Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay is not obseved. Unde assumptions weake than independent pivate values, the joint distibution of bidde valuations is not identified see Athey and Haile 2002, so the expected evenue at a positive eseve pice, and the eseve pice that would maximize expected evenue, ae not uniquely pinned down. In a sepaate pape, Quint 2008, I calculate tight uppe and lowe bounds on these two measues fo the symmetic affiliated pivate values case; the uppe bounds coincide with the values achieved unde the special case of independent pivate values. Hee, I give an illustative example and seveal extensions. 2 Model A selle has one indivisible object to sell, and values it at. Thee ae n potential buyes, with pivate values v,..., v n whose joint distibution f is symmetic and has bounded suppot [v, v] n. Let v v 2 v n be the ode statistics of the values, and F i the cumulative distibution function 2 of v i. I conside a stylized vesion of an ascending auction: the selle announces a eseve pice, and as long as at least one buye s valuation exceeds this pice, the object is sold to the buye with the highest valuation, at a pice which is the geate of the eseve pice and the second-highest valuation. I assume that the distibution of this second-highest valuation F 2 is known exactly, but that no futhe infomation is available about F. 3 Note that expected selle pofit can be witten as π = F 2 F + Given the distibution F 2, define H implicitly by v df 2 v H F 2 = nn s n 2 sds 2 0 7428 Social Science Bldg., 80 Obsevatoy D., Madison WI 53706, United States; dquint@ssc.wisc.edu 2 Cumulative distibution functions in this pape exclude any mass at the point being consideed, that is, F i Pv i <, not Pv i. 3 The evenue assumption, and pecise knowledge of F 2, would hold exactly fo second-pice sealed-bid auctions and button auctions, and up to a bid incement fo fist-pice auctions with poxy bidding and any ascending auction without jump bids.
o, equivalently, F 2 = nh n n H n, and define π v df 2 v π F 2 H n + v df 2 v 3 The main esult fom Quint 2008: Theoem. Suppose biddes have pivate values which ae symmetic and affiliated.. Fo any >, expected evenue π is bounded above by π and below by π, and both bounds ae tight 2. Suppose in addition that π is continuous, diffeentiable, and stictly quasiconcave; let I be its maximize. Then the optimal eseve pice is bounded above by I and below by, and both bounds ae tight 3 An Example With A Paamete Fo Coelation Let ɛ, ɛ 2,..., ɛ n be i.i.d. daws fom the unifom distibution on [0, ], and let ɛ ɛ 2... ɛ n be thei ode statistics. Let biddes i s pivate value be v i = ɛ 2 + ɛ i 4 Since v 2 = ɛ 2, the obseved distibution F 2 does not depend on ; thus, this example allows us to paameteize the coelation between bidde values while holding fixed the data that would be obseved. = 0 coesponds to the case of independent pivate values, while = would be pefectly coelated values. Fo simplicity, let = 0. Result. Fo <, expected evenue is π = n n n+ n+ n+ + n n+ fo n n n+ n+ n+ + n n+ + + n n fo > 5 and the evenue-maximizing eseve pice is = n n + n+ 6 both of which ae stictly deceasing in. Since = 0 coesponds to independent pivate values, both π and ae bounded above by thei value unde IPV, and both ae deceasing in the degee of unobseved coelation. 2
4 Relaxing Affiliation Theoem 2. Suppose that v, v 2,..., v n ae conditionally independent 4 but not necessaily affiliated.. The same evenue bounds hold: π π π, with both bounds being tight. 2. The lowe bound on is still, and still tight. 3. It is not necessaily tue that I = ag max π. An uppe bound not tight on is povided by H n F 2 vdv 7 Thus, the fist pat of Theoem extends to conditionally independent values. In fact, a sufficient condition fo the evenue bounds is that fo any v [, v], Pv i < v is inceasing in the numbe of othe biddes with values v j < v. Howeve, the second pat of Theoem does not fully extend to conditionally independent values: in the appendix, we give an example whee > I. Equation 7 is still a nontivial uppe bound on, since as v appoaches v, v H n v appoaches v and F 2 v does not. 5 What If Losing Bids Ae Obseved Above, I assumed that the distibution of v 2 was known exactly, but that no othe infomation was available about the joint distibution f of values. Hee, I conside the infeences that can be made fom othe losing bids. Let b i denote bidde i s bid, and b i the i th highest bid. As in Haile and Tame 2003, I do not intepet a losing bids as an exact indication of that bidde s willingness to pay, only as a lowe bound on it. Thus, no obsevations will be able to falsify pefect coelation of bidde values, which is used to pove the lowe bounds on both π and. These lowe bounds, theefoe, ae unchanged if losing bids ae obseved. On the othe hand, if losing bids ae sufficiently high close enough to v 2, they may falsify the assumption of independence, in which case a tighte uppe bound on π will follow, which may in tun lead to a tighte uppe bound on. As a demonstation, conside the case of symmetic, affiliated pivate values when the distibution of the thid-highest bid b 3 is obseved along with F 2. Simila esults will hold fo othe losing bids. Let G 3 be the obseved distibution of b 3, and note that by assumption, v 3 b 3, and theefoe F 3 G 3. Then unde symmety and affiliation, F n F 2 F n F 2 F nc 2 F 3 F 2 n F 2 F nc 2 G 3 F 2 The fist inequality is fom the poof of Theoem in Quint 2008. Simplifying gives F F 2 F 2 n 2n G 3 F 2 8 9 4 Conditionally independent values satisfy fv, v 2,..., v n = E θ {fv θfv 2 θ fv n θ} fo some family of distibutions f θ. 3
Since the left-hand side is stictly inceasing in F, Equation 9 gives a lowe bound on F, which gives an uppe bound on π. As we saw above, an uppe bound π π imposes an uppe bound π π on ; if the losing bids ae high enough, this bound may be lowe than I. 6 Auctions With Enty Fees Results fo auctions with enty fees ae simila to the esults fo auctions without. Fist, conside auctions whee potential biddes must pay an enty fee e befoe leaning thei pivate values and paticipating in the auction. That is, playes lean e and but not v i, decide simultaneously whethe to pay e and paticipate, lean v i, and then the auction is held. I efe to this as an ealy enty fee. It is easy to show that in such an auction with symmetic biddes, the selle maximizes expected evenue by setting = and using the enty fee to extact all expected suplus fom the selles by setting e = e v v df v π 0 n Let e I denote the value of e when bidde values ae independent that is, substituting H n v fo F v in Equation 0. Theoem 3. Suppose bidde values ae symmetic and affiliated o conditionally independent. In an auction with an ealy enty fee, the optimal eseve pice is ; the optimal enty fee is bounded below by 0 and above by e I, with both bounds being tight. Finally, conside the hade poblem of auctions with an enty fee which is paid afte biddes lean thei valuations. That is, biddes lean e,, and v i, decide simultaneously whethe to paticipate, and then the auction is held among those who ente. The esults ae not as complete, but I do offe the following bounds on the evenue-maximizing paametes: Theoem 4. Suppose bidde values ae symmetic and affiliated o conditionally independent. In an auction with a late enty fee, the optimal eseve pice and enty fee, e ae not bounded away fom, 0; an uppe bound on + e is given by Refeences +e v dh n v π, 0. Athey, S. and P. Haile 2002, Identification of Standad Auction Models, Econometica 70.6 2. Haile, P. and E. Tame 2003, Infeence with an Incomplete Model of English Auctions, JPE. 3. Quint, D. 2008, Unobseved Coelation in Pivate-Value Ascending Auctions, Economics Lettes 00.3 4
Appendix Poof of Result We begin by calculating F, the cumulative density function of v : F = Pv < = Pɛ 2 + ɛ < Since we know by constuction the distibution of ɛ 2, F 2, we can ewite this as F = Pɛ 2 + ɛ < ɛ 2 = xdf 2 x = 0 Pɛ < ɛ2 ɛ 2 = xdf 2 x Now, the distibution of ɛ, conditioned on a given value of ɛ 2, is simply the distibution of ɛ conditional on being above that value. That is, knowing that ɛ 2 = x makes the conditional distibution of ɛ the unifom distibution on [x, ]. So P ɛ < ɛ2 ɛ2 = x = if x < + + x/ x if x [ x 0 if x >, ] Plugging this into the integals above gives 0 F = x x df 2x F 2 + + + / x x df 2x if if > Since v 2 = ɛ 2, F 2 x = nx n n x n ; plugging in, integating, and simplifying then gives Case : > F = n n + n n As noted in equation, when = 0, expected evenue is When >, this is π = F 2 F + xdf 2x = if if > π = F 2 F + xdf 2 x n n n n n + + n n + = n n n n+ n+ + + n n + nn = n n n n+ n+ + + n n + n xn x= x nn x n 2 x dx x n x n dx xn+ nn n+ = n n n n+ n+ + + n n + n n n nn n+ + nn n+ n+ = n n n n n n+ + nn n+ n+ n+ + + n n = n n n+ n+ n+ + + n n + n n+ 5 x= + n nn n+
and so π = n n n n n+ n + + n n + n+ n n = n n n n n + + n = + = + + n n + n n n n + n + + n + n n + n n n + n + + n + + + n < n + n n + + n n + + n Since = + > 0, > + ; the inequality then follows, since n > + n and + n > + + n. Then π < n < n < n < 0 + + + n 2 n + + n n 2 + n 2 + + n Since π < 0 fo >, we know that [0, ]. To show that π is deceasing in, ewite expected evenue as n 2 + + n { } x π = F 2 min, df 2 x + xdf 2 x 0 x { } x which is stictly deceasing in by inspection. This simply equies that min, x on a positive-measue w..t. F 2 subset of [0, ]; this is the case on [ +, ]. Case 2 : Fo, F = n, and so π = n n n + n+ n+ + n n + which is deceasing in by inspection. Diffeentiating, π = n n n n n + n Note that π has the same sign as n n + n+, so π is stictly quasiconcave. Thus, the fist-ode condition gives us the maximize, which is = n n + n+ n 6
Poof of Theoem 2 As in the poof of Theoem in Quint 2008, we show F H n ; then π π fo. Define ψ, ψ 2 : [0, ] [0, ] by ψ x = x n and ψ 2 x = nx n n x n. Fo a given distibution H of one vaiable, then, ψ H and ψ 2 H ae the distibutions of the highest and second highest, espectively, of n independent daws on H. If values ae conditionally independent, let {H θ } be the set of distibutions fom which values may be independently dawn. It is easy to show that F 2 = E θ ψ 2 H θ and F = E θ ψ H θ Next, we show that ψ ψ 2 is convex. This is because ψ ψ 2 ψ s = ψ 2 s ntn = s nn t n 2 t = ψ 2 ψ 2 t n t whee t = ψ2 s; since this is inceasing in t, and theefoe s, ψ ψ2 is inceasing so ψ ψ2 is convex. Recall also that H was defined by F 2 = ψ 2 H. Applying Jensen s inequality, Fom Equations and 3, then, F = E θ ψ H θ = E θ ψ ψ2 ψ 2 H θ = E θ ψ ψ2 ψ 2 H θ = E θ ψ ψ2 ψ 2 H θ ψ ψ2 E θ ψ 2 H θ = ψ ψ2 F 2 = H n π π = H n F 0 and the bound is tight because IPV is a special case of conditionally independent pivate values. The lowe bound on π, as well as on, is poved the same way as in Quint 2008. To show that is not necessaily lowe than ag max π, we offe a counteexample. Let n = 3, = 0, and suppose that θ takes the values 0, with equal pobabilities and when θ = 0, bidde valuations ae i.i.d. U[3, 9] when θ =, bidde valuations ae i.i.d. U [[0, 3] [9, 0]]. Fo i {, 2}, F i x = 2 ψ x i 4 2 ψ 3 i 4 2 ψ x 6 i 4 fo x 3 + 2 ψ x 3 i 6 fo x 3, 9 + 2 fo x 9 This allows us to calculate a closed-fom if messy expession fo π. While we don t have a closed-fom expession fo H = ψ 2 F 2, we can calculate it, and theefoe π, numeically. 7
Figue : An example with CIPV whee π π but ag max π > ag max π. It tuns out that while π π eveywhee as equied by Theoem 2, π is maximized at = 6.09, and π at I = 5.37, as shown in Figue. In Figue, π is not quasi-concave, so this example is not an exact contadiction of Theoem in Quint 2008 when affiliation is elaxed. Howeve, we can eliminate the lip in π nea = 9 without changing the esult. If athe than the unifom distibution on [3, 9], the CDF of each bidde s value when θ = 0 is x 3 6 fo x between 3 and 9, then π is stictly quasiconcave, and ag max π is still stictly geate than ag max π. As fo the new uppe bound on, π π π π the middle inequality is the optimality of, the fist and thid ae simply the bounds on π. π π can be witten as F 2 H n Integating the ight-hand side by pats and simplifying gives Poof of Theoem 3 H n v df 2 v F 2 vdv An auction with a eseve pice = and enty fee e = max{v n E, } max{v 2, } 2 8
achieves efficiency and extacts all bidde suplus; thus, it must be optimal. It is not had to show that in nondegeneate cases, this auction is uniquely optimal. Since max{v, } is an inceasing function of v, its expectation, and theefoe e, ae inceasing functions in the distibution F with espect to fist-ode stochastic dominance. We agued above that F F 2 eveywhee, with equality being attained fo the pefectlycoelated joint distibution. Thus, e max{v n E 2, } max{v 2, } = 0 foms a tight lowe bound. Similaly, we showed that F F I eveywhee, and since independent values ae a special case of symmetic affiliated values and conditionally independent values, equality is attainable, so e v max{v, }df I v E max{v 2, } = e I n foms a tight uppe bound. v Poof of Theoem 4 Lowe Bound We again use the pefectly-coelated values example consistent with the obseved distibution F 2, and claim that fo any, e, 0, π, e < π, 0. If e = 0 and, all playes ente, and the expected evenue is v df 2 v < v df 2 v = π, 0 Suppose, theefoe, that e > 0 fo the est of the poof. Fo a given value of v, let π v, e be the expected evenue including enty fees fom the auction with eseve pice and enty fee e when all biddes have the pivate value v fo the good, so that π, e = E v π v, e = v π v, edf 2 v We assume that each playe has an independent enty stategy τ i : [v, v] [0, ] giving thei pobability of enteing fo each ealization of thei pivate value v. Note that no playe will eve ente when v < e +, so π v = 0 fo v < e +. Fo a given v, we conside two cases: when only one playe consides enteing τ i v = 0 fo all i but at most one, and when moe than one conside enteing. In the fist case, letting x be the playe who may ente, π v, e = τ x ve + ; since τ x is zeo when v < e +, we know that π v, e max{0, e + v e+ } In the second case, note that the evenue fom the auction, excluding the enty fees, is 0 when nobody entes, when one playe entes, and v when at least two ente; thus, we can 9
expess total expected evenue as π v, e = i τ iv e + i τ i v j i τ jv + i τ iv i τ i v j i τ jv v Now, enteing biddes get no suplus fom an auction if any othe biddes ente, since the pice paid is equal to thei pivate value; so equilibium play equies that fo each i, eithe τ i v = 0, o e + j i τ jvv 0. In eithe case, τ i ve τ i v j i τ j vv Plugging this into the expession fo π v, e and simplifying gives π v, e τ i v v i Now, if moe than one playe consides enteing, let y be the playe with the second-highest value of τ i v. By assumption, τ y v > 0, so equilibium play equies o j y τ jv e v know that τ y v τ x v biddes conside enteing, e e j y τ j vv v. Letting x again be the playe with the highest value of τ iv, we e v, so i τ iv e 2; v thus, when moe than two π v, e e 2 v v Thus, we have now shown that given equilibium play by the biddes, { } e 2 π v, e v e+ max 0, e +, v v This expession is eveywhee weakly less than max{0, v }, and stictly less than v on, v] {e + }. Thus, π, e = v π v, e < v df 2 v = π, 0 Since this agument holds fo any, e, 0, it follows that, e =, 0 must be optimal. Uppe Bound Fo the uppe bound on + e, note that fo any, e, the maximum possible suplus to both the selle and the buyes is +e v df v since nobody will ente when v < + e. Since in equilibium, biddes must have nonnegative expected payoff, π, e +e v df v +e v df I v The last inequality is because v v>+e is a nondeceasing function of v, so its expectation is inceasing with espect to fist-ode stochastic dominance, and we showed in the poof of Theoem that F F I eveywhee. Optimality of, e implies π, e π, 0 = π ; combining the inequalities gives +e v df I v π, completing the poof. 0