Implementation with Extensive Form Games: One Round of Signaling Is Not Enough

In situations of complete information, when a social choice function can be implemented using an extensive form game it is always the case that there is an equilibrium which does not go beyond the first stage (that is, an equilibrium with "one round of signaling"). We show that this is not true in the incomplete information case. We provide an example in which a social choice function cannot be implemented with an extensive form game if the mechanism has an equilibrium with one round of signaling and we put some restrictions on the way in which out-of-equilibrium beliefs are formed. The social choice function can, however, be implemented using an extensive form mechanism, but the equilibrium has to reach the second stage. Journal of Economic Literature Classification numbers: C72, D70.