AuctionsStrategic SituationReview: Second Price AuctionsReview: English AuctionsWhat to BidRevenuesOrder StatisticsOrder Statistics of Uniform DistributionsTwo DrawsKey ObservationGeneralizingFirst Price AuctionsSetting Up the ProblemBidder’s ProblemWhat is Pr(B > b(v))?Conjectures about b(v)Bidder’s Problem RevisitedOther ValuesEquilibriumUncertainty about my RivalBayes-Nash EquilibriumCommentsSlide 23Revenue EquivalenceMore RivalsOptimal BiddingProbability of WinningBidder 1’s optimizationSlide 29Slide 30Slide 31Comparing RevenuesRevenue Equivalence TheoremImplicationsOther Strange Auction formsOptimal AuctionsOne Bidder AuctionsMonopolyMonopoly ProblemSlide 40Slide 41Back to AuctionsAuction/Monopoly ProblemSlide 44Demand CurveSlide 46OptimizationSlide 48So what is Marginal Revenue?Uniform CaseRecipe for Optimal AuctionsConclusionsMore ConclusionsAuctionsStrategic SituationYou are bidding for an object in an auction.The object has a value to you of $20.How much should you bid?Depends on auction rules presumablyReview: Second Price AuctionsSuppose that the auction is a second-price auctionHigh bidder winsPays second highest bidSealed bidsWe showed (using dominance) that the best strategy was to bid your value.So bid $20 in this auction.Review: English AuctionsAn English (or open outcry) auction is one where bidders shout bids publicly.Auction ends when there are no higher bids.Implemented as a “button auction” in JapanImplemented on eBay through proxy bidding.What to BidAgain, suppose you value the object at $20.Dominance says to drop out when bid = value.The fact that bidding strategies are the same in the two auction forms means that they are strategically equivalent.RevenuesHow much does the seller earn on the auction?Depends on the distribution of values.Suppose that there are 2 bidders and values are equally likely to be from $0 to $100.The seller earns an amount equal to the expected losing bid.Order StatisticsThe seller is interested in the expected value of the lower of two draws from 0-100. This is called the second order statistic of the distribution. We will sometimes write this as E[Vk(n)] where the k denotes the order (highest, 2nd highest, etc.) of the draw and (n) denotes the number of draws. So we’re interested in E[V2(2)]Order Statistics of Uniform DistributionsThere order statistics have simple regularity propertiesThe mean of a uniform draw from 0-100 is 50. Note the mean could be written as E[V1(1)].010050Two DrawsNow suppose there are two draws.What are the first and second order statistics?03310066Key ObservationWith uniform distributions, the order statistics evenly divide the number line into n + 1 equal segments.Let’s try 3 draws:01002550751st2nd3rdGeneralizingSo in general, E[Vk(n)] = 100* (n – k + 1)/(n + 1)So revenues in a second price or English auction in this setting are:E[V2(n)] = 100 * (n – 1)/(n + 1)As the number of bidders grows large, the seller’s revenues increaseAs the number of bidders grows unbounded, the seller earns all the surplus, i.e. 100!First Price AuctionsNow suppose you have a value of $20 and are competing with one other bidder in a first-price auctionYou don’t know the exact valuation of the other bidder.But you do know that it is randomly drawn from 0 to 100. How should you bid?Setting Up the ProblemAs usual, you want to bid to maximize your expected payoffBut now you need to make a projection about the strategy of the other bidderPresumably this strategy depends on the particular valuation the bidder has. Let b(v) be your projection for the bid of the other bidder when his valuation is v.Bidder’s ProblemChoose a bid, B, to maximize expected profits.E[Profit] = (20 – B) x Pr(B is the highest bid)What is Pr(B is the highest bid)?It is Pr(B > b(v))What is Pr(B > b(v))?vb(v)BI winI loseb-1(B)Conjectures about b(v)Suppose that I believe that my rival’s strategy is to bid a constant fraction of his valueThen b(v) = avWhere a is some fractionI win wheneverB >= avOr, equivalentlyv <= B/aSo Pr(B > b(v)) becomes:Pr( v <= B/a) = B/100aBidder’s Problem RevisitedSo now I need to choose B to maximizeE[Profit] = (20 – B)(B/100a)Optimize in the usual way:(1/100a) x (20 – 2B) = 0Or B = 10So I should bid 10 when my value is 20.Other ValuesSuppose my value is V?E[Profit] = (V – B)(B/100a)Optimize in the usual way:(1/100a) x (V – 2B) = 0Or B = V/2So I should always bid half my value.EquilibriumMy rival is doing the same calculation as me.If he conjectures that I’m bidding ½ my valueHe should bid ½ his value (for the same reasons)Therefore, an equilibrium is where we each bid half our value.Uncertainty about my RivalThis equilibrium we calculated is a slight variation on our usual equilibrium notionSince I did not exactly know my rival’s payoffs in this gameI best responded to my expectation of his strategyHe did likewiseBayes-Nash EquilibriumMutual best responses in this setting are called Bayes-Nash Equilibrium.The Bayes part comes from the fact that I’m using Bayes rule to figure out my expectation of his strategy.CommentsIn this setting, dominant strategies were not enoughWhat to bid in a first-price auction depends on conjectures about how many rivals I have and how much they bid.Rationality requirements are correspondingly stronger.RevenuesHow much does the seller make in this auction?Since the high bidder wins, the relevant order statistic is E[V1(2)] = 66.But since each bidder only bids half his value, my revenues are½ x E[V1(2)] = 33Notice that these revenues are exactly the same as in the second price or English auctions.Revenue EquivalenceTwo auction forms which yield the same expected revenues to the seller are said to be revenue equivalentOperationally, this means that the seller’s choice of auction forms was irrelevant.More RivalsSuppose that I am bidding against n – 1 others, all of whom have valuations equally likely to be 0 to 100. Now what should I bid?Should I shade my bid more or less or the same?In the case of second-price and English auctions, it didn’t matter how many rivals I had, I always bid my valueWhat about in the first-price auction?Optimal BiddingAgain, I conjecture that the