A buyer has value vb for a potential acquisition and believes the seller’s reservation price has the cumulative probability distribution F(v). The buyer chooses P to maximize its expected profit:
πb = (vb - P)Pr(P accepted) = (vb - P)F(P).
Find the buyer’s marginal profit and set it equal to zero. Show that the buyer’s optimal price satisfies P = vb - F(P)/f(p), where f(v) = dF(v)/dv is the associated density function. The buyer shades down its value in making its optimal bid.