Group learning and opinion diffusion in a broadcast network

We analyze the following group learning problem in the context of opinion diffusion: Consider a network with M users, each facing N options. In a discrete time setting, at each time step, each user chooses K out of the N options, and receive randomly generated rewards, whose statistics depend on the options chosen as well as the user itself, and are unknown to the users. Each user aims to maximize their expected total rewards over a certain time horizon through an online learning process, i.e., a sequence of exploration (sampling the return of each option) and exploitation (selecting empirically good options) steps. Different from a typical regret learning problem setting (also known as the class of multi-armed bandit problems), the group of users share information regarding their decisions and experiences in a broadcast network. The challenge is that while it may be helpful to observe others´ actions in one´s own learning (i.e., second-hand learning), what is considered desirable option for one user may be undesirable for another (think of restaurant choices), and this difference in preference is in general unknown a priori. Even when two users happen to have the same preference (e.g., they agree one option is better than the other), they may differ in their absolute valuation of each individual option. Within this context we consider two group learning scenarios, (1) users with uniform preferences and (2) users with diverse preferences, and examine how a user should construct its learning process to best extract information from others´ decisions and experiences so as to maximize its own reward. Performance is measured in weak regret, the difference between the user´s total reward and the reward from a user-specific best single-action policy (i.e., always selecting the set of options generating the highest mean rewards for this user). Within each scenario we also consider two cases: (i) when users exchange full information, meaning they share the actual rewar- s they obtained from their choices, and (ii) when users exchange limited information, e.g., only their choices but not rewards obtained from these choices. We show the gains from group learning compared to individual learning from one´s own choices and experiences.