A Deterministic Protocol for Sequential Asymptotic Learning

In the classic herding model, agents receive private signals about an underlying binary state of nature, and act sequentially to choose one of two possible actions, after observing the actions of their predecessors. We investigate what types of behaviors lead to asymptotic learning, where agents will eventually converge to the right action in probability. It is known that for rational agents and bounded signals, there will not be asymptotic learning. Does it help if the agents can be cooperative rather than act selfishly? This is simple to achieve if the agents are allowed to use randomized protocols. In this paper, we provide the first deterministic protocol under which asymptotic learning occurs. In addition, our protocol has the advantage of being much simpler than previous protocols

Additive Conjugacy and the Bohr Compactification of Orthogonal Representations

We say that two unitary or orthogonal representations of a finitely generated group G are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of G a topological action that is a complete additive conjugacy invariant: the action of G by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that G is amenable if and only if there is a nonzero homomorphism from L²(G) into R/Z that is invariant to the G-action

Equitable voting rules

May's Theorem (1952), a celebrated result in social choice, provides the foundation for majority rule. May's crucial assumption of symmetry, often thought of as a procedural equity requirement, is violated by many choice procedures that grant voters identical roles. We show that a weakening of May's symmetry assumption allows for a far richer set of rules that still treat voters equally. We show that such rules can have minimal winning coalitions comprising a vanishing fraction of the population, but not less than the square root of the population size. Methodologically, we introduce techniques from group theory and illustrate their usefulness for the analysis of social choice questions.Comment: 43 pages, 5 figure

The speed of sequential asymptotic learning

In the classical herding literature, agents receive a private signal regarding a binary state of nature, and sequentially choose an action, after observing the actions of their predecessors. When the informativeness of private signals is unbounded, it is known that agents converge to the correct action and correct belief. We study how quickly convergence occurs, and show that it happens more slowly than it does when agents observe signals. However, we also show that the speed of learning from actions can be arbitrarily close to the speed of learning from signals. In particular, the expected time until the agents stop taking the wrong action can be either finite or infinite, depending on the private signal distribution. In the canonical case of Gaussian private signals we calculate the speed of convergence precisely, and show explicitly that, in this case, learning from actions is significantly slower than learning from signals

Equitable Voting Rules

A celebrated result in social choice is May's Theorem (1952), providing the foundation for majority rule. May's crucial assumption of symmetry, often thought of as a procedural equity requirement, is violated by many choice procedures that grant voters identical roles. We show that a modification of May's symmetry assumption allows for a far richer set of rules that still treat voters equally, but have minimal winning coalitions comprising a vanishing fraction of the population. We conclude that procedural fairness can coexist with the empowerment of a small minority of individuals. Methodologically, we introduce techniques from discrete mathematics and illustrate their usefulness for the analysis of social choice questions

Equitable Voting Rules

A celebrated result in social choice is May's Theorem (1952), providing the foundation for majority rule. May's crucial assumption of symmetry, often thought of as a procedural equity requirement, is violated by many choice procedures that grant voters identical roles. We show that a modification of May's symmetry assumption allows for a far richer set of rules that still treat voters equally, but have minimal winning coalitions comprising a vanishing fraction of the population. We conclude that procedural fairness can coexist with the empowerment of a small minority of individuals. Methodologically, we introduce techniques from discrete mathematics and illustrate their usefulness for the analysis of social choice questions

Essays on Social Learning and Social Choice

This dissertation contains three essays, two which contribute to the study of social learning (Chapters 1 and 2) and one which contributes to the study of social choice (Chapter 3). In Chapter 1, I introduce a fully rational model of social learning on networks with endogenous action timing. I show that the structure of the network can play an important role in the aggregation of information. When the social network contains high-degree vertices, agents can be arbitrarily likely to make good choices. In contrast, when the social network is linear, there is a bound on how likely agents are to make good choices which holds regardless of how patient they are. The main contribution of this chapter is the identification of a novel mechanism through which strategic behavior can substantially impede the flow of information through a social network. In Chapter 2, co-authored with Vadim Martynov and Omer Tamuz, we study the asymptotic rate at which the probability of taking the correct action converges to 1 in the classical sequential learning model with unbounded signals. We provide a characterization of the asymptotic law of motion of the public belief, and we use this characterization to show that convergence occurs more slowly than when agents directly observe private signals, and that the expected time until the last incorrect action can be finite or infinite. In Chapter 3, co-authored with Laurent Bartholdi, Maya Josyula, Omer Tamuz, and Leeat Yariv, we introduce equitability as a less stringent alternative to symmetry for modeling egalitarianism in voting rules. We then use techniques from group theory to show that equitable voting rules can have minimal winning coalitions comprising a vanishing fraction of the population, but they cannot be smaller than the square root of the population size.</p