Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.
We study a sequential-learning model featuring a network of naive agents with Gaussian information structures. Agents apply a heuristic rule to aggregate predecessors' actions. They weigh these actions according the strengths of their social connections to different predecessors. We show this rule arises endogenously when agents wrongly believe others act solely on private information and thus neglect redundancies among observations. We provide a simple linear formula expressing agents' actions in terms of network paths and use this formula to characterize the set of networks where naive agents eventually learn correctly. This characterization implies that, on all networks where later agents observe more than one neighbor, there exist disproportionately influential early agents who can cause herding on incorrect actions. Going beyond existing social-learning results, we compute the probability of such mislearning exactly. This allows us to compare likelihoods of incorrect herding, and hence expected welfare losses, across network structures. The probability of mislearning increases when link densities are higher and when networks are more integrated. In partially segregated networks, divergent early signals can lead to persistent disagreement between groups. , and three anonymous referees for useful comments.
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers α so that either (α, α 2 ) or (α, α − α 2 ) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u ′ ) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u ′ ) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.
Agents learn about a state using private signals and the past actions of their neighbors. In contrast to most models of social learning in a network, the target being learned about is moving around. We ask: when can a group aggregate information quickly, keeping up with the changing state? First, if each agent has access to neighbors with sufficiently diverse kinds of signals, then Bayesian learning achieves good information aggregation. Second, without such diversity, there are cases in which Bayesian information aggregation necessarily falls far short of efficient benchmarks. Third, good aggregation requires agents who understand correlations in neighbors' actions with the sophistication needed to concentrate on recent developments and filter out older, outdated information. In stationary equilibrium, agents' learning rules incorporate past information by taking linear combinations of other agents' past estimates (as in the simple DeGroot heuristic), and we characterize the coefficients in these linear combinations.
We provide a framework for determining the centralities of agents in a broad family of random networks. Current understanding of network centrality is largely restricted to deterministic settings, but practitioners frequently use random network models to accommodate data limitations or prove asymptotic results. Our main theorems show that on large random networks, centrality measures are close to their expected values with high probability. We illustrate the economic consequences of these results by presenting three applications: (1)In network formation models based on community structure (called stochastic block models), we show network segregation and differences in community size produce inequality.Benefits from peer effects tend to accrue disproportionately to bigger and better-connected communities.(2) When link probabilities depend on spatial structure, we can compute and compare the centralities of agents in different locations. (3) In models where connections depend on several independent characteristics, we give a formula that determines centralities 'characteristic-by-characteristic'. The basic techniques from these applications, which use the main theorems to reduce questions about random networks to deterministic calculations, extend to many network games.In many settings of economic interest, agents benefit from connections to others. These peer effects depend on network structures, and better positioned agents can benefit much more than their less central counterparts. In education, for example, students form networks of friends and these connections affect academic achievement through group study, as a source of motivation, etc. Empirical evidence suggests that the impact of these peer effects on outcomes is approximated well by measures of network centrality such as Katz-Bonacich centrality (Calvó-Armengol, Patacchini, and Zenou (2009), Hahn, Islam, Patacchini, and Zenou (2015)). More generally, measures of centrality and related quantities are crucial to understanding economic models from peer effects and quadratic games on networks to social learning models such as DeGroot updating. 1 While there is a large literature on Katz-Bonacich and other centrality measures in deterministic settings, relatively little is known about centrality measures on stochastic networks.But in many applied settings, precise data about the full network is not available. Researchers instead use statistical models of network formation where links form with probabilities depending on agent characteristics. As a very simple example, one could model the social network in a school with black and white students by assuming two students of the same race are friends with probability 50% while two students of different races are friends with probability 25% (where all connections form independently). Moreover in theoretical work, varying parameters in models of random network formation often provides more insight than comparing particular deterministic networks.The current paper gives a framework for determining how central eac...
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