Presentation On Auction Theory By Simon Herrmann

Introduction to Auctions

A short history of auctions

• • • • • •

500 B.C. Babylon: women were auctioned off as wives (Herodot) ~150: Roman soldiers sold war plunder at auction. 1744: Sotheby's was created and Christie's in 1766. (mainly art) 1887: Sale of fruits and vegetables in the Netherlands through auctions 1995: eBay was launched 2000/01: UMTS - G3 auction for radio spectrum in Europe (revenue of >50bn EUR)

Use of auctions today An auction is a market institution with an explicit set of rules determining resource allocation and prices on the basis of bids from the market participants. (McAfee, McMillan, 1987) e.g. used for

• • • • • •

Treasury bills Drilling rights on oil fields Privatisation of firms and other assets Takeover battles Procurement and many more

Why selling through auctions •

Price Setting - difficult to determine the highest willingness to pay, however lower search cost for customer

•

Lottery: not efficient since not necessarily the participant with the highest value for the good wins

•

Contract awarding (Beauty Contest): might easily yield a political outcome and might lower the chance of getting new entrants to the market.

Four basic types of auctions

• Ascending bid (English) auction

the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction

the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Strategical equivalent auctions I

• Ascending bid (English) auction

the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction

the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Strategical equivalent auctions II

• Ascending bid (English) auction

the price is successively raised until only one bidder remains, and that bidder wins at the final price.

• Descending bid (Dutch) auction

the price is lowered continuously; the first bidder calling out wins the object at this price.

• First Price sealed bid auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder who makes the highest bid; at a price of his bid.

• Second Price sealed bid (Vickrey) auction

each bidder independently submits a single bid, without seeing others’ bids, the object is sold to the bidder with the highest bid at the second highest bid.

Truth telling a dominant strategy Second Price Auctions price

Strategic under-bidding (three cases)

Strategic over-bidding (three cases)

v

my bid

opponents bid

my loss

my profit

Truth telling a dominant strategy Truth Telling is the dominant strategy for Second Price auctions => bid shading is not a good (dominant) strategy. What happens in a First Price auction? Truce telling can not be a optimal strategy since: for a bid b=vi the expected revenue is 0; E[r]=0 Therefore the optimal bid b* must be smaller than bidders i true value vi => bid shading is a good (dominant) strategy.

Strategy for first price auction Recall the first price auction: all players state a bid, and the winner is the player with maximum bid, and has to pay his bid as price. Assumptions: - there are n>=2 bidders - every bidder draws (independently) a valuation v from [0,1] - if bidder i submits the highest bid b* she realizes a economic gain of vi-b* Claim: best strategy for bidder with value vi is to bid:

" bn % Proof: probability of winning for bid b is: $ ' # n !1 &

n !1

vi

(n !1) n

" bn % and the expected value: (v i ! b)$ ' # n !1 &

First derivative with respect to b (set equal to 0 to find the maximum) gives:

" bn % !$ ' # n !1 & which reduces to:

n !1

" n % + (v i ! b)(n !1)b n !2 $ ' # n !1 &

n !1

=0

!b + (v i ! b)(n !1) = 0 b = vi

(n !1) n

n !1

Strategy for first price auction In a First price auction the dominant strategy is given by: optimal bid is

b = vi

(n !1) n

in case of n=2

1 bi = v i 2

According to the Revenue Equivalence Theorem (here without proof): Second and First Price auctions generate the same revenue for bidder and auctioneer. If so, why bother about auction design at all?

Auction design needs to consider

• Available time to sell the goods (fish, flowers, tobacco) =>Dutch auction

• Risk of collusion (limited amount of buyers) => First Price auction

• Predatory behavior (a strong bidder participating might discourage new entrants) => Sealed bid auction

Collusion First Price Auction

Second Price Auction loss

price

v profit

agreed price my bid

opponents bid

agreed price my loss

my profit

The Winners Curse Interdependent valuation

• Bidding on drilling rights on oil fields • All bidders form their estimate value v

i

according to their own geological evaluation

• Each bidder makes a single bid • If you win you probably overestimated the true value of the oil field

Underlying assumptions

• All bidders are risk neutral • private values are independent drawn from a uniform distribution

The marriage market