Secure Implementation in Production Economies

One thing that has been assumed for a long time is that whenever there is dominant strategy equilibrium in the game form of any mechanism and the outcome corresponding to that strategy profile is socially optimal, people will play that particular equilibrium strategy pro?le. The theory has been silent on why they will play that particular strategy pro?le when there are other (Nash) equilibria. The Nash/Bayes' Nash implementation being a possible solution to this problem suffers from the drawback of either the requirement of the designer knowing the (common) prior (in case of Bayes' Nash implementation) or the requirement of the players predicting the actions of other players and collaborate without pre-talk (in case of Nash implementation with absence of dominant strategy or unique Nash). Secure implementation [Saijo et al. (2007)] is a relatively new concept in the theory of mechanism design and implementation. This requires double implementation in Dominant Strategy Equilibrium and Nash Equi- librium by the same Mechanism. This concept has worked well in some particular environments and has been tested on data [Cason et al. (2006)]. Unsurprisingly, being stronger than both the two above said concepts of implementation, there are many impossibility results in specific environ- ments with richer domains. We look for secure implementability in pro- duction economies with divisible goods. We find that a very broad gener- alization of "Serial" Social Choice Function (SCF) [Moulin and Shenker (92)] as defined in [Shenker (92)] is securely implementable. We call such functions as Generalized Serial SCF (GSS). We also find that under cer- tain conditions the Fixed Path SCFs are special cases of GSS and thus they are also securely Implementable. We conjecture that these are the only securely impleme