  A double auction game with an infinite number of buyers and sellers is
introduced. All sellers posses one unit of a good, all buyers desire to buy one
unit. Each seller and each buyer has a private valuation of the good. The
distribution of the valuations define supply and demand functions. One unit of
the good is auctioned. At successive, discrete time instances, a player is
randomly selected to make a bid (buyer) or an ask (seller). When the maximum of
the bids becomes larger than the minimum of the asks, a transaction occurs and
the auction is closed. The players have to choose the value of their bid or ask
before the auction starts and use this value when they are selected. Assuming
that the supply and demand functions are known, expected profits as functions
of the strategies are derived, as well as expected transaction prices. It is
shown that for linear supply and demand functions, there exists at most one
Bayesian Nash equilibrium. Competitive behaviour is not an equilibrium of the
game. For linear supply and demand functions, the sum of the expected profit of
the sellers and the buyers is the same for the Bayesian Nash equilibrium and
the market where players behave competitively. Connections are made with the
ZI-C traders model and the $k$-double auction.
