Auctions PDF

Games of Incomplete Information: Auctions Gibbons Chs.3&4; B&F, Ch.14; P&R, Ch.13.8 Incomplete Information: At least one player does not know at least one other’s payoff for certain Don’t know rival’s true type, e.g. cost, ability, health, risk, WTP, etc. player may try to ‘signal’...

Games of Incomplete Information: Auctions Gibbons Chs.3&4; B&F, Ch.14; P&R, Ch.13.8 Incomplete Information: At least one player does not know at least one other’s payoff for certain Don’t know rival’s true type, e.g. cost, ability, health, risk, WTP, etc. player may try to ‘signal’ their true ‘type’ to others, or possibly hide it (Rational) player maximises expected payoff, based on own type and beliefs about others’ types → refined equilibrium solution concept: Bayesian Nash Equilibrium No player deviates from their strategy, given own type and beliefs about the types of all other players Auctions Many ‘items of value’ are traded via auctions Mobile phone licences TV rights Works of art Public infrastructure projects and contracts Online selling What type of auction can be used? Is auction design important to maximise revenue? Are valuations (max WTP) private or common across buyers? (i) English auction: oral, ascending-bid, first price Begin at zero or a reserve (minimum) price ends when no one willing to exceed current highest bid all bids are common knowledge (perfect info) Winner pays their bid Used for antiques, paintings, property, horses, etc. (ii) Dutch auction: oral, descending-bid, first price Begin at price above expected winning bid If no bids, reduce price until 1st bid made Winner pays their bid Used in Dutch flower market, Sydney fish market (iii) first-price, sealed-bid Players ‘simultaneously’ submit sealed bids highest bid wins, winner pays their bid Used for TV rights, licences, tenders, etc. (iv) second-price, sealed-bid as above, except winner pays 2nd highest bid Common trade-off in all auction formats: Higher bid increases probability of winning, but reduces payoff (surplus) if win, and vice versa Valuation = max WTP for ‘object’ (a) private: each bidder only knows own valuation for certain, valuations can differ across bidders Players may form beliefs of other players’ valuations (may learn through behaviour in English auction) usually applicable to items of personal ‘taste’ (b) common: item has (approx) same expected value for all bidders, but estimate of this differs across bidders, e.g. jar of coins can apply to items that can be profitably exploited e.g. TV rights, contracts for transport, construction, exploration, etc. Valuations, bids and payoffs: (a/s two bidders, i and j) valuations Vi & Vj (max WTP), bids bi & bj, payoffs Ui & Uj    Vi  bi  English & Dutch: if bi   b j , then U i   (i ≠ j)    0  1st price sealed-bid: a/s if equal bids, winner decided by coin toss (i ≠ j) In all of above, any bi < Vi weakly dominates any bi ≥ Vi, for any bj 2nd price sealed-bid: a/s if equal bids, winner decided by coin toss (i ≠ j) bi ≤ Vi weakly dominates bi > Vi, given any bj Benchmark: Auction outcomes with complete information: a/s 2 bidders (Tom & Anne), private values are common knowledge a/s VT = 21, VA = 14, denote bids as bT and bA What are the Nash equilibrium bids in each type of auction? a/s bids must initially change in increments (e) of 3 (i) English auction: (VT = 21, VA = 14, e = 3) Both know Anne will not bid more than 12 Tom bids 15 and wins  UT = 21 – 15 = 6 , UA = 0 the lower the bid increments (e → 0), bA → 14-e and bT → 14 Given common knowledge, Anne bids (approx.) her valuation but Tom bids less than his valuation Nash equilibrium: highest valuation wins, pays 2 nd highest valuation (ii) Dutch auction: (VT = 21, VA = 14, e = 3) a/s initial price is 40, reduces by 3 Tom knows Anne will bid when price reaches 13 Tom bids when price reaches 16 and wins  UT = 21-16 = 5, UA = 0 as increments get smaller (e 0), Anne will bid when price reaches 14-e, so Tom bids 14 and wins Nash equilibrium: Highest valuation wins, pays 2 nd highest valuation (iii) 1st price sealed-bid: (VT = 21, VA = 14, e = 3) expected payoffs: Tom bids 9 12 15 18 9 2.5, 6 0, 9 0, 6 0, 3 Anne 12 2, 0 1, 4.5 0, 6 0, 3 Unique Nash equil bids are (bA*, bT*) = (12, 15) Tom’s iterated weakly dominant strategy is to bid 15 and win Again, as increments e → 0, Tom’s bid → 14 Nash equilibrium: highest valuation wins, pays 2 nd highest valuation (iv) 2nd price sealed-bid (VT = 21, VA = 14, e = 3) expected payoffs: Tom bids 9 12 15 18 9 2.5, 6 0, 12 0, 12 0, 12 Anne 12 5, 0 1, 4.5 0, 9 0, 9 bidding 12 is weakly dominant for Anne Two Nash equil, including where both bid (approx) true valuations As bid increments → 0, Tom wins by bidding any amount > 14 and pays Anne’s bid of 14-e Nash equil: highest valuation wins, pays 2nd highest valuation Given common knowledge of valuations (complete info), outcome is identical in each type of auction format Winner has highest valuation, pays 2nd highest valuation Seller gets same value for ‘object’, irrespective of auction format ‘Revenue equivalence theorem’ In real life, we see different auction formats, due to incomplete info a