Condorcet's Theorem Research Articles

Rescuing Collective Wisdom when the Average Group Opinion Is Wrong

The total knowledge contained within a collective supersedes the knowledge of even its most intelligent member. Yet the collective knowledge will remain unaccessible to us unless we are able to find efficient knowledge aggregation methods which produce reliable decisions based on the behavior or opinions of the collective's members. It is often stated that simple averaging of a pool of opinions is a good and in many cases the optimal way to extract knowledge from a crowd. The method of averaging has been applied to analysis of decision-making in very different fields such as forecasting, collective animal behavior, individual psychology and machine learning. Two mathematical theorems, Condorcet's theorem and Jensen's inequality, provide a general theoretical justification for the averaging procedure. Yet the necessary conditions which guarantee the applicability of these theorems are often not met in practice. Under such circumstances, averaging can lead to suboptimal and sometimes very poor performance. Practitioners in many different fields have independently developed procedures to counteract the failures of averaging. We review such knowledge aggregation procedures and interpret the methods in the light of a statistical decision theory framework to explain when their application is justified. Our analysis indicates that in the ideal case, there should be a matching between the aggregation procedure and the nature of the knowledge distribution, correlations and associated error costs. This leads us to explore how machine learning techniques can be used to extract near-optimal decision rules in a data-driven manner. We end with a discussion of open frontiers in the domain of knowledge aggregation and collective intelligence in general.

Consensus Decision Making by Fish

Decisions reached through consensus are often more accurate, because they efficiently utilize the diverse information possessed by group members [1-3]. A trust in consensus decision making underlies many of our democratic political and judicial institutions [4], as well as the design of web tools such as Google, Wikipedia, and prediction markets [5, 6]. In theory, consensus for the option favored by the majority of group members will lead to improved decision-making accuracy as group size increases [2, 4]. Although group-living animals are known to utilize social information [7-10], little is known about whether or not decision accuracy increases with group size. In order to reach consensus, group members must be able to integrate the disparate information they possess. Positive feedback, resulting from copying others, can spread information quickly through the group, but it can also result in all individuals making the same, possibly incorrect, choice [8, 11, 12]. On the other hand, if individuals never copy each other, their decision making remains independent and they fail to benefit from information exchange [4]. Here, we show how small groups of sticklebacks (Gasterosteus aculeatus) reach consensus when choosing which of two replica fish to follow. As group size increases, the fish make more accurate decisions, becoming better at discriminating subtle phenotypic differences of the replicas. A simple quorum rule proves sufficient to explain our observations, suggesting that animals can make accurate decisions without the need for complicated comparison of the information they possess. Furthermore, although submission to peers can lead to occasional cascades of incorrect decisions, these can be explained as a byproduct of what is usually accurate consensus decision making.