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# Introduction to Mathematical Economics ECON C103

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Econ C103 2003 Daniel McFadden Symmetric Nash Equilibrium Bid Functions for Various Auctions Consider a symmetric twobidder privatevalue sealedbid auction for a single item in which the highest bidder wins the item the first player pays Cb1b2 and the second player pays Cb2b1 with the function c specified by the auction rules Assume that c satisfies the condition c0b 0 so that it is consistent with individual rationality This class of cost functions includes standard first and second price auctions as well as first and second price allpay auctions in which a bidder pays respectively his own bid or the smaller of the two bids Suppose the bidders values are drawn from a known common distribution Gv that has a density gv In a symmetric Nash equilibrium each player will use a bid function b Bv To analyze such a Nash equilibrium suppose bidder 2 usesthe bid function bidding b2 Bv2 Then the payoffto bidder 1 from a bid b1 is v11b1gtBv2 Cb1BV2 Suppose the bid function b Bv is increasing and differentiable and let v Vb denote its inverse Then the event b1 gt Bvz occurs with probability GVb1 The expected payoff to bidder 1 is then V139GVb1 L Cb1BV2gV2dV2 Bidder1 chooses b1 to maximize this expected payoff Then b1 satisfies the firstorder condition 0 V1399Vb1V b1 L Cb1yBV2gV2dV2 1 In a symmetric Nash equilibrium the b1 that solvesthis condition must be b1 Bv1 Substituting this in and using the identities VBv1 v1 and V Bv1 1B v1 gives d w B V1 V1 gV1d bl 0 Cb1yBV2gV2dV2b140 lfthis differential equation can be solved for Bv with the boundary condition BO 0 then this characterizes a symmetric Nash equilibrium forthe specified auction mechanism The expected revenue to the seller is R 2 L 0 L 00BV1BV29V19V2dV1dV2 2 fv 0 fv 00BV1BV29V19V2dV1dV2 The revenue equivalence theorem implies that all auctions ofthe form given above will have the same R since they have the same assignment function awarding the item to the higher value bidder with probability one and have expected cost zero at value zero The following table gives some cases Standard 15 Price Standard 2 d Price 15 Price AllPay 2 d Price AllPay Cblib2 bi391bigtb2 b2391lbigtb2 bi minblib2 foibisivaigivadw bi39GMbi f quot quotBvgvdv bi f m BVgVdV 0 0 0 brliGVbil d Givibi Bivibiiigivibiiivtbi 1 1Gvbi EL CstWMVJdW bigivibiiivtbi evaluated at b1Bvi FOC at b1 Bv1 B v Bv v B v vgv B v vgv1Gv VBV9VGV equ igiibnmetbiid39i rfclion fovsgisidsGM V fovsgisids fovisgS1Gsds Bv Seller Expected Revenue R 2fvo1 Gvgvdv 2fvo1 Gvgvdv 2 V 01Gvgvdv 2 V o1Gvgvdv Bv when Gv v 0ltvlt1 v2 V v22 og1v v Bv when Gv ie Va avae Va1e Va V avae Wa v22a Verify as an exercise that the bid from a standard 2quotd price auction is greater than the bid from a standard lSI price auction the bid from a 2quotd price allpay auction is greater than the bid from a 1st price allpay auction and the bid from a standard lst price auction is greater than the bid from a lSI price allpay auction Show that the bid from a 2quotd price allpay auction may be larger or smaller than the bid from a 1st price standard auction depending on G and V Econ C103 2003 McFadden Existence of Walrasian Equilibrium Theorem GrandmontMcFadden 1972 De ne the closed unit simplex U pelRm p 2 0 and 1p 1 and the open unit simplex U0 p U pgtgt0 Suppose there exists a set U with U0 Q U Q U and an excess demand correspondence that maps U into nonemply subsets of JRquot and satisfies a C is bounded below ie there exists b e Rm such that b g x for all x e Cp p e U b For each p e U Cp is a convex set and px O for all x e Cp c C is upper hemicontinuous on U ie the graph px UgtltRm xeCp is a closed subset of Rmem Then there exists a p e U and a x e Cp such that x g 0 Proof Let Uk peU p 2 1mk1mk then the Ukare convex and compact and their union is U Let Xk denote the closed convex hull of Cp peUk Property a property b that px O for all x e Cp and the de nition of Uk imply that Xk is bounded and hence compact For xp e kaUk define a mapping n into nonempty subsets of XquotgtltUk by nxp x p ekaUk x eCp and p x 2 p x for all p eUk The maximands of a linear function p x on the compact convex set Uk form an upper hemicontinuous convex valued correspondence Together with properties b and c of C this impliesthat n is an upper hemicontinuous convexvalued correspondence on kaUk A fixed point theorem of Kakutani 1941 then guarantees that there exists xkpk such that xkpk e nxkpk Then xk e Cpk and O 2 pkxk 2 pxk for all p e U Consider the sequence xkpk k 12 Property a and p1xk 0 imply that this sequence is bounded Hence it has a subsequence converging to a limit point x p Property c implies x e Cpo while the property 0 2 pxk for all p e Uk implies O 2 px0 for p e U since each p e U0 is contained in Uk for k sufficiently large This in turn implies 0 2 x D Econ C103 2003 Daniel McFadden THE WINNER S CURSE Consider an auction for a single item whose value to a buyer is not known with certainty but must be estimated Each player s bid will be based on his estimate which in turn will be based on his own information and any information that he can obtain from others that he considers reliable This in turn introduces new strategic elements in the game If a player knows that in an extensive game his actions may influence the beliefs of others then he may modify his actions to convey information that gives himselfthe most benefit A very simple circumstance in which the implications of an estimated value can be seen is a secondprice sealed bid auction for a single item This item in truth has a common value v the same for any buyer which is unknown to the buyers Suppose each bidder j 1 has an estimate t of v Suppose that these estimates are independent and identically distributed with a cumulative distribution function Ftjv We will assume that higher v will shift the distribution of t1 upward so that when v is large t1 will tend to be larger Technically we will require that Ftv be decreasing in v the condition that Ftv1 2 Ftv2 whenever v1 lt v2 with strict inequality at some arguments is called stochastic dominance An implication of stochastic dominance is that the mean of Ftv2 is larger than the mean of Ftv1 and all the quantiles of Ftv2 such as the median are at least as large as the corresponding quantiles of Ftv1 For example if the estimates are uniformly distributed on the interval 02v then Ftv 12tv for O lt t lt 2v which satis es the stochastic dominance property and has Et v Note that if the family Ftv displays stochastic dominance as v increases so does the family FtvJ for any positive integer J Suppose for the moment that the players bid their estimates ofthe value of v ie they behave as ifthe t were independent private values and follow the bidding strategies that would be optimal underthese circumstances in a second price auction lfwe let to denote the jth highest estimate then the auction will award the item to the buyer with the estimate ta at the price ta Now the cumulative distribution function oft is Ft1vJ the probability that all J estimates are no higherthan this value The cumulative distribution function of ta is the probability that all J estimates are no higher than this value plus the probability that one ofthem is or Ht2V Ft2IV I J Ft2IVJ1139Ft2IV These formulas can be found in any probability theory text under order statistics The expected revenue to the seller is then E ta 0quotwath j1Htvdt where the last formula is obtained by integration by parts When this expectation is no greaterthan one the winner on average pays less than the value ofthe item and is a net gainerfrom the auction However when the expectation exceeds v the winner on average pays more than the value of the item This is called the winner s curse Due to the properties of order statistics the winner s curse will tend to occur when J is moderately large even if the individual estimates of value are on average correct Consider the uniform distribution example Ftv 12tv which yields Htv t2vJ Jt2vJ3911t2v Then E ta L2Vt2vJ Jt2vJ3911t2vdt 2vJ1J1 For J 2 E ta 2v3 and for J 3 E ta v so there is no winner s curse However for J gt 3 E ta gt 1 and the winner s curse is operating Rational players facing uncertainty about value will be aware ofthe winner s curse and will respond to it by shaving their bids in situations where it would othenNise operate Then the bids in a secondprice sealed bid auction where players are uncertain about the value ofthe item will tend to be lower than their estimates of the value To solve for the Nash equilibrium bid function it is necessary to consider the bidders beliefs about v Before obtaining their estimates t the bidders have a prior beliefthat v is distributed with a density gv Assume this is the same for all bidders and all bidders know this Suppose b Bt is the bid that a player will make given his estimate t ofthe value and assume that B is increasing and differentiable Let t Vb denote its inverse Consider bidder 1 The maximum T ofthe remaining bids given v has the cumulative distribution function FTvJ391 The payoff to bidder 1 from bid b1 given v and T is VBT1b1gtBT VBT1VbgtT The joint density ofv and T is gvJ1FTvJ392fTv Hence the expected payoffto bidder 1 is fiofT fgquotvBTgvJ1FTvJ392fTvdev This is maximized when b1 satisfies the firstorder condition 0 V b1fvoVBVb1gVJ1FVb1IVJ392fVb1IVdV In a symmetric Nash equilibrium the firstorder condition is met at b1 Bt1 or Vb1 t1 implying Bt 10vgvFtvJ392ftvdvfv gvFtvquot392ftvdv But gvFtvquot392ftvdvf 39 gvFtvquot392ftvdv isjust the posterior conditional distribution of v0 v given an estimate tthat winsthe auction and Bt is therefore the mean ofthis posterior conditional distribution given an estimate t thatwinsthe auction The firstorder stochastic dominance of Ftv with increasing v ensures that the mean ofthe posterior conditional distribution is increasing in t satisfying the assumption made earlier that Bt was increasing in t As an example assume again that Ftv t2v for O lt tv lt 2 and assume that the prior beliefs have an exponential density gv e39V Hu which has mean u We need to evaluate integrals ofthe form fquot vigvJ1Ft1vquot392ft1vdv f vi e39V J1t12vJ392dv2vu v0 t2 J391t1J3922lJJ391Hiri2Jt12l1 where Fkc fcax39k391e39xdx is an incomplete gamma function1 Then the bid function for J gt 3 satisfies Bt uF3Jt2uF2Jt2u u uJ3FOt2ut2uquot393e 2 3213 1quot39439 nt2pquot39 394 J3l1 When t2u or is fixed so that the estimate t is at the or percentile ofthe distribution of estimates then the bid function is homogeneous of degree one in the mean value u For fixed u when or is near zero Bt is approximately t2 and when or is near one Bt is approximately u Thus forJ gt 3 there is substantial bid shaving compared to each bidder s unbiased estimate ofvalue 1Using integration by parts kc satisfies the recursion kc c ke clk Ak1ck which can be iterated to give 39kc 390c e433 1 nc Finally 390c has a continued fraction expansion 390c e 11c2c3c Econ C103 2003 Daniel McFadden THE THEORY OF FIRSTPRICE SEALEDBID AUCTIONS 1 Within the class of firstprice sealedbid auctions there are a number of possible variations in environment information and rules 1 The number of potential bidders is either known or unknown with a distribution that is common knowledge 2 There may be no reservation price so that the item will definitely be sold or there may be a reservation price which is announced or unannounced in advance ofthe auction lfthere is a reservation price that is unannounced then it may be made known afterthe auction is complete eg as a losing bid or the item may simply be withdrawn from the auction If there are ties there may be a tiebreaking mechanism or subsequent rounds of bids between those tied 3 The winning bid may be announced or may be private information shared by the winner and seller so that nonwinners have only the unveri ed information that their bid was lower Nonwinning bids may be announced or may be private information shared by a losing bidder and the seller Finally a third party acting as an agent forthe seller and buyers may designate a winner or announce the withdrawal of the item from the auction without revealing the amounts of bids to either seller or buyers or revealing the reservation price Of course if an item is sold then the winning bid is known at least to the buyer and seller These information differences are not germane in a oneshot auction but are relevant if there is a possibility of resale or reauction 4 There may or may not be an opportunity for negotiation afterthe auction One alternative is that bids are binding Another is that they may contain contingencies whose value and prospects for clearing offer players the opportunity for postauction negotiation If such contingencies are allowed then there must be a mechanism forterminating negotiations with the winning bidder if the contingencies are not satisfied and negotiating with a lower bidder or restarting the auction An obvious consequence of allowing contingencies in bids is that setting contingencies becomes part of the bidder s strategy where a high bid with contingencies offers the opportunity to engage in bilateral negotiations with less effective competition from potential rival buyers Finally if one player not necessarily a bidder has a right of rst refusal RFR then all bidders must take into account the possibility that this right will be exercised so that the RFR holder preempts the winning bidder and acquires the item at the winning bid and that the item may subsequently be available following the auction through contracting with the party holding the RFR 5 lfthe item is not sold there may or may not be opportunities forthe item to be reauctioned If the item is sold there may or may not be opportunities for resale 6 The value placed on the item by a potential buyer may be known to this buyer may be unknown with a known distribution or may be unknown with a distribution that is not completely known eg in a class known up to location This is also true forthe seller No player knowsthe value ofany other player but each player has beliefs about the distributions from which other player39s values are drawn These beliefs might be symmetric among all buyers or not and might be rational or not rational the sense that each value is in fact drawn from the distribution that other players believed to hold The distribution of beliefs about a given player may be common to all other players or may be specific to each other player If there is uncertainty about values it may be independent across players or may contain common uncertainties eg different buyer39s values for oil leases are influenced by common uncertainties about the future price ofoil as well as individual uncertainty about the capacity ofthe lease The information situation on buyer39s values is critical ifthere is a possibility that the item will be withdrawn and subsequently reauctioned to the same potential buyers 2 The simplest case is J gt 1 buyers with J common knowledge no reservation price so the item will definitely be sold to the highest bidder with ties broken by random assignment among those tied Later when a reservation price is introduced assume that in case ofties all bidders have priority over the seller Bids are binding without contingencies There are no resale possibilities Buyers know their own values with certainty They do not know the values of other bidders but all know that these values are independent draws from a distribution Gv that is common knowledge Under these assumptions buyer k will have a mixed strategy depending on his value v described by a cumulative distribution function CDF Fbv with a support Bv lfthere is a pure strategy then Fbv has unit mass at the singleton Bv When Bv is a singleton call it the w function Define F bv supbltbFb v to be the probability ofa bid strictly less than b Hb FbvGdv to be the expected probability of a bid strictly less than b Hb FbvGdv to be the expected probability of a bid less than or equal to b and h b H b Hb to be the point mass at b Then Hb is the probabilitythat bidder kwill make a bid no largerthan b and hb isthe probability that he will make a bid of b In the symmetric information case assumed here this will describe the strategies of all rivals of bidder k as well Since Hb is nondecreasing there are at most a countable number of values of b at which it can have jumps at which hb gt 0 Taking account ofthe tiebreaking mechanism the probability of buyer k winning with bid b is Pb Hbquot1 Hb39J392hbJ12 HbJ39139mhbmJ1lm1lJ1ml hbJ391J HbJ HbJJHb Hb When hb 0 Pb Hb This is also the limit ofthe last formula as Hb Hb hb approaches Hb Note that Pb is always nondecreasing It is continuous at any b where hb 0 and jumps at any b where hb gt 0 At a jump it is a proper weighted average of its left and right limit HbJ391 lt Pb lt Hb The payoff to buyer k is his expected profit vbPb ln Nash equilibrium each b e Bv maximizes this payoff The figures below plot Pb and contours ofthe form p Avb for various A and v Bv consists ofthe points of contact of Pb and the northwestmost contour p Avb which touches Pb This has some general implications A bid b is in Bv only if Pb is at least as steeply sloped as the tangent contour to the immediate left ofthe tangency and no more steeply sloped than the contourto the immediate right ofthe tangency This rules out the possibility of points in Bv with the property that Pb Pb for b to the immediate left ofb orthe property that b is a jump point of P As v increases the contours rotate clockwise and as a result the contact points will necessarily roll to the right or remain fixed Figure 1 shows a case in which Bv is a singleton that is increasing in v Fig1 1 08 06 04 7 02 Pb o o 20 4o 60 80 100 Figure 2 illustrates a case where Pb has a kink resulting in Bv being fixed for an interval of v But there is then a positive probability of a tie at b and by the previous analysis Pb must then have a jump at b Then the situation in Figure 2 is impossible Consequently Pb cannot have kinks and instead must be differentiable at each b Fig 2 08quot 04 02 7 Pa Figure 3 shows a situation in which Bv contains an interval for some v Fig 3 Pb 4O 60 80 100 Fig 4 08 W 06 0477 2 Pb 02 7 0 l l 0 20 40 60 80 100 Figure 4 depicts a situation in which Bv is not a singleton for some v containing isolated points When Pb is sufficiently convex in some regions this outcome will typically occur for some v and the range 8 of Bv will not be an interval The following argument formalizes the properties of Bv that are obvious from the geometry Suppose bb are maximizing forv and v vA respectively with A gt 0 Then vbPb 2 vb Pb and vAb Pb 2 vAbPb Adding these inequalities APb Pb 2 0 This implies b 2 b Therefore Bv is nondecreasing in v For every value ofv for which Bv is not a singleton points in Bv bracket an open interval Since the number of disjoint open intervals is countable there are at most a countable number of v for which Bv is not a singleton Since G has a bounded density it follows that Bv is a singleton with probability one Next suppose a bid b at which hb gt 0 and let or Hb39Hb and B hbHb Th n PbHbJ391 orquot39139mBmJ1lm1lJ1ml m0 lt HbJ391 orquot39139mBmJ1lmlJ1ml HbJ391 m0 Since the number ofb values with hb gt O is countable there exist b x b with Pb Hb J391 2 HbJ391 Therefore lim vb Pb gt vbPb In this case a maximum does not exist Nevertheless it contradicts the supposition ofa bid b at which there is a positive probability of a tie and hence implies that when a maximum is achieved Pb is differentiable Summarizing Bv is increasing in v and is a singleton with probability one It can jump so that its range 8 is not necessarily an interval It has an inverse v Vb that is strictly increasing on 8 Note that the results so far could also have been established without the symmetry assumption ofa common probability Pb of winning However with symmetry Hb GVb b e B or HBv Gv The preceding results establish that Bv almost surely satisfies a rstorder condition for maximization of vbHbquot 1 0 E 39PBV V39BVP39BV From the condition PBv 2 HBvJ391 2 Gvquot 1 in the absence ofties this implies P BvB v J1Gvquot G v and hence Gvquot 1 vBvJ1Gvquot 2G vB v This is a differential equation in B B vvBv J1G vGv that has the solution Bv fl tJ1GtJ392G39tdtGVJ391 But this isjust the Conditional Second Value CSV the expected maximum value among bidder k39s rivals conditioned on this maximum value being no greaterthan bidder k39s value v This is the primary result How can the seller expect to do in this auction The density ofthe maximum value among all bidders is JGv391G v and hence the expected revenue from the sale is f BvJGvquot 1G vdv f f tJ1GtJ392G tdtJG vdv v0 10 vt f tJJ1GtJ392G t1Gtdt i0 To illustrate these solutions consider the concrete case Gv v6 for O lt v lt 1 where 6 is a positive parameter In this case Bv v6J116J1 is proportional to v with a proportion that is closer to one when J andor 6 are larger The expected revenue to the seller is JG1 J6J1 61 J1 6 This gets close to one when J andor 6 are large Question Are the theoretical results above completely general or can they fail ifthe smoothness of B and connectedness of the range 8 of B fail Note that a condition like Hb concave is sufficient to make 8 an interval The objective function vbHb 391 is maximized when vb J391Hb is maximized But the term vb quot391 is positive concave and decreasing so that if Hb is positive and increasing the product is concave On the other hand consider examples like Gvg v2 for O lt v lt1 Do the Bv 2J1v12J1 and Hb b212J12J1 that solve the differential equation also solve the original Nash problem 3 There is no dif culty in making J unknown with a distribution that is common knowledge This changes the Pb function to a mixture over the possible values of J In particular when the distribution of J is geometric there is a nice simplification Suppose the event J 2 1 and consider the strategy of buyer 1 Let m be the unconditional probability of J bidders Given the event that there is at least one bidder the first bidder sees the conditional probability n J nJ1 no of J1 rival bidders Each additional potential buyer will have a value v distributed with the CDF Gv The probability that the highest value among other potential buyers is no greater than v is given by QV f GVJ1nJEPBV39 J1 Recall that for fixed J Bv is strictly increasing in v This followed from the monotonicity of Pb and the geometry of the payoff function and remains true when Pv is obtained bythe mixture overJ given above Then bidder1 will win the auction only ifhis value v1 exceeds the maximum value ofall other potential buyers The probability ofthis event is Qv1 when ties are ruled out Substitute this into the firstorder condition 0 2 PBv vBvP Bv to get Qv vBvQ vB v This has the solution Bv L tQ tdtQv Then the optimal strategy for bidder 1 is to bid the expected highest value of all remaining potential buyers conditioned on the event thatthis value is lessthan the value of bidder 1 simply taking account the probability of various numbers of bidders The formulas above simplify when the number of potential buyers has a geometric distribution 11 1J Then 11J 1quot1 and Qv 11Gv Note that when A is small QO is large and Bv will be near zero As an example consider Gv v for O lt v lt 1 which in the case of fixed J leads to Bv vJ1J In this case one obtains Bv v log1Av 4 Suppose that the auction rules are changed to have an announced reservation price r For the analysis of this case again assume a fixed number of potential buyers J gt 1 From the standpoint of a potential buyer it is worth entering the auction ifthis buyer has a value greater than r With common knowledge on the distribution of values Gv this implies there is a probability 1Gr that a potential buyer will enter the auction and if a buyer enters he will bid between r and his value Suppose there is a Nash equilibrium at a symmetric bid function Bv that is differentiable and invertible and let Vb denote its inverse Then as before the probability of winning at a bid b gt r is Pb GVbquot 1 This takes account fully of the fact that some rivals may not bid at all Then as before we obtain the differential equation for Bv B vvBv J1G vGv The one difference is that we now have the boundary condition that Br r with a potential buyer not bidding if his v lt r The solution with this boundary condition for v gt r is Bv 0maxrtJ1Gtquot392G tdtGvquot391 The seller s tradeoff in fixing r is that increasing r lowers the probability that the item will be sold but raises the expected bid ifit is sold The probability of no bids above the reservation price is 1Grquot The expected revenue to the seller is then f BvJGvJ391G vdv fr f JJ1rGtJ392G tG Vdth f f JJ1tGtJ39ZG tG vgtdvdt 10 vt 39 quot4 fr JJ1rGtJ2G t1Gtdt f JJ1tGtquot392G t1Gtdt 10 39 rJGrJ391 rJ1Gr I f JJ1tGtquot392G t1Gtdt An additional issue is whether in the extensive game where the seller receivesthe bids and then declares a winner he is obligated to stick to his declared reservation price Unless he can by some mechanism as a binding legal contract precommit to this reservation price he has an incentive in the end game to accept any maximum bid above his true value for retaining the object Then absent pre commitment potential buyers expect the stated reservation price to be meaningless and the de facto reservation price to be the seller strue value which generally would be unknown but perhaps with a commonly known distribution see the next section However when the seller is able to precommit to the reservation price r then he can choose rto maximize expected revenue 5 Suppose the situation is as in 4 but the reservation price is not announced in advance ofthe auction From the standpoint of potential buyers the seller in this case actsjust like another buyer in effect putting in a bid and buying the item back for himself if his bid exceeds the others that are submitted Then what matters are the potential buyers beliefs about the distribution of reservation prices ofthe seller and whether this is common knowledge Assume that the seller s distribution of reservation prices is known to be Mb It will be convenient to define a distribution Mv MBv the apparent distribution of values for the seller which if he followings the same bidding strategy as the buyers Bv produces his actual M distribution The probability that a potential buyer will win at a bid b when all other buyers are using a bid function Bv is absent ties Pb GVbJ39139MVb A bidder will choose b to maximize vbPb leading to the firstorder condition Pb vbP b lf Bv is a Nash equilibrium bid function then GVJ391MV VBVJ1GVJ392G VMV GVJ39139V VB V This differential equation has the solution Bv fv tJ1GtJ392MtG tGtJ391M tdtGvJ391Mv 10 Since Mv MBv this relationship given G and M defines Bv implicitly and it will not be possible to obtain an explicit form for Bv in most cases 6 Asmmetric auctions Consider a firstprice sealed bid auction ofa single item with two bidders B and C Suppose that there is an announced reservation price of 300 Suppose resale is prohibited Bidder B attaches value 400 to the item Bidder C attaches value 300 with probability onehalfand value 800 with probability onehalf so his expected value is 550 Assume a tiebreaking rule that the item goes among those tied first to bidder B second to bidder C and last to the seller 7 In a firstprice sealed bid auction with buyer s values drawn from the same probability distribution that is common knowledge we know that Nash equilibrium strategies are for each buyerto bid his Conditional Second Values CSV ie given his own value nd the conditional expectation ofthe highest value ofthe remaining bidders conditioned on its being less than his value In the asymmetric value auction described above the CSV of B is the expectation of bidder C39s value given it is less than the value 400 of bidder B This CSVB is 300 The CSVCVC of C is 400 at value v0 800 and is 300 at value v0 300 the reservation price lfthe bidders follow CSV strategies then with probability 12 vC 800 and C wins with bid 400 and payoff 400 and with probability 12 vC 300 and B wins with bid 300 and payoff 100 so that the expected payoff to B is 50 and the expected payoff to C given vC 300 is zero and given vC 800 is 400 The expected payment to the seller is 350 However these strategies are not a Nash equilibrium NE Since B bids 300 with probability one if C bids 300 if vC 300 and 301 if vC 800 C s expected payoff given v0 300 is still zero but C s expected payoff given vC 800 improves to 499 This shows that in an asymmetric auction CSV bidding is not necessarily a NE 8 We claim that the NE strategies for the bidders are mixed with cumulative distribution functions FBb 450800b for 300 lt b g 350 with probability FB300 09 of a bid at 300 Fcb300 1b2300 when vC 300 FCb800 b300400b for 300 g b g 350 when vC 800 The forms ofthese distributions come from the proposition that a mixed strategy is NE when the strategies of others is given only ifthe payoff to a player is constant on the support ofthe player s strategy and no higher outside the support Thus the payoffto player C if v0 300 is 300bCFBbC391 b02300 where the first term is the difference between value and bid the product ofthe second and third terms is the probability that the reservation price is reached and the seller will sell and B will not win This expression is maximized at zero when bC 300 and is negative for higher bids lf v0 800 the payoff is 800bCFBbC391b02300 Before considering the set of points on which this is maximized form the similar payoff for B 400bB quot2 12FCbB8001 b02300 where the second term isthe probability that the item will be sold and C will not win The question is to find a pair FB and FC800 which make these expressions constant over some intervals and no larger elsewhere But clearly the first expression is constant over an interval only if FB has the form K1800b on this interval where K1 is a constant and the second expression is constant over an interval only if 12 12Fcb800 is inversely proportional to 400b on this interval or FCb800 has the form KZ400b 1 where K2 is a constant Since C loses ties it will never assign positive probability to 300 when vC 800 so FC300800 0 This gives K2 100 and hence FCb800 b300400b Since FCb800 1the upper limit ofthe support is 350 Verify that on 300350 B has expected payoff 50 from any bid and that the expected payoff from any bid 10 above 350 is less than 50 Finally if C39s bids never exceed 350 then B can never gain from bidding more than 350 and hence K1800b must be one at b 350 This establishes K1 450 We now verify directly our claim that the mixed strategies we have given are a NE With these strategies B always wins if v0 300 receiving expected payoff f35 400bFBdb 9100 450 f 35 400b800b2db 300 300 50 450 n109 9741223 lf vC 800 then the expected payoff to B is 35 400bFCb800FBdb 450 f 35 b300800b2db 50 450 ln109 300 300 25878 The expected payoff to B the average ofthese payoffs is 50 lfvC 800 then the expected payoff to C is f35 800bFBbFCdb 450 35 FCdb 450 300 300 We now show these mixed strategies are a NE Consider C39s strategy when vC 300 In this case C knows that B playing FB will always win so that C39s payoff to any bid between zero and 300 is zero and to any bid above this is negative Hence the bid of 300 weakly maximizes his payoff Next consider C39s strategy when vC 800 Bids at or below 300 never win and bids above 350 always win so the support of C39s strategy is contained in 300350 If C plays a mixed strategy G then its payoff is f 800bFBbGdb450 f3 Gdb450 300 300 for any Gb with this support so that FCb800 is weakly maximizing Finally consider B39s strategy Bids below 300 never win and bids above 350 always win so the support of B39s strategy is contained in 300350 Suppose B plays a mixed strategy G with this support Either v0 300 so that C39s bid is 300 and B always wins or vC 800 and C39s bid has distribution b300400b for 300 g b g 350 The expected payoff to B is then 2 350 400bGdb12 faso b300Gdb5o faso Gdb50 300 300 300 Then FBb which achieves this expected payoff is weakly maximizing This proves that FB FC are a mixed strategy Nash equilibrium Note that a bid of350 for B wins with probability one with payoff 50 and no higher bid can yield a payoffthis high this is the condition that determinesthe upper limit 350 ofthe support At the Nash equilibrium one has the following properties ofthe equilibrium bids B B B n I I Eb 35 bF db 350 35 F bdb 350 450i 109 302 5878 300 300 EbCvC800 f bFC bdb 350 35 FCbvC800db 300 300 300 100 ln2 3306853 Ebc 123oo EbCvC800 3153426 E maxltbBbc 12iEbB 35 bdltFBbFcb 1 300 1zEbB V4350 35 FBbFCbdb 300 14350450 ln109 350 450 323 b300800b400 b db 12350 450 ln109 350 450ln2 11252ln259 316937 Consider as an alternative mechanism a secondvalue sealed bid auction Suppose C bids his value Then any bid from B from 301 to 799 is equally good giving a payoff of 1 00 with probability 12 B s value of 400 is weakly optimal in this range Suppose B bids his value Then C s optimal response when vC 800 is to bid his value or any bid from 401 to 800 giving a payoff of 400 Hence his value of 800 in this case is weakly optimal The expected revenue to the seller is 350 A completed ascending bid auction with public bidding will start with bids of 300 from B and C If C does not bid again the item goes to B at bid 300 given the tie breaking rule If C does bid again say with bid 301 then this establishes that he has v0 800 and will continue the bidding until B stops before or at 400 Then B has no incentive to either continue or stop bidding and the auction could end at the bid 301 or continue until B bids 400 and C wins with 401 The final bids 349 for B and 350 for C are weakly optimal Thus revenue equivalence fails in general and the firstprice sealed bid auction is forthe seller inferiorto the secondprice auction Summarizing the optimal bidding strategies in the firstprice sealedbid auction are asymmetric The average bid for B is 3025878 and the average bid for C is 300 if v0 300 and 3306853 if v0 800 an overall average of3153426 Thus B bids higher than its CSV of 300 while C bids substantially less than its CSV of400 in the case v0 800 The expected revenue to the seller is 316937 versus the expected CSV of 350 that it could attain in a secondprice sealedbid auction Thus this auction fails to meet the hypotheses of the revenue equivalence theorem The firstprice sealed bid auction is not efficient since bidder B wins the auction with a positive probability when C has value 800 9 Now drop the assumption that no resale is possible Suppose that resale ofthe item from B to C is possible but because of the auction rules eg a standstill agreement by B resale ofthe item from C to B is not possible Also exclude the possibility ofresale to third parties This isjusti ed because ifthird parties had high value they would have incentives similarto C to participate in the primary auction Then the ex ante value of the item to B before the auction is the larger of his own value for holding the item and the expected price at which it could be resold in the event that he wins the auction In general the possibility of profit from resale will give B some incentive to bid more aggressively while the possibility of acquiring the item through resale will give C some incentive to bid less aggressively The strength of these incentives and their impact on bidding will depend substantially on the bargaining power ofthe two players in the resale market Assume for concreteness that in the event ofa resale from B to C the item is traded at the average oftheir values This is the Nash bargaining solution1 when each player own value establishes its threat point First consider player C s options lfvC 300 C cannot resell and cannot pro t and consequently will submit the minimum bid of 300 lfvC 800 then the expected pro t of C at bid b is 800 bFBb 2001 FBb 600 bFBb 200 where the rst term isthe expected profit obtained through winning the auction and the second term is the expected profit obtained from losing the auction and then obtaining the item through resale The expected pro t of B at bid b isthe sum ofthe probability ofvC 300 timesthe expected payofffrom winning and holding the item and the probability of v0 800 times the probability of winning and reselling at the average of vB and v0 800 or 124oo b 2600 bFCb800 2600 bFCb800 1 100 In a Nash equilibrium these payoffs must be constant on a common support and 1 The term Nash bargaining solution is a different concept than Nash equilibrium and is one proposed solution to a cooperative game between two agents each of which has a monetary payoff with a threat level that each can achieve if no bargain is made Under some plausible axioms on behavior a bargain will be struck at a division that maximizes the product of the excess payoffs that a bargain gives relative to the players threat points larger than the payoffs in other bid ranges We show that this is satisfied by FBb 150 600b for 300 g b g 450 and FCb3OO 1b2300 and FCb800 b 300600b for 300 lt b lt 450 First at this FB the payoff of C when v0 800 is 350 for 300 lt b lt 450 and bids above 450 always win and yield lower payoff Second at this FCb800 the payoff of B is 50 for 300 lt b lt 450 and bids above 450 win with certainty and have an expected payoff less than 50 Therefore the pair above is proven to be a Nash equilibrium Note that in this NE B will with some probability bid higherthan his own value and hence with some probability will incur a loss Some features ofthis equilibrium are EbB 450 45 FBbdb 450 15o In2 3460279 300 Ebc800 450 332 FCbdb 3920558 Ebc 3690428 EmaxbBbC 12 45 bFBdb 450 bdFBbFCb800 300 300 1245o L220 FBbdb 45o L220 FBbFCb800db 34602792 225 12 45 FBbFCb800db 375 300 Then the payoff to the seller is substantially higher when resale by B is permitted than in the market where resale was prohibited Bidder B gains nothing from the availability of resale receiving an expected payoff of 50 as in the previous case so that the seller gains all ofthe rents from resale through its impact on bidding in the primary auction Further the expected payoff to the seller exceeds the expected payoff from a secondprice sealed bid auction where resale is prohibited showing that in the absence of symmetric buyer values this auction format is not revenue maximizing for the seller 10 Now suppose resale from B to C but not from C to B is possible due to a standstill agreement signed by B and suppose there are no other potential buyers So far this is the same as the previous case However now suppose a third player D has a rightof rstrefusal RFR meaning that this player can supplant the winning bidder in the primary auction paying the winning bid to the original seller and then either hold the item or sell it to C Suppose that D has value zero for holding the item and that it will exercise its RFR ifand only if B wins and C agrees to buy at a price at which D has a positive profit From the state in which B wins with bid b considerthe extensive game in which C offers D a conditional purchase agreement at price b8002 the Nash bargaining solution when vC 800 and D s threat point is its cost to exercise the RFR and D then decides to exercise when b8002 gt b or b lt 800 Then C will make a purchase offer ifvC 800 and B wins at any bid b lt 800 the RFR will be exercised and C will attain a payoff 800b2 Now the expected payoff to C from a bid b is 800bFBb 800b21FBb 800b1FBb2 On the other hand bidding 300 is a dominant strategy for B since it then wins with maximum payoff 100 if v0 300 and B in any case receives payoff zero if v0 800 Given this the optimal strategy for C when v0 800 is to bid 301 win and attain payoff 499 Thus there is a NE in which the RFR is not exercised the expected payment to the seller is 3005 B has expected payoff 50 C has expected payoff zero if v0 300 and expected payoff 499 if v0 800 and D has payoff zero In this case B neither gains nor loses compared to the cases of no resale or resale available to B and the RFR holder does not gain any positive rent Nevertheless the presence of the RFR holder essentially eliminates the ability of the seller to utilize its market power to garner rents above its reservation price The analysis in cases 3 and 4 where resale is possible and wherein case 4 there is an RFR holder depends critically on the solution to the bilateral bargaining game between C and the winning bidder or the RFR holder The solution also depends critically on what information is available to the RFR holder regarding C s value and the outcome of bargaining at the time the RFR must be exercised For example an assumption at one extreme is that the RFR holder can exercise the RFR and then make a take it or leave it ultimatum to any bidder winner or loser In this case the RFR holderwill indeed exercise the RFR make the ultimatum price of799 to C and the ultimatum price of399 to B ifC does not accept Then neither B or C have any incentive to bid above the 300 level necessary to put the item in play and the RFR holder gets essentially all the rents available in the market An assumption at the other extreme isthat the RFR holder must decide to exercise or not without knowing who the winner is or what resale contracts might be possible say because the auction rules state that it must submit its own RFR reservation price in advance Assume further that it is C as potential buyer that makes an ultimatum take it or leave it offer of p gt 0 to the RFR holder when the RFR is exericsed and v0 800 Then by backward recursion a dominant strategy for the RFR holder is to accept an offer of p if the RFR is exercised receiving an expected payoff of p2 b and hence to not exercise the RFR unless p gt 2b Then I claim that primary auction bids of 300 from B and 300vC 800 from C followed if B wins by a repurchase offer of b1 from C in case vC 800 is a subgame perfect NE in which B will win with maximum payoff if vC 300 C wins with payoff 499 othenNise the RFR is not exercised and the initial seller expected revenue is 3005 In this case the rents go essentially to C 11 The cases considered are contrasted in the table below EbB Ebc EmaxbBbc NO RESALE Conditional Second Values not a NE 300 350 350 FirstPrice Sealed Bid 3025878 3153426 316937 SecondPrice Sealed bid 300 350 350 RESALE FROM B TO C PERMITTED FirstPrice Sealed Bid 3460279 3920558 3690428 RFR HELD BY D SALE TO C PERMITTED FirstPrice Sealed Bid 300 300 300 Econ C103 2003 Daniel McFadden MECHANISM DESIGN DIRECT SELLING MECHANISMS EFFICIENT AUCTIONS The theory of mechanism design provides some general insights into the construction of resource allocation mechanisms that achieve specified objectives One of its tools is to establish a correspondence between possibly complex auction designs and relatively straightfonNard but somewhat abstract constructs called direct selling mechanisms Then theoretical properties of direct selling mechanisms can be translated into propositions about auction designs Further the properties of direct selling mechanisms may be directly useful in suggesting the form of implementations that accomplish speci c purposes A primitive of mechanism design is the environment in which the allocation mechanism is considered For current discussion consider an environment in which one indivisible item is available fortrade This item is owned by one seller indexedj 0 and there are J potential buyers indexed j 1J Assume that all the players have independent private values for the item and denote these v0 v1 vJ Each value v is known to its holder with certainty but is unknown to all other players However it is common knowledge to all players that the value v of player is drawn from a cumulative probability distribution function G with GJO 0 Another possible environment which we will not consider here is that players are uncertain about their value ofthe item and the values are jointly distributed across the different players A direct selling mechanism is de ned by 1 assignment probabilities pjv0v1vJ that are nonnegative and sum to one and give the probabilities that the item will be assigned to each player 01 J 2 cost functions cjv0v1vJ that give the amount paid by each player 1J to the seller 0 3 a revelation step in which each player knowing his value v and knowing the functions pk and ck for k 0J sends a message to an auctioneer with a reported value rj and 4 an execution step in which an assignment ofthe item is made using the probabilities pjr0rJ and payments cjr0rJ are made to the seller Note that in a direct selling mechanism payments may be required whether or not a player wins the item Example 1 A secondprice sealedbid auction in which the seller also acts as a bidder has the assignment probabilities pjv0v1vJ 1vj gt maxigjvi for 0J and the cost functions cjv0v1vJ 1v gt maxvmaxgvi for 1J Example 2 A symmetric firstprice sealed bid auction in which the seller has value zero and all buyers have the same value distribution G and each bids the conditional second value CSVvJ Emaxigjvvj gt maxigjvi has for 1J the assignment probabilities pjv0v1vJ 1vj gt maxigjvi and the cost functions cjv0v1vJ 1vj gt maxigjvCSVv These examples show that secondprice and symmetric rstprice sealed bid auctions map into direct sales mechanisms that have the same assignment probabilities and payoffs as the original auctions and hence have equivalent expected payoffs This mapping is not special to these auctions In fact any auction mechanism no matter how complex with players who select Nash equilibrium strategies will produce assignment probabilities and cost functions and these in turn define a direct selling mechanism with the same expected payoffs This correspondence allows properties of direct selling mechanisms to be translated into corresponding properties of families of auction mechanisms Definition A direct selling mechanism is incentive compatible or truthrevealing if each buyer nds it optimal to report his own value truthfully when all other players are doing so A direct selling mechanism is individually rational if each player has a nonnegative expected payoff from participating Definition For player 1 the expected probability of winning is 1 P1V1 f p1V01V11V2quot39IVJG0dV0G2dV2mGJdVJ v20 vJ0 and the expected cost is 2 C1V1 f C1V01V11V2iVJG0dV0G2dV239quotGJdVJ39 v10 VJ0 The expected payoff from reporting value r1 when the truth is v1 is then 3 U1r1yV1 P1r139V139 C101 lncentivecompatibility requires that U1r1v1 U1v1v1 for all v1 and individual rationality requires that maxU1rv1 2 0 There are analogous expressions for each of the other players Example 1 continued The secondprice sealed bid auction has P1V Lvoquot39fvvoGodVoG2dV239quotGJdVJ HM GiVy k 01vl fva o max vi GodeGZdeGdv csv1vP1v Theorem A direct selling mechanism is incentivecompatible and individually rational if and only if a PJv is nondecreasing for OJ b CJO O for 1J and c the expected cost satisfies 4 Cv CJO Pjvv Flo Pjsds Remark This theorem shows that the requirements of incentive compatibility and individual rationality completely determine the cost function once the assignment probabilities are established Then two mechanisms with these properties the same assignment probabilities and the same costs at value zero will necessarily yield the same revenue to the seller Proofofthe theorem First assume that conditions ac hold Then 4 Ujrv c0 Pjrvr ffoPJsds c0 fjoPJsds f PjsPjrds The first two terms on the righthandside of 4 are nonnegative The last term is zero at r v and nonpositive othenNise Check the cases v lt r and v gt r separately and use the fact that Ps is nondecreasing in s Hence Ujrv Ujvv implying incentive compatibility and UJvv 2 0 implying individual rationality To prove the only if part ofthe theorem assume that the mechanism is incentivecompatible and individually rational Then 5 Pjrv CJr Pjvv CJv when v is true 6 PJvr CJv PJrr CJr when r is true Adding these inequalities Plv Pjrv r 2 0 Then v gt r implies PJv 2PJr and a holds Next break the interval 0v into subintervals vk1KvkK for k 1K The inequality 5 applied to the end points ofthese subintervals gives 7 CJvkK CJvk1K g PJvkK Pjvk1KvkK PjvkKvkK Pjvk1Kvk1K Pjvk1KvK Similarly the inequality 6 applied to the end points ofthese subintervals gives 8 CjvkK Cjvk1K 2 PjvkK Pjvk1Kvk1K PjvkKvkK Pjvk1Kvk1K PJvkKvK Adding the inequalities 7 over k 1K and similarly adding the inequalities 8 gives 9 Pjvv 1 PjvkKvK g cv 00 g Pjvv 5 Pjvk1KvK Letting K a co both ends of 9 converge to Pjvv fv Pjsds Hence this establishes 0 that Cjv Cj0 Pjvv fv Pjsds Cj0 fv PjvPjsds 2 Cj0 implying c s0 F0 The individual rationality condition 0 maxUJr0 Cj0 implies b D Theorem Revenue Equivalence lftwo individually rational incentivecompatible direct selling mechanisms have the same assignment probabilities then they yield the same revenue to the seller Corollary All auctions that are ef cientie assign the item to the highestvalue bidder with probability one are revenue equivalent Then in particular the four standard auctions with symmetric bidders which have this assignment probability are revenue equivalent Suppose that the value distribution GJv for buyer has a density gJv so that GJdv gjvdv and consider the case where the seller has with certainty value zero for the item The seller s expected revenue from buyer in an individually rational incentive compatible direct selling mechanism is 10 Rj onvGjdv 00o Pjvijdvo onsdsGJdv 00 0 Pjvijdv 0 stsdsGJdv 00 0 Pjvijdv 0Pjv1 Gjvdv 00 0 Pjvv 1GjvgvGdv 010 fmoquotquotfmopjV11VJVj 39 139GjVjgjVjG1dV1quot39GJdVJ vl v The seller s total expected revenue R is the sum of 10 over 1J or 11 R 211 00 from pv1vv 1Gv gv1lt31civilGrew v1 v This is maximized by setting pjv1vJ 1 for the j that maximizes vj 1Gjvjgjvj with the seller retaining the item when all the terms vj 1 Gjvjgjvj are negative Since CJO 0 it is revenuemaximizing to set CJO 0 The assignment probability just de ned will by construction be consistent with incentive compatibility and individual rationality provided that the expected assignment probabilities PJv are all nondecreasing in v A sufficient condition for this is that v 1 Gvgv be non 4 decreasing in V This is a condition on the value distribution that is satisfied by many but not all common probability distributions on the nonnegative real line The table below gives a few examples Distribution Gv v 1Gvgv Condition satisfied Exponential 1 e39 v gt 0 a gt 0 v 1a Yes Power va 0 lt v lt 1 a gt O 1 1av v139aa Yes ifa21 No ifalt1 Ratio vav vgt0 agt0 1 11av Yes ifa21 No ifalt1 When the consistency condition above is satis ed it is possible to implement the revenueoptimal direct selling mechanism above as a form of sealed bid auction with rules for the winner and the payment matching those in the direct selling mechanism This is somewhat cumbersome as it involves the value distribution of each player We assumed these are known to everyone so it is valid to have the seller use them in setting the auction rules However in reality a seller may be less than completely confident that these are known exactly There is considerable simpli cation however when the buyers value distributions are symmetric In this case one can have the inequality vj 1 G1vjg1vj gt vi 1 G1vg1v only if v gt v so that ifthe item is sold it goes to the buyer with the highest value Suppose we attempt to implement the optimal revenue mechanism in this symmetric case via a secondprice sealed bid auction with a reservation price p that satisfies 0 p 1G1pg1p As in our previous analysis of a secondprice sealed bid auction each bidder has a dominant strategy of stating his true value so the highest value bidder will win provided this value exceeds p and the winner pays the larger of p and the second highest value One can show that the expected revenue from this auction design is precisely the expected revenue from the revenueoptimal individually rational incentive compatible direct selling mechanism Then the secondprice sealed bid auction with this reserve price is revenueoptimal among all possible mechanism designs meeting these conditions

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