Nash implementation and tie-breaking rules

I analyze Nash implementation when agents might use an exogenous tie-breaking rule to choose among the messages they are materially indifferent between. If the planner is endowed with the knowledge of the rule, this might expand or shrink the set of implementable social choice correspondences depending on the particular rule used by the agents. If the planner is not endowed with the knowledge of the rule, then the problem of implementation is almost equivalent to double implementation in Nash and strict Nash equilibrium.