Here is a problem at has been in the back of my mind for a long time.  What is the second best dominant-strategy mechanism (DSIC) in a market setting?

For some background, start with the bilateral trade problem of Myerson-Satterthwaite.  We know that among all DSIC, budget-balanced mechanisms the most efficient is a fixed-price mechanism.  That is, a price is fixed ex ante and the buyer and seller simply announce whether they are willing to trade at that price.  Trade occurs if and only if both are willing and if so the buyer pays the fixed price to the seller. This is Hagerty and Rogerson.

Now suppose there are two buyers and two sellers.  How would a fixed-price mechanism work?   We fix a price p.   Buyers announce their values and sellers announce their costs.  We first see if there are any trades that can be made at the fixed price p.  If both buyers have values above p and both sellers have values below then both units trade at price p.  If two buyers have values above p and only one seller has value below p then one unit will be sold: the buyers will compete in a second-price auction and the seller will receive p (there will be a budget surplus here.) Similarly if the sellers are on the long side they will compete to sell with the buyer paying p and again a surplus.

A fixed-price mechanism is no longer optimal.  The reason is that we can now use competition among buyers and sellers and “price discovery.”  A simple mechanism (but not the optimal one) is a double auction.  The buyers play a second-price auction between themselves, the sellers play a second-price reverse auction between themselves. The winner of the two auctions have won the right to trade. They will trade if and only if the second highest buyer value (which is what the winning buyer will pay) exceeds the second-lowest seller value (which is what the winning seller will receive.)  This ensures that there will be no deficit.  There might be a surplus, which would have to be burned.

This mechanism is DSIC and never runs a deficit.  It is not optimal however because it only sells one unit.  But it has the viture of allowing the “price” to adjust based on “supply and demand.”  Still, there is no welfare ranking between this mechanism and a fixed-price mechanism because a fixed price mechanism will sometimes trade two units (if the price was chosen fortuitously) and sometimes trade no units (if the price turned out too high or low) even though the price discovery mechanism would have traded one.

But here is a mechanism that dominates both.  It’s a hybrid of the two. We fix a price p and we interleave the rules of the fixed-price mechanism and the double auction in the following order

  1. First check if we can clear two trades at price p.  If so, do it and we are done.
  2. If not, then check if we can sell one unit by the double auction rules.  If so, do it and we are done.
  3. Finally, if no trades were executed using the previous two steps then return to the fixed-price and see if we can execute a single trade using it.

I believe this mechanism is DSIC (exercise for the reader, the order of execution is crucial!).  It never runs a deficit and it generates more trade than either standalone mechanism: fixed-price or double auction.

Very interesting research question:  is this a second-best mechanism?  If not, what is?  If so, how do you generalize it to markets with an arbitrary number of buyers and sellers?

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