Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 15 October 28 2008 When we did independent private values and revenue equivalence one of the auction types we mentioned was an all pay auction All pay auctions are generally not used in actual auction settings but are used to model lobbying efforts or research and development races or other winner take all scenarios involving sunk costs Another variation is an all pay version of an ascending auction called a war of attrition These are sometimes used to model industry shakeouts lots of firms trying to compete in an industry or market that is a natural monopoly or will only support a small number of firms The idea is that while the war of attrition is going on firms operate at a loss over time firms give up and drop out and eventually the remaining firms are able to operate profitably In entry scenarios like this firms which drop out stop incurring costs at the time they drop out Bulow and Klemperer The Generalized War of Attrition give another example of a war of attrition this time in a political setting The 1993 budget passed by the Clinton administration was hugely unpopular politically Democrats did not want to see the new Democratic president lose the fight but also did not want to support the bill if they could avoid it The bill passed the House 218 216 and the Senate 50 50 with the tiebreaker both after long delays as one member after another reluctantly fell into line NY Times 7 7 93 Here the prize the congressmen were competing over was the ability to not vote for the bill knowing that the game could not end until sufficiently many gave up on the prize agreed to support it Bulow and Klemperer point out that wars of attrition are also useful in modeling situations where firms which give up continue to incur costs until the game ends The example they use for this a competition to decide on a standard Suppose there are three firms with similar technologies competing to establish a standard the whole industry will use One firm may give up on trying to push through their own standard but as long as the other two firms are still battling none of them can start selling products based on the standard so even the firm that has dropped out still incurs costs until the war is over Bulow and Klemperer offer a model that nests both of these and prove some nice results 1 Model There are N K players competing for N prizes so K players must drop out for the game to end Each bidder has a private value vi for winning a prize the prizes are identical independently drawn from a distribution F with support V V with V 0 Bidders incur a cost of 1 per unit time as long as they remain in the competition and then a cost c 0 per unit time after dropping out until the game ends The c 0 case is the usual entry game they consider the limit as c 0 since when c 0 the game does not have a symmetric equilibrium in pure strategies They generally interpret c as being less than 1 but allow for it to be greater Dropouts are visible so strategies can condition on how many other players are still active They focus on symmetric perfect Bayesian equilibrium which they show is unique Results Like in an ascending auction a strategy is a plan for when to drop out in a given stage of the game Since there are private values a continuation game does not depend on the exact history of when players have dropped out only the number of players left and the lowest type that could still be in the game given history They show that each continuation game has a unique symmetric PBE Their first lemma is that at any point in the game higher types plan to wait longer before dropping out than lower types and therefore that the allocation of winners is efficient at any point in the game the probability you win is the probability you are among the highest N types remaining The logic is the same single crossing differences intuition we had in auctions The proof that equilibrium is unique is by induction on the number of players left Basically given a number N k of players remaining and a knowledge of the unique equilibrium and therefore payoffs in a continuation game with N k 1 players expected payoffs are uniquely determined by the Envelope Theorem and therefore uniquely determine expected payments in this stage which lead to a unique determination of strategies This leads to a simple characterization of the unique symmetric equilibrium of the game In the stage with N k players left when v is the lowest possible type still in the game given equilibrium strategies a player with type v plans to drop out if nobody else has in time Z v f x k 1 T v v k c Nx dx 1 F x v 2 Note that for c 1 this means dropouts occur more rapidly while there are still lots of players left in the game then slow down as there are fewer and fewer players left The intuition for T first consider the game when k 1 the next player to drop out ends the game Consider a player with type v who s supposed to drop out at time T v v 1 but who decides to wait till time T v dv v 1 instead Conditional on the game not ending before T v and for dv small he pays T 0 v dv more and in return wins a prize worth v with additional probability N f v dv 1 F v the probability that one of the other N players has a type between v and v dv T being a best response requires these to be equal integrating up to v gives the result for k 1 For k 1 the argument is a bit different Conditional on reaching time T v with k 1 too many players left waiting till T v dv for dv small still doesn t give you any chance of winning What quitting a later gains you is this we said that dropouts happen faster when there are more players in the game So if you stay in a little bit longer it speeds up the game for the other players so it reduces how long you ll have to pay the cost c for They show that for indifference to hold T 0 v v k cT 0 v v k 1 The LHS is what you pay by staying in a little bit longer the RHS is what you gain by shortening the game Repeated k times this gives T 0 v v k ck 1 T 0 v v 1 ck 1 N f v dv 1 F v Again for c 1 dropouts happen fast early in the game when k is large then slow down as k …