Auction Theory Yosuke YASUDA | Osaka University
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Introduction
October, 2016
Variety of Auctions Selling single objects or multiple objects.
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Our lectures exclusively focus on single object auctions.
Open auctions: all bids are publicly observable.
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English (ascending-price) auction Dutch (descending price) auction
Sealed-bid auctions: only seller can observe bids.
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First price auction Second price (Vickrey) auction
There are many other auction formats.
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Introduction October, 2016
Assumptions Private values ⇔ Common (interdependent) values
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Each bidder knows her valuation, i.e., type = value of the good.
Independent values ⇔ Affiliated values
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Type (valuation) for each player is independently drawn.
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Seller’s valuation is normalized to be 0.
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No secondary market (no other resale possibility).
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Introduction October, 2016
Second Price Auction }
The winner is the bidder with the highest bid while she will pay the second highest bid.
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Equivalent to English auction under the assumptions of private and independent values.
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Is there (weakly) dominant strategy? }
Yes, it is optimal for each bidder to bid her own valuation. }
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Verify that overbidding or underbidding can never be profitable.
In English auction, it is optimal for each bidder to stay until the price reaches her own valuation. Introduction October, 2016
First Price Auction }
The winner is the bidder with the highest bid while she will pay her own bid. }
Truthful-bidding CANNOT be optimal
⇒ no gain from winning ⇒ bid-shading.
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Strategically equivalent to Dutch auction.
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Suppose there are n bidders; each bidder’s value (type) is uniformly and independently distributed between 0 and 1.
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What is the Bayesian Nash equilibrium? } 5
In an unique BNE, each bidder bids n-1/n times her valuation. Introduction October, 2016
First-Price Auction (1) Equilibrium of First
Price Auction (1)
Bidders simultaneously and independently submit bids b1 and b2 . I I
The painting is awarded to the highest bidder i⇤ with max bi , who must pay her own bid, bi⇤ .
To derive a Bayesian Nash equilibrium, we assume the bidding strategy in equilibrium is i) symmetric, and ii) linear function of xi . That is, in equilibrium, player i chooses (xi ) = c + ✓xi .
(1)
Now suppose that player 2 follows the above equilibrium strategy, and we shall check whether player 1 has an incentive to choose the same linear strategy (1). Player 1’s optimization problem, given she received a valuation x1 , is max (x1 b1
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b1 ) Pr{b1 > (x2 )}.
(2) Introduction October, 2016
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First-Price Auction (2) Price Auction (2) Equilibrium of First Since x2 is uniformly distributed on [0, 1] by assumption, we obtain Pr{b1 > (x2 )} = Pr{b1 > c + ✓x2 } ⇢ b1 c b1 c = Pr > x2 = . ✓ ✓ The first equality comes from the linear bidding strategy (1), the third equality is from the uniform distribution. Substituting it into (2), the expected payo↵ becomes a quadratic function of b1 . max (x1 b1
b1 )
b1
c ✓
Taking the first order condition, we obtain du1 1 c x1 = [ 2b1 + x1 + c] = 0 ) b1 = + . db1 ✓ 2 2 Comparing (3) with (1), we can conclude that c = 0 and ✓ = constitute a Bayesian Nash equilibrium. 7
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Introduction October, 2016
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Revenue Equivalence Theorem We already verify that (for each possible combination of types), second price = English, first price = Dutch. } Comparing expected revenues, 2nd price = 1st price. }
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In the first price auction, it can be shown that the winner’s payment is equal to the expected second highest value.
Four auction formats yield the same expected revenue!
(General) Revenue Equivalence Theorem } Any auction formats which induce (1) the same allocation and (2) the same expected payoff for the lowest value type, yields identical expected revenue. 8
Introduction October, 2016
Revenue Equivalence: Illustration
Open Dutch (Descending)
Sealed-Bid Strong
First-Price
Revenue Equivalence Theorem
English (Ascending) 9
Weak
Second-Price Introduction October, 2016
Bilateral Trade }
A buyer names a asking price (pb) and a seller names an offer price (ps). } }
If pb ≥ ps, then trade occurs at price p = (ps + pb) / 2. If pb < ps, then no trade occurs.
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The buyer’s valuation (vb) and the seller’s valuation (vs) are private information, and these are independently and uniformly distributed on [0, 1].
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Trade is efficient if and only if vb > vs.
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Introduction October, 2016
Impossibility of Efficient Trade }
A buyer (/seller) has an incentive to understate (/ overstate) the value in hopes of increasing her surplus.
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In a (linear) Bayesian Nash equilibrium, } } }
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There are other BNE with non-linear strategies. }
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Buyer’s strategy: 2/3∙vb + 1/12 Seller’s strategy: 2/3∙vs + 1/4 Profitable trade may not happen (with positive probability).
But efficiency cannot be achieved in any equilibrium.
What is an optimal selling mechanism for the seller?
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Introduction October, 2016
