Optimising collective accuracy among rational individuals in sequential decision-making with competition

Abstract

Theoretical results underpinning the Wisdom of Crowds, such as the Condorcet Jury Theorem, point to substantial accuracy gains through aggregation of decisions or opinions, but the foundations of this theorem are routinely undermined in circumstances where individuals are able to adapt their own choices based after observing what other agents have chosen. In sequential decision-making, rational agents use the choices of others as a source of information about the correct decision, creating powerful correlations between different agents' choices that violate the assumptions of independence on which the Condorcet Jury Theorem depends. In this paper I show how such correlations emerge when agents are rewarded solely based on their individual accuracy, and the impact of this on collective accuracy. I then demonstrate how a simple competitive reward scheme, where agents' rewards are greater if they correctly choose options that few have already chosen, can induce rational agents to make independent choices, returning the group to optimal levels of collective accuracy. I further show that this reward scheme is robust, offering improvements to collective accuracy across of wide range of competition strengths, suggesting that such schemes could be effectively implemented in real-world contexts to improve collective wisdom.

Figures (2)

• Figure 1: Characterising the response to social information in sequential decisions under binary rewards. (A) the probability that an agent will choose option A when that is the correct choice, conditioned on the number of previous decisions for options A and B, averaging over all sequences consistent with those aggregate number of decisions. In this example the environmental noise is $\epsilon = 2.32$, giving an individual choice accuracy of $q=2/3$; the red contour line indicates this probability. (B) The probability for $n_A$ agents to select option A when that is the correct choice, averaged over a full sequence of decisions in a group of $n=25$ agents. The blue bars indicate the probability when rational agents are subject to binary rewards, red bars indicate the probability if all agents select independently. The dashed lines indicate the mean of each probability distribution. Agents responding rationally to binary rewards have a higher average number of individually-successful decisions, but a lower probability of a correct majority decision.
• Figure 2: The effect of competition on collective accuracy. (A) With $q=2/3$, across different group sizes ($n$) collective accuracy increases with increasing competition ($\beta$) up to an optimal value given by $\beta = Q$ (indicated by the dashed red line), where collective accuracy matches that predicted by the Condorcet Jury Theorem. Higher levels of competition reduce collective accuracy, with sufficiently high values of $\beta$ leading to lower collective accuracy than under binary rewards ($\beta = 1$, indicated by the dashed black line) . Negative values of $\beta$, indicating rewards for conformity, always lead to lower collective accuracy; (B) Individual accuracy is maximised at values of $\beta$ close to one, indicating weak positive competition, and increases with group size. At the optimal competition for collective accuracy, individual accuracy is the same for all group sizes as agents choose independently; (C) The collective accuracy under optimal competition (solid line) compared to that achieved under binary rewards (dashed line) as a function of group size; (D) The maximum value of $\beta$ for which competitive rewards outperform binary rewards, for varying group size and as a function of $q$ (representing the probability for a solo agent to choose correctly). The dashed line shows the optimal value of $\beta=Q$ for comparison. The range of effective competition values (those that improve on binary rewards) is greater for easier decisions and in larger group sizes. Note the logarithmic scale on the y-axis.