Some Auction Theory:

1
Some Auction Theory Independent Private
valuations
Prof. dr. M.C.W. Janssen March 9 2009
2
Difference between Auction and Beauty Contest

• Players bid in both mechanisms
• Players qualities can be assessed in both
  mechanisms possibly ex ante
• In Beauty Contest, money is not part of the bid?
• In Auctions, only monetary bids?
• Once Auction design is fixed, subjective
  judgments do not play a role (algorithm).
  Subjectivity essential to B.C.

3
Different notions of efficiency

• Market Efficiency TS CS PS
• Efficient firms cost efficiency
• Operating cost?
• Including entry cost?
• Asymmetry entrant/incumbent
• Efficiency of allocation mechanism object(s)
  gets in the hands of those players who value them
  the most

4
Introduction to auction theory

• Auction rules

- Who can bid?
- What bids acceptable? (reserve price)
- How are bids submitted?
- What info is made public?
- When does auction end?
- Who is the winner?
- What price is paid?

• Auction environment

- Population of potential bidders
- Their values on object being auctioned
- Their attitudes towards risk
- Information they possess
5
Auction rules

• Sealed-bid auction vs. oral auction

• Ascending, descending, simultaneous ascending

• first-price auction vs. second-price auction

English auction ascending, first-price, oral
auction
Dutch auction descending, first-price, oral
auction

• single-unit auction vs. multi-unit auction vs.
  multi-object

6
Auction environment
1. Auctions with Independent Private Values
(IPV)
2. Interdependent values and Common Value
auctions (CV)
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Value determination

• All values are drawn from a certain distribution
  F(.)
• IPV model each vi is drawn independently of
  others players only observe own vi
• Affiliated values each player observes a signal
  si (from F) and vi is a function of si and s-i
  players only observe own si
• Correlated (common) value is special case
• What type of game is such an auction?

8
Auctions with IPV (1)
First-price, sealed bid

• N players

• Bidders have valuations uniformly between 0 and v

• Strategy Bid b(vi) (nonnegative number)

• Pay-off

v(i) b(vi) If you have the highest bid
0 If someone else has higher bid
? If youre among the highest bidders

• In general Individuals bid (N-1)/N times their
  private valuation

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Derivation optimal bid
10
Auctions with IPV (2)
Second-price, sealed-bid

• N players

• Strategy Bid b(i) (nonnegative number)

• Bidders have valuations between 0 and 10

• Pay-off

v(i) b(i, 2nd highest) If you have the
highest bid
0 If someone else has higher bid
? If youre among highest bidders

• What is optimal bid? (Vickrey 1961, Nobel prize
  1996)

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Optimal bid in 2nd price sealed-bid

• (weakly) Dominant strategy to bid your valuation,
  b(i)v(i)
• Three possibilities
• Max b(j) gt v(i). If you bid b(i)v(i), then
  pay-off is 0 but any other bid gives negative
  pay-off (if you bid more than b(j)) or pay-off of
  0 (otherwise)
• Max b(j) lt v(i). If you bid b(i)v(i), then
  pay-off is v(i) - max b(j) gt 0 any bid larger
  than max b(j) gives same pay-off, any lower bid
  yields pay-off 0
• Max b(j) v(i). You always get a pay-off of 0,
  whether you win object or not

12
Auctions with IPV (3)

• English auction is auction where bidders bid
  against each other in sequential fashion (name
  your own bid, or auctioneer gives bid)

• English auction is strategically equivalent to a
  second-price sealed bid auction and gives
  exactly the same result

• Dutch auction is a clock (going down) auction
  where the bidder who first stops the clock wins
  the object and the price given by the clock at
  moment it stops

• Dutch auction gives exactly the same result as a
  first-price sealed bid auction

13
Revenue Equivalence Theorem I

• Sellers point of view
• Bidders bid more in 2nd price auction (they dont
  shade their bid as in 1st price auction)
• Seller receives not bid, bid second-highest bid?
• Which effect dominates?

14
More general formulation (RET II)

• Bidders have valuations independently distributed
  between 0 and v according to some F and
• Second-price auction bid your valuation
• Expected payment for valuation x is FN-1(x)E(max
  xj given xj lt x)
• First-price auction
• bid the expected value of max xj given xj lt x
• Expected payment for valuation x is FN-1(x)E(max
  xj given xj lt x)
• Revenue equivalence all four auction types
• Expected value identical, but variation in
  second-price auction is larger

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Crucial assumptions of RET

• Risk-averse bidders (symmetric players)
• With second-price auction nothing changes
• Under first-price auction, bidders will increase
  bids
• Budget constraints (each player has a w(i))
• With second-price auction, bid min(v(i),w(i))
• Under first-price auction, and some conditions,
  bid min(n-1)/n)v(i),w(i)
• Budget constraint softer under first-price
  auction
• Asymmetries between players
• One players value is drawn from 0,v other
  players value drawn from 0,v with v gt v
• With second-price auction nothing changes
• With first-price auction (see next slide)
• Do these three assumptions hold in the type of
  context we discussed last week?

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Asymmetries in first price auction

• Uniform distribution of values F1 and F2
• Bidding functions ß1(v(1)) and ß2(v(2))
• p(1) F2(ß-12(b(1)))v(1)-b(1)
• Suppose inverse bidding function of player 2 is
  as in Krishna (4.25) with k1 -k2 as (4.26),
  then p(1) 2b(1)v(1)-b(1)/(1- k1 b2(1)
• Maximizing gives that this is indeed an
  equilibrium (in class)
• Interesting feature inefficiency due to
  asymmetry
• Revenue comparison with 2nd price auction may go
  either way.

17
Bidding function where player 1 is stronger
18
Reserve price and Entry fees

• Of interest for our examples with aftermarkets?
• Reserve price price below which seller commits
  not to sell
• Setting a low reserve price r (above the lowest
  possible value of the buyer) is revenue
  increasing (similar for entry fees)
• To see, suppose valuations between 0,v. Chance
  you want sell due to reservation price rgt0 Fn(r)
• Chance you increase your revenue nFn-1(r)(1-
  F(r))
• For r close to 0, F(r) close to 0 and second
  expression much larger than first expression.