-— general results on auctions
to “ online
— we saw some specific connections drawn pairwise between auctions
— is there something we can say more generally about the space of single-good auctions?
— yes; we will discuss briefly two examples
• revelation principle
• revenue equivalence

—— .,,,..
revelation principle
online
— private value setting
— definitions:
• in a direct auction means bidders simply declare a value
• a direct auction is truthful if it’s an equilibrium for all to declare truthfully
— theorem:
any auction can be converted to an equivalent truthful auction.
— “proof”: the auctioneer “lies for you”. see figure on next slide.

“proof” of the revelation principle
— thëóry-
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example: applying the principle
• •et4t — garnç “
theory-:
online
— consider a firstprice auction with n bidders with ipvs drawn i.i.d. from [0,1001.
— as you know, it is a bne when an agent with value x bids x(n-1)/n.
— define a a modified auction in which, when you bid x, the auctioneers considers it as x(n-1)/n and runs a first-price auction.
— it is now an equilibrium to bid truthfully.

the revenue equivalence theorem i 9
— — ‘. ‘‘ online
— suppose you wantto sell your used camera, and maximize the sale price.
— which auction type should you use?
— in particular, which is better, first-price or second-price auction?

revenue comparison: gbè.
- .
first vs. second price auction ..
• n bidders each bidding for a painting
• each player has a value 0 drawn from the uniform distribution over [o,$loom] for the painting
• equilibrium in first price auction: bid o (n-1)/n
• equilibrium in second price auction: bid e

revenue comparison:
first vs. second price auction
• winning payment first price: max1 0, (n-1)/n winningpaymentsecondprice: second highest 0,
game ,.:: thëór-
online

• revenue comparison:
first vs. second price auction •.
• winning payment first price: max1 0, (n-1)/n
• winning paymentsecond price: second highest o

winning paymentsecond price: second highest e,
gánië
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revenue comparison:
first vs. second price auction
winning payment first price:
max1 0, (n-1)/n

revenue comparison:
first vs. second price auction
• winning payment first price: max, 8 (n-1)/n e[max1 o] = 100 n/(n+1)
game theory
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revenue comparison:
first vs. second price auction
• winning payment first price: max, 8, (n-1)/n e[max1 o j = 100 n/(n+1)
gàmë ;: théáry-
line

—
• —
revenue comparison: gamç .. •
first vs. second price auction
• winning payment first price: max, 8 (n-1)/n e[max1 o] = 100 n/(n+1)

revenue comparison:
first vs. second price auction
• winning payment first price: max e, (n-1)/n e[max1 0 1 = 100 n/(n+1)
i s.i.i-1.r. .,.,,,. — game thót
online

revenue comparison:
first vs. second price auction
• winning payment first price: max1 e (n-1)/n e[max1 0, ] = 100 n/(n+1)
e[rev] = 100 (n-1)/(n+1)
• winning payment second price: second highest 0 e[second highest 0] = 100 (n-1)/(n+1)
—— game ,
‘• line

revenue comparison:
first vs. second price auction
• winning payment first price: max, 8 (n-1)/n e[max1 o j = 100 n/(n÷1)
e[rev] = 100 (n-1)/(n+1)
• winning payment second price: second highest 8, e[second highest o] = 100 (n-1)/(n+1)
e[rev] = 100 (n-1)/(n+1)
— game — thëó

reve n u e corn pa ri son: gaie :
j - —.
first vs. second price auction
• winning payment first price: max e (n-1)/n e[max1 0 j = 100 n/(n+1)
e[rev] = 100 (n-1)/(n+1)
• winning payment second price: second highest 0 e[second highest 0,] = 100 (n-1)/(n-i-1)
e[rev] = 100 (n-1)/(n+1)

revenue comparison: game
first vs. second price auction
• winning payment first price: max, 8 (n-1)/n e[max1 o) = 100 n/(n÷1)
e[rev] = 100 (n-l)/(n+1)
• winning payment second price: second highest e e[second highest o] = 100 (n-1)/(n+l)
e[rev] = 100 (n-1)/(n+1)
• is this a coincidence? no...

game —
revenue eq u iva len ce thëy•2
online
theorem (vickrey (1961), myerson (1981))
in ipv setting with lid values, all single-item auctions in which
— item goes to bidder with highest (true) value
— bidder with value 0 pays 0
have the same expected revenue
so: true in particular for first price, second price, english and dutch auctions

Original on youtube.com

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Professor Matthew O. Jackson & Yoav Shoham are offering a free online course on Game Theory starting in March 23, 2012. https://class.coursera.org/gametheory/

Offered by Coursera: https://www.coursera.org/