Large-Scale Decision-Making via Small Group Interactions : the Importance of Triads 1

We study a framework for large-scale decision-making through small group interactions. In this framework, a crowd of participants interact with each other through a sequence of small group interactions, the composition of which are chosen by an algorithm designer, or in some settings, by nature. We consider the problem of finding the wisdom of the crowd, which we take to be the generalized median, in two settings: opinion formation and strategic bargaining. In both cases, we find a significant difference between groups of two and three. When the small groups are of size two, we find that there is no sequence of pairwise interactions which can always converge to a non-trivial approximation of the generalized median so long as the small group interactions satisfy a natural property we call local consistency. This holds even in the simple case when participants come from a line. In contrast, when the small groups are of size three, we find that the generalized median can be tightly and efficiently approximated when participants come from R under the l1 norm or when they come from any median graph, a class of graphs including squaregraphs, trees, and grids. Specifically, suppose that participants of each small group either update their opinions to, or come to consensus on, the generalized median of the group as a result of their interaction. Then by simply choosing the sequence of triads uniformly at random, the process is able to find a (1+O( √ lnn n ))-approximation of the global generalized median with high probability. Moreover, this occurs after each participant has only participated in an average of O(log n) small group interactions. In the strategic setting, we also design a mechanism for the entire extensive form game which implements this behavior under a Nash equilibrium. When participants treat each small group interaction as a separate game, we show that this can be improved to a strong Nash equilibrium.