A double auction

Auction

An auction is a process of buying and selling goods or services by offering them up for bid, taking bids, and then selling the item to the highest bidder...

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.

Game theory approach to modelling double auctions

A double auction can be analyzed as a game. Players are buyers and sellers. They have some valuations of a good that is traded in an auction. 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 and the valuation of a player.

Equilibrium strategies of simple double auction

Consider a double auction with a single buyer and a single seller. Suppose that the valuation of a buyer is v and the valuation of a seller is c (e.g. the cost of producing the product). And v, c .Submitted bid of a seller is , and bid of a buyer is . Let .

Suppose an auctioneer sets the price:

if ≤ . And if trade does not occur.

Consumer surplus of buyer is
if ≤ and 0 if

Producer surplus of a seller is
if ≤ and 0 if

In a complete information (symmetric information) case when the valuations are common knowledge it can be shown that the continuum of pure strategy efficient Nash equilibrium

Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

s exists with

In an incomplete information (asymmetric information) case a buyer and a seller know only their own valuations. Suppose that these valuations are uniformally distributed over the same interval. Then it can be shown that such a game has a Bayesian Nash equilibrium

Bayesian game

In game theory, a Bayesian game is one in which information about characteristics of the other players is incomplete. Following John C. Harsanyi's framework, a Bayesian game can be modelled by introducing Nature as a player in a game...

 with linear strategies. That is there is an equilibrium when both players' bids are some linear functions of their valuations. It is also the equilibrium that brings the highest expected gains for the players than any other Bayesian Nash equilibrium

See also

• Other topics:
  • Game theory
    Game theory
    Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
    
    
  • Auction Theory
    Auction theory
    Auction theory is an applied branch of economics which deals with how people act in auction markets and researches the properties of auction markets. There are many possible designs for an auction and typical issues studied by auction theorists include the efficiency of a given auction design,...
    
    
  • Sealed first-price auction
    Sealed first-price auction
    A first-price sealed-bid auction is a form of auction where bidders submit one bid in a concealed fashion. The submitted bids are then compared and the person with the highest bid wins the award, and pays the amount of his bid to the seller...
    
    
  • Vickrey auction
    Vickrey auction
    A Vickrey auction is a type of sealed-bid auction, where bidders submit written bids without knowing the bid of the other people in the auction, and in which the highest bidder wins, but the price paid is the second-highest bid. The auction was created by William Vickrey...