Understanding the formation and evolution of opinions is of broad interdisciplinary interest. Many classical models for opinion formation focus on the impact of different notions of locality , e.g., locality due to network effects among agents or the role of the proximity of opinions. In practice, however, opinion formation is often governed by the interplay of local and global influences. In this paper, we study these influences with a model for opinion formation of agents embedded in a social network. Each agent has a static intrinsic opinion as well as a public opinion that is updated asynchronously over time. Moreover, agents have access to a global aggregate (e.g., the outcome of a vote) of all public opinions. We focus on the popular median voting rule and show that pure Nash equilibria always exist. For every initial state of the dynamics, a pure equilibrium can be reached. The set of reachable equilibria forms a complete lattice, and extremal equilibria can be computed in polynomial time. We show that by uniformly increasing the influence of the global median we can enforce that the median opinion is the same in every reachable equilibrium. We can compute the increase scheme that achieves this property in polynomial time. In contrast, when we can increase the influence of the global median for a set of at most k agents, finding the set that leads to a unique median opinion in every reachable equilibrium is NP -complete.

Abstract We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5‐competitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 − o(1), and an algorithm with ratio at least 0.5459 for all n ≥ 1. We extend all algorithms and ratios to k ≥ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r′ log(n/r′)), where r′ is the minimum rank of the matroid and the dual matroid.

We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).