Quorum responses and consensus decision making

Abstract

Animal groups are said to make consensus decisions when group members come to agree on the same option. Consensus decisions are taxonomically widespread and potentially offer three key benefits: maintenance of group cohesion, enhancement of decision accuracy compared with lone individuals and improvement in decision speed. In the absence of centralized control, arriving at a consensus depends on local interactions in which each individual's likelihood of choosing an option increases with the number of others already committed to that option. The resulting positive feedback can effectively direct most or all group members to the best available choice. In this paper, we examine the functional form of the individual response to others' behaviour that lies at the heart of this process. We review recent theoretical and empirical work on consensus decisions, and we develop a simple mathematical model to show the central importance to speedy and accurate decisions of quorum responses, in which an animal's probability of exhibiting a behaviour is a sharply nonlinear function of the number of other individuals already performing this behaviour. We argue that systems relying on such quorum rules can achieve cohesive choice of the best option while also permitting adaptive tuning of the trade-off between decision speed and accuracy.

Figure 1

Condorcet's theory. The probability that the majority of individuals are correct (for odd numbers of individuals) when each is correct with probability p=0.6.

Figure 2

Examples of empirical quorum responses in the decisions of migrating insects. (a) Cockroaches. Crosses indicate measured leaving times, dashed line is fit given by Ame et al. (2006) of$\frac{\theta }{1+\rho {\left(\frac{x-1}{S}\right)}^{\alpha }}.$with parameter values S=40, θ=0.01, ρ=1667 and α=2 and solid line is the best fit of the equation$\varphi +\frac{\theta }{1+\rho {\left(\frac{x-1}{S}\right)}^{\alpha }},$with parameter values S=40, φ=0.00051, θ=0.0067, ρ=1667 and α=1.73. This second fitted line allows for the fact that the probability of leaving does not go to zero with the number under the shelter. (b) A quorum rule governs the probability of a Temnothorax scout switching from tandem run recruitment of fellow scouts to faster transport of the bulk of the colony. Crosses show proportions of scouts choosing transport over tandem runs at different populations under high urgency. Open circles show corresponding data under low urgency. Solid and dashed lines, respectively, show a Hill function fit to these data: probability of transport=x^k/(x^k+T^k), where x is the new site population.

Figure 3

Commitment to an option as a function of the number of conspecifics that have already chosen it (x). The dashed line shows the purely linear response given by equation (4.2). The solid lines show nonlinear responses given by equation (4.1), for different values of k. For k>2 equation (4.1) gives a quorum response: that is, the probability of committing is less than the linear response for x<T and greater than or equal to the linear response for x≥T. Other parameters are p_x=1, T=10, a=0.1 and m=0.9.

Figure 4

Simulations of a simple quorum response model, for (a,c) shallow (k=1) and (b,d) steep (k=9) thresholds. (a,b) plot the change in the number of individuals committed to options X, solid line; and Y, dotted line for one simulation with k=1 and k=9, respectively. (c,d) show the distribution taken over 1000 simulation runs of the proportion of individuals choosing X after everyone has decided. Other parameters are r=0.02, p_x=1, p_y=0.5, T=10, a=0.1 and m=0.9.

Figure 5

Speed and accuracy of decision making for the simple quorum response model. Predicted effects of the parameters a, T and k on (a–d) the time until all individuals have made a decision and (e–h) the accuracy of that decision. In each image, a and T are varied for different threshold steepness, k. The plots show mean duration (time steps of the model) and accuracy (proportion of individuals choosing the less attractive option Y) over 1000 simulations for each parameter combination. (a) k=1; (b) k=2; (c) k=4; (d) k=9; (e) k=1; (f) k=2; (g) k=4; (h) k=9.

Figure 6

Speed–accuracy trade-off for the simple quorum response model. For fixed k and for a fixed minimum requirement for decision accuracy, we searched over all parameter values of a and T which give the fastest possible average time until all individuals have made a decision. This was done repeatedly for different minimum accuracy requirements to give a speed versus accuracy trade-off. Solid line, k=9; dashed line, k=4; dashed-dotted line, k=2; dotted line, k=1.

Figure 7

Trade-off of decision speed and accuracy for an agent-based model of Temnothorax emigrations. Predicted effects of the parameters accept, quorum threshold and k on the (a–d) duration and (e–h) accuracy of emigrations. In each image, the intrinsic accept rate and quorum threshold are varied for different threshold steepness, k. The plots show mean duration and accuracy over 32 simulations for each parameter combination. All other parameters are set to values estimated as described in Pratt (2005). Accept gives the recruitment initiation rate at good nests; the rate for mediocre nests was obtained by multiplying by the factor 0.52 (the ratio of observed values of accept for mediocre and good nests). (a) k=1; (b) k=2; (c) k=4; (d) k=8; (e) k=1; (f) k=2; (g) k=4; (h) k=8.