The (maximum/minimum) price at which they are willing to buy/sell.
The quantity they are willing to buy/sell (optional, in multi-unit double auctions).
The double auction is a mechanism designed to find a price at which the quantity demanded equals the quantity offered and is, therefore, closely related to the standard economic analysis of equilibrium in competitive markets (where supply equals demand). For example, treasury auctions in the US are an example of a double auction, with the specific feature that there is just one seller on the supply side of the market.
In this exercise, you are asked to characterise the Nash equilibria of the simplest possible version of the double auction with just one good, one seller and one buyer. This special case of a double auction is also often referred to as bilateral trade or bilateral bargaining. The details of this strategic interaction are as follows: the buyer values the good for sale at vB, while the seller values it at vS. Suppose that vS
If s>b then no trade occurs as the seller asks for more money than the buyer is willing to pay.
If s≤b then trade occurs at price p=(b+s)/2.
The buyer’s net payoff is 0 if no trade occurs, and vB−p when trade occurs. Likewise, the seller’s net payoff is 0 if no trade occurs, and p−vS when trade occurs.
Your task is to:
Set up the normal-form game that captures the strategic interaction described above.
Derive separately for each player (i.e., the buyer and the seller) the way their payoff varies with their bid in order to identify their payoff-maximal bids.
In a b-s-diagram, depict graphically both players’ best responses (similar to what we’ve done in this week’s lecture when characterising all Nash equilibria of the second price auction) and then highlight and interpret your results. Are any of the Nash equilibria you found inefficient? Explain!