Auction is a commonly used way of allocating indivisible

Transcription

1 Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 16. BIDDING STRATEGY AND AUCTION DESIGN Auction is a commonly used way of allocating indivisible goods among interested buyers. Used cameras, Salvator Mundi, and spectrum auctions. Online platforms (Amazon, ebay) have greatly increased popularity of auctions in the modern digital economy. 1

2 16.1 Types of auctions Open outcry versus sealed bid. Best known open outcry: English, Dutch auctions. First-price versus second-price. In sealed bid auctions, highest bidder wins but price depends on rule. Private value versus common value. Distinction in auction environment rather than rules. 2

3 16.3 Bidding strategies Second price, sealed bid auctions with private values. Each bidder i, i = 1,..., N, values an object for sale at v i ; each i knows own valuation v i, but not any other v j, j = i; each i submits a bid b i independently; bidder i wins the auction if b i is higher than all other b j, j = i, wins with equal probability if b i is among the highest, and otherwise loses; payoff to each i is v i p if i wins, where p is the highest losing bid, and 0 otherwise. 3

4 This is a complex Bayesian game. To set it up, we will need to specify what each bidder i knows about how each v j, j = i, is distributed. Type of each bidder i is own valuation v i. A bidding strategy of each bidder i specifies the bid b i depending on v i. Regardless how we specify the Bayesian game, there is a weakly dominant strategy for each bidder i: bidding b i = v i weakly dominates all other bids. 4

5 Bidding one s own valuation is a weakly dominant strategy. Fix any bidder i, and fix any valuation v i. Denote as b the highest outstanding bid; this is the price i pays if i wins the auction. Bidding b i > v i is weakly dominated by b i = v i : they give the same payoff when b < v i, when b = v i, and when b > b i, but b i > v i is strictly worse than b i = v i when v i < b b i. Similarly, bidding b i < v i weakly dominated by b i = v i. 5

6 First price, sealed bid auctions with private values. Each bidder i, i = 1,..., N, values an object for sale at v i ; each i knows own valuation v i, but not any other v j, j = i; each i submits a bid b i independently; bidder i wins the auction if b i is higher than all other b j, j = i, wins with equal probability if b i is among the highest, and otherwise loses; payoff to each i is v i b i if i wins. 6

7 There is no weakly dominant bidding strategy. Bidding one s own valuation is weakly dominated by bidding below it; so is bidding above it. Equilibrium bidding strategy involves shading the bid, i.e., bidding below one s own valuation. To analyze how much one should shade the bid, we need to specify the Bayesian game in greater detail. 7

8 A Bayesian game. Suppose that N = 2. Each bidder i, i = 1, 2, privately and independently draws valuation v i from same uniform distribution over interval [0, 1]. From bidder i s perspective, for any v between 0 and 1, probability that v j, j = i, lies on interval [0, v] is equal to v, and probability that v j lies on interval [v, 1] is 1 v. 8

9 A Bayesian Nash equilibrium: each bidder i uses bidding strategy b i = 1 2 v i. Fix any bidder i, and fix any valuation v i. Any bid b i wins when b i > b j = 1 2 v j, i.e. when v j < 2b i, so b i wins with probability 2b i, with expected payoff 2b i (v i b i ). The expected payoff is maximized by setting b i = 1 2 v i. 9

10 16.2 Winner s curse Second price, sealed bid auctions with common values. Each bidder i, i = 1, 2, receives a private estimate s i between 0 and 1 of the value of the objective for sale; each i observes s i, and believes that s j, j = i, is uniform between 0 and 1; each i s valuation v i = s i + αs j, with α a known constant between 0 and 1. Parameter α represents the strength of common value component in valuation. 10

11 Winner s curse. Suppose each bidder i bids expected valuation given own signal: b(s i ) = s i + α 1 2. Fix i and s i. Probability of winning is s i. Expected valuation conditional on winning is s i + α 1 2 s i. Expected price paid conditional on winning is 1 2 s i + α 1 2. Expected payoff conditional on winning is difference, which is negative for s i < α/(1 + α). 11

12 A Bayesian Nash equilibrium: b(s i ) = s i + αs i. Fix i and s i, and consider any b i. Probability of winning is b i /(1 + α). Expected valuation conditional on winning is given by s i + α 1 2 b i/(1 + α). Expected price paid conditional on winning is 1 2 b i. Expected payoff, which is probability of winning times the difference of expected valuation and expected price conditional on winning, is maximized at b i = (1 + α)s i. 12