Auctions Lirong Xia Sealed-Bid Auction One item A set of bidders - PowerPoint PPT Presentation

Auctions Lirong Xia

Sealed-Bid Auction Ø One item Ø A set of bidders 1,…, n • bidder j ’s true value v j • bid profile b = ( b 1 ,…, b n ) Ø A sealed-bid auction has two parts • allocation rule: x ( b ) ∈ {0,1} n , x j ( b )=1 means agent j gets the item • payment rule: p ( b ) ∈ R n , p j ( b ) is the payment of agent j Ø Preferences: quasi-linear utility function • x j ( b ) v j - p j ( b ) 2

Second-Price Sealed-Bid Auction Ø W.l.o.g. b 1 ≥ b 2 ≥ … ≥ b n Ø Second-Price Sealed-Bid Auction • x SP ( b ) = (1,0,…,0) (item given to the highest bid) • p SP ( b ) = ( b 2 ,0,…,0) (charged 2 nd highest price) 3

Example $ 10 $10 Kyle $ 70 $70 $70 Stan $ 100 $100 Eric 4

Incentive Compatibility of 2 nd Price Auction Ø Dominant-strategy Incentive Compatibility (DSIC) • reporting true value is the best regardless of other agents’ actions Ø Why? • underbid ( b ≤ v ) • win à win: no difference • win à lose: utility = 0 ≤ truthful bidding • overbid ( b ≥ v ) • win à win: no difference • lose à win: utility ≤ 0 ≤ truthful bidding Ø Nash Equilibrium • everyone bids truthfully 5

First-Price Sealed-Bid Auction Ø W.l.o.g. b 1 ≥ b 2 ≥ … ≥ b n Ø First-Price Sealed-Bid Auction • x FP ( b ) = (1,0,…,0) (item given to the highest bid) • p FP ( b ) = ( b 1 ,0,…,0) (charged her reported price) 6

Example $ 10 $10 Kyle $ 70 $100 $70 Stan $71? $ 100 $100 Eric 7

Nash Equilibrium of 1 st Price Auction Ø Complete information • max bid = 2 nd bid + ε Ø Not sure about other bidders’ values? • winner’s curse 8

Games of Incomplete Information for auctions Ø Bidder j ’s type = her value θ j (private) • quasi-linear utility functions Harsanyi Ø G : joint distribution of bidders’ (true) values (public) Ø Strategy: s j : R à R (from type to bid) Ø Timing Ex ante • 1. Generate ( θ 1 ,…, θ n ) from G , bidder j receives θ j Interim • 2. Bidder j reports s j ( θ j ) Ex post • 3. Allocation and payments are announced 9

Bayes-Nash Equilibrium Ø A strategy profile ( s 1 ,…, s n ) is a Bayes-Nash Equilibrium (BNE) if for every agent j , all types θ j , and all potential deviations b j ’, we have other agents’ bids unilateral deviation E θ -j u j ( s j ( θ j ) , s - j ( θ -j )| θ j ) ≥ E θ -j u j ( b j ’ , s - j ( θ -j )| θ j ) your bids conditioned on j ’s information • s - j = ( s 1 ,…, s j- 1 , s j +1 ,…, s n ) 10

BNE of 1 st Price Auction Ø Proposition. When all values are generated i.i.d. from uniform[0,1], under 1 st price auction, the strategy profile where for all j , s j : θ à !"# ! θ is a BNE Ø Proof. • suppose bidder j ’s value is θ j and she decides to bid for b j ≤ θ j • Expected payoff ( θ j - b j ) × Pr( b j is the highest bid) = ( θ j - b j ) × Pr(all other bids ≤ b j | s -j ) = ( θ j - b j ) × Pr(all other values ≤ ! !"# b j ) = ( θ j - b j )( ! !"# b j ) n -1 • maximized at b j = !"# ! θ j 11

BNE of 2 nd Price Auction Ø b j = θ j Ø Dominant-Strategy Incentive Compatibility 12

Desirable Auctions Ø Efficiency in equilibrium (allocate the item to the agent with the highest value) • 1 st price auction • 2 nd price auction Ø Revenue in equilibrium 13

Expected Revenue in Equilibrium: 1 st price auction Ø Expected revenue for 1 st price auctions with i.i.d. Uniform[0,1] when b j = !"# ! v j !"# ! % 𝑐 × Pr(highest bid is 𝑐)𝑒 θ 5 # 789 7 θ × Pr(highest value is θ )𝑒 θ = ∫ 5 7 θ × 𝑜 θ !"# 𝑒 θ # 789 = ∫ 5 = 789 7?9 14

Expected Revenue in Equilibrium: 2 st price auction Ø Expected revenue for 2 st price auctions with i.i.d. Uniform[0,1] when b j = v j # % 𝑐 ×Pr(2 nd highest bid is 𝑐)𝑒𝑐 5 # = 𝑜(𝑜 − 1)∫ θ × 1 − θ θ !"D 𝑒 θ 5 = 𝑜(𝑜 − 1)∫ θ !"# − θ ! 𝑒 θ # 5 = 789 7?9 = expected revenue of 1 st price auction in equilibrium 15

A Revenue Equivalence Theorem Ø Theorem. The expected revenue of all auction mechanisms for a single item satisfying the following conditions are the same • highest bid wins the items (break ties arbitrarily) • there exists an BNE where • symmetric: all bidders use the same strategy • does not mean that they have the same type • increasing: bid increase with the value Ø Example: 1 st price vs. 2 nd price auction 16

Ad Auction keyword Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 winner 1 winner 2 winner 3 winner 4 winner 5 17

Ad Auctions: Setup Ø m slots • slot i gets s i clicks Ø n bidders • v j : value for each user click • b j : pay (to service provider) per click • utility of getting slot i : ( v j - b j ) × s i Ø Outcomes: { (allocation, payment) } 18

Generalized 2 nd price Auction (GSP) Ø Rank the bids • W.l.o.g. b 1 ≥ b 2 ≥ … ≥ b n Ø for i = 1 to m , • give slot i to b i • charge bidder i to b i+ 1 pay per click Ø Example • n =4, m =3; s 1 =100, s 2 =60, s 3 =40; v 1 = 10, v 2 = 9, v 3 = 7, v 4 = 1. • bidder 1 utility • HW: show GSP is not incentive compatible 19