Research ArticleSOCIAL SCIENCES

Conflicts of interest improve collective computation of adaptive social structures

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Science Advances  17 Jan 2018:
Vol. 4, no. 1, e1603311
DOI: 10.1126/sciadv.1603311
  • Fig. 1 The error rate of a pair using Nash equilibrium thresholds increases as the weight given to either decision time or decision preference increases.

    (A to D) The lines show the Nash equilibrium thresholds for a pair of individuals as a function of the probability that the stronger animal wins, c, with blue indicating the stronger individual and red the weaker individual. When c = 0.5, the individuals are equally matched. In (C), when c = 0.5, the Nash thresholds are symmetric: Either individual could use either strategy. The optimization weights for each panel are indicated in the simplex with the corresponding letter. (E) The color in the simplex indicates the error rate of a decision made by a pair making a difficult decision (c = 0.55) using Nash equilibrium thresholds, as a function of the optimization weights w1, w2, andw3. In the lower left corner of the simplex, only error rate matters (w1 = 1). In the upper corner, only decision time matters (w2 = 1). In the lower right corner, only preference matters (w3 = 1). Parameters: in (A), w1 = 1, w2 = 0, and w3 = 0; in (B), w1 = 0, w2 = 0, and w3 = 1; in (C), w1 = 0, w2 = 0.2, and w3 = 0.8; and in (D), w1 = 0, w2 = 1, and w3 = 0; in all panels, b = 1, r = 1, and l = 0.1.

  • Fig. 2 The mutual information of the power scores computed by a group using Nash thresholds increases as the weight given to decision preference increases, as long as there are nonzero waiting costs.

    (A) The points show the average Nash thresholds as a function of position in the group for different optimization weights, where 1 is the strongest individual and 20 is the weakest individual. The shaded region around each curve shows the average plus or minus the SD across 1000 draws. The optimization weights for each line are indicated in the simplex with the corresponding letter. (B) The color in the simplex indicates the average error rate of all decisions made by members of a group using Nash thresholds, as a function of the optimization weights w1, w2, and w3. In the lower left corner of the simplex, only error rate matters (w1 = 1). In the upper corner, only decision time matters (w2 = 1). In the lower right corner, only preference matters (w3 = 1). (C) The points show the average error rate of decisions made by all individuals, as a function of the weight given to decision preference, w3, when w2 = 0.1 [this is also shown by moving from left to right in the second row of hexagons from the bottom in (B)]. (D) The points show the mutual information between the power scores computed using each algorithm, as a function of the weight given to decision preference, w3, when w2 = 0.1. Each algorithm is shown in a different color: eigenvector centrality in blue, unweighted in-degree in green, weighted in-degree in red, and entropy in purple. Parameters: i indicates w1 = 1, w2 = 0, and w3 = 0; ii indicates w1 = 0, w2 = 0, and w3 = 1; iii indicates w1 = 0.9, w2 = 0.1, and w3 = 0; and iv indicates w1 = 0, w2 = 0.1, and w3 = 0.9; in all panels, n = 20, b = 1, r = 1, and l = 0.1.

  • Fig. 3 The average skewness of the distribution of eigenvector centrality is maximized at intermediate waiting costs.

    The color indicates the average skewness of the distribution of power computed by a group using Nash thresholds, as a function of the optimization weights w1, w2, and w3. In the lower left corner of the simplex, only error rate matters (w1 = 1). In the upper corner, only decision time matters (w2 = 1). In the lower right corner, only preference matters (w3 = 1). Parameters: n = 20, b = 1, r = 1, and l = 0.1.

  • Fig. 4 The best measure of consensus in the decision network depends on the average error rate and the types of errors being made.

    (A) The color indicates the most informative consensus measure to apply to a network constructed by a group using Nash thresholds, as a function of the optimization weights w1, w2, and w3. In the lower left corner of the simplex, only error rate matters (w1 = 1). In the upper corner, only decision time matters (w2 = 1). In the lower right corner, only preference matters (w3 = 1). (B to D) We show how the mutual information of each consensus algorithm changes over time as the network forms. The optimization weights for each panel are indicated in the simplex with the corresponding letter. Parameters: in (B), w1 = 0, w2 = 0, and w3 = 1; in (C), w1 = 0, w2 = 0.2, and w3 = 0.8; and in (D), w1 = 0, w2 = 0.4, andw3 = 0.6; in all panels, n = 20, b = 1, r = 1, and l = 0.1.

  • Table 1 Variables in the model and their interpretations in social and neural systems.
    VariableModel definitionSocial interpretationNeural interpretation
    aiValue of component iFighting ability of individual iHow much the part of the visual stimulus
    to which population i responds is present
    in the input.
    bChange due to new evidence
    cStrength of inputDegree of asymmetry between opponentsCoherence of moving dots
    lLeak rate
    rRate at which evidence appearsRate at which fights occurRate at which dots appear
    T1 and T2Decision thresholds
    w1Error rate weightCost to both individuals when the stronger
    individual incorrectly signals, for example,
    because the relationship is less stable
    Cost of making incorrect decision,
    imposed by experimenter
    w2Decision time weightCost of prolonged fighting, for example,
    injury and opportunity cost of time
    Penalty for taking a long time
    w3Decision preference weight,
    captures strength of conflict
    Cost of emitting a signal and agreeing to be
    subordinate, for example, because of limited
    access to resources
    X1 and X2Decision variablesEvidence accumulated about relative dominanceFiring rates of neural populations

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/e1603311/DC1

    section S1. Background on models of decision-making

    section S2. Study system

    section S3. Skewness of DSP

    section S4. Dimensionality and initial conditions

    section S5. Analogous model of neural decision-making

    section S6. Derivation of partial differential equations for decision time, error rate, and probability of reaching decision preference

    section S7. Nash equilibria

    section S8. A notion of correctness for biological computation

    section S9. Calculation of mutual information

    section S10. Most informative measures of consensus

    section S11. Tuning waiting costs

    section S12. Comparison of our model to previous studies of animal conflict

    section S13. War of attrition

    section S14. Supplementary table

    section S15. Supplementary figures

    table S1. Examples of collective computation.

    fig. S1. Error rate decreases as decision time increases, as long as the initial conditions are not biased toward the correct decision.

    fig. S2. The mutual information of the power scores computed by a group using Nash thresholds increases as the weight given to decision preference increases, as long as there are nonzero waiting costs.

    fig. S3. The average skewness of the distribution of eigenvector centrality is maximized at intermediate waiting costs.

    fig. S4. The best measure of consensus in the decision network depends on the average error rate and the types of errors being made.

    fig. S5. Schematic of the model.

    fig. S6. The error rate of a group using Nash thresholds decreases as the weight given to decision preference increases, regardless of the size of the group.

    fig. S7. Pairs with similar and high abilities always take as long or longer to make a decision than any other pairs do.

    fig. S8. The mutual information of each consensus algorithm is a decreasing function of the average pairwise error rate.

    fig. S9. The average skewness of the distribution of unweighted in-degree is maximized at intermediate waiting costs.

    fig. S10. The average skewness of the distribution of consensus scores from each measure is maximized at intermediate waiting costs.

    fig. S11. The average skewness of the distribution of consensus scores from each measure is maximized at intermediate waiting costs.

    fig. S12. When a pair of animals have equal fighting abilities, c = 0.5, there are asymmetric Nash equilibrium thresholds.

    References (5559)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Background on models of decision-making
    • section S2. Study system
    • section S3. Skewness of DSP
    • section S4. Dimensionality and initial conditions
    • section S5. Analogous model of neural decision-making
    • section S6. Derivation of partial differential equations for decision time, error rate, and probability of reaching decision preference
    • section S7. Nash equilibria
    • section S8. A notion of correctness for biological computation
    • section S9. Calculation of mutual information
    • section S10. Most informative measures of consensus
    • section S11. Tuning waiting costs
    • section S12. Comparison of our model to previous studies of animal conflict
    • section S13. War of attrition
    • section S14. Supplementary table
    • section S15. Supplementary figures
    • table S1. Examples of collective computation.
    • fig. S1. Error rate decreases as decision time increases, as long as the initial conditions are not biased toward the correct decision.
    • fig. S2. The mutual information of the power scores computed by a group using Nash thresholds increases as the weight given to decision preference increases, as long as there are nonzero waiting costs.
    • fig. S3. The average skewness of the distribution of eigenvector centrality is maximized at intermediate waiting costs.
    • fig. S4. The best measure of consensus in the decision network depends on the average error rate and the types of errors being made.
    • fig. S5. Schematic of the model.
    • fig. S6. The error rate of a group using Nash thresholds decreases as the weight given to decision preference increases, regardless of the size of the group.
    • fig. S7. Pairs with similar and high abilities always take as long or longer to make a decision than any other pairs do.
    • fig. S8. The mutual information of each consensus algorithm is a decreasing function of the average pairwise error rate.
    • fig. S9. The average skewness of the distribution of unweighted in-degree is maximized at intermediate waiting costs.
    • fig. S10. The average skewness of the distribution of consensus scores from each measure is maximized at intermediate waiting costs.
    • fig. S11. The average skewness of the distribution of consensus scores from each measure is maximized at intermediate waiting costs.
    • fig. S12. When a pair of animals have equal fighting abilities, c = 0.5, there are asymmetric Nash equilibrium thresholds.
    • References (55–59)

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