Double auction
A double auction is a process of buying and selling goods when potential buyers submit their bids and potential sellers simultaneously submit their ask prices to an auctioneer, and then an auctioneer chooses some price p that clears the market: all the sellers who asked less than p sell and all buyers who bid more than p buy at this price p. Buyers and sellers that bid or ask for exactly p are also included.
As well as their direct interest, double auctions are reminiscent of Walrasian auction and have been used as a tool to study the determination of prices in ordinary markets.
A simple example of a double auction is a bilateral trade scenario, in which there is a single seller who values his product as S (e.g. the cost of producing the product), and a single buyer who values that product as B.
From an economist's perspective, the interesting problem is to find a competitive equilibrium - a situation in which the supply equals the demand.
In the simple bilateral trade scenario, if B≥S then any price in the range [S,B] is an equilibrium price, since both the supply and the demand equal 1. Any price below S is not an equilibrium price since there is an excess demand, and any price above B is not an equilibrium price since there is an excess supply. When B<S, any price in the range (B,S) is an equilibrium price, since both the supply and the demand equal 0 (the price is too high for the buyer and too low for the seller).
In a more general double auction, in which there are many sellers each of whom holds a single unit and many buyers each of whom wants a single unit, an equilibrium price can be found using the natural ordering of the buyers and sellers:
Every price in the range [max(sk,bk+1),min(bk,sk+1)] is an equilibrium price, since both demand and supply are k. It is easier to see this by considering the range of equilibrium prices in each of the 4 possible cases (note that by definition of k, bk+1 < sk+1):
A double auction can be analyzed as a game. Players are buyers and sellers. Their strategies are bids for buyers and ask prices for sellers (that depend on the valuations of buyers and sellers). Payoffs depend on the price of the transaction (determined by the auctioneer) and the valuation of a player. The interesting problem is to find a Nash equilibrium - a situation in which no trader has an incentive to unilaterally change their bid/ask price.
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