Vickrey (2nd-price) auction: Winner is highest bidder; pays second-highest bid. Dominant strategy: bid truthfully (b_i = v_i) regardless of others' bids. Proof: bidding less than v_i can only cause you to lose when you should win; bidding more can only cause you to win when you'd lose, paying too much. Revenue Equivalence Theorem (Myerson 1981): Any auction where (1) winner has highest value, (2) loser gets 0 payoff, yields the same expected revenue. First-price sealed bid, Vickrey, ascending all give same expected revenue with risk-neutral bidders and IID values. Myerson Lemma: A mechanism is DSIC (dominant-strategy IC) iff its allocation rule is monotone and payments = critical threshold prices. The optimal mechanism: ironing virtual values φᵢ(vᵢ) = vᵢ − (1-F(vᵢ))/f(vᵢ).