When Being the Fifth Wheel Pays Off: Wisdom of the Crowds with Costly Information
Matteo Michelini

Abstract:
The Condorcet Jury Theorem (CJT) is considered one of the cornerstones of the wisdom of the crowd, i.e. the idea that large groups of people are better at tracking the truth than small ones. However, such a result is often criticized on the ground that its assumptions are unrealistic. In this thesis, we provide a new perspective on the truth-tracking potential of a group by introducing in the CJT setting a more realistic point of view. In particular, we consider a group whose members are interested in finding the right answer to a question, but where being competent is costly. Consequently, agents can either acquire information by spending effort and express a vote, or abstain. We model this scenario game-theoretically and characterize its equilibria. In addition, we also look at the social welfare of the game thus defined. We use our formal results to show that some of the claims of the CJT do not hold under our proposed assumptions. For example, we prove that larger groups do not necessarily produce better outcomes if the group size surpasses a certain threshold.
Next, we extend this basic setting in three different directions. Firstly, we study the possibility for agents of having different stakes in the matter, i.e. to be less or more interested. We prove that a group can be as accurate as possible only if the agents with the higher stakes vote, and similarly we show which groups are more efficient in terms of social welfare. Secondly, we describe the game in the case where agents may reach different levels of competence. In this scenario, we study the pure equilibria, and we also introduce other possible ways of characterizing this scenario further. We also discuss the notion of weight of a voter and the possibility of delegating.
Lastly, we consider the possibility for the agents of being connected through a network. We characterize equilibria on distinct classes of networks and we comment these results. We integrate our analytical results with a computer simulation of best response dynamics in those games.