Ordinal status games on networks

We consider a modification of ordinal status games of Haagsma and von Mouche (2010). A number of agents make scalar choices, e.g., their levels of conspicuous consumption. The wellbeing of each agent is affected by her choice in three ways: internal satisfaction, expenses, and social status determined by comparisons with the choices of others. In contrast to the original model, as well as its modifications considered so far, we allow for some players not caring about comparisons with some others. Assuming that the status of each player may only be "high" or "low," the existence of a strong Nash equilibrium is shown; for a particular subclass of such games, the convergence of Cournot tatonnement is established. If an intermediate status is possible, then even Nash equilibrium may fail to exist in very simple examples