beyond the first-price auction
• other sealed-bid auctions
• open outcry auctions
game
i w4
rirtheory..i
online
caution:
- this is just a sample of some of the main types of auctions for a single good
- even for a single good, the space of auction types is infinite
- for more than a single good, the space is even larger...

• • gnië,
second price (sealed-bid) auction
jnhine
• n bidders, a single good
• sealed bids: each bidder offers a price
• second price rule: highest bidder wins, pays second highest bid

formally
gánie theoly
online
• each agent i has a type o, say in [0,100]; for now we will not care about the distribution from which the types are drawn
• private values: agent i payoff depends on the auction outcome (a function of the bids) and his type o but not the other types o..
• the mechanism
— actions: each agent bids a1 in [0,100]
— define a(2) = second highest bid
— payoffs: u1 (a1, a.1 °) = 01— a(2)
=0
of (a1, ... , a) if a> a(2)
if a,< a(2)
= (of— a(2))/y if tied with y-1 other bidders for highest bid

analysis
online
• we will see that agents have a dominant strategy:
u(a1,a1e1)>=u(a’,a1e) foralia’,a1
• this means that we get a dominant strategy equilibrium
• the dominant strategy: bid your true value, or a1 =
• such auctions are called truthful. they are rare.

why is second-price auction truthful?
• online
• u (a, a.1 0) = 0 — a(2) if a> a(2)
= 0 ifa<a(2)
(01—a(2))/y if tied
• 0— a(2) is positive if and only if 0> a(2)
• so, want to have a1> a(2) if and only if 01> a(2)
• a = o guarantees win if and only if get positive payoff from winning

0
—.
z
r-i•
cd,
c1 1<
<_0
cc
(0•’
cd
—.
(1)
cd ‘

case 1: suppose you bid over
your
value
your bid
[iil --
theoiyuui
oni me

case la:you lost
gamç, theory
online
->
your value
your bid
highest rival
bid
well, had you bid your value your payoff would have been the same

case ib: you won and highest rival bid gan’
iii i li
is below your value
online
highest
rival your your
bid value bid
again, had you bid your value your payoff would have been the same

gànié ;
i i, i_ii
theory
ill a.
online
case ic: you won
and highest rival bid
is
above
your value
highest
your rival
value bid
*
your
bid
boy, now your really wish you’d bid your value ©

case2: supposeyou bid under
your bid
your
value
oiil
subcases left as an exercise...

other sealed-bid auctions
online
- fij iji iliz dullil)l i
• second-price auction
• k’th price auction for k>2
— highest bidder wins and pays the k’th highest bid
• all-pay auction
— highest bidder wins and everyone pays their bid
ii•

open outcry auctions gfrec&y
online
• english auction
• dutch auction

english / ascending price auction
online
— auctioneer raises price until all but one bidder drop out, last bidder wins and pays price at which second to last bidder dropped out
— surprisingly, for independent values, equivalent to second-price auction
• dominant strategy to stay in until your value is exceeded
• winner pays second-highest value ______
i
ii

dutch i descending price auction
online
— auctioneer starts at high price and lowers it until someone agrees to pay that price
— surprisingly, strategically equivalent to first-price auction
• strategy space: choose a number
• auction outcome: the person who chose the highest number wins, and pays that number
• true regardless of where the values come from; in particular, not limited to the independent value case

Original on youtube.com

You need Flash player 8+ and JavaScript enabled to view this video.

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/