Auction Theory Yossi Sheffi Mass Inst of Tech Cambridge, MA ESD.260J/1.260J/15.770J 12 Draft Outline 1st 2nd Revenue equivalence Other auctions curse © Yossi Sheffi, MIT Introduction to auctions Private value auctions price auctions price auctions Reservation price Interdependent values and the winner’s Extensions 23 Draft Fixed price is only 100+ years old © Yossi Sheffi, MIT Auctions - Examples As old as the hills… John Glenn – Rocket and module built by the lowest bidder Babilonian times (500 BC) sale of women In 193 AD the Pretorian Guard auctioned off the entire Roman Empire. Winning bid: 25,000 sesterces/man. The winner: Didius Julianus was declared emperor but as broke and was beheaded after two month (winner’s curse) 34 Draft to pay. Anonymity Auctions © Yossi Sheffi, MIT Auctions – What and Why? An auction is an allocation pricing mechanism An auction determines: Auctions elicit information about how much buyers are willing Universality The framework: Each bidder has a value for the item If he wins his surplus is the price paid minus the value. Avoid dishonest “smoke-filled-room” dealings Determine the value Give it to the buyer who wants it most (efficiency) 45 Draft Simple Auctions (Single Item) Open bids: English auction until one bidder left. Bidder pays the price at that point (Japanese auction). Sealed bids: First price © Yossi Sheffi, MIT – bidder calls increasing price Dutch auction – bidder starts high and lower price. First bidder to call gets the item – highest bid wins Second price – highest bid wins but pays the second-highest bid 56 Draft Information distribution Both buyers and seller are uncertain what the value of the item sold is. Private values value to himself (no bidder knows the will not affect the self valuation) Common values real value becomes known later) Both common and private elements © Yossi Sheffi, MIT – each bidder knows the valuation of other bidders; in any case it – the value is the same for all bidders (example: mineral rights – the Interdependent values – bidders modify their estimate during the bidding process. 67 Draft Equivalent auctions English 1st PriceDutch 2nd Price =PV CV =PV © Yossi Sheffi, MIT eBay is second price. The automatic bidders can be set to bid in set increments until a certain point. This means that one bids the value but pays the bid (value) of the next highest bidder (within the accuracy of the increment) 78 Draft Auction Metrics auctioneer wants the highest Efficiency the bidder who values the item the most ex post. In most procurement auctions there is no costs Time and effort © Yossi Sheffi, MIT Revenue (expected selling price) – the – make sure that the winner is secondary markets Secondary markets involve extra transaction Simplicity 89 Draft Assumptions n bidders F(V) with f(V) Risk neutrality No collusion or predatory behavior © Yossi Sheffi, MIT Private values i.i.d. values from (symmetric, independent bidders) 910 Draft 2nd Price – Bidding Strategies V ∆V Bid ∆V VBid nd 1 b-maxb-max 2 b-max 3 2 b-max 1 b-max 3 b-max © Yossi Sheffi, MIT Dominant strategy in 2 price (and English) auctions: Bid your value In English auctions a bidder should raise his bid “by a little” as long as the current price is lower than his valuation. The outcome is that the person with the highest valuation wins but pays the valuation of the second highest bidder (plus “a little”) If bidding higher than value – may win and pay a price higher than the value. Or may lose at a price still lower than the value (should have gone higher) 1011 Draft 2nd Price – How Much will the Winner Pay? n bidders, iid F(v f(v V1, V2, …, Vn,} V(1), V(2), …, V(n),}. [ ] [ ]() !( ) ( ) ( ) 1 ( )( 1)! ( )! k k nfv fv k n k − = − − − i i i Density of kth lowest: [ ]1 () !( ) 1 ( 1)! ( )! nkk k nfv v v k −−= − − − i i Density of U(0,1): ()[ ] 1k kEv n = + kth order statistic: 2nd order statistic: ( 1) 1[ ] 1n nEv n − − = + © Yossi Sheffi, MIT ) with density ), PV: Bidders’ values: {Order statistics: {n k Fv Fv n k Mean value of Mean value of (expected revenue for the auctioneer) To see this result: 1112 Draft 1st Price – Bidding Strategy E[winning]=(v-b)•P(b) v b P(b b The optimal bid, b* solves: * *() ( ) ()0vb db − − =i When the valuation are drawn from U(0,1) i.i.d. distributions: * 1nb v n − = i © Yossi Sheffi, MIT – valuation of the object by the bidder –The bid ) – Probability of winning with bid dP b P b “shading” by 1/n to account for the need to get surplus To develop the strategy, assume that each bidder wants to maximize the expected value of winning, which is the surplus associated with winning times the probability of winning: E[winning]=(v-b)•P(b) Where: V = the bidder’s valuation of the object auctioned off b = the bid P(b) = the probability that a bid b will win the auction. The optimal bid is the one that maximizes the expectation in Eq. . Thus the optimal bid is the solution to: Consider the case with n bidders whose valuations are drawn from independently from identical distribution. The probability that b is the winning bid is the probability that (n-1) valuations will be lower than b[1] or: p(b) = [F(b)]n-1. In the special case in which the valuations are drawn from a U(0,1) density function, P(b) = bn-1 the optimal bid of a participant whose value is v would be: Thus each bidder has to “shave” his bid by (1/n)⋅v. [1] No bidders would bid higher than his value in a first price auction since this will cause him to have negative surplus in case he wins. 1213 Draft The winning (highest) bid is the bid of the person with the highest order statistic: V(n). 1st Price – The expected Payment () 1 [ ]n n EV n − i For U(0,1), this person bids: In this case: ()[ ] 1n nEV n = + So the payment is: 1 1 n n − + © Yossi Sheffi, MIT Same result as before (!) 1314 Draft Revenue Equivalence Theorem In 2nd highest losing bid In 1st : All auction that allocate the item Risk neutrality No collusion © Yossi Sheffi, MIT price participants bid their value and pay the
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