Nash implementation and tie-breaking rules

I study Nash implementation when agents might use a 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 (SCC) depending on the particular rule used by the agents. The effect might be considerable. For instance, there exists a tie-breaking rule under which any SCC is implementable in the presence of three or more 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. A characterization is provided and it is shown that this severely limits the set of implementable SCCs. JEL classification: C72; D01; D70; D78