[1] Potential buyers submit their bids and potential sellers submit their ask prices to the market institution, and then the market institution 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.
[2] For the mathematical modelling of satisfaction level Euclidean distance is used, where the offer and demand are treated as vectors.
From an economist's perspective, the interesting problem is to find a competitive equilibrium - a situation in which the supply equals the demand.
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.
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.
Consider the bilateral trade scenario, in which the buyer submits a bid of b and the seller submits s. Suppose an auctioneer sets the price in the following way: The utility of the buyer is: The utility of the seller is: In a complete information case when the valuations are common knowledge to both parties, it can be shown that the continuum of pure strategy efficient Nash equilibriums exists with
Then it can be shown that such a game has a Bayesian Nash equilibrium with linear strategies.
It is also brings higher expected gains for the players than any other Bayesian Nash equilibrium (see Myerson–Satterthwaite theorem).
Each buyer pays the lowest equilibrium price max(sk,bk+1), and each seller receives the highest equilibrium price min(bk,sk+1), as in the following table: Similar to the bilateral trade scenario, the mechanism is IR, IC and EE (optimizes the social welfare), but it is not BB - the auctioneer subsidizes the trade.
The uniqueness-of-prices theorem[3] implies that this subsidy problem is inevitable - any truthful mechanism that optimizes the social welfare will have the same prices (up to a function independent of the ask/bid prices of each trader).
If we want to keep the mechanism truthful while not having to subsidize the trade, we must compromise on efficiency and implement a less-than-optimal social welfare function.
Babaioff, Nisan and Pavlov[5] generalised the trade reduction mechanism to a market that is spatially-distributed, i.e. the buyers and sellers are in several different locations, and some units of the good may have to be transported between these locations.
The parameter p controls the tradeoff between EE and BB: In a variant of this mechanism,[6] after the bids are submitted, the k-1 cheap sellers trade with the k-1 expensive buyers; each of them receives/pays the expected payment of the original mechanism, i.e. each buyer pays
The downside is that now the mechanism is truthful only ex-ante; i.e., a risk-neutral trader cannot gain in expectation by misreporting his value, but after he knows the results of the lot, he might feel regret for not reporting otherwise.
Segal-Halevi, Hassidim and Aumann[7] present a trade-reduction mechanism that is SBB, in addition to being IR and IC and attains (1-1/k) of the optimal GFT.
Dütting, Roughgarden and Talgam-Cohen[8] proposed a modular approach to the design of double auctions.
An immediate consequence of this framework is that classic double auction mechanisms such as the trade reduction mechanism are not only strategyproof but also weakly group-strategyproof (meaning that no group of buyers and sellers can benefit by a joint misreport of their preferences).
The basic double auction model involves two categories of traders: buyers and sellers.
Babaioff and Walsh[9] extended it to handle markets in an arbitrary directed acyclic graph.
Gilor, Gonen and Segal-Halevi[10] study a multilateral market, with a set G of agent categories.
Each trade in the market involves rg agents of category g, for each g in G. The standard double auction market is a special case in which there are two categories (buyers and sellers), and the recipe is r=(1,1).
They present algorithms that are SBB, IC, IR and attain (1-1/k) of the optimal GFT.
They present randomized ascending-price mechanisms that are universally IR, obviously-IC, SBB, and attain an asymptotically-optimal GFT.