Non-Bayesian Mechanism Design

Suppose we have a mechanism where a finite number of agents possess private information that is not drawn from a probability distribution.

The agents' types are given and fixed but agents only know their own type. As there is no probability distribution, players do not form any beliefs.

The agents' behavior is modeled as utility-maximizers and the utility functions are quasilinear in nature.

Does it make sense, from the above, to use as a solution concept the non-Bayesian Nash equilibrium (Nash, 1950)? If yes, could you please point me out to the right literature (some paper references)?

Thank you in advance.