# 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. 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.

## 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 $\in [0,1]$.Submitted bid of a seller is $b_1$, and bid of a buyer is $b_2$. $b_1,b_2 \in [0,1].$ Let $v>c$.

Suppose an auctioneer sets the price:

$p= \frac{(b_1+b_2)}{2}$ if $b_1$$b_2$. And if $b_1 > b_2$ trade does not occur.

Consumer surplus of buyer is $u_1=v-p$ if $b_1$$b_2$ and 0 if $b_1 > b_2$

Producer surplus of a seller is ${u_2}=p-c$ if $b_1$$b_2$ and 0 if $b_1 > b_2$

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 equilibriums exists with $b_1=b_2=p \in [c,v].$

In an incomplete information (asymmetric information) case a buyer and a seller know only their own valuations. Suppose that these valuations are uniformly distributed over the same interval. Then it can be shown that such a game has a Bayesian Nash equilibrium 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[1]