Casting votes of antecedents play a key role in successful sequential decision-making

Abstract

Aggregation of opinions often results in high decision-making accuracy, owing to the collective intelligence effect. Studies on group decisions have examined the optimum weights for opinion aggregation to maximise accuracy. In addition to the optimum weights of opinions, the impact of the correlation among opinions on collective intelligence is a major issue in collective decision-making. We investigated how individuals should weigh the opinions of others and their own to maximise their accuracy in sequential decision-making. In our sequential decision-making model, each person makes a primary choice, observes his/her predecessors' opinions, and makes a final choice, which results in the person's answer correlating with those of others. We developed an algorithm to find casting voters whose primary choices are determinative of their answers and revealed that decision accuracy is maximised by considering only the abilities of the preceding casting voters. We also found that for individuals with heterogeneous abilities, the order of decision-making has a significant impact on the correlation between their answers and their accuracies. This could lead to a counter-intuitive phenomenon whereby, in sequential decision-making, respondents are, on average, more accurate when less reliable individuals answer earlier and more reliable individuals answer later.

Fig 1. Summary of the region of optimum weights w_n for individual n (n = 1, 2, 3) in the simultaneous decision-making involving three individuals.

The largest triangle exhibits the 2-simplex {w = (w₁, w₂, w₃); w₁ + w₂ + w₃ = 1}, where W₁, W₂, and W₃ denote (1,0,0), (0,1,0), and (0,0,1), respectively. Each smaller triangle exhibits the region of optimum weights when each inequality for $r_1^{*}$, $r_2^{*}$, and $r_3^{*}$ holds. For example, when the ability of individual 1 is so high that $r_1^{*}>r_2^{*}+r_3^{*}$ is satisfied, the optimum weights should satisfy w₁ > 0.5, i.e., w₂ + w₃ < 0.5, which corresponds to the upper red triangle, and these weights correspond to the expert rule governed by individual 1.

Fig 2. Optimal behaviours of the second and third individuals.

The diagram on the right-hand side shows the resulting influential relationship among three individuals by the arrows, each pointing to the individual that one follows. For example, the top diagram (a1) reveals that the second individual incorporates the answer by the first individual, and the third individual decides without referring to others.

Fig 3. Mean performance of the third individual E[π₃(S₂, T₂, R₂, X₃)] in sequential decision-making.

(a) E[π₃(S₂, T₂, R₂, X₃)] in sequential decision-making when p₃ = 0.7 is exhibited by contours. The horizontal and the vertical axes show the abilities p₁ and p₂ of the first and second individuals, respectively. The brighter colour stands for the higher mean performance. (b) E[π₃(S₂, T₂, R₂, X₃)] is shown by red text. The horizontal and vertical axes are the same as that in panel (a). Each region corresponds to each case of a1) to b3) in Fig 2. The coordinates of α and β are $\sqrt{p_3}/\left(\sqrt{p_3}+\sqrt{1-p_3}\right),\sqrt{p_3}/\left(\sqrt{p_3}+\sqrt{1-p_3}\right)$ and (p₃, p₃), respectively.

Fig 4. Mean performance versus order.

Simulation (blue circles) and analytical (red triangles) results of mean performance E[π_n] (ordinate) versus order n (abscissa) are exhibited for each individual ability p in each panel. The horizontal grey line shows the value of π_max(p).

Fig 5. Mean performance E[π_n,q] of the expert with ability q when the expert believes that his/her ability is p.

Each panel exhibits E[π_n,q] as the other individuals have the ability p = 0.6, 0.7, 0.8 or 0.9. The colour of the plots stands for the value of q. The horizontal line (grey) shows the value of π_max(p).