Understanding the formation of opinions on interconnected topics within social networks is of significant importance. It offers insights into collective behavior and decision-making, with applications in Graph Neural Networks. Existing models propose that individuals form opinions based on a weighted average of their peers' opinions and their own beliefs. This averaging process, viewed as a best-response game, can be seen as an individual minimizing disagreements with peers, defined by a quadratic penalty, leading to an equilibrium. Bindel, Kleinberg, and Oren (FOCS 2011) provided tight bounds on the "price of anarchy" defined as the maximum overall disagreement at equilibrium relative to a social optimum. Bhawalkar, Gollapudi, and Munagala (STOC 2013) generalized the penalty function to non-quadratic penalties and provided tight bounds on the price of anarchy. When considering multiple topics, an individual's opinions can be represented as a vector. Parsegov, Proskurnikov, Tempo, and Friedkin (2016) proposed a multidimensional model using the weighted averaging process, but with constant interdependencies between topics. However, the question of the price of anarchy for this model remained open. We address this by providing tight bounds on the multidimensional model, while also generalizing it to more complex interdependencies. Following the work of Bhawalkar, Gollapudi, and Munagala, we provide tight bounds on the price of anarchy under non-quadratic penalties. Surprisingly, these bounds match the scalar model. We further demonstrate that the bounds remain unchanged even when adding another layer of complexity, involving groups of individuals minimizing their overall internal and external disagreement penalty, a common occurrence in real-life scenarios.
We study resource allocation in two-sided markets from a fundamental perspective and introduce a general modeling and algorithmic framework to effectively incorporate the complex and multidimensional aspects of fairness. Our main technical contribution is to show the existence of a range of near-feasible resource allocations parameterized in different model primitives to give flexibility when balancing the different policymaking requirements, allowing policy designers to fix these values according to the specific application. To construct our near-feasible allocations, we start from a fractional resource allocation and perform an iterative rounding procedure to get an integer allocation. We show a simple yet flexible and strong sufficient condition for the target feasibility deviations to guarantee that the rounding procedure succeeds, exhibiting the underlying trade-offs between market capacities, agents' demand, and fairness. To showcase our framework's modeling and algorithmic capabilities, we consider three prominent market design problems: school allocation, stable matching with couples, and political apportionment. In each of them, we obtain strengthened guarantees on the existence of near-feasible allocations capturing the corresponding fairness notions, such as proportionality, envy-freeness, and stability.
