Harbingers of Artin's Reciprocity Law. III. Gauss's Lemma and Artin's Transfer

We briefly review Artin's reciprocity law in the classical ideal theoretic language, and then study connections between Artin's reciprocity law and the proofs of the quadratic reciprocity law using Gauss's Lemma

Elementary proofs of Berndt's reciprocity laws

Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz

Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials

In the first article of this series we have presented the history of auxiliary primes from Legendre's proof of the quadratic reciprocity law up to Artin's reciprocity law. We have also seen that the proof of Artin's reciprocity law consists of several steps, the first of which is the verification of the reciprocity law for cyclotomic extensions. In this article we will show that this step can be identified with one of Dedekind's proofs of the irreducibility of the cyclotomic polynomial

Measuring Indirect Reciprocity: Whose Back Do We Scratch?

This paper presents an experimental investigation of strong indirect reciprocity. We examine both generalized indirect reciprocity (if A helps B then B helps C) and social indirect reciprocity (if A helps B then C helps A), in a setting where reciprocal behavior cannot be explained by strategic motivations. We also consider a treatment for direct reciprocity, as a benchmark, and use a variant of the strategy method to control for di®erences in ¯rst movers' actions across treatments. We ¯nd evidence of strong reciprocity within each treatment, both for strategies and decisions. Generalized indirect reciprocity is found to be signi¯cantly stronger than social indirect reciprocity and, interestingly, direct reciprocity. This ¯nd- ing is interpreted as re°ecting the relevance of ¯rst movers' motivation for second movers' reciprocal behavior.Reciprocity, Experimental Economics.

Reciprocity in Social Networks with Capacity Constraints

Directed links -- representing asymmetric social ties or interactions (e.g., "follower-followee") -- arise naturally in many social networks and other complex networks, giving rise to directed graphs (or digraphs) as basic topological models for these networks. Reciprocity, defined for a digraph as the percentage of edges with a reciprocal edge, is a key metric that has been used in the literature to compare different directed networks and provide "hints" about their structural properties: for example, are reciprocal edges generated randomly by chance or are there other processes driving their generation? In this paper we study the problem of maximizing achievable reciprocity for an ensemble of digraphs with the same prescribed in- and out-degree sequences. We show that the maximum reciprocity hinges crucially on the in- and out-degree sequences, which may be intuitively interpreted as constraints on some "social capacities" of nodes and impose fundamental limits on achievable reciprocity. We show that it is NP-complete to decide the achievability of a simple upper bound on maximum reciprocity, and provide conditions for achieving it. We demonstrate that many real networks exhibit reciprocities surprisingly close to the upper bound, which implies that users in these social networks are in a sense more "social" than suggested by the empirical reciprocity alone in that they are more willing to reciprocate, subject to their "social capacity" constraints. We find some surprising linear relationships between empirical reciprocity and the bound. We also show that a particular type of small network motifs that we call 3-paths are the major source of loss in reciprocity for real networks

Empathy, Asymmetrical Reciprocity, and the Ethics of Mental Health Care

I discuss Young’s “asymmetrical reciprocity” and apply it to an ethics of mental health care. Due to its emphasis on engaging with others through respectful dialogue in an inclusive manner, asymmetrical reciprocity serves as an appropriate framework for guiding caregivers to interact with their patients and to understand them in a morally responsible and appropriate manner. In Section 1, I define empathy and explain its benefits in the context of mental health care. In Section 2, I discuss two potential problems surrounding empathy: the difficulty of perspectivetaking and “compassion fatigue.” In Section 3, I argue that these issues can be resolved if examined through the lens of an ethics of care. Reciprocal relationships between patients and caregivers are an important element in the development of an ethics of care. In Section 4, I introduce two models of reciprocity that can be applied to a health care context: Benhabib’s symmetrical reciprocity and Young’s asymmetrical reciprocity. In Section 5, I demonstrate how asymmetrical reciprocity cultivates empathy and, in Section 6 and Section 7, I show how it overcomes the objections of empathy and improves therapeutic relationships

Social Reciprocity

We define social reciprocity as the act of demonstrating one's disapproval, at some personal cost, for the violation of widely-held norms (e.g., don't free ride). Social reciprocity differs from standard notions of reciprocity because social reciprocators intervene whenever a norm is violated and do not condition intervention on potential future payoffs, revenge, or altruism. Instead, we posit that social reciprocity is a triggered normative reponse. Our experiment confirms the existence of social reciprocity and demonstrates that more socially efficient outcomes arise when reciprocity can be expressed socially. Too provide theoretical foundations for social reciprocity, we show that generalized punishment norms survive in one of the two stable equilibria of an evolutionary game with selection drift.reciprocity, norm, experiment, public good, learning, evolution