Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 5 Today mechanisms direct revelation mechanisms and Roger Myerson 1981 Optimal Auction Design Intro So far we ve looked at a few types of auctions and shown that from the point of view of a risk neutral seller they re equivalent But we certainly haven t considered everything since the seller need not always give the object to the bidder with the highest type and need not always give away the object at all he may only consider bids above some minimum price or have some unusual allocation rule we haven t thought of yet Today choosing among every conceivable auction what is the best the auctioneer could do That is how does the auctioneer maximize his expected profit To answer this we ll introduce the mechanism design approach then prove the main results from Myerson s 1981 paper Optimal Auction Design 1 Mechanism Design Basically a mechanism is a theoretical abstraction to model situations where a planner a mechanism designer designs a game to extract private information and solve an allocation problem Broadly speaking mechanism design takes the environment as given the players their type distributions and their preferences over the different possible outcomes and designs a game for the players to play in order to select one of the outcomes Outcomes can be different legislative proposals different allocations of one or more objects etc Formally denote the environment as N i i N X ui X i N and probabilities over A mechanism is a strategy set S1 S2 SN for each player and a mapping from strategy profiles to outcomes S1 SN X Actually given a set of strategies played the outcome selected by the mechanism can be deterministic or stochastic so really S1 SN X For example first price auction as a mechanism each player s strategy set is the positive reals mechanism allocates the good to whoever plays the highest strategy and demands a payment from that player equal to his strategy no transfers to or from anyone else Note that like with auctions the mechanism designer is assumed to be able to commit to the mechanism Just like after a second price auction auctioneer can t approach highest bidder and demand he pay his bid In general we consider two types of implementation Dominant Strategy Implementation where every player plays a weakly dominant strategy Very nice but in our setting this generally boils down to generalizations of the secondprice auction Bayesian Implementation where the players play a Bayesian Nash Equilibrium This is the one we ll be focusing on in this class A performance is a mapping of types to outcomes we say that a given performance is implemented by a mechanism if that mechanism has an equilibrium where the players equilibrium strategies lead to that mapping of types to outcomes 2 For example in a symmetric IPV world the first price auction is a mechanism that implements the efficient allocation the highest type being allocated the object along with a particular transfer from that player In general in mechanism design we don t worry about there being multiple equilibria just that the one that we want is indeed one of the equilibria Implicitly we re kind of assuming that in addition to setting up the game the mechanism designer gets to select the equilibrium played if there is more than one By using this approach we re also kind of assuming that all that matters to the players is what outcome is reached not the exact process by which it is chosen Direct Revelation Mechanisms and the Revelation Principle Much of the time we are able to restrict our attention to a particular class of mechanisms called direct revelation mechanisms Informally a direct revelation mechanism consists of the mechanism designer specifying a mapping from types directly to outcomes and asking each player in private to tell him their type Formally this is just a mechanism where the strategy set for each player matches his type space and given the mapping of types to outcomes we expect there to be an equilibrium where every player reveals his type truthfully A mechanism is feasible if it promises a possible outcome at every strategy profile or reported type profile that is if there is no combination of actions types at which it promises the same object to multiple players or anything stupid like that Lemma 1 The Revelation Principle Given any equilibrium of any feasible auction mechanism there exists an equivalent direct revelation mechanism in which truthful revelation is an equilibrium and in which the seller and all the bidders get the same expected utilities Proof is basically this take for example the first price auction Instead of running the firstprice auction the mechanism designer approaches each of the players and says tell me your type and then I ll calculate the equilibrium bids of you and the other bidders in the BNE of the firstprice auction and if yours is highest I ll give you the object and charge you that much So any deviation to reporting another type in the direct revelation mechanism would be a deviation to that type s equilibrium bid in the first price auction By the same logic a direct revelation mechanism can implement any equilibrium of any feasible auction For this reason we will focus only on direct revelation mechanisms since they re easier to analyze Again we do not require truthful revelation to be the only equilibrium just that it be an equilibrium 3 Myerson Optimal Auctions Jump back to our IPV setting N bidders each with independent type ti drawn from distribution Fi which is strictly increasing on its support ai bi Myerson does not assume symmetry but does assume independence Seller values the object at t0 An outcome is a choice of who if anyone gets the object and transfers to and from each player so we can write any direct revelation mechanism in this world as a set of mappings pi T1 T2 TN 0 1 and xi T1 T2 TN where at a profile of reported types t player i gets the object with probability pi t and pays in expectation xi t When the allocation is nondeterministic it doesn t matter whether each player pays only when he gets the object or not since the buyers are risk neutral all that matters is their expected payment at each profile Actually Myerson uses a generalization of private values by allowing for a bunch of additive adjustment functions the value of the object to bidder i is vi t ti X ej tj j6 i and the seller values the object at v0 t t0 X ej tj j N However these adjustment functions don t really