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UT ECO 304K - Auction Credibility Problem Set

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Conference 8Intermediate Microeconomics - Fall 201822/10/2018Discussion: Auction credibilityRead Stanford Business Insights’s article “Why top bidders don’t always feel like winners”.What issue with second price auctions does the article underline? How are first price andascending auctions different?Recall our discussion on first/second price auctions: second price auctions are preferable inthe sense that they are strategy proof: players’ dominant strategy is to bid their true value,whereas first price auctions are not strategy proof.The Stanford Business Insights’s article uncovers a new problem with second price auction:they are not credible. It means that auctioneers can cheat (i.e. tell the winner the secondhighest bid was x when it was actually y < x), and that bidders are aware of it, which mightlead them to leave the auction. Actually Akbarpour and Li (the researchers mentionned in thearticle) show that none of the mechanisms among first, second and ascending price auctionsare credible, static and strategy proof at the same time, as reflected in the graphic from theirpaper below:Problem 1: Reserve price in SPA. Consider a second price auction auction where biddershave values v1, v2drawn independently from a uniform distribution on [0, 1] and submit b1, b2?The auctioneer sets a reserve price R > 0 and imposes the following rules:if bi, bj< R, there is no sale.1if bi> R and bj< R, player i gets the good at price Rif bi, bj> R the second price auction unfolds as ususal.1. Show that it is still a dominant strategy for player to bid their true valuation.We use the same reasoning as in Problem 1, but duplicate it to differentiate betweenvi> R and vi< RIf vi> R:Suppose bi> vi.- If b−i> biplayer i gets 0 and therefore is indifferent between bidding bi> viandbi= vi- If b−i< viplayer i wins and gets vi− b−i> 0 or vi− bR> 0 whether she bidsbi> vior bi= vi. Hence she is indifferent.- If vi< b−i< biplayer i wins and gets vi− b−i< 0. But she would rather lose byplaying bi= viand get 0.Suppose bi< vi.- If b−i< biplayer i wins and gets vi− b−i> 0 or vi− R > 0 whether she bidsbi> vior bi= vi- If b−i> viplayer i loses and is indifferent between bidding bi> viand bi= vi- If bi< b−i< viplayer i loses and gets 0. Had she chosen bi= vishe would havewon and gotten vi− b−i> 0If vi< R:Suppose R > bi> vi. Then player i gets 0 and is indifferent between bi> viandbi= viSuppose bi> R > vi.- If b−i> biplayer i gets 0 and therefore is indifferent between bidding bi> viandbi= vi- If b−i< viplayer i wins and gets vi− bR< 0. She would have been better offbidding bi= vi.- If vi< b−i< biplayer i wins and gets either vi− b−i< 0 or vi− R < 0. But shewould rather lose by playing bi= viand get 0.Suppose bi< vi. Then player i gets 0 no matter what because it implies bi< R.Therefore she is indifferent between bi> viand bi= vi.2. What is the auctioneer’s expected revenue if players play their dominant stragegies? Usethat E[min(v1, v2|min(v1, v2) > R)] =1+2R3The auctioneer’s expected revenue depends on R and on players’ valuations. It is equal to:2Pr(v1, v2< R) × 0 + Pr(v1> R, v2< R) × R + Pr(v1< R, v2> R) × R+ Pr(v1, v2> R) × E[min(v1, v2| min(v1, v2) > R)]= Pr(v1< R) Pr(v2< R) × 0 + P r(v1> R) Pr(v2< R) × R + Pr(v1< R)P r(v2> R) × R+ P r(v1> R)P r(v2> R) × E[min(v1, v2| min(v1, v2) > R)]= R2× 0 + R(1 − R) × R + R(1 − R) × R + (1 − R)2×1 + 2R3= 2(1 − R)R2+ (1 − R)21 + 2R3=1 + 3R2− 4R333. Find the reserve price that maximizes the auctioneer’s expected revenue. Does the expectedrevenue increase of decrease compared to a classic second price auction?The auctioneer solves maxR1+3R2−4R33. FOC is:13[6R − 12R2] = 0⇔ 2R − 4R2= 0⇔ 2 − 4R = 0⇔ R =12Therefore the reserve price that maximizes auctioneer’s revenue is R =12. The correspond-ing expected revenue is:1 + 3(12)2− 4(12)33=13[1 +34−48]=13[88+68−48]=13×54=512>13The auctioneer’s expected revenue increases compared to classic SPA.Problem 2: VCG in politicsA fisherman, a farmer, an ecologist and an retiree are part of the city council and need to decidewhether their municipality should get a wind turbine installed, and where to place it. Theyhave a choice between building it on land (policy L) or at sea (policy S). Not to get a windturbine at all is policy N. Citizens’ valuations for each of the policies are the following:3N L SFisherman 1 3 -10Farmer 2 -5 5Ecologist -10 10 10Retiree 7 -5 -51. What is the efficient choice?Add all utilities for each choice and select the choice that yields the highest utility:N: 1 + 2 − 10 + 7 = 0L: 3 − 5 + 10 − 5 = 3S: −10 + 5 + 10 − 5 = 0Therefore the efficient choice is to build the wind turbine at land.2. Suppose the city council decides to implement a Vickrey auction to decide about the windturbine: transfers dictated by the VCG mechanism will be taxes paid or compensationreceived by the four citizens. What will these transfers be?The VCG mechanism outputs the efficient choice L. Transfers in the VCG mechanism areequal to other players’ total surplus if the player is present minus other players’ total surplusif he is not (taking into account the possible change in efficient choice). For all four citizens,this gives:Fisherman: (3 − 5 + 10 − 5) − (5 + 10 − 5) = −10Farmer: (3 + 10 − 5) − (3 + 10 − 5) = 0Ecologist: (3 − 5 − 5) − (1 + 2 + 7) = −16Retiree: (3 − 5 + 10) − (3 − 5 + 10) = 0Therefire the fisherman and ecologist should respectively pay 10 and 16 taxes with a VCGmechanism.Problem 3: Construction bidding.Paris City Hall has three large construction projects to carry out: a new metro line (projectA), an extension to CDG airport (project B) and building a large start up incubator (projectC). The projects are attributed to four construction companies (Bouygues, Eiffage, Vinci andSaint-Gobain) through a Vickrey auction. We will assume that Paris City Hall can choosewhich packages to put for sale. For example it could put only A, B and C for sale, or A, andB&C only, or A,B, C and A&B&C. Companies’ valuation for various construction projects (inmillion euros) are given in the following table:A B C A&B A&C B&C A&B&CBouygues 2 7 10 9 12 17 19Eiffage 6 5 9 12 15 14 21Vinci 13 10 5 0 18 15 0Saint-Gobain 1 1 2 2 3 3 44Note that all companies’s valuations for bundles with several projects are simply the sum oftheir valuations for


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