Hi, I have a question on mechanism design. It reads as follows

A buyer and a seller have to decide whether or not to trade and at what

price to trade if they do trade. Each agent knows only his/her own value but

it is commonly known that the buyer and sellers values are independently

drawn from the uniform distribution on [0,1]. Consider the mechanism of double auction in which the seller submits a sealed bid or asking price ps and simultaneously the buyer submits a sealed bid or offer price pb. If pb ps, they trade at a price p=kpb+(1-k)ps, Where k is (0,1).

My question is how to determine the probability of trade, and how to find the value of k which maximizes this.

I've done the work to get the BNE for this game, in linear strategies, such that our linear BNE is given by

pb(vb)=(2/3)vb+(1/12)

pb(vb)=(2/3)vs+(1/4)

I know that (2/3)vb+1(/12)>=(2/3)vs+(1/4) in order for trade to occur.

I'm just unsure of how to determine the k part, and the probability of trade. Any help is greatly appreciated!