Making Consensus Tractable

We study a model of consensus decision making, in which a finite group of Bayesian agents has to choose between one of two courses of action. Each member of the group has a private and independent signal at his or her disposal, giving some indication as to which action is optimal. To come to a common decision, the participants perform repeated rounds of voting. In each round, each agent casts a vote in favor of one of the two courses of action, reflecting his or her current belief, and observes the votes of the rest. We provide an efficient algorithm for the calculation the agents have to perform, and show that consensus is always reached and that the probability of reaching a wrong decision decays exponentially with the number of agents.Comment: 18 pages. To appear in Transactions on Economics and Computatio

Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit

We consider an auction of identical digital goods to customers whose valuations are drawn independently from known distributions. Myerson's classic result identifies the truthful mechanism that maximizes the seller's expected profit. Under the assumption that in small groups customers can learn each others' valuations, we show how Myerson's result can be improved to yield a higher payoff to the seller using a mechanism that offers groups of customers to buy bundles of items.Comment: 11 pages, 1 figure. After posting the first version of this paper we learned that much of its mathematical content already appears in the literature, for example in "Multiproduct nonlinear pricing" by Armstrong (1996), although in a slightly different context of bundling products, rather than customer

Stabilizer Rigidity in Irreducible Group Actions

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.Comment: 25 page

Symbolic dynamics on amenable groups: the entropy of generic shifts

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that for every entropy value $c \in [0,\log |A|]$ there is a weakly mixing subshift of $A^G$ with entropy $c$. We also show that the set of strongly irreducible shifts does not form a $G_\delta$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space

Efficient Bayesian Social Learning on Trees

We consider a set of agents who are attempting to iteratively learn the 'state of the world' from their neighbors in a social network. Each agent initially receives a noisy observation of the true state of the world. The agents then repeatedly 'vote' and observe the votes of some of their peers, from which they gain more information. The agents' calculations are Bayesian and aim to myopically maximize the expected utility at each iteration. This model, introduced by Gale and Kariv (2003), is a natural approach to learning on networks. However, it has been criticized, chiefly because the agents' decision rule appears to become computationally intractable as the number of iterations advances. For instance, a dynamic programming approach (part of this work) has running time that is exponentially large in \min(n, (d-1)^t), where n is the number of agents. We provide a new algorithm to perform the agents' computations on locally tree-like graphs. Our algorithm uses the dynamic cavity method to drastically reduce computational effort. Let d be the maximum degree and t be the iteration number. The computational effort needed per agent is exponential only in O(td) (note that the number of possible information sets of a neighbor at time t is itself exponential in td). Under appropriate assumptions on the rate of convergence, we deduce that each agent is only required to spend polylogarithmic (in 1/\eps) computational effort to approximately learn the true state of the world with error probability \eps, on regular trees of degree at least five. We provide numerical and other evidence to justify our assumption on convergence rate. We extend our results in various directions, including loopy graphs. Our results indicate efficiency of iterative Bayesian social learning in a wide range of situations, contrary to widely held beliefs.Comment: 11 pages, 1 figure, submitte

Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem

Arrow's theorem implies that a social choice function satisfying Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are allowed, a dictatorial social choice function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not provide a complete characterization of all social choice functions satisfying Transitivity, the Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow's theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow's and Wilson's result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Pareto Principle). Additionally, we derive formulas for the number of functions satisfying these conditions.Comment: 11 pages, 1 figur

There are no monotone homomorphisms out of the convolution semigroup

We prove that there is no nonzero way of assigning real numbers to probability measures on R in a way which is monotone under first-order stochastic dominance and additive under convolution

Opinion Exchange Dynamics

We survey a range of models of opinion exchange. From the introduction: "The exchange of opinions between individuals is a fundamental social interaction... Moreover, many models in this field are an excellent playground for mathematicians, especially those working in probability, algorithms and combinatorics. The goal of this survey is to introduce such models to mathematicians, and especially to those working in discrete mathematics, information theory, optimization, probability and statistics."Comment: 62 pages. arXiv admin note: substantial text overlap with arXiv:1207.589

Unimodularity of Invariant Random Subgroups

An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups.Comment: 23 pages, one figur