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.

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 b₁, and bid of a buyer is b₂. $b_1,b_2 \in [0,1].$ Let v > c.

Suppose an auctioneer sets the price:

$p= \frac{(b_1+b_2)}{2}$ if b₁ ≤ b₂. And if b₁ > b₂ trade does not occur.

Consumer surplus of buyer is u₁ = v − p if b₁ ≤ b₂ and 0 if b₁ > b₂

Producer surplus of a seller is u₂ = p − c if b₁ ≤ b₂ and 0 if b₁ > b₂

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 uniformally 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]

See also

• Other topics:
  • Game theory
  • Auction Theory
  • Sealed first-price auction
  • Vickrey auction

References

Fudenberg, Drew; Tirole, Jean (1991), Game theory, MIT Press, ISBN 978-0-262-06141-4

Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press, ISBN 978-0-691-00395-5

Footnotes

1. ^ Myerson, Roger B.; Mark A. Satterthwaite (1983). "Efficient Mechanisms for Bilateral Trading". Journal of Economic Theory 29: 265–281.