WuhanUniversity:Introduction & ReviewModern Auction TheoryProfessor Paul MilgromSeptember 200413Various Auction MechanismsGames & MechanismsIn normal form, a gameis In normal form, a mechanismis Static auctionsa triple consisting ofa triple consisting of¾Sealed tenders¾A set of players¾A set of players¾Second price auctions¾A strategy set for each player¾A strategy set for each player¾Priority auctions¾A payofffunction mapping ¾An outcomefunction mapping ¾All-pay & two-pay auctionsstrategy profiles to payoff strategy profiles to outcomesvectors.¾Uniform price v discriminatory auctionsMechanism theory:¾Package auctions¾Payoffs are determined by outcomesDynamic Auctions¾In principle, mechanisms can ¾Dutch descending auctionsbe designed.¾English ascending auctions¾Clock vsopen outcry auctions¾…45
Dutch & “1st-Price” Auctions“Dutch ≡1st Price”Initial formulation.Theorem. Suppose outcomes are identified by A “Dutch Auction” is an auction in A “sealed tender” or “first-price the assignment of the item and the price paid which auction”is a sealed bid auction for it. Then, in reduced normal form, the Dutch in which ¾the auctioneer starts at a high price and reduces it continuously ¾the object is assigned to the auction and the sealed tender auction are the highest bidder ¾the auction ends when some bidder same “Mine!” to claim the item at ¾the price is the corresponding the current pricebid.Proof. In each, a (reduced) strategy is a single StrategiesStrategiesnumber, the item is assigned to the bidder ¾A strategyin the Dutch auction game ¾A strategyin the sealed tender specifies, for each price, whether to auction mechanism is a number, who names the largest number, and the price shout “Mine!”representing the amount bid. ¾A “reduced strategy” is a numberpaid is that number. QED specifying the highest price at which to shout “Mine!”67English & “2nd-price” Auctions“Simplified English ≡2nd Price”A “simplifiedEnglish auction” is a A “second price auction” is a Theorem. Suppose the outcome is identified mechanism in whichsealed bid auction in which as the assignment of the item and the price ¾the auctioneer raises the price ¾the item is awarded to the highest continuously bidder, but paid for it. Then, in reduced normal form, the ¾bidders observe onlythe current ¾the price is set equal to the “simplified” English auction and the second-pricehighest bid among the remaining bidders.¾at each price, bidders decide price auction are the same to become (permanently) inactive. Proof. In each, a (reduced) strategy is a single ¾the auction ends when only one A “reduced strategy”in this bidder is activesimplified English auction specifies number, the item is assigned to the bidder who ¾the last active bidder gets the the price at which to become names the largest number, and the price paid item for the final the “second highest” number. QED89
English Auctions GenerallyStrategy sets are larger in the English auction ndthan in the 2price auction.That matters because…Review Material:¾Enforcement of collusive agreements¾Attacking a competitor’s bid budgetVickrey and Hotelling¾Value inferencesValue assumptions¾Private value models¾Interdependent value models1011Review: Vickrey Auction RulesReview: Dominant StrategiesBids and allocationsTruthful reporting is a dominant strategy for ¾One or more goods of one or more kindseach bidder. ¾Each bidder i makes bids b(x) on all bundles¾Indeed, the payoff to any other bid is:i*¾Auctioneer chooses the feasible allocation x∈Xthat **ˆˆˆπ(b)=vx(b−maxb(x+bx(b()(∑)∑()iiiiijjjjij≠ij≠ix∈Xmaximizes the total bid accepted≤−maxb(x)+maxv(x)+b(x)(∑)()jijj≠ijiVickrey auction (Vickrey-Clarke-Groves “pivot x∈Xx∈X=Payoff to Truthful Reportingmechanism”) payments for each bidder i are:*p=maxb(x)−b(x)(∑)∑jjDominant strategy equilibrium j≠ij≠ix∈X¾Simplicity saves bidding costs, ¾Reduces error¾Improves predictability of results1213
Review: Hotelling’sLemmaReview: Producer SurplusLWith production set X⊂R, a price-taking firm’s indirect profit function expresses its maximum Priceprofits as a function of the prevailing prices: pπ(p)=maxpixx∈XProducer Surplus(p).Lemma(Hotelling): If π() is differentiable at p, *.then x ( p ) = ∂ π / ∂ p for all j. Moreover, if π() is jjabsolutely continuous, thenQuantityp1π(p)=π(0,p)+π(s,p)ds−1∫−1The shaded area between a firm’s 0supply curve and the vertical is the p1*=π(0,p)+x(s,p)ds1∫1firm’s producer TheoremsEnvelope theorems deal with the properties of the value function: V(t)≡maxf(x,t)x∈XTopic # 1: MWG textbook intuition. ¾If the relevant functions are differentiable, then using the chain rule,Envelope TheoremV(t)=f(x(t),t)′′V(t)f(x(t),t)x(t)+f(x(t),t)12and Related Auction Theory′=0⋅x(t)+f(x(t),t)2=f(x(t),t)2¾In the special case where f(x,p)=p⋅x, this becomes Hotelling’s lemma.But what if f(x,t) is not differentiable in x, for example, if x is a “message” and does not lie in a metric space?1617
The Derivative FormulaThe Integral Formula*LetV(t)≡maxf(x,t),X(t)=argmaxf(,t)Theorem 2(A). Let X be the choice set and x∈Xx∈X[0,1] the parameter set. Suppose Theorem 1. Take t ∈[0,1] and x∈X*(t), and suppose that f(x,t) 2exists. ¾For all t,X*(t) ≠∅¾If t<1 and V’(t+) exists, then V’(t+) ≥f(x,t).2¾for all (x,t), f(x,t) exists2¾If t>0 and V’(t-) exists, then V’(t-) ≤f(x,t). 2¾V(t) is absolutely continuous. ¾If t∈(0,1) and V’(t) exists, then V’(t) = f(x,t).2Then for any selection x(t) from X*(t),Proof:tV(t)=V(0)+f(x(s),s)∫0Vf(x,t)f(x,t)Note similarity to the derivative formula:V′(t)=f(x(t),t)2tt1819Proof of Theorem 2(A)Multi-Dimensional ParametersSince V is absolutely continuous, it is differentiable The same theorem can be applied to paths almost a multidimensional parameter space.By Theorem 1, where the derivative exists, it Sample objective: f(x,t)Nsatisfies V’(t)=f(x(t),t).¾Let t(⋅) be a smooth path through [0,1].2¾Define g(x,s)=f(x,t(s)), where sŒ[0,1].By the Fundamental Theorem of Calculus, an ¾Applying the envelope theorem and the chain rule,absolutely continuous function is the integral of its V(t(s))=V(s)=V(0)+g(x(t(r)),r)drg2∫0s=V(0)+′f(x(t(r)),t(r))⋅t(r)∫02021
Absolute ContinuityProof of Theorem 2(B)DefinetTheorem 2(B). Suppose that B(t)=b(s)ds∫.0¾f(x,)is absolutely continuous for all x∈X. .¾there exists an integrable function b(t) such that |f(x,)| ≤2Then:b(t) for all x∈Xand almost all t∈[0,1]. ′′′′′′|V(t)−V(t)|≤sup|f(x,t)−f(x,t)|Then Vis absolutely continuous and |V′(t)| ≤b(t) x∈Xfor almost all t∈[0,1].′′′′tt=supf(x,t)dt≤supf(x,t)dtt∫∫x∈XxX′′t′′′≤b(t)dt=B(t)−B(t)∫′t2223The Integrable BoundThe bounding function bis indispensableLet X=(0,1] and f(x,t)=g (t/x), where g is smooth and single-peaked with unique maximum at 1.¾V(0)=g(0), V(t)=g (1): Vis discontinuous at Applications¾The example has no integrable bound: 1tt1′′supf(x,t)=supg()=supzg(z)()2txxtx∈(0,1]x∈(0,1]z∈(0,∞)Vg(1)f (x,t)g(0)2425t
Dominant StrategiesHolmstrom’s lemmaiAn outcome is a pair (z,p) where z is a Letting Sdenote a strategy from some finite set and p is a vector Fixing the strategies played by others:of cash −iii−iiSuppose payoffs can be written in the quasi-V(t,σ)=maxu(x(σ,σ),t)iσ∈Silinear form:iiiiiiiiitu(z,p,t)=v(z,t)−p=v(t)⋅z−pi−ii*i−i=V(0,σ)+ux(σ(s),σ),sd()2∫Applying the envelope theorem leads to:0Holmstrom’s lemma: In this problem, ii−iii*ii−ii*ii−iV(t,σ)=v(t)⋅z(σ(t),σ)−p(σ(t),σ)itiii*ii′=V(0,σ)+v(s)⋅z(σ(s),σ)ds∫02627Proof IdeaGreen-Laffont-Holmstrom TheoremBy Holmstrom’slemma, once the outcome function is Under certain conditions, the VCG mechanism fixed, a player’s payoffs as a function of its type are fixed is the onlyway to implement efficient up to a constant that depends on the others’ types. outcomes in dominant strategies. Once the outcome and a player’s payoffs are fixed, its Theorem. Suppose that for each bidder i, the payment is fixed. type space is compact and smoothly path-So, the set of payment functions for player i in an efficient and v() is continuously dominant strategy mechanisms is a family of functions that differentiable. Then, any direct mechanism that varies up to a constant that depends on the others’ types. implements the efficient outcome in dominant The set also includes all the VCG payment mechanisms, strategies entails the same payments as some but that leaves no room for anything -Clarke-Groves mechanism. 2829
Proof, 1Proof, 2Let p be the payment rule of some mechanism i*ii−i−ii−i−iii*ii−i−ip(σ(t),σ(t)=−V(0,σ(t)+v(t)⋅z(σ(t),σ(t)that implements the efficient decision itperformance and let V be its value function.*i−i−i′−v(s)⋅z(σ(s),σ(t))ds∫*Let pbe a VCG payment function with value 0*i-ifunction Vand with h(t) chosen so that and*i−ii−ii*ii−i*i−iiii−iV(0,t)≡V(0,σ(t)).p(t,t)=−V(0,t)+v(t)⋅z(t,t)itApplying Holmstrom’slemma twice,−i′−v(s)⋅z(s,t)ds∫0i*ii−i−i*ii−ip(σ(t),σ(t)≡p(t,)By inspection, QED3031Bayes-Nash EquilibriumAssumptionsDefinition. A strategy profile σis a Bayes-Nash Payoff assumptionsiequilibriumof Γif for all types t, ¾Quasi-linear −i−ii¾Risk neutrality.σ(t)∈argmaxEuωσ ,σ(),t()iiσ ∈SGBelief assumptionsii−i−ii−ii=argmaxuωσ ,σ(t),tdπ(t|).()∫ii¾Identical beliefs.σ ∈SNotation.¾Types independentlydistributed. iσ=i's strategyTechnical assumptionsω=outcome function¾Conditions of envelope theorem ¾Types distributed uniformlyon [0,1]. If vis u(ω,t)=i's payofincreasing, theniπ=i's beliefsiii−1Pr{v(t)≤γ}=(v)(γ)3233
Myerson’s lemma“Revenue Equivalence”Theorem. In the standard symmetric auction model Using the assumptions, for any player i:with a single indivisible good for sale and atomless ii−iii−iiiiii−iiV(t;σ)=maxEzσ ,(t)iv(t)pσ ,(t)i()()σ type distribution, if the outcome of a mechanism is Gii−i−iefficient and the lowest type bidder always pays Myerson’s lemma: Let z ( t ) ≡ z σ ( t ) , σ ( t )()zero, then the seller’s expected revenue is the Then, at equilibrium, the expected payoffs expectation of the second highest buyer value. satisfy: τiG¾Moreover, every type of every bidder has same dviiiiV(τ)=V(0)+E[z(t)|t=s]⋅dsconditional expected payoff, given its type, as in the ∫ds0second-price −iwhere we have written V(τ) for V(τ;σ)Proof Outline. By Myerson’s lemma, the expected payoff of each type of each bidder is the same as in the Vickrey second-price auction. QED3435Vickrey’s ExamplesTwo-Person BargainingM items for sale. N bidders. Each bidder wants only one A seller has a value s and a buyer has value b, item. both distributed on [0,1]. Auction designs:In the VCG “pivot” mechanism, trade iffb>s. ¾Each of the M highest bidders pays the amount of its own ¾Seller then receives price .¾Each of the M highest bidders pays the lowest winning bid. ¾Buyer then pays price s. st¾Each of the M highest bidders pays the M+1highest bid ¾Each type of each bidder has expected payoff equal (the “pivot mechanism”).to expect total surplus, given its information. Vickrey’s surprise: all lead to the same average price!¾Explained by Myerson’s lemma. 3637
Myerson-SatterthwaiteFCC Auction ApplicationTheorem. Any mechanism that (1) supports FCC sells the licensesefficient trade in the two-person bargaining ¾Efficient outcomes are theoretically implementablein the independent, private values environmentsproblem at Bayes-Nash equilibrium and (2) ¾Theory: Vickrey auction theorementails no payments when there is no trade incurs an expected loss equal to the total FCC uses a lottery among biddersexpected gains from trade. ¾Initial misallocation may be uncorrectable by any incentive compatible mechanism in an independent Proof. private values environment¾By Myerson’s lemma, any mechanism that implements ¾Contrary to Coase theoremefficient trade with V(0)=0 and V(1)=0 has the same bs¾Theory: Myerson-Satterthwaite theoremexpected payoffs for each type of the buyer and (separately) for each type of the seller. Conclusion¾The loss incurred by the pivot mechanism is equal to ¾Getting the initial allocation right can matter, as US and the gains from experience confirms 3839Ex Post ImplementationMath ReviewA strategy profile σis an ex postequilibrium if for all type profiles, Lemma. Let f,g:[0,1]Æ[0,1] satisfy the strategy profile σ(t)is a Nash equilibrium: Giiii−i−iσt∈uσ ()argmax(,σ(t),)iσ ttIf ∀t∈(0,1)f(s)ds=g(s)ds()∫∫00-i¾σmaps ontothe set of opposing strategy profiles and¾the private values assumption applies, that is,Gii−i−iii−i−iiuσ σ(t),t)≡u(σ (,,σ(t),t)Then, f=galmost everywhere. then, an ex post the condition is “nearly”a dominant strategy equilibrium.Sometimes argued to be an “appropriate”extension of dominant strategies to implementation environments with general payoffs. 4041
Interdependent ValuesImpossibility (Ex Post)Suppose any bidder’s value for a good may depend on Theorem(Jehiel-Moldovanu, 1999). In the what others know. model described above, no player’s probability iii¾Its type is t=(t,...,t)1Nof acquiring an item can depend, at ex postii−i¾Its value ist+v(t)equilibrium, on its knowledge about the other players’ values. Can i’sinformation about j’svalue be used to improve the allocation at ex postequilibrium? Corollary. Efficient outcomes cannot “generally” iiii−ii−i−iV(t,t,)=maxz(σ,σ(t)⋅(t+v(t)+p(σ,σ(t)i−iibe implemented in dominant strategies in this iσienvironment. tiiii*iiii=V(0,t,t)+z(σ(s,t),σ(t)ds−i−i∫¾“Counterexample”: 1 always has the lowest value and 0i knows the values of players 2 and 3. So, by the lemma, the integrand does not depend on t-ii¾Idea: you can’t give me incentives for truthful reporting (except possibly on a set of tof zero measure.)iunless I’m already indifferent. 4243Impossibility (Bayesian)ProofWith private values, mechanisms besides the Observe that:VCG mechanism can sometimes achieve iiiii−i−iii−iii−i−iiV(t,t)=maxEzσ,ˆσ(tt+v(tpσ,ˆσ(t|t()()()i−iiσˆBayesian implementation of the efficient ioutcome. does not depend on t.-iTheorem(Jehiel-Moldovanu). Suppose that By the envelope theorem, Giitˆthe allocation is z ( t ) and that the conditional iiiiiiiii−i−iiV(t,t)−V(0,t)=Ezσs,t)σ(tdsexpectation depends non-trivially on t . Then ()i−i−i−i∫−i0the allocation is not implementable by any Bayesian mechanism. iSo the integrand depends only trivially on t. -i4445
ExerciseAuction RevenuesExtended Revenue Equivalence TheoremSuppose thatiii¾Suppose that payoffs are quasi-linear and the value to ¾value of good to bidder iis v(t) where each vis ii-ia bidder of winning the item is v(t,t). increasing and differentiablei¾Show that if vis continuously differentiable, then any ¾types distributed independently, uniformly on [0,1]two mechanisms such that (i) the maximum value satisfies V(0)=0 and ¾An augmented mechanism is voluntaryif the (ii) at equilibrium, the highest type bidder always winsiimaximal payoff V(t) is non-negative have the same average revenue for the seller. ¾The expected revenue from an augmented mechanism is: Ni11NR(S,ω,σ)=Ep(σ(t),...,σ(t))∑i=14647Revenue Max ProblemTotal & Marginal RevenueTypes uniform on [0,1]. Temporarily assume a single bidder with value i-1¾Values distributed according to (vv(s), where v is increasing and s is uniformly ).distributed on [0,1].ProblemmaxR(S,ω,σ)S,ω,σIf the seller fixes a price v(s), it sells when the buyer’s value is higher, which happens with Observe that for any feasible mechanism:probability 1-s. G¾Then, 1-s is like the “quantity” sold. ix(t)≥0 for all i∈N−0¾Expected total revenueis (1-s)v(s).Gix(t)≤1∑¾Marginal revenueis the derivative of total revenue i∈N−0with respect to (1−s)v(s)d(1−s)v(s)m(s′)==−=v(s)−(1s)v(s)d(1−s)ds4849
Revenue Max TheoremProof Outline: Calculate!Define the “marginal revenue” functions:Use envelope theorem to express bidder profits as a iiiiiiilinear function of the allocation probabilities (s)≡v(s)−(1−s)dv/dsExpress expected profits as a linear function of x, iTheorem. Suppose that for all i, v(0)=0 and that the gathering coefficients (by reversing the order of marginal revenue functions are non-decreasing. Then, integration). an augmented mechanism is expected revenue imaximizing if (i) each V(0)=0 (“no subsidies”) and (ii) Total value is also a linear function of x. the good is allocated according to bidder iexactly when iijjm(t)>max0,maxm(t).Furthermore, at least one Seller revenue is total value minus bidder expected ()j≠isuch augmented mechanism , a linear function of x.Maximize the linear function, subject to the constraint that only one bidder can win!5051ProofProof Continued, 1Bidder 1’s maximal payoff satisfies:Reverse the order of integration and re-express…τ1dv1111−1111V(τ)−V(0)=E[x(s,t)|t=s]ds1τ1111∫dvds11111N2N10E[V(t)]−V(0)=...x(s,...,s)ds...dsdsdτ1∫∫∫∫τ111ds0000dv11N2N1=...x(s,...,s)ds...dsds11111∫∫∫dvds11N1N000=...dτx(s,...,s)ds...ds1∫∫∫ds1So, the bidder’s expectedpayoff is:00s111111111dvE[V(t)]−V(0)=E[V(t)−V(0)]111N1N=...(1−s)x(s,...,s)ds...ds1∫∫1τ111ds00dv11N2N1=...x(s,...,s)ds...dsdsdτ1∫∫∫∫ds00005253
Proof Continued, 2Proof Continued, 3GGTotal payoff is x(tA mechanism that implements this decision )⋅v(t)performance in dominant strategies is the Total expected revenue is therefore:direct mechanism with this payment function:GGNiiR(S,ω,σ)=E[x(t)⋅v(t)]−E[V(t)]∑i=1ii−1jjiG11v(m)max(0,maxm(t) if x(t)=1()−j≠iiiiNiNii1Nip(t)=p(t)=...x(s,...,s)v(s)ds...ds−E[V(t)]∑∫∫=10 otherwise0011iNdvNiNiiiNi...x(s,...,s)v(s)(1s)ds...ds−V(0)∫∫ii=1i=1ds0011i1Nii1Ni=...x(s,...,s)m(s)ds...ds−V(0)∑∑∫∫=10011ii1N≤...max(0,maxm(s))ds...ds∫∫005455Relation to Monopoly TheoryWeak Cartels (McAfee-McMillan)Bulow-Roberts “interpretation”:Given a mechanism, the corresponding random allocation ignores the types and The “expected quantity” sold to a bidder is the assigns the good to bidder i with probability bidder’s probability of -1¾At price p, the bidder buys “quantity” 1-vx=E[x(t)].(p).¾For quantity 1-s, the bidder pays v(s) per Theorem. Suppose that for each i, ( 1 − t ) d v / d tTotal revenue for quantity 1-s is (1-s)v(s).iis decreasing and each v(0)=0. Then, any “Marginal revenue” is then:mechanism for a weak cartel (hence with iV(0)=0 for all i) is ex ante Pareto dominated d[(1−s)v(s)]d[(1−s)v(s)]iim(s)==−(for cartel members) by its corresponding d[1−s]dsrandom allocation. iiiii=v(s)−(1−s)dv/ds5657
Majorization InequalitiesProofUsing a majorization inequality: If f,g:\→\ are non-decreasing functions,iand E[f(X)] and E[g(X)] exist, then 11τdviiiiE[V(t)]=V(τ)dτ=x(s)dsdτ∫∫E00[f(X)g(X)]≥E[f(X)]iE[g(X)].dsi111dvdvidτx(s)ds=(1−s)x(s)dsIf f,g:\→\ with f non-decreasing and g ∫∫∫s0dsdsnon-increasing and E[f(X)] and E[g(X)] exist, i11dvi<(1−s)dsx(s)ds∫∫then E[f(X)g(X)]≤E[f(X)]iE[g(X)].00dsi1dvi=(1−s)xds=...=E[V(t)].∫0ds5859Weber’s Martingale TheoremProofBackground assumptions and notationLet Ibe the information available when item n+1 is n¾Standard symmetric, independent private values be sold. By revenue equivalence, the expected ¾k items sold sequentially, with each bidder eligible to win average payments by winners of the last k-n items, just one. (k+1)E[v(t)|I].is n¾Idenotes the information available to each bidder after nthe sale of item n.The expected average payment at round n for items sold in rounds starting at n+1 is Theorem. If each bidder’s bid at any round is an increasing function of his type, then (k+1)(k1)EE[v(t)|I]I=E[v(t)I]n+1nn(k+1)E[p|I]=E[v(t|I].nn−1n−1so that price must also be expected for round the auction is a first or second-price auctions with Notice that the price formula describes a martingale prices publicly announced, then the sequence of if pis adapted to forms a
Math ReviewLattices¾SublatticesTopic # 3: ¾Supermodularity¾AffiliationAsymmetric Auction Models “Single crossing lemma”with Private or Common ValuesOther single-crossing related theorems6263NLatticesSupermodularity on ℜDefinitions. Definition. A function fon a lattice is supermodularif for all lattice elements x andy, f(x)+f(y) ≤f(x∧y)+f(x∨y). ¾A lattice(X,≥)is a set with partial order such that for every x,y∈X, the following exist in X:NTheorem. f:(ℜ,≥)→ℜis supermodular if and only if for all »x∨y ≡inf{z∈X: z≥x and z≥y}, called the “join”of x and y1≤i≤N and all x>y, the function ∆(x)≡f(x,x)-f(y,x) is iii-ii-ii-i»x∧y ≡sup{z∈X: x≥z and y≥z}, called the “meet”of x and y nondecreasing in x.-i¾A sublatticeis a subset of X closed under meet and join.Note: For twice continuously differentiable functions, an 2Example. Product order on ℜ: x≥y ⇔[x≥yand x≥y]1122equivalent condition is that:2∂f≥0 for all i≠jx∨y∂x∂xxijyx∧y6465
Proving NecessityProving SufficiencyLet x=x∨y and x=x∧y. Then,Remark: Project onto any two-dimensional f(x)−f(x)subspace. Rearranging the supermodularity Ninequality to f(x)-f(x∧y)≤f(x∨y)-f(y)implies that f(x,...,x,x,...,x)−f(x,...,x,x,...,x)∑1ii+1N1i−1iNi=1∆(x)≡f(x,x)-f(y,x) is nondecreasing. Ni-ii-ii-i≥f(y,...,y,x,x,...,x)f(y,...,y,x,x,...,x)∑1i−1ii+1N1i−1ii+1Ni=1Ni componentf(y,...,y,y,x,...,x)f(y,...,y,x,x,...,x)∑1i−1ii+1N1i−1ii+1Ni=1=f(y)−f(x)x∨yxRemarks:¾The inequality follows from monotonicity of ∆i¾The middle equation follows by noticing that for each i, either yx∧yx=xor y=xand that the equality holds in both cases. iiiij component6667AffiliationProofWe treat the case of just two variables. For t >t’ & s >s’, Affiliationis the condition that the log density ln(f) is using the assumed increasing differences of ln(f(s,t)), supermodular. sf(r,t)Theorems. If ln(f(t,s)) is supermodular and has support on dr∫f(r,t)drs′2F(s|t)F(s|t)−Fs′(|t)′∫f(s,t)[0,1], then ln(F(t|s)) is supermodular. s′−1===s′s′f(r,t)Fs′(|t)F(s|t)f(r,t)drdrRemarks:∫0∫0′f(s,t)¾The theorem asserts “conditional stochastic dominance,” that s′f(r,t)is, the following conditional distribution is decreasing in sfor dr∫s′F′′′′′′(s|t)−F(s|t)F(s|t)f(s,t)all t>t’.≥==−1s′′f(r,t)F′′′′(s|t)F(s|t)F′(t|s)/F(t|s)=Pr{X≤t′|X≤t,X=s}112dr∫0′′f(s,t)¾With t =1, it asserts unconditional stochastic dominance, that is, the following is decreasing in s:∴F′′(s|t)F(s|≥′′t)F(s|t)F(s|t)∴′′F′′lnF(s|t)+lnF(s|t)≥′lnF(s|t)+′lnF(s|t) QED(t|s)=Pr{X≤t|X=s}126869
Affiliation of SubvectorsProofLet fby the density for Zand let gbe the density for Z. Suppose x>yand x>y. Then, -N11-1N-1NTheorem. If Z=(Z,…,Z) is affiliated, then Z1N-N=(Z,…,Z) is also (x,x)−lng(y,x)1N-1()()1−1N1−1Nf(x,x,s)ds1−1Nf(x,x,s)f(y,x,s)∫1−1N1−1N=ln=lnds∫f(y,x,s)f(y,x,t)dtf(y,x,t)dt1−1N1111∫f(x,x,s)(,y,)1−1N1−1Nlnf(s|y,x)ds≥lnf(s|y,y)ds1−1N1−1N∫∫f(y,x,s)f(y,y,s)−1N−1N=...=lng(x,x)−lng(y,x)()()1−1N1−1N7071Math: Single Crossing LemmaLemma. Suppose that f:ℜ→ℜis a differentiable function with the property that for all t, either f(t) > 0 or f′(t) > 0. Then fhas the strict single Single Crossing Theorems:crossing property. Monotonicity and SufficiencyRemarks: The lemma comes in several versions.¾“Weak”version of the lemma: if for all t, either f(t) > 0 or f′(t)≥0, then fhas weak single crossing property. ¾We use this lemma to repeatedlyto make revenue comparisons. Proof Sketch: Apply the mean value theorem. 7273
“Single Crossing”Monotonic Selections*:ℜ→ℜhas the single crossing propertyif for all x>y, For X⊂ℜ,define X(t,X)=argmaxf(x,t)x∈(y)>0⇒f(x)>0 and (y)≥0⇒f(x)≥0. (“strict”single crossing adds that f(x)>0)Monotonic Selection Theorem. The following two :ℜ→ℜhas the (strict) single crossing differences are equivalent:propertyif for all x>y, f(t)≡g(x,t)-g(y,t) has the (strict) ¾for all finite X, every selectionx(t) from X*(t;X) is non-single crossing :ℜ→ℜhas the smoothsingle crossing differences ¾f satisfies the strict single crossing differences , in addition to single crossing differences, it satisfies g(x,t)=0⇒(∀δ>0)g(x,t+δ)≥0≥g(x,t−δ)1117475ProofIncreasing DifferencesLet x(t) be a selection from X*(t,X) that is not A simple case of single crossing occurs when nondecreasing: for some t<t, x(t)=x>x=x(t). the differences are strictly monotonic. 010011Then Definition. The function f(x,t) has increasing (x,t)-f(x,t)≥0 0010differencesif (x,t)-f(x,t)≤00111′xx⇒′>f(x,t)−f(x,t) is increasing in contradicts strict single crossing differences.Theorem. The function fhas increasing Conversely, suppose single crossing differences differences if and only if f(x,t)+g(x) has the fails. Then for some t<tand x>x, inequalities 1 0101strict single crossing differences property for and 2 hold. Then taking X={x,x}, x(t)=xand 0100 all g:ℜ→ℜ.x(t)=x,we have a selection from X*(t,X) that is 11decreasing. QED7677
ProofPicture ProofThe relevant difference function is: ′′h(t)=f(x,t)−f(x,t)+[g(x)−g(x)]′f(x,t)−f(x,t)+2∆g′f(x,t)−f(x,t)+∆Since gis arbitrary, hsatisfies strict single g′f(x,t)−f(x,t)crossing for allfunctions gif and only if ′f(x,t)−f(x,t)+∆satisfies the property for all real numbers ∆, which holds if and only f(x,t)-f(x′,t) is increasing in t. QED7879Finite Sufficiency TheoremProof SketchLet X={x,…,x} and assume (1)-(3). Theorem. Suppose that f(x,t) has single differences. Suppose x:[0,1]→Xx() nondecreasing & onto ⇒there exist 0=t,…,t=1 with 0Nx(t)=xfor t∈(t,t). [0,1] ontoa finiteset X, nondecreasing, and Integral formula ⇒f(x(t),t) continuous ⇒for k=1,…,n, the envelope formula:f(x,t)=limf(x(t),t)=limf(x(t),t)=f(x,t)kkk+1ktt↑tt↓tkf(x(t),t)−f(x(0),0)=f(x(s),s)ds2∫f(x,t)=f(x,t) and single crossing of differencesk+1kkk0⇒for t>t, f(x,t) ≥f(x,t) and for t<t,f(x,t) ≤f(x,t)Then x(t) is a selection from X*(t,X). kk+1kkk+1k⇒for t∈(t,t), f(x,t)≥f(x,t)≥f(x,t)... n-1nnn−1n−2andf(x,t)≥f(x,t)f(x,t)...+1+2QED8081
General Sufficiency Theorem“Constraint Simplification”Theorem. Suppose that g(x,t) has smooth Theorem. Let f(x,t) be continuously differentiable .single crossing differences. Suppose x() with smooth single crossing differences and [0,1] onto Xsatisfy the (“integrable bound”) condition of the , nondecreasing, envelope theorem. Let x([0,1])=X⊂ℜbe the sum a jump function and an absolutely continuous is the sum of a jump function and an absolutely .*continuous function, and , x() is a selection from X(t,X) if the envelope formula:and only if:t.(1)x() is nondecreasing andg(x(t),t)−g(x(0),0)=g(x(s),s)ds2∫t0f(x(t),t)−f(x(0),0)=f(x(s),s)ds(2)2.∫0Then x() is a selection from X*(t,X). 8283Proof SketchFOC⇒Envelope FormulaThat (1) and (2) are necessary follows from the Theorem. Suppose that f(x,t) and x(t) are both envelope theorem and the monotonic selection continuously differentiable and that for all theorem. t∈[0,1], f(x(t),t)=0. Then, the envelope integral 1formula holds:The converse follows from the general sufficiency tU(t)−U(0)≡f(x(t),t)−f(x(0),0)=f(x(s),s)dstheorem. 2∫0QEDProof. This follows from the Fundamental Theorem of Calculus and the observation that the total derivative of f(x(t),t) is:d′f(x(t),t)=f(x(t),t)x(t)+f(x(t),t)=f(x(t),t)dt122QED8485
DiscussionMirrlees-Spence Condition3Applies to U(x,y,t):ℜ→ℜ. Myerson’s original approach¾Assume Uexists and is nowhere ¾Use constraint simplification to characterize feasible ¾Assume in a class (“piecewise differentiable x”)1¾The condition is that for all x and y, the following ratio is ¾Optimize over that class of mechanismsnon-decreasing in t: A refinementU(x,y,t)1¾Observe that the constraint simplification conditions |U(x,y,t)|2are necessary, even in the whole class of mechanisms. ¾Optimize. ¾Verify feasibility using sufficiency theorem. 8687M-S Single CrossingApplications of Constraint Simplification3Verifying equilibriumTheorem. Suppose that h(x,y,t):ℜ→ℜis twice ¾Useful to verify that a candidate equilibrium strategy derived continuously differentiable with h≠0 and |h| 21from first-order conditions or envelope formula is actually a 4bounded and for every (x,x’,y,t)∈ℜthere exists best reply for the player. y’∈ℜsuch that h(x,y,t)=h(x’,y’,t). Then, the Incentive Theoryfollowing are equivalent:¾Useful to replace incentive constraints in optimal mechanism analyses. (Optimal auction example to follow.)¾hsatisfies the Mirrlees-Spence condition¾Note, in particular, that models with quasi-linear payoffs of ¾For every continuously differentiable function f, the form y+f(x,t) where fg(x,t)=h(x,f(x),t) satisfies the smooth single crossing »y is monetary transferdifferences conditions. »fis strictly supermodular function of outcome (., “effort” or“quantity”) and typequalify for constraint
Variant of Maskin-Riley ModelBidder j with value function v:[0,1]Æℜ, j=1,The value functions are increasing & differentiable, Auctions with Weak and the reserve price r is in the range of both functions.& Strong BiddersConsider increasing strategies βsatisfying jβ(r) = β(r) = Defining the “matching function”−1m(t)=β(β(t))219091First-order conditionsUnique EquilibriumBidder 1 of type tequivalently chooses its Theorem (Maskin-Riley).The system of probability sof winning by solvingequations below has a unique solution (β,β),12and it describes the unique equilibrium of the maxsv(t)−β(s)()12sauction game:and bidder 2 solves a similar problem. At −1m(t)=β(β(t))21equilibrium, we must have s=m(t), so the first-order −1conditions for bidders 1 and 2 are:′0v(m(s)−β(s)−sβ(s)122−1′0=v(m(t)−β(t)tβ(t)′0=v(m(s)−β(s)−sβ(s)211122−1−rβ(v(r))=(v(r))′0=v(m(t)−β(t)−tβ(t)1122211β(1)=β(1)129293
Ranking Bid DistributionsRanking valueÆbid functionsTheorem. Suppose that for all t∈(0,1), v(t)>v(t). Then, Theorem. Suppose that values are drawn from 12for all t∈(0,1), β(t)>β(t). •12the distributions F(|0) for the “weak” bidder •Proof. Let ()=β(1−)−β(1−).and F(|1) for the “strong” bidder, where 12For any tsuch that h(t)=0, β(1−t)−β(1−t) & m(t)=(F(v|s)) is supermodular. Then for each Hence, possible value, the strong bidder bids less than h′(t)=β′(1−t)−β′(1−t)21the weak bidder. 1=(v(1−t)−v(1−t)>0Proof. Exercise: Use the first order conditions 121−tto apply the single crossing lemma to the following function:Since h(0)=0, h(t)>0 for all t>0 (a single crossing lemma).QEDh(v−t)=β(t)−β(t)019495Ranking ProfitsTheorem. Under the hypotheses of the previous theorem, the equilibrium expected profit of a “strong” bidder with any value v is higher in the second-price auction than in the first price auction. The reverse The Drainage Tract Modelinequality holds for the weak bidder. Proof. The strong bidder’s probability of winning is lower in the first-price auction than in the second-price auction. Apply Myerson’s lemma.¾A symmetric argument applies for the weak to Drainage Tract Model9697
Winner’s Curse: IntroductionTypes of TractsModel and Idea:Wildcat drilling¾The value V of a certain item, say the right to extract oil from¾Drilling in a previously unexplored tract, is the same for all bidders¾Each bidder j makes an unbiased estimate Xof V. Drainage tractsj¾Each bidder bids the same increasing function of its estimate.¾Wells near previously drilled wells. ¾Then, the winner’s estimate is the highest of the ¾Drilling experience provides superior information ¾A selection bias results for the winner: about geologic structure, likelihood of finding [max(X,...,X)|V]>maxE[X|V],...,E[X|V]=V()We now study Wilson’s drainage tract model. Directions for analysis¾Bidding strategy to anticipate the winner’s curse is subtle. ¾Information policy can influence bids, efficiency, and profits. 9899Formulation & EquilibriumProof: EquilibriumA “common value” model of a sealed tender auction in which the Verifying that 2 is playing a best reply. 1“neighbor” observes a signal tabout Vand the “non-neighbor” ¾By construction of 1’s strategy, 2 cannot earn more than an uninformative signal t. ¾Bidder 2 earns zero from this strategy. 12Without loss of generality, we let tandtbe uniformly distributed 1Verifying that 1 is playing a best reply. on (0,1), take v(s)=E[V|t=s], and assume that v(s) is nondecreasing.¾We apply the sufficiency theorem. »Single crossing differences is verified for the usual reason.Theorem: This game has an essentially unique Nash equilibrium. 1211»The bid function is nondecreasing. The equilibrium strategies are β(s)=β(s)=E[v(t)|t<s].»Letting F(v) be the distribution of 1’s value estimate.¾“Essentially unique” means that the distribution of bids and thevpayoffs for each type are the same at all equilibria.π(v)=F(v)v−EV|V<v=F(s)ds[])∫¾Surprise!!Bid distributions match!!0»Thus, the envelope formula is satisfied. QED100101
Proof Sketch: UniquenessWhy Distribution Matching?Neighbor’s strategy is uniquely determined, because:A surprising feature of the equilibrium is that it ¾non-neighbor must expect zero profit from the lowest bid in appears “symmetric,” with both players having the the support of its strategy, (because…) same distribution of bids. Why would they?¾non-neighbor must earn zero expected profit from every bid Answer: in the support of its mixed strategy, (because…) ¾study the model in which there is actual symmetry, with ¾the supports of the two bid distributions must “essentially” 12bidders observing equally informative signals tand t. coincide, (because…) 12¾First-order condition for β(t) and β(t) depends on the 12Therefore, the unique equilibrium strategy for player values in the event of a tie: v(t,t) and v(t,t). 111121 is β(s)=E[v(t)|t<s] . ■¾In the symmetric case, v(t,t)=v(t,t)¾In the present case, despite asymmetry, we still have 12v(t,t)=v(t,t). So, the approach to the symmetric case also applies to this case. 102103N Uninformed biddersEmpirical Success?Theorem. Suppose the seller sets a reserve r Surprising predictionsaccording to distribution G satisfying bid distribution for informed and uninformed 1biddersG(r)>Pr{β(t)<r}. Then the profile (β,F,…,F) is bid independent of the number of uninformed a Nash equilibrium of the model with N biddersuninformed bidders if and only if βis the pure strategy given by the data well for higher bids above the range of 11β(s)=E[v(t)|t<s] reserve prices (~$2 million), but not for lower the “distribution matching” condition holds: a regression test to predict winning bids, coefficient of number of non-neighbors is close to zero. …1F(b)F(b)G(b)=Prob{β(t)<b}1N104105
Hendricks-Porter-WilsonMath ReviewDefinition. “A is as risky as B” means that for every convex function f, E[f(A)]≥E[f(B)].Theorem: E[V|X,Y] is riskier than E[V|X].Proof. By the law of iterated expectations and Jensen’s inequality, for every convex function f, EfE[V|X,Y]EEfE[V|X,Y]|X()={()}≥EfEE[V|X,Y]|X[]=EfE[V|x]{()}Application: Let F and G be the cdf’sof E[V|X] and E[V|X,Y]. Consider the convex function f(v)=max(0,v-x). Then, for all x, ∞∞∞(s−x)dG(s)≥ (s−x)dF(s)⇒(s−x)d(G−F)(s)≥0∫∫∫x∞x⇒(G(s)−F(s))ds≤0⇒(G(s)−F(s))ds≥0∫x0106107The Value of PublicityProof…to the informedbidder—the neighbor.Let Gdenote the distribution of E[V|X,Y] and Fthe distribution of E[V|X]. By the envelope Let A be the neighbor and B the non-neighbor. We endogenize A’s information , when the estimate is v, profits in the two cases are:Theorem. Consider A’s profit when A:vv(i) Observes X only and has B know thatG(z)dz and F(z)dz∫∫00(ii) Observes X and Y and has B act as if only X was observed.(iii) Observes X and Y and has B know thatBy the math review application (“second-order stochastic dominance”), the first of these is A’s expected profits are higher for option (ii) than for option (i). Option (iii) has a higher conditional expected payoff than larger. (ii) for every realizationof E[V|X,Y].Suggested Intuition:A more severe winner’s curse causes the non-neighbor to bid “less aggressively” (in a relevant, but very particular, sense).108109
The Value of SecrecySeller’s Expected Receipts…to the uninformedbidder—the non-neighbor.Theorem. For the uninformed bidder, equilibrium expected profits are zero. Let Fbe the distribution Theorem. Let A observe X and Y and consider 1of the value v(t). Then the informed bidder’s ex B’s profit when it…anteexpected profits are:(i) observes nothing and A knows that.∞(ii) observes X and A knows that.π=F(z)1−F(z)dz()∫(iii) observes X+ε, where εis independent noise, and A 0knows the seller’s expected receipts areE[v(t)]-π.(iv) observes X but A bids as if B observed nothing. B’s expected profits are zero under options (i) and (ii) and positive under options (iii) and (iv).Interpretation: The non-neighbor cannot scare off its competitor and prefers to hide what it Royalty Reports-1With value v, the neighbor maximizes F(β(b))(v-b).Suppose A observes X and Y and the reports X to the seller (. a royalty report). Hence, by the envelope theorem, the neighbor’s equilibrium expected profits with value estimate vare:Theorem. The policy of revealing X reduces A’s vexpected profits (and raises the seller’s F(z)dz∫0expected receipts rise by an equal amount).The bidder’s expected profits are∞v∞F(z)dzf(v)dv=f(v)dvF(z)dz()()∫∫∫∫000z∞=F(z)1−F(z)dz()∫0The last assertion of the theorem is immediate. ■112113
ProofMore Profit ExpressionsDefine functions, make assumptions:The neighbor’s expected profit is:¾h(x)=E[V|X=x]. Assumeh’(x)>0.∞∞EF(z|X)(1−F(z|X)dz=E[F(z|X)(1−F(z|X)]dz¾k(x,y)=E[V|X=x,Y=y]. Assumek>0.∫∫
x00↑ in F(z|X)↓ in F(z|X)A neighbor of type x who bids as if it were of type z earns ∞(h(x)-b(z))F(z). At equilibrium, z*=x. Hence, by the envelope ≤Ex[F(z|X)](1−E[F(z|X)])dz∫0theorem, the expected profits of type xare:∞xF′(s)h(s)ds=F(z)1−F(z)dz()X∫0∫0Ex anteexpected profits are therefore:¾The inequality can step follows from majorization (alternatively, ∞x∞Jensen’s inequality applied to the concave function ϕ(y)=y(1-y) F′′(s)h(s)dsf(x)dx=f(x)dxh(s)F(s)ds(X)X(X)X∫∫∫∫evaluated at the random variable F(z|X)).000s∞¾The last step follows from the law of iterated expectations. ■′=1−F(sF(s)h(s)ds()XX∫0114115Information RevelationDecomposing EffectsAnalogously, if the seller observes and announces that Y=y, Suppose the neighbor observes X,the seller expected profits are thenobserves Y, and the seller’s policy is to report xinformation, the neighbor’s expected profits change F(s|y)k(s,y)dsXx∫0by an amount ∆=W+Pthat reflects two different When the seller’s policy is to announce Y, ex anteexpected effects:profits are¾W: the weighting effect--Yreduces (or, if negative, vE1−F(s|YF(s|Y)k(s,Y)ds()increases) the weight of the private information Xin the YXXx∫0“multiple regression” estimate of V. Denote this effect on Define ∆to be that expected profit minusthe expected profit profits by the seller reports no information:¾P: the publicity effect--Yconveys information about X, vmaking Xless private and reducing A’s information rents. ∆=E1−F(s|YF(s|Y)k(s,Y)ds()YXXx∫0Denote this effect on profits by P.v′−1−F(sF(s)h(s)ds()XX∫1161170
Two EffectsExample: “Neutral Information”Define the weighting effect W and the publicity effect P by:Suppose V=X+Ywhere Xand Yare independent.∆=W+PThen, ∞P=F(x)(1−F(x))−E[F(x|Y)(1−FxYh′(|))(x)dx≥0[]¾W=0: revealing Ydoes not effect the weight accorded to XXXX∫0
Integrand is positive!Xin estimating V, that is, h´(x)≡k(x,y)≡∞¾P=0: revealing Yconveys no information about X. W=Eh′(x)−k(,Y)F(x|Y)1−F(x|Y)dx()()xXX∫0Therefore, revealing Ydoes not affect expected =E(h(X)−k(X,Y))(2F(X|Y)−1)[]Xprofits or expected revenues.¾The second expression for Wcomes from integrating by parts. Discuss economic ideas captured by the decomposition. 118119Informational SubstitutesProof that W ≥0Suppose that Xand Yare distributed according to a We already know that P≥0. Calculating,joint density fwith log(f) strictly supermodular:W=E(h(X)−k(X,Y))(2F(X|Y)−1)[]X2∂logfx,y()()>0 everywhereEEhX=(()−k(X,Y))(2F(X|Y)−1)|X∂x∂yX
Decreasing in YDecreasing in YTheorem. Assume that k>0, k>0 and log(f)is xy>EE[h(X)−k(X,Y)|X]E[(2F(X|Y)−1)|X]Xstrictly supermodular. Then, P>0 and W>0.=E0⋅E[(2F(X|Y)−1)|X¾Revealing information then reduces the informed bidder’s []Xexpected profits and increases the seller’s expected =0receipts by P+W.The inequality is by majorization. ■120121
Informational ComplementsSuppose X=V+Y, where V and Y are independent.Then, ¾It is obvious that the reported information is useless to Topic #4:the uninformed bidder. Hence, the situation can formally be mapped into that covered by our results. Hence, the Costly Entryinformed bidder’s profits rise from the revelation of Y.¾Evidently, P<0, so W>0. 122123Costly Sequential Entry, 1Costly Sequential Entry, 2The model (McAfee-McMillan) is as follows:¾valuations are drawn iidfrom a distribution F, but are not Theorem. If the reserve price is zero and a freely known to -price auction is used, the number of ¾the seller commits to auction rules and a reserve price Nat the “sequential entry equilibrium” ¾bidders make entry decisions in sequence, each knowing is the number that maximizes expected total the rules and the past history of entry decisions. surplus net of entry costs. ¾a bidder who enters incurs entry cost cto learn its own valuation. ¾consider a “sequential entry equilibrium” in which the first bidders enter and learn their values as long as expected net profits are non-negative, while other potential entrants stay out. 124125
Costly Sequential Entry, 3Costly Sequential Entry, 4Continuation of Proof...Proof. There are several steps to the proof, as follows.When the reserve is zero, a bidder who expects to be the last entrant has expected profit from the Lemma. The incremental expected contribution nd(2-price) auction equal to his expected toexpectedsurplus is declining in the number of contribution to expected social .¾By inspection of the surplus ¾The kentrant’s contribution to total surplus when its value is xand the highest opposing value is y is given When the reserve is zero, a bidder enters if and by (x-y)1, which is a nonincreasing function of if its expected profit from the auction exceeds {x>y}the entry cost.¾The maximum order statistic from a sample of size k is everywhere weakly larger than the maximum order ■statistic from the subsample. (1)EX−X1¾Hence, ( ) (1 ) is decreasing in k. kk−1{X>X}kk−1126127Costly Sequential Entry, 5Costly Simultaneous Entry, 1Theorem. Let Nbe the optimal number of entrants. The model (Levin-Smith) is as follows:Then, the second price auction with zero reserve and ¾valuations are drawn iidfrom a distribution F, but are not freely known to fee(1)(2)¾the seller commits to auction rules and a reserve price =EX−X−cNNN¾the K potential bidders make their entry decisions simultaneously, knowing the only rules of the auction and is an auction that maximizes the seller’s expected total the reserve. revenue at a sequential entry equilibrium.¾each entrant incurs entry cost cto learn its own valuation. thProof. With that fee, the Nentrant has expected net profits of ¾after entry, bidders learn (alternately, do not learn) the zero. Then, by definition, there are Nentrants at the sequential number of entrants entry equilibrium, so total surplus is maximized. By symmetry, all ¾consider a “symmetric simultaneous entry equilibrium” in bidders have expected net profits of zero. Hence the seller’s which each bidder enters with probability pexpected revenue equals the maximum expected total surplus. ■128129
Costly Simultaneous Entry, 2Costly Simultaneous Entry, 3Theorem. In the symmetric simultaneous entry Theorem. The seller’s expected revenue is reduced by equilibrium, the expected-revenue-maximizing a mean-preserving spread in the number of bidders. reserve price is zero. At this price, the seller captures Proof. By the lemma of the sequential entry section, the the entire social surplus is a concave function of the number of Proof. At a mixed strategy equilibrium, the bidders’ expected net participants. The result then follows by the Rothschild-profits must be zero, so the seller captures the total surplus. Stiglitz theorem. ■At a reserve price of zero, a bidder enters if and only if its ¾Thus, random participation is bad for sellers. expected profit, which is the same as its expected contribution to ¾Do we see mechanisms to reduce it?total surplus, exceeds its entry cost. So, the probability of entry p that maximizes expected total surplus is an equilibrium probability of entry. Since this zero-profit equilibrium is unique, the equilibrium p maximizes expected total surplus and hence seller revenue. ■130131A Search Theory ResultAuction Entry as SearchSequential auction entry model.Sequential search model.¾auctioneer controls entry process¾infinitely many items¾entrant incurs cost cto learn its value¾each item searched costs c¾bidder’s value is distributed as F¾value of each item searched is distributed as F¾only one item may be sold¾only one item may be takenTheorem (Riley & Zeckhauser, 1983).Let V* be the optimal Theorem. Let V* be the optimal value of this value of the searchproblem. Then, the maximum expected search problem. Then, the optimal policy is to revenue in the auction problem is also V*. The optimal policy issearch sequentially until an item of value at to set a reserve equal to V* and to sell at that price to the first least V* is found and then to take that item. entrant willing to pay it.Proof. By inspection. ■Proof. By
Two-Stage ProceduresModeling “Indicative Bidding”Exploratory ModelIn auctions for business assets, bidders incur costs ¾each bidder j=1,…,Nlearns its value v¾“due diligence,” investigating the condition of the ¾bidders make indicative bids b(v)j¾analysis, evaluating business plans that use the assets¾seller selects the top n≥2 bidders to proceed to stage 2The sale is often conducted in two stages:¾those nbidders each incur a due diligence cost c¾in stage #1, potential bidders are identified and make ¾those nbidders participate in a second price auctionpreliminary bids to “indicate interest.” Two Common Questions:»these “indicative” bids are used to “screen” bidders ¾Why does the seller want to limit the number of bidders ¾in stage #2, bids represent binding stage 2?»buyers who are invited to stage #2 are offered extensive ¾Why does a bidder not bid an infinite amount at the access to voluminous, confidential business data indicative stage?134135Can Indicative Bidding Work?DiscussionYe further shows that even if there is information Theorem(Lixin Ye, 2000). In the exploratory learned before the second stage, equilibrium model, there exists no pure symmetric generically fails to bidding strategy.Reason: pure increasing equilibrium strategies Proof. We show here only that there exists no exist only if for all v, a marginal entrant of type v increasingequilibrium bid function b. Suppose is indifferent about entering. While this condition otherwise. Then, a bidder of type v does better to bid may hold for certain specialized models, it fails b(v-c) at the indicative stage, because such a bid generically. loses only when there is some other bidder with value v'∈(v-c,v), and in that case a successful bid would incur a loss in the continuation game. ■136137
Uniform Price RulesAuctioneer sets supply Q(p)¾Initially, assume an inelastic supply Q. Each bidder j Multi-Item Auctions¾Has a value function V(q) for goods acquiredj¾Bids a schedule of prices and quantities (p,q), jkjkk=1,2,…K. jPseudo-Vickrey rules: The auctioneer ¾Allocates goods to the Q highest bids¾Sets the price equal to the highest rejected bid.¾(Using lowest accepted bid leads to similar results).139140Example & lessonsRole of the “Shoe”Based on work by David McAdams (2002)Rules: 10 items for sale; seller takes 10 Suppose the seller increases supply, but makes it elastic. highest bids, reserve price = promises to ¾Sell 10 if the price is at least 1. Bidders: 10 bidders, each with a value of ¾Sell 11 if the price is at least 40. 100+εfor as many items as it can get. ¾Sell 12 if the price is at least 70.¾Sell 13 if the price is at least 85.A “collusive-seeming” Nash equilibriumEvery equilibrium has Q = 13 and price ≥85.¾Each bidder bids 100 for a single item and 1 Elastic supply eliminates bad each additional item. ¾Equilibrium price = highest rejected bid = 1Effect is strategic: increasingsupply increasesthe price!! ¾There are many other equilibria, and all are robust to model
General Lessons Ausubel & Cramton: “Every uniform price auction encourages some form of demand reduction.”¾Idea: same as traditional monopoly theory. Making supply elastic can drastically reduce strategic price manipulation in uniform price auctions. 143
