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.
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.
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