CC BY
2Maria Angeles de Frutos1 and Maria Paz Espinosa2
1 Universidad Carlos III de Madrid Departamento de Economda C./ Madrid, 126, 28903 Getafe, Madrid, Spain E-mail: frutos@eco.uc3m.es 2 Universidad del Pats Vasco Departamento de Fundamentos del Andlisis Economico II Facultad Ciencias Econdmicas y Empresariales Avda. Lehendakari Aguirre 83, 48015 Bilbao, Spain E-mail: mariapaz.espinosa@ehu.es
Abstract. This paper analyzes auctions for big facilities or contracts where bidders face financial constraints that may force them to resell part of the property of the good (or subcontract part of a project) at a resale market. We show that the interaction between resale and financial constraints changes previous results on auctions with financial constraints and those in auctions with resale. The reason is the link between the resale price and the auction price introduced by the presence of financial constraints. Such link induces a potential loser to modify the auction price in order to fine-tune the winner’s resale offer, which may require forcing the winner to be financially constrained.
Keywords: auctions, big facilities, resale, financial constraints, subcontracting
Introduction
The allocation of a big contract or facility usually involves a small number of qualified buyers who assign a large value to the good and face financial constraints. One specific example is the allocation problem of the European Spallation Source that had to be allocated to a single country or location but whose property can be shared after the initial allocation, to alleviate the winner’s financial constraints.1 Similarly, operating licences (e.g., in the telecommunications sector) are awarded to one firm, and (some or all of) the actual services can be subcontracted.2 Partial resale or horizontal subcontracting is also a common assumption in two-stage contract games (Kamien et al., 1989; Spiegel, 1993; Chen et al., 2004; and Meland and Straume, 2007).
* We acknowledge comments and suggestions from participants at the M-SWET 2009 and GTM 2010. We gratefully acknowledge financial aid from MICINN (SEJ2007/04339/001 and EC02009-09120) and Gobierno Vasco, DEUI (IT-313-07).
1 On September 10, 2008, the European Strategy Forum on Research Infrastructures (ESFRI) published its report on the three candidates to host the European Spallation Source (ESS): Lund in Sweden, Bilbao in Spain, and Debrecen in Hungary. A decision was reached in May 2009 in favour of Lund. In June 2009, Spain and Sweden agreed to collaborate in order to build the European Spallation Source; a facility for manufacturing some accelerator components will be built in Bilbao.
2 There is horizontal subcontracting or horizontal outsourcing whenever parts of the final production of a good are subcontracted to rival firms.
Previous work on auctions with resale relies mostly on the potential inefficiencies of the auction allocation mechanism to provide the basis for resale. An inefficient allocation may result from noisy signals at the time of the auction, as in Haile (2000, 2001, 2003), from asymmetries between bidders when the auction is conducted as first price, as in Gupta and Lebrun (1998) or Hafalir and Krishna (2008), or from the presence of speculators who value the object only by its resale price, as in Garratt and Troger (2006). In contrast, in our model the auction is a second price auction and there are no pure speculators, all participants value the good, and the resale market is justified by the presence of financial constraints which may force the winner of the auction to sell part of the property of the good. That is, if bidders were wealthy enough the resale market would be inactive and it is the presence of financial constraints which makes participants aware that there will be partial resale.
The main purpose of the paper is to check the robustness of previous results of auctions with resale and auctions where bidders face financial constraints. This is particularly important when auctioning big facilities or contracts. Our results are driven by the link between auction price and resale price, which induces a potential loser to increase the auction price in order to fine-tune the winner’s resale offer, which may require forcing the winner to be financially constrained. We first compare the outcomes of the auction with resale and financial constraints, to those without resale or without financial constraints, and find that the loser’s incentives to modify the resale price may preclude a truth telling behavior and also bidding the minimum between valuation and wealth as part of the equilibrium. We also find that the presence of financial constraints may eliminate the speculative equilibria a la Garrat and Troger in auctions with resale.
The paper is organized as follows. Section 1 presents the model. In Section 2 we solve the resale stage. Section 3 presents the bidding stage under private information and the main results of the paper. Section 4 concludes.
1. The model
A government wants to auction the location of a facility, or to assign a big project to one of two potential risk-neutral buyers, buyer A and buyer B. The worth of the auctioned good may be large compared to the buyers’ wealth, so that default may occur. Each buyer i has a budget or wealth wi. As in Zheng (2001), a buyer wealth represents both her liquidity constraint and her liability. Thus, wa and wb will set the maximum amount by which buyers can be penalized if they default. We assume that wa and wb are known.
Buyer i has use value vi when i is the solo owner of the good.3 If i obtains a fraction z of the property of the good then her use value will be zvi. Use values va and vB are private information. Each vi has distribution Fi with associated density fi and supports Vi] for v^, with vA > vB and —v{ = 1. We will denote by hi(vi) the hazard rate of Fi, i.e. hi(x) = 1^pX^ and we will assume that it is increasing. The hazard rate represents the instantaneous probability that the valuation of buyer i is vi given that it is not smaller than vi. Monotonicity of the hazard rate implies
3 Due to the possibility of resale, and following Haile (2003), we will distinguish between buyers’ use value of the object which is exogenously determined and buyers’ valuation -the value players attach to winning the auction- which will be endogenously determined.
that the virtual use valuations = Vi — 1 ^Fi, i = A, B) are strictly increasing,
and it is equivalent to assume log-concavity of the reliability function.4 We further assume vAtiA(vA) > 1 which is the necessary condition for no buyer A exclusion through a positive reserve price being profitable for the seller.
We will define buyers ex-ante financial situation by the relationship between their use values and their wealth. We will say that buyer i is ex-ante financially constrained if vi > wi, and that i is ex-ante unconstrained otherwise.
An important assumption of the model is the inability of the initial seller to prohibit resale. Because of this, buyers participate de facto in a two-stage selling game; in the first stage, they compete for the object at the auction, and in the second stage the auction winner can make a take-it-or-leave-it offer to the auction loser for the entire or for part of the property of the object. Player i can always guarantee himself wi, by not participating in the selling game.
At the first stage, the good is sold through a second-price auction and assigned to a single buyer. Bids are denoted b = (6a,bB). We will denote by p the price to be paid by the winner, i; this payment will take place at the end of the game. The loser does not pay anything to the auctioneer. We will denote by UWj the utility of player i, i = A, B, when the auction winner is j, j = A, B. At the end of this first stage the auction price is announced publicly.
At the second stage, the winner of the first stage auction, i, must decide whether to keep the object or to resell it, and if so, at what price and which fraction. We will assume that the winner has all the bargaining power.5 Thus, resale takes place via monopoly pricing - the winner of the auction makes an offer to the loser after updating her prior beliefs based on her winning and on the information revealed by the auction price. A resale offer by bidder i, Oi, is a pair [ri, zi] which comprises a resale price ri and a fraction of the good zi. Keeping the object is dominated by reselling it if the auction winner does not have wealth enough to cover the auction price, i.e., if wi < p.6 We will denote the option of keeping the object by the offer Oi = [0, 0]. If the winner is unable to pay p after resale, she defaults and loses all her wealth. We will denote defaulting by an empty offer, i.e., by Oi = 0. Note that if p > wa + wb then Oi = 0 no matter the identity of the auction winner. The auction loser must decide whether to accept or reject the resale offer. It can be easily verified that the auction loser j will accept to buy zi at a price ri if and only if the following two conditions simultaneously hold: 1) ri < vj and 2) rizi < Wj.
We search for the Perfect Bayesian Equilibria of the selling game (PBE, for short). A strategy for a player must hence specify a first round bid, a second round offer if the player is the auction winner, and a second round acceptance decision if the player is the auction loser. Furthermore, posterior beliefs are determined by Bayes rule whenever possible and the resale offer is optimal given the posterior beliefs and the first round bids. Regarding the off-equilibrium beliefs, we will assume that the
4 See Bagnoli and Bergstrom (2005) for the class of distributions satisfying this property.
5 Similar assumption is adopted in Zheng (2002) to characterize the optimal auction with resale, and can also be found in Hafalir and Krishna (2008). In contrast, Pagnozzi (2007) assumes that bidders bargain in the resale market, so that the outcome is given by the Nash bargaining solution.
6 An alternative interpretation is that the winning bidder can sell equity to finance a portion of his bid as in Rhodes-Kropf and Viswanathan (2005). But here the equity provider is the losing bidder and not the equity market.
posteriors will coincide with the priors whenever the first stage bid is outside the range of equilibrium bids.7 Finally, we will only consider rationalizable equilibria or equilibria which survive the elimination of (weakly) dominated strategies.
2. The resale stage
The optimal offers at the resale stage will depend on the winner’s use value, on the first round price, on bidders’ wealth and on the information revealed at the first stage. Since bidders’ posteriors depend on the first round bids, different information structures can exist at the second stage. At one extreme, after the auction and before the resale stage, the loser’s use value becomes common knowledge (perfectly revealing first round bids or separating equilibrium); at the other extreme, the posterior and the prior on the loser’s use value coincide (perfect pooling equilibrium). The resale offers in a first-stage fully separating equilibrium coincide with those described in previous section as there will be perfect information at the resale stage. We concentrate here on the case of incomplete information at the resale stage.
To solve the resale stage, we assume that bidding strategies are non-decreasing, so that the behavior of the winner at the resale stage must take into account what she learns from the auction price.
Lemma 1. If the auction loser, bidder j, followed a non-decreasing bidding strategy at the first stage such that p = bj (vj) for all vj G [vjp, vj^] and p = bj (vj) for vj G [vj^vj^], the winner, bidder i, has updated beliefs given by Fj (x) =
Pr (vj < x\ vj G [vL,vH]), where
j
0 if x <vL
with p = bj (vj') = bj (v^1), and Vj < Vj < v^ < Vj.
If v^ = v_j and Vj1 = Vj, then the updated distribution coincides with the prior distribution, i.e., Fj (x) = Fj (x). Conversely, if vL = vjH then the updated distribution is a point distribution.
At the resale stage, the auction winner will set the resale offer that maximizes her expected payoff given her posterior beliefs about the loser’s use value. The objective function for player i is hence:
max {[wi - p + rizi + vi (1 - zi)] (1 - Fj (n)^ + Ki(p)Fj (ri)}
where
Ki(p) =
0 if p > wi
vi + Wi - p if p < Wi
Ki(p) stands for the utility when buyer i keeps the object because the resale offer is rejected. It coincides with the utility in an auction without resale and it hence
7 We will do so whenever these off-equilibrium beliefs satisfy Cho and Kreps Intuitive Criterion.
takes on the positive value vi + wi — p when there is no risk of default (p < wi) and
0 when there is risk of default.
The key difference between the strong buyer and the weak buyer behavior at the resale stage lies in the shares they put up for sale. Whereas buyer A resells the minimum fraction needed to cover the auction price, i.e., = min , oj,
buyer B may resell the entire object if wa > vA. Resale offers will hence depend on the identity of the winner.
2.1. Resale offers by the strong buyer
If the winner is player A, she will not resell if p < wa as va > vB. Thus, if p < wa then Oa = [0, 0]. At the other extreme, A will always default, Oa = 0, if p > min(wB,vH) + wa.
For intermediate prices, p G (wa, min(wB, vH) + wa], player A resells the minimum fraction needed to cover the first round price, i.e., za G (0,1] : zArA = p — wa. Note that the loss from increasing out-weights its gain as va > vB > > va,
so that zarA = p — wa follows. Thus, buyer A will set rA G \yB, vH) to maximize her expected utility given by
max | ^4 ^1 - ) + rAzA +WA-pj (l - FB (>a)) + 0 FB (ta) |
where FB (rA) is the probability that A’s offer be rejected. Since the objective function is concave, the unconstrained optimal resale price must solve the first order condition, which after some simplifications, can be written as
* (1)
\p - WA j hb (rA )
where hB(ra) = I^fr^y • Let r*A(p) be implicitly defined by +1-
Since t ■h1^ t -j + 1 is decreasing in ta for any regular cdf, a necessary condition for
(yL )2jB/ L )
the unconstrained maximum to satisfy r*A > vk is that B , B) fl < p — wa-
J A - B i+vLfB(vL) A
1+^L fB (yL )
Denoting MB(v%) = , LB2 B\ BL{ , buyer A’s optimal resale offers whenp — WA <
min(wB ,vH) are hence: OA(p - wa) =
(vLL) fB(yL):
0,0] if p - wa < 0
if 0 < p - WA <
L P-wa if n ^ ^ 1
, VB
(p—wa)
if mb\v^) <P~wa< min(wB, wf)
Discussion above is summarized in the next lemma.
Lemma 2. At the resale stage, player A will set Oa = O*A(p - wa) if p - wa < min(wB ,vH), and she will default, Oa = 0, if p - wa > min(wB ,vH).
If the first stage bids are weakly non-decreasing, so that FB is a truncation of FB, then OA(p - wa) is non-decreasing in p, and rA(p) is increasing in p. Player B’s behavior at the first stage auction will take this into account if he were to set p.
r
2.2. Resale offers by the weak buyer
Assume now that winner is player B and that p < min(wA, vH) + wb so that he is not forced to default.
Denoting by (rB ,z°B) the optimal pair, some basic properties of the optimal offer are first derived.
Lemma 3. An optimal offer (rB, zB) must satisfy the following properties: (i) vH > rB > vA and rB zB < wa ; (ii) If rB z% < wa, then zB = 1; (iii) If zB < 1, then r°Bz°B = wa, and (iv) If wa < vA, then zB < 1.
Buyer B can either sell the entire object (zB = 1) or only a part of it (zB < 1). When zB = 1, rB must solve
max{[wB -p + rB] (1 - Fa (rB)) + Kb(p)Fa (rB)}
In an interior solution, the optimal resale price when zB = 1, denoted r*B (p), is implicitly defined by:
rB ~ (p +KB(p)-wB) = -—j(2)
hA (rB)
This equation yields the optimal price when the solution satisfies (a) r*B > vA (see Lemma 3 (i)), which requires the LHS to be larger than the RHS when evaluated at the minimum possible price vA and (b) rB < wa (from zB = 1 and Lemma 3 (i)), which requires the LHS to be larger than the RHS when evaluated at wa . Note also that, from Lemma 3 (iv), zB = 1 may only be optimal when wa > vA. Trivially, if wa > vH then the condition rB < wa will never bind as r*B < vH holds by Lemma
3 (i). Finally, there can also exist corner solutions with resale offers [vA, 1] if (a) fails, and [wa, 1] if (b) fails.
When selling only part of the object, zB < 1, then rBzB = wa (Lemma 3 (iii)) and rB must solve
max
TB
, , 11 WA
WB - p + WA + VB 1-----------------------
rB
1 - Fa (rB)) + Kb (p)Fa (rB)
In an interior solution, when zB < 1 the optimal resale price rB (p) is implicitly defined by:
rB 1
[rB(wA - (p + KB(p) - wB) + vB) - vBwA] = y—-—- (3)
vB WA hA (rB)
and z*B = pgr. The equation for r*B yields the optimal price when the solution
satisfies r*B > max{vA,wa} (from Lemma 3 (i)) r*B > vA, and from Lemma 3 (iii) r*B > wa is necessary for r*B zB = wa and zB < 1 to hold simultaneously). When wa < vA, the condition rB > vA requires that the LHS of equation above be larger than its RHS when evaluated at vA, whereas when wa > vA the relevant condition is rB > wa , which requires that the LHS of equation above be larger than its RHS when evaluated at wa. We may also have corner solutions with resale offers vA, and [wa, 1] when the aforementioned conditions fail (rj* < v\ and
V A J
rB < WA, respectively).
The optimal resale offers [rB, zB] for player B when p < min(wA, vH) + wb are summarized below:
O°B (p)
„L
,
[rB, 1]
[wA, 1]
VR
WAhA(wA) .
** WA
rB , r« r*B, 1] .
VL VA’
** WA
rB > rV
if WA > vA, p + Kb (p) - WB < '<Pa(vA)
if v2 > wa > vA, p + Kb (p) - wb € ^rtpA(vA),rtpA(wa)
if vH > wa > vA, p + Kb (p) - wb € 4>a(wa),wa-
if vH > wa > vA, p + Kb (p) - wb > wa -
WAhA(wA)
if v2 < wa, p + Kb(p) - wb > ^a(vA)
if wa <vA, p + Kb (p) - wb < wa + vb - vbwaMa(vA)
if wa <vA, p + Kb (p) - wb > wa + vb - vb wa Ma(vA )
where V’a(x) stands for the virtual valuation of player A when her use value is x,
i.e., 1pA(x) = x -
Ha{x)
and Ma(vA) =
_ ( 1 +vA^A (va)
(vA) 'ha(vA)
Lemma 4. At the resale stage, player B will set Ob = OB (p) if p < min(wA, vH) + wb , and she will default, Oa = 0, if p > min(wA,vH) + wb .
Note that B’s offers depend on the value of p + Kb (p) - wb , which is
p + Kb (p) - wb
p - wb if p > wb vB if p < wB
In words, B’s offers depend on the amount to be covered at resale to avoid bankruptcy, p - wb , when there is risk of default, and they depend on B’s use value for the good, vB, when there is no risk of default.
Regarding zB, when all the A types are financially constrained (wa < vA) B will always set zB = < 1, whereas when A is wealthy he may set zB = 1 but he
may set zB = ^ < 1 when p — wB or vB are high. In the first case, if wa < vA, the highest price B can set if he were to sell the entire object is wa so that his payoff will be wa + wb - p which is lower than what he can get by setting a higher resale price such as vA and getting wa from player A while additionally using part of the good.
Regarding rB, if wa < vA then resale prices satisfy vA < r*B < wa < r*B. Furthermore, note that p + K(p) is constant with p when p < wb and increasing in p when p > wb , so that the optimal resale prices set by B are non-decreasing in p in (wb , to) and are constant in p in [0, wb). However, resale prices may decrease with p at p = wb since K (p) = 0 at p > wb and K (p) = vB at p = wb .
Detailed calculations and proofs for the claims above may be found in the appendix.
Losers’ expected utilities
When the strong buyer wins, her optimal resale offers will depend on p - wa. From Lemma 2, straightforward calculations allow to derive B’s expected utility when he loses the auction, UWA(p - wa). Trivially, UWA(p - wa) = wb when B either does not receive any resale offer or he rejects it. No resale offer will be
vB
1
made whenever A has wealth enough to cover p or whenever A defaults because p - wa > min(wB, vH)). For the remaining cases UWA(p - wa) is given by
WB+WA-p + VB (E1fL) ifO<p-WA<-^TJ
max | w£ +wa -p + vB (7*^7) if Mb\v^) <P~Wa^ min(wB, vf).
When the weak buyer wins, his optimal resale offers take into account his rival wealth, particularly whether wa > vA or wa < vA. Because of this, in what follows, when computing A's expected utility when losing the auction we treat the two cases separately.
If wa > vA then UWB(p) is given by
wa + va - vA if wa <vH, p + Kb (p) - wb < 1a (vA)
max {wa + va - r*B ,wa } if wa < vH, p + Kb (p) - wb € ( 1a (vA ),1a (wa ))
max{«i,wa} if wA < vH, P + KB (p) - wB G t1>a{wa),wa-
WA hiA (wa )
max{vA ,«m} if wa < vH, P+ KB(p) -wB > wA - wJ*{wa) max {wa + va - r*B ,wa } if wa > vH, p + Kb (p) - wb > 4>a (vA)
If A faces a financially unconstrained seller because p < wb then p + Kb (p) - wb = vB and B’s resale prices will only depend on his type. If this is the case, A is better off the larger is the difference between her use values and that of her rival, being maximal if vB < i^a(vA) as she will then be offered rB = vA. If she faces a financially constrained bidder then p + Kb (p) - wb = p - wb and A can affect the resale price by affecting p. Since vA < r*B < wa < r*B, she is always better off, conditional on losing, if B is financially constrained unless vB < ip a (vA) holds.
If wa < vA then UWB(p) is given by
VA (ff) ifp + KB(p) -wB <WA + VB -wavbMa{vH), and
max ,wa} if p + KB(p) - wB >wA + vB - wavbMa{va)■
Buyer A's utility is maximal if her rival i2s financially unconstrained but has very low use values, i.e., if vB < M \ l\ = TT> as then r*B = v\. When p > wB
( A ) +VA A (VA )
then r*B* = vA iff vB < > which always holds for a sufficiently low wa or
if p is sufficiently large as compared to the total resources, wa + wb .
From the above expressions, next result follows.
Lemma 5. (*) In an equilibrium in which B loses: if Ba > wa + M | L) then bB > wa + M | Ly with equality if F concave. Similarly, ifbA > wa then bB > wa-
(ii) If wa > v'A and wb < 'Va, in an equilibrium in which A loses: if bB > wb and Vb > 1a (vA) hold, then bA > wb .
(iii) If wa < v'A and wb < va, in an equilibrium in which A loses: if bB > wb and vB > Ma\vL) hold, then bA G (wB,wB +wA{ 1 - vbMa(va)) + vB\.
Results in Lemma 5 stem from the loser’s preference for facing a financially constrained winner. Buyer B will not receive a resale offer unless A is financially constrained, which explains part (i). Similarly, if the loser is buyer A, to induce the best possible resale offer she will make her rival financially constrained (parts (ii) and (iii)). Because of this, whenever A is always ex-ante financially constrained a strategy profile in which A bids no more than her wealth can only be part of a PBE if it results in A losing.
The importance of bidders’ use-values asymmetries is also highlighted in Lemma 5. If A loses but her rival has very low use-values as compared to hers, then she will always be offered rB = v1A. When this is the case, a loser buyer A will make no attempt to manipulate the auction price (other than avoiding default). The conditions for rB = vA are
vb < 1a{va) if WA > vA, and vB < if wA < vA.
Ma(vA )
If asymmetries are less severe (as would be the case for example with vb = vA) then those inequalities will not hold for, at least, vA = vA.
3. Results
The possibility of default due to the presence of financial constraints creates a link between the resale price and the auction price. This link can affect equilibrium behavior at the auction stage, an issue we explore next. We have seen in the previous section that the presence of financial constraints may induce a potential loser to increase the auction price in order to fine-tune the winner's resale offer. Similarly, the presence of a resale market affects the possibility of default as the auction winner can get extra resources from reselling.
To disentangle the impact of resale from that of financial constrains, in what follows we first explore if the equilibria when there is resale among unconstrained buyers remain equilibria under financial constraints. We then analyze whether the equilibria in no-resale auctions with financial constraints remain equilibria when there is resale. Finally, we explore the equilibria emerging from the interplay between resale and financial constraints.
3.1. The resale equilibria
Garrat and Troger (2006) have shown that in second-price and English auctions resale creates a role for a speculator -a bidder with zero use value for the good on sale. Because of this, in a second price auction with resale the truth-telling equilibrium coexists with a continuum of inefficient equilibria in which the speculator wins the auction and makes positive profits. In our set-up no buyer is a pure-speculator as their use values are strictly positive. Nevertheless if vB is sufficiently low, buyer B could be considered as a speculator.
We first show that financial constraints preclude the existence of a truth-telling equilibria.
Proposition 1. Truth-telling at the bidding stage is part of a PBE of the auction with resale if and only if wa > va.
Proposition above shows that when there are potential financial constraints, truth-telling fails to be an equilibrium. Moreover, as soon as there exist “potential” financial constraints, a separating perfect revealing equilibrium fails to be an equilibrium. The reason is that whenever the strong buyer can be made financially constrained the weak one takes advantage of it by raising the price at least up to wa + £■ Thus, a strategy by bidder A such as bA = va > wa can only be part of an equilibrium if bB > wa. Recall from Lemma 5 that If b = (bA, bB) is such that bA(vA) > wa and bA(vA) > bB (vb ) for some pair (va, vb ), then p > wa for b to be part of a PBE equilibrium.
We study next if a profitable “speculative-like” equilibrium a la Garrat and Troger can exist when the strong buyer is financially constrained. Garrat and Troger (2006) show that the following strategies constitute an equilibrium of a second-price auction with resale among two buyers: a non-speculator buyer i with use values in the range [0,1] bids 0 if v^ < 0* and bids her use value (be (v*) = Vj) if v^ > 0*, and a speculator with zero use value bids 0*. They show that there is an equilibrium in this family for each 0* G [0,1]. Thus, there is a continuum of inefficient speculative 0* -equilibria.
Considering the weak buyer as a speculator, and considering the strategy profile in which buyer A bids vA if Vi < 9* and bids her use value (bf (vi) = v{) if Vi > 9*, and buyer B bids 0* no matter his type (this candidate to equilibrium will be referred to as the speculative 0* -equilibrium, as displayed below)
'■ V-A „-----------------,0\---------------------VA
"V'' "V''
VA VA
bb ■vB----------------------------------------------vB
'-------------------v---------------------'
e*
We next show that a “speculative-like” 6*-equilibrium can only exist if 0* = va and wB > va- For any other 9* G (vA,va), a speculative 0*-equilibrium (in which buyer A bids vA if Vi < 9* and bids Vi if Vi > 9*, and buyer B bids 9* ) fails to exist due to the combination of budget constraints and the fact that vA> vB >0.
Proposition 2. (i) A speculative 0*-equilibrium, 0* < va, does not exist for any wa < va-
(ii) The speculative va~equilibrium with bids (vAj va) pari of a PBE in which the low valuation bidder wins (and makes positive profits if vB > iPa(v A)f ifwB > va > wa > V_a, while it is not otherwise.
Proof. See Appendix.
Note that financial constraints destroy any speculative 0*-equilibrium other than 0* = va (Proposition 2 part (i)). To see this, note that if 0* < va then in a speculative 0* -equilibrium some types of buyer A are bidding her value while being left financially unconstrained as vA < p = 9* < wa so that buyer B is better off deviating to force his rival to be financially constrained. When wa < vA a speculative 0*-equilibrium ( 0* < va) fails to exist as A types above 0* win the
8 Trivially, it is an equilibrium for buyer B to bid sufficiently high, say va, and for buyer A to bid y_A, provided that at the resale stage B sets tb = v_A which will be the case if vb < ip a (v_a ) • At such an equilibrium B breaks even.
auction but face the risk of default and are hence better off by deviating to lose and then buying at the resale market.
For 0* = Va at the speculative Va-equilibrium buyer B always wins and if wb > va holds, then he cannot be made financially constrained. His only possible deviation is to lose which is unprofitable if wa > vA- In contrast, if wB < va and vB > iPa(v.a) the strategy profile (va,va) cannot be part of an equilibrium, given that, conditional on losing, buyer A is always better off forcing his rival to be constrained.
Note that if VB is sufficiently low so that B could be considered a speculator, when he wins he prefers to resell the entire good. At p = vA, he makes a loss from any fraction he gets to consume as p > vb . Because of this, whenever he offers a resale price larger than the auction price, his offer is unaccepted with positive probability, and he will be forced to consume the good. For some types of buyer B to find it profitable to set rB > vA it must be the case that vB > V’aC^’a) holds ( which will be the case if vB = vA). In Garrat and Troger’s 0*-equilibrium p = vb = 0 so that the disutility from a price higher than the use value does not arise. Furthermore, vB > iPa(va) does always hold in their set-up as vA = vB = 0 and V’a(O) = -7^0) < °-
In sum, financial constraints destroy all the resale equilibria unless one of the buyers be wealthy enough so that Wi > Va holds for some i. When it is the strong buyer, then truth-telling is the unique resale equilibria that survives the introduction of financial constraints. When it is the weak one then the speculative 0* -equilibrium with 0* = va is the only surviving equilibrium.
3.2. The financial constraints equilibria
In a static one-round SPA with budget constraints it is a dominant strategy to bid min(wi,vi) (see Che and Gale, 1998). The reason is that if the second highest bid is above the winner’s budget, he will renege, will not get the object and will pay the fine, resulting in a negative surplus. With the possibility of resale this argument breaks down if the winner can resell the good and, by doing so, can get more than the auction price. This is the case here as long as the potential buyer at the resale market does not follow dominated strategies.
Arguably, truth-telling is not an equilibrium in the auction with resale as it is not an equilibrium either in an auction without resale among financially constrained bidders. Recall that the equilibrium bids in undominated strategies of the auction without resale requires each buyer to bid bi = min {vi,wi} (in what follows we refer to this as the no-resale equilibrium). Acknowledging this fact, we explore next the conditions needed for the equilibrium in the auction without resale to be an equilibrium under resale.
The possible realizations of bidders’ wealths with their corresponding bidding strategies in the no-resale equilibrium are displayed in table below.
wealths VB <WB < V_A wB > vA
vB < wA < vA (wA^m\Y{wB,vB}) (wA,vB)
V_A < WA (min {wa, va\ , min {wB, vB}) (min{wA,vA} ,vB)
The no-resale equilibrium is not an equilibrium under resale in the first row, i.e., if wa < v_Ai because any loser bidder B with vB < wa is better off deviating so as to win and then reselling part of the good. Note that whenever buyer A wins, since
wa > min {vB ,wb } for A to win, she will make no resale offer so that S’s utility will be wB- By deviating to wa + e, B wins and by setting [rs, zb] = ku, — ] he
—A
gets a larger expected utility since he recovers the price wa and is able to consume a fraction of the good.
If wa > v_a (second row) then at the no-resale equilibrium buyer B follows a weakly dominated strategy as he is bidding below his lowest possible endogenous valuation vA- His endogenous valuations takes into account the overall surplus from winning and reselling the object. Since wa > vA buyer B can always get vA from the resale market so that bs = v_A weakly dominates bg = min {wb ,vb}- If either winning or losing with the two of them they yield the same profits. But if winning with bs = vA while losing with min {wb , vb } then the former yields larger expected profits as p < vA <rs so that
Next proposition summarizes this discussion.
Proposition 3. (i) If wA < rtiA then bidding bi = min {«;*, Vi} i = A, B at the first stage is never part of a PBE.
(ii) If wA > vA then bidding bi = min{uij,Wj} i = A, B at the first stage can never be part of a PBE in which bidders do not employ weakly dominated strategies.
3.3. Resale and Financial constraints
We have seen that buyers’ attempts to make their rivals financially constrained give rise to profitable deviations which may destroy the equilibria under resale. Similarly, we have seen that the existence of a resale market can generate buyers’ valuations above the use values making some strategies weakly dominated.
Since buyers will anticipate their rivals’ incentives to make them financially constrained, they can incorporate these incentives in their bidding. Similarly, bids must account for the prospects at the post-auction resale, which, in turn, depend on wealth. We next characterize equilibrium bids depending of the strong buyer wealth.
High budget buyer A: wA > vA
We consider that player A has a relatively high budget, wA > vA , while maintaining the assumption that she is at least potentially financially constrained so that wa < Va.
By weak dominance arguments, any bs < vA is now weakly dominated. Consider the family of cut-off strategies bA (va ), parameterized by A* with A* G [HA, va), consisting of strategies for which any bidder A with a use value below A* bids her value, while any bidder with use value above A* bids wa, i.e.,
U^B (bA,vA) >U!B = U^A (bA,mm{wB,vB}) •
bA ■ Ha
A*
-vA-
Note that A* = va generates the truth-telling strategy, A* = wa gives the equilibrium strategy of the auction without resale,9 and A* = vA generates the pooling strategy bA (va) = wa.
We show that the strategy profile (b\ ,vA) allows to generate a continuum of equilibria, in which buyer A wins at the first stage and there is no resale, if one of the two conditions below hold (this requirement will be referred to as condition (w))
(i) wb G
(ii) wa G
wAhA(wA)
,wa
Ha{wa) {wa —vb)’ WB 1
Condition (w) ensures that player B will not raise his bid to win and profit from resale: either wb is not too low so that the risk of bankruptcy is a serious threat, or wa (and hence p) is high enough leaving no profits to be made at resale. It is worth noting that with symmetric wealths (wa = wb ) or when they are close enough, condition (w) always holds. When wa < wb , part (ii) may be rewritten as vb < Ma(WA) and when wA > wb, part (i) may be expressed as vB < ^(wa) '
Proposition 4. If condition (w) holds, then for any A* G \Ha-> wa\ there exists a Perfect Bayesian equilibrium of the SPA with resale in which players bid b = (bA ,vA) at the auction stage and they follow the resale strategies described in Lemma 2 and Lemma 4, respectively.
Proof. See Appendix.
To further clarify the role of (w) take A* = wa so that bA = min{vA,WA}- At the candidate equilibrium buyer A wins and does not resell since p < wa so that UB = wb . Deviations that will keep B as a loser while forcing A to be financially constrained are not feasible, in other words, Lemma 5 (i) does not bite as bA < wa. The only payoff relevant deviations for B involve winning by bidding more than wa (if winning at any p < wa he learns va so that rB = p = va and UB = wb ). Consider hence deviations to b'B > wa.
If wb < wa , he will need to resell to avoid defaulting as p = wa > wb . For any resale price rB > wa = p he earns (rB — p) when his offer is accepted but loses wb when unaccepted. For the gains to outweigh the loses it must be the case that wb is sufficiently small. Thus, there is a profitable deviation if and only if wB < WAhA{wA) holds. The reason for low-budget bidders to find it profitable to deviate is reminiscent of the results in Zheng (2001). A low-budget bidder B has less to lose from bankruptcy than a high-budget bidder. That is why bidder B must have a sufficiently low budget to find it profitable to bid high as to win and then resale while facing the risk of default.10
If wb > wa when buyer B deviates to win then he can always recover the auction price by setting rB = vA = wa whenever p = wa . Thus, he is indifferent between bidding vA and losing or bidding b'B > wA winning and reselling at rB = vA = wA. Furthermore, the deviation is worth it if vB > Ma(wa) holds, (i.e., if
9 The assumption wa > vA guarantees that these bidding strategies are weakly increasing.
If wa < vA then b\ is weakly decreasing unless A* = vA or A* = va-
10 In a model without resale Zheng (2001) does also find that the symmetric Bayes Nash equilibrium bidding strategy is not monotonic as a function of the bidder’s budget .
wA < hA(wA){wA—vb) )' ^or a ^arSe WA l°ses when a resale offer is rejected, given by (vB — wa), are too large, making it less attractive to become the winner.
Remark: A* < wa is needed for the above strategies to constitute a PBE. If A* > wa instead, from Lemma 5 (i), there will be a profitable deviation as p < wa.
We have provided a set of equilibria when wa > vA which yield (despite the multiplicity of equilibrium strategies) a unique equilibrium outcome characterized by the strong buyer winning, and consuming the entire good. In equilibrium there is no resale and p = vA.
Whenever B is wealthier than A, the above equilibrium relies on B not deviating when he is indifferent between winning or losing. If he bids wb to take advantage of his (relative) strength it is a best response for buyer A to bid vA provided that vB < ma\wb) holds. Note that vB > iI>a{va) is necessary for the above strategies to yield a profitable outcome for at least some types of buyer B. But even if A is wealthier than B, buyer B can prevent his rival from making him financially constrained, by bidding wb . When he does so, no type of A wants to deviate so as to win provided that r*B < wb .
Proposition 5. Assume B is wealthier than A so thatwB > wa > v_A. The strategy profile (vA,wB) is pari of a PBE (in which B wins) if vB < ' If A is
wealthier than B with wa > wB > vA then the strategy profile (vA, wB) is pari of a PBE (in which B wins) iff Vb < 1a(wb ).
Proof. See Appendix.
When B is wealthier than A and vB < Ma(wa ) we have found two equilibria both with p = v a which differ in the identity of the winner. If Ma(wa) <vb< ma(wb) equilibria in which A wins disappears and if vB > Ma^wb^ both equilibria disappear. Given that Ma^ is increasing in w with a minimum value of 0 at w = vA, and tends to va when w tends to va, we have that vB = Ma^ f°r some w G (vA, va) and for higher levels of wealth the condition vB < ^^wuld hold. Thus, these equilibria disappear when wealths are close to vA.
Low budget buyer A: wA < vA
At the resale market buyer B will get at most wa from buyer A. Because of this, now weak dominance arguments only eliminate bids bB < wa.
Whenever wB >vA> wa buyer B can take advantage of his (relative) strength (his larger wealth) and behave more aggressively incorporating in his bid his expected profits from resale (he can always get wa by setting rB = vA). Furthermore, he can prevent his rival from making him financially constrained, by bidding wb . Since the best response by the strong buyer to a sufficiently high bid is to hide her type, a candidate to equilibrium is b = (bA, bB) = (wa, wb ).
Proposition 6. If wB > vA > wA, the strategy profile (wA, wB) is pari of a PBE (in which B wins and makes positive profits).
4. Conclusion
This paper has analyzed the effect of two important features related to auctions for big facilities or contracts: partial resale and binding budget constraints. In these auctions bidders usually face severe financial constraints that may force them to resell part of the property of the good (or subcontract part of the project) at a resale market. We show that the interaction between resale and financial constraints changes previous results on auctions with financial constraints and those in auctions with resale. The reason is the link between the resale price and the auction price introduced by the presence of financial constraints. Such link induces a potential loser to increase the auction price in order to fine-tune the winner’s resale offer. We also characterize equilibria of the auction with resale and financial constraints that do not involve speculative behavior by the low use value player and yield efficient outcomes.
References
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5. Appendix
Proof of lemma 3 (i) Since buyer A will never pay more than her use value, vH > rB, and since any type will pay at least vA then rB > vA. Finally, buyer A total payment cannot exceed her wealth, rBz°B < wa.
(ii) If rBzB < wa it is possible for B to increase his utility by increasing zB and keeping rB constant, so that rBz°B < wa still holds. Since his utility increases (rB > vA > vB and the probability of acceptance does not change) it must be the case that z°B = 1 if r°B z°B < wa . The same arguments show (iii).
(iv) Assume not, z°B = 1. Then rBB = r°Bz°B < wa < vA, which contradicts (i).B
Proof of Lemma 4 We derive the optimal resale offers by player B.
Case 1. p < wb .
If wa > vA bidder B can set zB = 1 and rB G \yA, min(wA,vH)) to maximize her expected utility,
max {wb — p + rB Pr (rB < va/p) + vb Pr (rB > va/p)} .
rB
Solving the maximization problem above, B’s optimal resale price rB is the solution to equation below:
rB -vB= J—^—- (4)
hA (rB)
which has a unique solution as the standard hazard rate condition holds. Since the LHS of (4) is strictly increasing while the RHS is strictly decreasing a necessary condition for r*„ to be interior is that vh — vB < -—t—tt- Finally, the solution to
B A hA(v%)
equation above fails to be lower than wa if wa — vB > -—y—The constrained
hA(WA )
maximum in then rB = wa.
Alternatively, B can set zB be such that zBrB = wa, so that rB G [vA,vH] if
wa < vA and rB G \_wa , vH] if wa > vA will solve
max jwe - p+ (wa + vB ^1 - (! “ (rB)j +vBFA (rB)|
= max|wB -p + vB +wA[l ~ (l -FA(rB)^ |
The optimal resale price r*B is the solution to:
— {rB-vB) = Y~^—r (5)
Vb Ha (rB)
For the same arguments as before, the optimal (unconstrained) resale price is vA if (vh — vB) — > -—fry. It will be larger than wa if — (wa — vB) < -—^—r.
V A B) VB hA(vA) A VB v A hA(wA)
By comparing the two offers it follows that [rB = rB, zB < 1] dominates [rB = WA, zb = 1] if WA < vA as Ub (rB = r*B, zb < 1) > Ub (rB = vA, zb < 1) by the optimality of rB, and the fact that
UB (rB = vA, zB < 1) > UB (rB = wa, zB = 1). The optimal resale offers by B
when wa < vA are hence
o B (p)
rB
A
if vb < if vB >
() 1+va^a{va) 1+v%hA(v%)
If wa > vA since rB > r*B (the LHS of (4) is steeper than the corresponding one in (5) while they are both equal to zero at rB = vB), the resale offer [rB = rB < wa, zB = 1] dominates the offer [rB = r*B, zB < 1]. But if r*B > wa (i.e.
if wa Ha (wa )
<
then [rB = r*B > wa, zB < 1] dominates
(wA-VB) '
[rB = rB = wa, zB = 1] as with both B gets the same resources, namely wa, whereas with the former he gets to consume part of the good in return for the risk of getting more often his offer rejected. Because of this wa cannot be too large. In sum, the optimal resale offers if B is unconstrained are
[rB = vA, zb = 1] [rB = rB*, zb = 1] [rB = WA, zb = 1]
if vb < 1a(vA)
if vb G i> a (vA ),4>a(wa )
if vb G
[rB
4> A (wa), wa —
> wa, zb < 1] if vb > wa —
wA hA (wa ) V B
wA hA (wa )
Case 2. p G (wb , min(wA,vH)+ wB\. Bidder B will maximize:
max] (wb — p + rB zb + vb (1 — zb )) (1 — Fa (rB)) \
r B ,ZB V \ / J
As in case 1, whenever wa > vA bidder B may sell the entire object,11 zb = 1, for a resale price rB G \yA, vH) which maximizes his expected utility,
max
rB
|(wb — p + rB) (1 — FA(rB))}
The optimal resale price r\* is the solution to equation below
rB — (p — wb ) =
1
Ha (rB)
(6)
The optimal resale offers by B when wa > vA and zB = 1 are as follows
Vb = vA, zb = 1 if p — wb < a(vA)
[rB = rB*(p), zb = 1] if p — wb G 1a(vA),1a(wa) [rB = wa, zb = 1] if p — wb > 1a(wa)■
11 Assume that the optimal (r*B, z*B) is such that z*B < 1.Then, if r*B zB < wA it is possible to increase utility by increasing zB and keeping rB constant, since rB > vb and the probability of acceptance does not change. If rBzB = wa, increasing zb requires decreasing tb and this changes the objective function with the sign of [— (1 — + — ^b)//('Tb)].
A
v
A
VB
VB
Alternatively, buyer B can set zB rB = wa and rB G [vA, vH) to maximize (wB - p + wA+vB ^1 - (l - Fa )) + 0
The optimal resale price rB* (p) is now the solution to equation below:
— (rB (wA + WB - p + vB) - WAVB) = , WA . (7)
vB hA (rB)
where r r* (p) > wa for wa > vh iff » — wB > wa-^—r, whereas for wa < vh
B A WAhA(WA) A
it always holds that r*B (p) > wa.
If wa < vA then the optimal resale offers are
rB = vla, zB = ^t
vA
if (va(wa+wb -p + vB) - WAVB) >
r'B = r*B*(p), zB = f£ if ^ (v\(wA + wB - p + vB) -wAvB) <
whereas when wa > vA they are as follows
V'b = vA, zb = 1 if p - wb < 1a(vA)
[rB = rB*, zb = 1] if p - wb G 1a(vA), 1a(wa)
[rB = wa, zb = 1] if p - wb G i}a(wa),wa -
[rB = rB* > wa , zb < 1] if p - wb > wa -
Combining the different cases, OB follows.■
WAhA(wA) .
VB
W A hA (wa ) '
Proof of Lemma 5 (i) Let us write for short p - wa as y. Since for any y <
(yL )2^/ L )
LfB^ Lj = ]Q—|—y the expected utility by loser B is strictly increasing in y as
U'bA(v) =wB + (vB ~ vs)^,
B B vBA
it trivially follows that y = ■j—|—j yields larger expected utility to buyer B than
any lower y. Thus, if b = (6a, bB) is such that 6a(va) > wa and 6a(va) > bB (vB) for some pair (va , vB), then p > wa for b to be part of a PBE equilibrium, as B is always better off if A is financially constrained. Note that if A wins and has wealth enough to cover the auction price she will not resell, so that Uwa(va,vB) = wb . In contrast, if B deviates to bidding bB = wa + e he will get an expected utility
larger than wB, as player A still wins but he must now resell As buyer
r A
A’s posterior will coincide with her prior, her optimal resale offer employs f B = fB and vB = v_B. By making e arbitrarily small so that e < |—y holds, it follows
that r*A = vB, and U1^a(va,wa + e) > wB.
We next show that for larger values of y bounded above by min(wBvH) there
exists vB > ra such that dUBdy ^ > 0 if vB > vB whereas dUB9y ^ < 0 otherwise, where
U^A(y) = max wB + (vB - r*A(y)) ,V. . ,wB \ so that for any vB > rA
{ rA(y) J
it follows that
dU”A(y)_vB (r*A -yr'X) - (r*AY
dy
Total differentiation of ta (yjj- —
hB (rA)
(rA )2
gives
(8)
dr A dy
r2A (hB(rA)
(2rA - y) (hB (rA)) + yh B (rA)
>0
where the sign of the denominator equals the sign of — 8 and is hence positive
by the concavity of Ua. Plugging its value into (rA - yr'A) and simplifying we get vBrA (l — — (rA)2 = rA (vb fl — — rA) so that the sign of (8) is given
by the sign of vB ( 1 — — rA, with
[hB (rA)) (rA - y) + yhB (rA^ [vb - rA] - rA (hB (rA))
vB 1
yr'A\ * _
~ r A —
rA )
hB (r A)) (2r* - y)+ yhB (r A)
a (y)
Since vB > va must hold for buyer B to accept it follows that —Bdy y ’ < 0
for vB = ra, whereas dUB9y ^ could be positive for vB sufficiently above rA. Since vB — r*A < vB — vB = 1 we have that
vr'A \ *
VB \l-—A)-rA
<
r- \2]
y hB (rA) - (hB (rA))
hB (rA)) (2rA - y)+ yhB (rA)
<0
hB (rA) (2rA - y)+ yhB (rA)
fn(rA )
1-Fb (rA)’
and
where the equality follows from the fact that h'B(rA) — (h'B(rA)) +■ the last inequality from the concavity of F.
(ii). Assume by contradiction that there exists an equilibrium in which A loses by bidding 6a < wb while bB > wb and vB > ^a(vA) hold. Since vA < rB < wa < r*B then A expected utility is bounded above by wa + va - rB. Since by deviating to 6a = wb + e <bB her expected utility is wa + va - vA > wa + va - rB, we have found a profitable deviation contradicting equilibrium behavior.
(iii). Assume by contradiction that there exists an equilibrium in which A loses by bidding ba < wB while bB > wB and vB > Ma^vl^ hold. At such an equilibrium
,L\wa
,rg U““‘' A> V\
the utility he would get if 6a = wb + e < bB. The same arguments apply for
her expected utility equals wa+(va~ r*B)f^ which is smaller than WA+ivA—VAt^r-
6a > wb + wa (1 - vB Ma(va)) + vB which shows our claim.I
1
2
y
2
y
2
2
Proof of Proposition 1 The if part was first shown in Haile (1999). For the only if part assume first that wa € [hatVa)- By following the truth-telling equilibrium strategies player B loses and sets the auction price at p = vB < vA- Player A wins and as she has wealth enough to cover the auction price she does not resell, so that UWa(va, vB) = wb . In contrast, if B deviates to bidding b'B = wa + e he will get an expected utility larger than wb .If va > wa + e then the result follows from Lemma 5 (i). If va < wa + e buyer B wins and can infer va from the auction price. He will optimally resell the entire object at a price va so that Uwb(va, wa + e) = wb . Summing up over all the possible types of bidder A it follows that b'B = wa + e constitutes a profitable deviation so that (va ,vB) do not constitute equilibrium bids. Assume next that wa < vA. By deviating to wa + e player B remains a loser (as with vB ) and gets larger expected profit as he will get to use part of the good that he will acquire at the resale market. Since for any wa < va there is a profitable deviation, the statement follows. Finally, note that at wa = Va we have UWb(va, wa + e) = UWa(va, vB) = wb so that the deviation does not yield larger profits. ■
Proof of Proposition 2 (i) At any 0*-equilibrium (0* < va) buyer B wins if facing types va < 0* and loses otherwise.
Assume first wa € [0 *, Va) so that buyer B gets utility wb if he loses as types va > 0* will not resell. In contrast, if deviating to bid b'B = wa + e he will get a larger expected utility. Profits from types va < 0* will coincide with the two strategies. If he remains a loser with b'B because va > wa + e then he now makes larger profits (the result follows from Lemma 5 (i)). If va < wa + e buyer B wins with the deviation, he infers va from the auction price. He will optimally resell the entire object at a price va so that U][B (va,wa + c) = wB if p = va > vA- Summing up over all the possible types of bidder A it follows that b'B = wa + e constitutes a profitable deviation.
Assume next wa € \v_Ai6*) so that types va > 0* are financially constrained when winning. We first note that va > wa + —y f°r some va € (0*, va) suffices
for B to find profitable to deviate to bid wa + —y (recall Lemma 5 (*)). Even if
va < wa + —y holds then high use-values types prefer to get the entire object
at the resale market rather than winning and being forced to resell as p > wa.
Let us finally consider the case wa < vA- A buyer A with type va > 0* wins and will have to resell so that her utility is bounded above by ^1 — j va where
p = 8* so that UYA < ^1 — 8 VA as va < vb.
If deviating to lose, by bidding so as to pool with types va < Q* she gets UYB = vtVAl wliere we are using the fact that rB < 6* so that > ^VA >
wa . The deviation is profitable as
wa f vb - 0* + wa A (vb - 0*) wa + 0* (0* - vb )
vb J 0*vb
= (0* - wA) (9* - vB)
e*vB
> 0.
(ii) For 0* = va at the speculative va-equilibrium buyer B always wins and as wb > va holds, he cannot be made financially constrained. Because of this, buyer A
0
*
is trivially best-responding. As for buyer B, if vB > 4>a(v.a) he will profitably resell,
i.e., p = vA = vA < rB. His only possible deviation is to lose which is unprofitable if wa > vA as then buyer A will not resell.
We next show that if wb < va and vB > 4>a(v.a) holds then the strategy profile (vA, va) cannot be part of an equilibrium. Conditional on losing, buyer A is always better-off forcing his rival to be constrained and to set the best offer from her viewpoint, i.e., zB = 1 and rB = vA. Buyer A can do so if by bidding bA = wb + v_a ~ hA(v )• ®ut then p > rB and B is better off deviating to lose.
Finally If wa < v_a> buyer B will partially resell getting wa at the resale market and consuming part of the good. Let vB be such that vB = ma\v ) B = ^B ^ vb < Ma£v -,)■ For any vb < vB we have r*B(vB) = v_A so that U^B(p = vA) =
wB ~ Ha + vb + wA (l - |< wB as ^ (vA - wA) < vA - wA- Buyer B with use value below vB is hence better-off deviating to lose. Since r*B (vB) is increasing in vb for types sufficiently close to vb such that r*B (vb ) > UA it remains true that buyer B prefers to deviate so that (vA, va) is not an equilibrium as claimed. ■
Proof of Proposition 4 At the purported strategies bidder A wins the auction, keeps the entire good and pays p = v_A so that UfA{b\ ,bs) = wb and UjA{b\ ,bs) = va + wa — V_A. Consider first deviations by player A. The only payoff-relevant deviations require losing the auction and buying the entire good at the resale market at a cheaper price which is unfeasible as r*B > vA = p■ It hence follows that A will not deviate. Consider now player B. If he deviates to any b'B < wa he will win the auction if va < A *, will pay the use value of its rival and will resell at a price equal to that use value so that his expected utility will be equal to that under bB, namely wb . If he deviates to any b'B > wa, he will win to any buyer A. Now if p < wa then his expected payoff will be wb as rB = p = va. If p = wa he will infer that vA = A *. The deviation will be profitable if and only if B finds optimal reselling part of the good at rB > wa (i.e., if rB = r*B)- For wB > p = wa since Fa (x) = Fa[^ > the deviation is unprofitable iff rB ^ rB*, or, iff (wa — vB) ^ > hA^WA^ (see the computations of B resale offers).12 Since the LHS is decreasing in vB then (wa — V_B) — > hA^WA^ must hold for (bA, bB) to be part of a PBE, i.e., vb < ma\wa)
For vB < wb < p = wa deviation is unprofitable iff rB* € [A*, wa] and zB* = 1
which is the case if ^wB > hA^WA^ as shown in Lemma 4. Consequently, since wb > vB and wahA (wa) > 1, the deviation is not profitable in this case either.■
Proof of Proposition 5 At the purported equilibrium profile (vA,wB), buyer B wins and pays p = vA. At the resale stage he will follow the resale offers in Lemma
4 for F B (va) = FB (va). The only types that can (potentially) profitably deviate are types of buyer A with va > wb .
12 If Fa is a truncation of F then Fa (x) = Fa(^ so that
1 F A \vA '
1 _ 1 - Fa (wa) _ 1 -Fa (wa) _ 1
Iia(wa) Ia(wa) Ia(wa) 1ia(wa)
If they deviate to win they will while wb > wa they will have to resell part of
the good so that U^A < ^1 — WBr,WA j va- Since ^1 — WBr,WA j va < VA~r*B +
wa
provided that r*B < wa + yBr*WA J va then vB < ^a(wa) guarantees that the deviation is unprofitable as rB < wa . Similarly, if she only consumes part of the good but rB* < wb player A will not deviate as
TjWBi \ jjWAi 7 N WA (rA~ (wB — Wa)\
UA (vA,wB)-UA (bA,wB) =VA— -VA[---------------------;-- '
rt,* v r\
r*A (wa - rB*) + (wb - wa ) rB*
> va
= VA
~B r A
r*A -r*B*) + (wb - WA) r*A r*Br*A
va
(wb - rB* )
> 0,
where the first inequality follows from r*A < vb < vA < r*B and the second one from rB* < wb- Note that r*B* < wb requires > WBkAA^™B^1 (see equation (5)
in the appendix). It hence follows that vb < y^h^tw^)+i = ma(wb) suffices f°r the deviation by the strong buyer to be unprofitable.
If wa > wb , the expected utilities are given by
ttWB , \ I wa if va < rB
UA (va,wb) = \ * , -c ^ *
\ VA - r*B + WA if VA > r*B
U'b3 [va, wb) > WB as r*B > vA = vA.
and
If B deviates the only payoff relevant deviation is to bid as to lose but then UWa = wb as A will not resale. Since U][b ( vA,wB) > wB the deviation is unprofitable.
Focus next on buyer A. If her use value is below wb she is better-off losing than winning so she will not deviate. If her use value is above wb she will deviate if and only if rB > wb so that vB < ^a(wb) is a necessary and sufficient condition for the above strategies to be an equilibrium whenever wb < wa. ■
Proof of Proposition 6 At the purported equilibrium profile buyer B wins and must pay p = wa. At the resale stage B will follow the resale offers in Lemma 4 for FB (va) = FB (va). B only payoff relevant deviation is to lose but then his expected utility will be wb as A will not resell. The only other deviation to check is that of types va > wb . Since if deviating to win, she will have to resell, the deviation is unprofitable. More precisely at the candidate equilibrium UYB = max{^va, »a}-If deviating to win, she will have to resell so that her utility is bounded above by
u
WA
A <
1
p - wa
va.
Since ^1 — WBr,WA j va < (l — WBy™A j VA < f£vA the result follows as r*B > wa implies
wb - wa\ _ ('Vb - rB*) wa + rB* (wb - vb)
wa
1
vb
(vb ~Va)wa + wa (wb - vb ) _ (wB ~ va) wA „ .
• -±. -±. — -±. -±. — • •
rB vvB
rB vvB
r
rB vvB
r
