1、,Introduction to Game TheoryLecture 7 Disclaimer: this presentation is only a supporting material and is not sufficient to master the topics covered during the lecture. Study of relevant books is strongly recommended.,/100,2,Applications of NE and SPNEAuctionsSecond-Price Sealed-Bid AuctionFirst-Pri
2、ce Sealed-Bid Auction,Today,Auctions Second Price First Price Summary,/100,3,Used to allocate:ArtGovernment bondsRadio spectrumForms:Sequential biddingBid placed in sealed envelopesApplication of Game Theoretic Approach:Find most effective design,Auctions,Auctions Second Price First Price Summary,/1
3、00,4,Rules:bidders sequentially submit increasing bidsperson that made current bid wins if no one wishes to submit a higher bid everybody knows their personal value of an object= before the bidding starts, every bidder knows their “maximal bid”this type of auction is called English Auction,Auction 1
4、,Auctions Second Price First Price Summary,/100,5,Bidder with the highest maximal bid winshe pays the second maximal bidonly two bidders matter for the outcome bidder with the highest maximal bid B1bidder with the 2nd highest maximal bid B2to win, B1 has to bid only “slightly” more than maximal bid
5、of B2if the bidding increment is small - we take the winning price to be equal to the 2nd highest maximal bid,Auction 1,Auctions Second Price First Price Summary,/100,6,Rules:all bidders submit their bids simultaneouslyall bidders place their bids in sealed envelopesbidder with the highest bid winsw
6、inning bidder pays the 2nd highest bidthis type of auction is calledSecond Price Sealed Bid Auction,Auction 2,Auctions Second Price First Price Summary,/100,7,Bidder with the highest maximal bid winshe pays the second maximal bidonly two bidders matter for the outcome bidder with the highest maximal
7、 bid B1bidder with the 2nd highest maximal bid B2to win, B1 has to bid only “slightly” more than maximal bid of B2price is equal to the 2nd highest maximal bid,Auction 2,Auctions Second Price First Price Summary,/100,8,Auction 1 and Auction 2 lead to the same outcome:winner is the samewinner pays th
8、e same price,Auction 1 and Auction 2,Auctions Second Price First Price Summary,/100,9,Auction as a strategic gamePlayers: n biddersActions: all possible bids (non negative numbers)Payoffs: difference between value and second highest bid if win, zero otherwise,Second Price Sealed Bid,Auctions Second
9、Price First Price Summary,/100,10,Notation:n players are ordered according to their valuations: v1v2v3vneach player i submits a bid biif bi is highest and bj second highest bid: bidder i gets vi-bjall other bidders get zero,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,11,Na
10、sh Equilibrium 1:Every bidder bids their value:(b1,b2,b3,bn) = (v1,v2,v3,vn)bidder 1, with value v1, wins and pays b2bidder 1 has payoff (v1-b2)all the other bidders get zero,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,12,Every bidder bids their value is NE:(b1,b2,b3,bn) =
11、 (v1,v2,v3,vn)Winner bidder 1: bid more nothing changesbid less lose and get nothing= winner has no incentive to deviateLoser k: bid less nothing changesbid more (less than v1) nothing changesbid more (bk v1) win, but earn vk - bk losers have no incentive to deviate,Second Price Sealed Bid,Auctions
12、Second Price First Price Summary,/100,13,Nash Equilibrium 2:First bidder bids his value, others bid zero:(b1,b2,b3,bn) = (v1,0,0,0)bidder 1, with value v1, wins and pays 0bidder 1 has payoff v1all the other bidders get zero,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,14,Fi
13、rst bidder bids v1, others bid zero is NE: (b1,b2,b3,bn) = (v1,0,0,0)Winner bidder 1:bid more nothing changesbid less nothing changes= winner has no incentive to deviateLoser k: bid less not possiblebid more (less than v1) nothing changesbid more (bk v1) win, but earn vk - bk losers have no incentiv
14、e to deviate,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,15,Nash Equilibrium 3:Bidders bid in the following way:(b1,b2,b3,bn) = (v2,v1,0,0)bidder 2, with value v2, wins and pays v2bidder 2 has payoff 0all the other bidders get zero,Second Price Sealed Bid,Auctions Second P
15、rice First Price Summary,/100,16,Bidders bidding in the following way is NE: (b1,b2,b3,bn) = (v2,v1,0,0)Winner bidder 2:bid more nothing changesbid less (still more than v2) nothing changesbid less (less than v2) lose and get nothing= winner has no incentive to deviateLoser k: bid less nothing chang
16、esbid more (less than v1) nothing changesbid more (bk v1) win, but earn vk - bk 0 = losers have no incentive to deviate,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,17,NE 3 - (b1,b2,b3,bn) = (v2,v1,0,0) : bidder 1 has to believe that bidder 2 will continue bidding up to v1,
17、 then bidding v2 is best responsebidder 2 is taking risk of negative payoff if bidder 1 bids more than v2still, given that all bidders bid according to NE 3, everybody is playing the best responsefor bidder two, bidding v1 is weakly dominated by bidding v2In general: in a second-price sealed-bid auc
18、tion, a players bid equal to her valuation weakly dominates all her other bids,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,18,Many Nash Equilibria, bud one is special:NE where every bidder bids her value(b1,b2,b3,bn) = (v1,v2,v3,vn)is the only one where every players actio
19、n weakly dominates all her other actions,Second Price Sealed Bid,Auctions Second Price First Price Summary,/100,19,all bidders submit their bids simultaneouslyall bidders place their bids in sealed envelopesbidder with the highest bid winswinning bidder pays her own bidthis type of auction is called
20、 First Price Sealed Bid Auction,Auction 3,Auctions Second Price First Price Summary,/100,20,Winner pays the price she bids, not the second highest priceWe assume games with perfect information, i.e. everybody knows value of all biddersPlayers: n biddersActions: all possible bids (non negative number
21、s)Payoffs: difference between value and bid if win, zero otherwise,First Price Sealed Bid,Auctions Second Price First Price Summary,/100,21,Nash Equilibrium 1:Bidders bid in the following way:(b1,b2,b3,bn) = (v2,v2,v3,vn)bidder 1, with value v1, wins and pays v2bidder 1 has payoff (v1-v2)all the oth
22、er bidders get zero,First Price Sealed Bid,Auctions Second Price First Price Summary,/100,22,Bidders bid in the following way is NE:(b1,b2,b3,bn) = (v2,v2,v3,vn)Winner bidder 1: bid more still win, pay morebid less lose and get nothing= winner has no incentive to deviateLoser k: bid less nothing cha
23、ngesbid more (less than v2) nothing changesbid more (bk v2) win, but earn vk - bk losers have no incentive to deviate,First Price Sealed Bid,Auctions Second Price First Price Summary,/100,23,First-price sealed-bid auction has many NEIn all of them, bidder 1 wins the auctionFirst-price sealed-bid auc
24、tion where bidders bid (b1,b2,b3,bn) = (v2,v2,v3,vn)yields the same outcome as Second-price sealed-bid auctionNote: usually we do not know the value of other bidders - we use expected value,First Price Sealed Bid,Auctions Second Price First Price Summary,/100,24,Game theoretic approach and concept o
25、f Nash equilibrium has many useful applicationsAuctions Nash equilibrium concept helps to determine the winner and the return for the owner of the object being soldThis allow us to compare different types of auctions in terms of price paidIn case of perfect information First- and Second-price sealed
26、-bid auction yields the same results,Summary,Auctions Second Price First Price Summary,/100,25,! Surnames starting A - L 18:00 ! ! Surnames starting M - Z 18:45 !Static games: actions, action profiles, iterative elimination of dominated strategies, NE, mixed strategies, dominated strategies in mixed strategies, MSNEDynamic games: Backward induction, strategies, NE in dynamic games, SPNEOsborne chapters 1,2,4,5,6,Midterm Exam,Auctions Second Price First Price Summary,
