The emergence of unshared consensus decisions in bottlenose dolphins

Abstract

Unshared consensus decision-making processes, in which one or a small number of individuals make the decision for the rest of a group, are rarely documented. However, this mechanism can be beneficial for all group members when one individual has greater knowledge about the benefits of the decision than other group members. Such decisions are reached during certain activity shifts within the population of bottlenose dolphins residing in Doubtful Sound, New Zealand. Behavioral signals are performed by one individual and seem to precipitate shifts in the behavior of the entire group: males perform side flops and initiate traveling bouts while females perform upside-down lobtails and terminate traveling bouts. However, these signals are not observed at all activity shifts. We find that, while side flops were performed by males that have greater knowledge than other male group members, this was not the case for females performing upside-down lobtails. The reason for this could have been that a generally high knowledge about the optimal timing of travel terminations rendered it less important which individual female made the decision.

Appendix 1. Contextual knowledge and its influence on the emergence of unshared consensus decisions

The following model is based on the Condorcet jury theorem. Its purpose is to show that, if the level of information is generally high within the group, any increase in information of the group member with the highest reach (in relative and in absolute terms) adds relatively little advantage to an unshared decision. On the other hand, if the level of information is generally low within the group, already a small difference in information between the highest reach member and other group members might make an unshared decision more profitable than a shared decision. This could explain the different observations in decisions about the initiation and about the termination of traveling bouts, if these two types of decisions are accompanied by different general levels of relevant information within the group.

We assume that the animal with the highest reach in a school has the probability, p _r, to get the decision about timing of traveling right (i.e., to time it optimally given their energy budget) and the probability 1−p _r to get it wrong (i.e., time the traveling not optimal). Furthermore, we assume that all other group members, which have lower reach, have a lower probability p _s (p _s < p _r) to get the decision right, if they made the decision individually. Thus, the median probability to get the decision right of all individuals is p _s (assuming group size is larger than two). This median probability p _s is, thus, a measure of ‘the general level of information within the group’.

We rewrite p _r as:

$$ p_{\text{r}} = p_{\text{s}} + \lambda \times \left( {1 - p_{\text{s}} } \right), $$

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with 0 < λ ≤ 1 (since p _r > p _s, and p _r is bounded at 1).

Hence,

$$ p_{\text{r}} - p_{\text{s}} = \lambda \times \left( {1 - p_{\text{s}} } \right) $$

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where λ is a measure of the difference in information between the most informed group member and the other members.The probability that a school would make a right decision if it followed the SF (or ULT, respectively) signal of the animal with highest reach would be p _r (unshared consensus decision). The probability that a school would make a right decision if it decided democratically (assuming for reasons of simplicity that school size n is uneven) would be:

$$ p_r \cdot \sum\limits_{{i = \frac{n - 1}{2}}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_s^i \cdot \left( {1 - p_s } \right)^{n - 1 - i} + (1 - p_r ) \cdot \sum\limits_{{i = \frac{n + 1}{2}}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_s^i \cdot \left( {1 - p_s } \right)^{n - 1 - i} $$

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Thus, a school would make a better decision by following the most knowledgeable individual, if:

$$p_{\text{r}} > p_{\text{r}} \cdot \sum\limits_{{i = \frac{n - 1}{2}}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{\text{s}}^i \cdot \left( {1 - p_{\text{s}} } \right)^{n - 1 - i} + (1 - p_{\text{r}} ) \cdot \sum\limits_{{i = \frac{n + 1}{2}}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{\text{s}}^i \cdot \left( {1 - p_{\text{s}} } \right)^{n - 1 - i}$$

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and the difference between an unshared and a shared consensus decision (diff_desp-dem) in terms of probability to make the right decision would be:

$$ \begin{array}{*{20}c} {{\text{diff}}_{{{\text{desp - dem}}}} = p_{r} - p_{r} \cdot \sum\limits_{{i = \frac{{n - 1}}{2}}}^{{n - 1}} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{s}^{i} } \\ { \cdot \left( {1 - p_{s} } \right)^{{n - 1 - i}} - \left( {1 - p_{r} } \right)} \\ { \cdot \sum\limits_{{i = \frac{{n + 1}}{2}}}^{{n - 1}} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right) \cdot p_{s}^{i} \cdot \left( {1 - p_{s} } \right)^{{n - 1 - i}} } } \\ \end{array}$$

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$$ \begin{gathered} = p_{\text{r}} - p_{\text{r}} \cdot \sum\limits_{{i = \frac{n - 1}{2}}}^{{\frac{n - 1}{2}}} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{\text{s}}^i \cdot \left( {1 - p_{\text{s}} } \right)^{n - 1 - i} + p_{\text{r}} \cdot \sum\limits_{{i = \frac{n - 1}{2} + 1}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{\text{s}}^i \cdot \left( {1 - p_{\text{s}} } \right)^{n - 1 - i} \hfill \\ - \left( {1 - p_{\text{r}} } \right) \cdot \sum\limits_{{i = \frac{n + 1}{2}}}^{n - 1} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)} \cdot p_{\text{s}}^i \cdot \left( {1 - p_{\text{s}} } \right)^{n - 1 - i} \hfill \\ \end{gathered} $$

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$$ \begin{array}{*{20}c} { = p_{r} \cdot \left[ {1 - \frac{{\left( {n - 1} \right)!}}{{\left( {\frac{{n - 1}}{2}!} \right)}} \cdot p_{s}^{{\frac{{n - 1}}{2}}} \cdot \left( {1 - p_{s} } \right)^{{\frac{{n - 1}}{2}}} } \right]} \\ { - \sum\limits_{{i = \frac{{n + 1}}{2}}}^{{n - 1}} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right) \cdot p_{s}^{i} \cdot \left( {1 - p_{s} } \right)^{{n - 1 - i}} } } \\ \end{array}$$

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$$ \begin{array}{*{20}c} { = \left( {p_{s} + \lambda \cdot \left( {1 - p_{s} } \right)} \right)} \\ { \cdot \left[ {1 - \frac{{\left( {n - 1} \right)!}}{{\left( {\frac{{n - 1}}{2}!} \right)^{2} }} \cdot p_{s}^{{\frac{{n - 1}}{2}}} \cdot \left( {1 - p_{s} } \right)^{{\frac{{n - 1}}{2}}} } \right]} \\ { - \sum\limits_{{i = \frac{{n + 1}}{2}}}^{{n - 1}} {\left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right) \cdot p_{s}^{i} \cdot \left( {1 - p_{s} } \right)^{{n - 1 - i}} } } \\ \end{array}$$

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Since diff_desp-dem is the difference between an unshared and a shared decision with respect to the probability of getting the decision right, it is a measure of the relative advantage of an unshared over a shared decision.

From Eq. 12, it follows that diff_desp-dem is correlated with the difference in information between the most informed group member and the other members (λ) as follows:

$$ \frac{{\partial {\text{diff}}_{{{\text{desp}} - {\text{dem}}}} }}{{\partial \lambda }} = \left( {1 - p_{\text{s}} } \right) \cdot \left[ {1 - \frac{{\left( {n - 1} \right)!}}{{\left( {\frac{n - 1}{2}!} \right)^2 }} \cdot p_{\text{s}}^{{\frac{n - 1}{2}}} \cdot (1 - p_{\text{s}} )^{{\frac{n - 1}{2}}} } \right] $$

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That is, the advantage of an unshared versus a shared decision increases with the information discrepancy λ with a slope given by Eq. 13. The size of this slope is always positive but decreases with p _s (Fig. 3). That means the larger the general level knowledge of all group members within the group (i.e., p _s), the less advantage (i.e., diff_desp-dem) does an increase in superior information of the member with the highest reach (i.e., λ) convey in an unshared decision and vice versa.

Fig. 3

The relationship between the median knowledge of members of the schools (p _s , i.e., excluding the individual with the maximum knowledge), school size, and $\frac{{\partial {\text{diff}}_{{{\text{desp}} - {\text{dem}}}} }}{{\partial \lambda }}$ which is the difference in correctness to take an unshared consensus decision as opposed to a shared consensus one (diff_desp-dem) given the discrepancy in knowledge between the most knowledgeable and others in the schools (λ). See “Appendix 1” for derivation