  The original Hegselmann-Krause (HK) model consists of a set of~$n$ agents
that are characterized by their opinion, a number in~$[0, 1]$.
  Each agent, say agent~$i$, updates its opinion~$x_i$ by taking the average
opinion of all its neighbors, the agents whose opinion differs from~$x_i$ by at
most~$\epsilon$.
  There are two types of~HK models: the synchronous~HK model and the
asynchronous~HK model.
  For the synchronous model, all the agents update their opinion simultaneously
at each time step, whereas for the asynchronous~HK model, only one agent chosen
uniformly at random updates its opinion at each time step.
  This paper is concerned with a variant of the~HK opinion dynamics, called the
mixed~HK model, where each agent can choose its degree of stubbornness and mix
its opinion with the average opinion of its neighbors at each update.
  The degree of the stubbornness of agents can be different and/or vary over
time.
  An agent is not stubborn or absolutely open-minded if its new opinion at each
update is the average opinion of its neighbors, and absolutely stubborn if its
opinion does not change at the time of the update.
  The particular case where, at each time step, all the agents are absolutely
open-minded is the synchronous~HK model.
  In contrast, the asynchronous model corresponds to the particular case where,
at each time step, all the agents are absolutely stubborn except for one agent
chosen uniformly at random who is absolutely open-minded.
  We first show that some of the common properties of the synchronous~HK model,
such as finite-time convergence, do not hold for the mixed model.
  We then investigate conditions under which the asymptotic stability holds, or
a consensus can be achieved for the mixed model.
