Auctions. Economics Auction Theory. Instructor: Songzi Du. Simon Fraser University. September 15, 2016

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1 Auctions Economics Auction Theory Instructor: Songzi Du Simon Fraser University September 15, 2016 ECON 383 (SFU) Auctions September 15, / 28

2 Auctions Mechanisms of transaction: bargaining, posted price, auctions Auction: take bids, allocate resource, and collect payments. Babylonian wife auction (500 BC) Auction of the Roman Empire by the Praetorian Guard (who had killed Emperor Pertinax in 193 AD). The winning bidder Didius Julianus was crowned Emperor; beheaded 9 weeks later (winner s curse). Google AdWords auction (revenue of USD$28 billion in 2010), ebay Financial auctions (treasury bills, settlement of credit default swap, stock exchange) ECON 383 (SFU) Auctions September 15, / 28

3 ECON 383 (SFU) Auctions September 15, / 28

4 Second price auction A single, indivisible good. Second price auction: 1 Every bidder submits a bid, simultaneously (sealed bid). 2 The highest bidder gets the object and pays the second highest bid; everyone else does not pay. Also known as Vickrey auction. Proxy bidding in ebay: a computer program that automatically and minimally increases your bid (up to your pre-specified maximum amount) to ensure that you are the top bidder. ECON 383 (SFU) Auctions September 15, / 28

5 Second price auction A single, indivisible good. Second price auction: 1 Every bidder submits a bid, simultaneously (sealed bid). 2 The highest bidder gets the object and pays the second highest bid; everyone else does not pay. Also known as Vickrey auction. Proxy bidding in ebay: a computer program that automatically and minimally increases your bid (up to your pre-specified maximum amount) to ensure that you are the top bidder. Bidder i has a value v i for the good (his private information), payoff of v i P i if he gets it, 0 if not. ECON 383 (SFU) Auctions September 15, / 28

6 Ascending bid auction Also known as English auction. The auction is carried out interactively in real time. The auctioneer gradually raises the price, starting from some reserve price (e.g., zero), bidders drop out until finally only one bidder remains, and that bidder wins the object at this final price. Variants of ascending bid auction: bidders shout out prices, or submit them electronically. ECON 383 (SFU) Auctions September 15, / 28

7 Ascending bid auction ECON 383 (SFU) Auctions September 15, / 28

8 Strategy in Second Price Auction Strategy: a function s i (v i ) that maps values to bids. ECON 383 (SFU) Auctions September 15, / 28

9 Strategy in Second Price Auction Strategy: a function s i (v i ) that maps values to bids. n bidders Payoff function: v i max(b 1,..., b i 1, b i+1,..., b n ) if b i > max(b 1,..., b i 1, b i+1,..., b n ) U i (v i, b 1, b 2,..., b n ) = 0 otherwise ECON 383 (SFU) Auctions September 15, / 28

10 Strategy in Second Price Auction Strategy: a function s i (v i ) that maps values to bids. n bidders Payoff function: v i max(b 1,..., b i 1, b i+1,..., b n ) if b i > max(b 1,..., b i 1, b i+1,..., b n ) U i (v i, b 1, b 2,..., b n ) = 0 otherwise Dominant strategy s i (v i ) satisfies: for every (s 1 ( ),..., s i 1 ( ), s i+1 ( ),..., s n ( )) and every (v 1, v 2,..., v n ), U i (v i, s 1 (v 1 ),..., s i 1 (v i 1 ), s i (v i ), s i+1 (v i+1 ),..., s n (v n )) U i (v i, s 1 (v 1 ),..., s i 1 (v i 1 ), b i, s i+1 (v i+1 ),..., s n (v n )) for every b i R. ECON 383 (SFU) Auctions September 15, / 28

11 Dominant Strategy in Second Price Auction ECON 383 (SFU) Auctions September 15, / 28

12 Why second price? Why not third price? Third price auction: the highest bidder gets the good and pays the third highest bid; everyone else do not pay. Is truthful bidding the dominant strategy? ECON 383 (SFU) Auctions September 15, / 28

13 Auction of two goods Auction of two indivisible, identical goods. Each bidder i wants only one good, has a value v i if he gets a good. As before, each bidder submits a bid. Third-price auction: the top two bidders each gets a good, and each pays the third highest bid; the rest do not pay. Is truthful bidding the dominant strategy? ECON 383 (SFU) Auctions September 15, / 28

14 Facts about uniform distribution Suppose n bidders, with values v i randomly and independently drawn from the uniform distribution on [0, 1]: for x s between 0 and 1. P(v i x) = x, P(v 1 x 1, v 2 x 2, v 3 x 3 ) = x 1 x 2 x 3, E[max(v 1, v 2,..., v n )] = n n + 1, E[max 2(v 1, v 2,..., v n )] = n 1 n + 1, E[max 3 (v 1, v 2,..., v n )] = n 2 n + 1,..., E[min(v 1, v 2,..., v n )] = 1 n + 1, where max 2 means second highest, max 3 third highest, etc. ECON 383 (SFU) Auctions September 15, / 28

15 Reserve price in second price auction Reserve price (r): the minimum bid that is considered in the (second price) auction, announced before the auction. 1 The good is sold to the highest bidder if the highest bid is equal or above r; otherwise, the good is not sold. 2 The winning bidder (if any) pays the maximum of the second-place bid and the reserve price. ECON 383 (SFU) Auctions September 15, / 28

16 Reserve price in second price auction Reserve price (r): the minimum bid that is considered in the (second price) auction, announced before the auction. 1 The good is sold to the highest bidder if the highest bid is equal or above r; otherwise, the good is not sold. 2 The winning bidder (if any) pays the maximum of the second-place bid and the reserve price. Why set reserve price? What is the role of reserve price in revenue? Suppose the seller has no value for the (single, indivisible) good that he is auctioning. There are n bidders, with values randomly and independently drawn from the uniform distribution on [0, 1]. What s the optimal reserve price when n = 1? n = 2? What happens when the seller uses a posted price? ECON 383 (SFU) Auctions September 15, / 28

17 Second price auction with reserve price r Let Rev(r) be the seller s revenue given a reserve price r [0, 1]. If there is only n = 1 bidder: Rev(r) = (1 r) r. Rev (r) = 1 2r Optimal reserve price r = 1/2 (from solving Rev (r) = 0). If there are n = 2 bidders: Rev(r) = 2r(1 r) r + (1 r) 2 Rev (r) = 2r(1 2r) ( r + 1 r ) 3 Optimal reserve price r = 1/2. ECON 383 (SFU) Auctions September 15, / 28

18 Second price auction with reserve price r An observation: the seller s revenue from the optimal reserve price and 1 bidder (1/4) is less than his revenue from zero reserve price and 2 bidders (1/3). This is a general theorem (Bulow and Klemperer). Setting the optimal reserve price is less profitable than simply attracting an additional bidder. ECON 383 (SFU) Auctions September 15, / 28

19 First price auction A single, indivisible good. First price auction: 1 Every bidder submits a bid, simultaneously (sealed bid). 2 The highest bidder gets the object and pays his own bid; everyone else does not pay. Bidder i has a value v i for the good (his private information), payoff of v i P i if he gets it, 0 if not. ECON 383 (SFU) Auctions September 15, / 28

20 Descending bid auction Also known as Dutch auction. The auction is carried out interactively in real time. The auctioneer gradually lowers the price from some high initial value until the first moment when some bidder accepts and pays the current price. Flowers have long been sold in the Netherlands using this procedure. ECON 383 (SFU) Auctions September 15, / 28

21 A Model of First Price Auction n bidders (n 2) Each bidder i (1 i n) has a private value v i for the good. 0 v i 1. The distribution of v i is the uniform distribution on [0, 1]. Identical and independent distribution for every bidder. Bidding strategy is a function s i (v i ) that maps values to bids. v i is bidder i s type. ECON 383 (SFU) Auctions September 15, / 28

22 Strategy in First Price Auction Payoff function: v i b i if b i > max(b 1,..., b i 1, b i+1,..., b n ) U i (v i, b 1, b 2,..., b n ) = 0 otherwise ECON 383 (SFU) Auctions September 15, / 28

23 Strategy in First Price Auction Payoff function: v i b i if b i > max(b 1,..., b i 1, b i+1,..., b n ) U i (v i, b 1, b 2,..., b n ) = 0 otherwise Bayesian Nash Equilibrium: strategy profile (s 1 (v 1 ), s 2 (v 2 ),..., s n (v n )) such that for every bidder i and every v i, E[U i (v i, s 1 (v 1 ),..., s i 1 (v i 1 ), s i (v i ), s i+1 (v i+1 ),..., s n (v n ))] E[U i (v i, s 1 (v 1 ),..., s i 1 (v i 1 ), b i, s i+1 (v i+1 ),..., s n (v n ))] for every b i R. ECON 383 (SFU) Auctions September 15, / 28

24 Solving for equilibrium (first price auction) We focus on symmetric equilibrium: s 1 = s 2 = = s n = s. What is bidder i s profit from bidding s(v i ), given that others also bid according to s? U i (v i ) = (v i ) n 1 (v i b(v i )) Bidder i of type v i maximizes (by bidding s(x)): max x n 1 (v i s(x)) 0 x 1 FOC: (n 1)x n 2 v i (n 1)x n 2 s(x) x n 1 s (x) = 0 x=vi ECON 383 (SFU) Auctions September 15, / 28

25 Solving for equilibrium (first price auction) FOC: (n 1)(v i ) n 2 v i (n 1)(v i ) n 2 s(v i ) (v i ) n 1 s (v i ) = 0 Rearrange: Guess: s(v i ) = A(v i ) k s(v i ) = v i v i n 1 s (v i ) A(v i ) k = v i v i n 1 Ak(v i) k 1. Clearly k = 1. Then A = 1 A n 1 n 1, i.e., A = n. Equilibrium bidding strategy: s(v i ) = n 1 n v i s(v i ) < v i. This is called bid shading. ECON 383 (SFU) Auctions September 15, / 28

26 All Pay Auction All pay auction: the highest bidder gets the good, everyone pays his/her bid. Everything else as before (a single good, simultaneous bids, private values, etc.) Example (bribery): in 2008, Governor Rod Blagojevich of Illinois tried to sell Barack Obama s senate seat to the highest bidder. Other examples: war of attrition, political campaign, Olympic game, etc. ECON 383 (SFU) Auctions September 15, / 28

27 Solving for equilibrium (all pay auction) We focus on symmetric equilibrium: s 1 = s 2 = = s n = s. What is bidder i s profit from bidding s(v i ), given that others also bid according to s? U i (v i ) = (v i ) n 1 v i s(v i ) Bidder i of type v i maximizes (by bidding s(x)): max x n 1 v i s(x) 0 x 1 FOC: (n 1)x n 2 v i s (x) = 0 x=vi ECON 383 (SFU) Auctions September 15, / 28

28 Solving for equilibrium (all pay auction) FOC: Guess: s(v i ) = A(v i ) k (n 1)(v i ) n 1 = s (v i ). (n 1)(v i ) n 1 = Ak(v i ) k 1. k = n and n 1 = Ak, i.e., A = n 1 n. Equilibrium bidding strategy: s(v i ) = n 1 n (v i) n ECON 383 (SFU) Auctions September 15, / 28

29 Average Price Auction Average Price Auction: the highest bidder gets the good, pays the average of all bids; everyone else does not pay. Everything else as before (a single good, simultaneous bids, private values, etc.) ECON 383 (SFU) Auctions September 15, / 28

30 Solving for equilibrium (average price auction, n = 2) Suppose n = 2. We focus on symmetric equilibrium: s 1 = s 2 = s. What is bidder i s profit from bidding s(v i ), given that the other also bids according to s? ( U i (v i ) = v i v i 1 ) 2 (s(v i) + E[s(v j ) v j v i ]), j i Bidder i of type v i maximizes (by bidding s(x)): max x 0 x 1 ( v i 1 2 (s(x) + E[s(v j) v j x]) ) ECON 383 (SFU) Auctions September 15, / 28

31 Solving for equilibrium (average price auction, n = 2) Guess: s(v i ) = Av i max 0 x 1 x ( v i 1 ( 2 Ax + Ax )) 2 FOC: v i 3 2 Ax x=vi = v i 3 2 Av i = 0 A = 2 3. Equilibrium bidding strategy: s(v i ) = 2 3 v i ECON 383 (SFU) Auctions September 15, / 28

32 Revelation Principle Bidder i s equilibrium strategy s i (v i ) is his agent. Bidder i tells the agent his true value, the agent bids on his behalf. No incentive to deviate from the strategy s i is equivalent to an incentive to report the true value to the agent. This is known as the revelation principle. Bidder i is not necessarily bidding truthfully with s i (v i ) (i.e., s i (v i ) needs not be v i ). ECON 383 (SFU) Auctions September 15, / 28

33 Comparing payments from auctions First price auction: s fp (v i ) = n 1 n v i. Second pay auction: s sp (v i ) = v i. All pay auction: s all (v i ) = n 1 n (v i) n. ECON 383 (SFU) Auctions September 15, / 28

34 Comparing payments from auctions First price auction: s fp (v i ) = n 1 n v i. Second pay auction: s sp (v i ) = v i. All pay auction: s all (v i ) = n 1 n (v i) n. In all of these auctions, the expected payment of a bidder i with value v i is n 1 n (v i) n. Same payment, i.e., revenue equivalence! Bidders respond strategically to the change in auction rule, un-do the intended change. ECON 383 (SFU) Auctions September 15, / 28

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