparison we used the Bayesian Information Criterion (BIC)

More intuitive than the direct

We used BIC to compare different versions of our model, and also to compare our model with those of references

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We acknowledge useful comments by Sara Arganda, Larissa Conradt, Iain Couzin, Jacques Gautrais, David Sumpter, Guy Theraulaz, Julián Vicente Page and COLMOT 2010 participants.