The (n,k) game with heterogeneous agents

The \((n,k)\) game models a group of \(n\) individuals with binary opinions, say 1 and 0, where a decision is made if at least \(k\) individuals hold opinion 1. This paper explores the dynamics of the game with heterogeneous agents under both synchronous and asynchronous settings. We consider various agent types, including consentors, who always hold opinion 1, rejectors, who consistently hold opinion 0, random followers, who imitate one of their social neighbors at random, and majority followers, who adopt the majority opinion among their social neighbors. We investigate the likelihood of a decision being made in finite time. In circumstances where a decision cannot almost surely be made in finite time, we derive a nontrivial bound to offer insight into the probability of a decision being made in finite time.