Collusion Agreements in Auctions: Design and Execution by an Informed Principal⇤

Alejandro Francetich† and Peter Troyan‡ VERY PRELIMINARY

Abstract The standard mechanism approach to mechanism design by an informed principal posits an uninformed, uninterested third party who conducts the mechanism. In auctions where collusion is illegal, collusion agreements are likely to be both designed and executed by the involved parties. When this is the case, the standard approach is inadequate; it neglects potential “information leakages” and minimizes frictions. We model collusion in a second-price auction as a contract- design problem by an informed principal who runs the mechanism herself. Bid- ders’ valuations are allowed to be interdependent and their private signals, affiliated. We show that all equilibria are monotonic and involve partial pooling at the top. In a pure common-value example, we find a first-mover disadvantage: the proposer is worse off than the receiver, though both are better off than without collusion. Outside competition for the bidders can also undermine the gains from collusion.

⇤We are grateful to Kyle Bagwell, Jeremy Bulow, Paul Milgrom, Michael Ostrovsky, Ilya Segal, Andy Skrzypacz and Robert Wilson for their guidance and invaluable feedback. We are also grateful to the participants of the Theory Lunch at the Economics Department at Stanford and to the participants of the 2012 Southwest Economic Theory Conference at UCSD. All remaining errors and omissions are our own. Previous versions of this paper circulated as “Collusion in Interdependent-Value Auctions: An Informed-Principal Approach.” †Department of Decision Sciences and IGIER, Bocconi University; alejan- [email protected]. ‡Department of Economics, University of Virginia; [email protected].

1 1 Introduction

Collusion is a well-documented phenomenon in auctions, one of first-order interest to sellers. In one-shot games, collusion agreements are typically analyzed using standard tools of mechanism design, invoking an uninformed third party that designs and enforces the contract.1 Even when bidders interact repeatedly in the same auctions and enforcement is provided endogenously (through continuation values), their communication is typically mediated by a “center.”2 Hendricks et al. (2008) are the first to use these tools to analyze the case of interdependent-value auctions, motivated by oil and gas lease auctions. Joint bidding was legal during the sampling period in Hendricks et al. (2008). However, many such agreements are now illegal and therefore unlikely to be designed and enforced by an impartial party. When collusion agreements are designed and executed by the involved parties, there is room for information leakages that may erode the benefits to collusion. Problems of mechanism design by privately involved parties are prevalent in Eco- nomics, from bilateral or multilateral trade with an informed seller, to contracting with privately informed firms. The literature stems from the seminal work of Myer- son (1983) and Maskin and Tirole (1990, 1992). More recent work includes the work of Mylovanov and Kovac (2012); Mylovanov and Troeger (2012). These recent papers characterize the problem of mechanism design by an informed principal, whose information is payoff-irrelevant to the agents (but is relevant in determining her behaviour in the mechanism). McAfee and McMillan (1992) introduce the “third-party approach” to the design of collusion agreements. To dispel with the third party in the design of the mechanism in collusion, Quesada (2005) introduces the informed-principal approach to a problem of collusion in production delegation. The tools of mechanism design by an informed principal have both parties simul- taneously reporting to the mechanism designed by the principal. In other words, the informed party designs a mechanism but then acts as another “agent”; the mechanism is executed or “run” by an uninformed, impartial outsider. When collusion is illegal, it is unlikely that the agreement can be implemented by an impartial enforcer. This leads the informed parties in charge of the execution as well as of the design of the mechanism. The literature has studied the design problem in detail, but the execution problem has been largely neglected. While execution is not an issue with formal contracts, it is an important aspect of informal contracts and illegal agreements. In this paper, we look at collusion in a second-price auction (SPA) with interdependent values and affiliated signals as a problem of contract design and execution by an informed principal. As Hendricks et al. (2008) point out, bidders in common-value

1See, for instance, McAfee and McMillan (1992). 2Aoyagi (2003, 2007).

2 auctions have an incentive to share information to determine if the asset in question is worth acquiring. To isolate the effect of information leakage and other frictions from enforcement issues, we analyze a one-shot interaction game.3 There are two bidders. Bidder 1 approaches bidder 2 with a collusion proposal. The agreement consists in sharing (unilaterally) information in exchange for a transfer and the commitment to staying out of the auction. In this setting, the informational content of bidder 2’s report turns out to be irrelevant; our results thus constitute an extension of Es˝oand Schummer (2004). The information shared becomes relevant when we introduce outside competition, in the form of a third bidder who is unaware of the possibility of collusion between the other two bidders. We identify conditions under which equilibria are monotonic and feature partial pooling at the top. Under some additional technical assumption, we characterize the unique continuous and non-constant equilibrium. The rest of the paper is organized as follows. Section 2 describes the model. Sec- tion 3 characterizes equilibria. Section 4 analyzes a pure common-value example and computes ex-ante expected payoffs, compared against different benchmarks. Section 5 re-computes these payoff under the presence of outside competition. Section 6 con- cludes.

2 The model

There are 2 agents, the bidders, as well as a seller. Bidder 1 (‘she”) is the “informed principal,” endowed with the bargaining power to approach bidder 2 (“he”) with a collusion proposal. The auction is a sealed-bid second-price auction (SPA). Before any social interaction, each bidder receives a private signal Si, which we normalize to lie in the interval [0, 1]. In addition, there is another signal Z which may influence the value of the object but is not observed by either bidder. Let F(z, s1, s2) be the joint cumulative distribution function of (Z, S1, S2). We assume that F is sym- 4 metric in (s1, s2), and that all of the variables are affiliated. In addition, we assume that F(z, s1, s2) has a continuous density function denoted f (z, s1, s2). Each bidder has a valuation function Vi = V(Z, Si, S i), where V is nonnegative, continuous, and nondecreasing in all variables. In addition, we assume that all necessary expectations exist and are finite. This formulation encompasses both the private-

3We can think of our game as the stage game in a repeated game, where we look at the cooperative interaction of bidders that can (presumably) be enforced by some form of future punishment. However, this extrapolation can only work if we take the roles of (alternatively, the allocation of bargaining power between) the two bidders as ﬁxed. A more proﬁtable collusion scheme might involve role rotating. This question is left for future research. 4The extreme case of independence is allowed.

3 5 value (Vi = Si) and common-value (Vi = Z) models. The timing of the game is as follows.

Stage 0: Nature draws signals z, s1 and s2; bidder i privately observes si.

Stage 1 (Collusion stage): Bidder 1 has the option of offering a collusion contract C to bidder 2, which consists of a message space and an outcome function (t˜, b˜ , b˜ ) : M 1 2 R R2 that speciﬁes: M! ⇥ + a transfer t˜(m) from bidder 1 to bidder 2 • bids (b˜ (m), b˜ (m)) that bidder 1 will submit (on behalf of both) in the auction. • 1 2 Bidder 2 either accepts or rejects the contract. If she accepts, she reports a message m and receives transfer t˜(m).

Stage 2 (Auction stage): If bidder 2 accepts the contract, she commits to staying out of the auction and bidder 1 executes the agreement. If bidder 2 rejects the contract, both agents bid non-cooperatively.

The payoff to bidder 1 with value V1 who faces a payment of m1 to the auctioneer and who pays bidder 2 a transfer of t is I(1 wins object)V m t. Bidder 2’s 1 1 payoff, when his value is V2, pays m2 to the auctioneer and receives t from bidder 1 is I(2 wins object)V m + t. 2 2

3 Characterizing Equilibria

3.1 Noncooperative auction

We begin by solving the auction stage in the case that collusion does not occur or the proposed collusion agreement is rejected. Let G1(s1) be any belief for bidder 2 over signal s1, and let G2(s2) be any belief for bidder 1 over s2. Deﬁne the function w(x, y)=E[V S = x, S = y]. Milgrom and Weber (1982) show that 1| 1 2 w(x, y) is nondecreasing in x under our assumptions; like them, we make the non- degeneracy assumption that this function is strictly increasing in x, and normalize w so that w(0, 0)=0.

Proposition 1. [Milgrom and Weber (1982)] The symmetric bidding proﬁle given by b(si)= w(si, si) is a Bayesian-Nash equilibrium, for any beliefs, in any continuation game following failure to collude.

5The set-up is very similar to that found in Milgrom and Weber (1982).

4 A shortcoming of this equilibrium, one that simpliﬁes the analysis, is that equilibrium behaviour “partially” disregards information that may be leaked through the proposal: while the outside options with respect to collusion contract , C si ui(si; ) := [w(si, s i) w(s i, s i)]dF i(s i ), C ˆ0 |C

6 obviously depend on the updated beliefs, equilibrium bids do not. Therefore, type s2 of agent 2 will accept a transfer t in collusion-proposal provided that t u (s ; ).7 C 2 2 C

3.2 Simplifying bidder 1’s strategies

We can now move back to the collusion proposal/acceptance stage of the game, and simplify bidder 1’s strategies. Recall that a collusion contract specifies a message C space and an outcome function (t˜, b˜ , b˜ ). By the revelation principle, we can focus M 1 2 on direct revelation mechanisms, and set =[0, 1].8 Additionally, if is accepted, M C bidder 1 will clearly submit bids on behalf of herself and bidder 2 such that she wins the object and pays nothing.9 With these simplifications, a collusion contract can be reduced to a menu of transfers t˜ : [0, 1] R[0,1]. Since the signals are non-verifiable, incentive compatibility ! implies that the function t˜ must be a constant. In effect, bidder 1 is “bribing” bidder 2 to staying out of the auction. Thus, our game reduces to an extension of Es˝o and Schummer (2004) that allows for interdependencies in valuations and affiliation of the private signals. A strategy for bidder 1 in the first stage can be identified by a

6As Milgrom (1981) shows, with two bidders there is in fact a family of ex-post equilibria, indexed by a strictly increasing continuous real function f ,

s + f 1(s ) s + f (s ) b (s )= 1 1 , b (s )= 2 2 . 1 1 2 2 2 2 For simplicity, we focus on the symmetric equilibrium where f is the identity function. This is the equilibrium commonly used in the auction literature, and, with 3 or more bidders, Bikhchandani and Riley (1991) show that the symmetric equilibrium is in fact the only equilibrium in strictly increasing and continuous strategies in which all bidders have a positive ex-ante probability of winning. There- fore, comparability of the results with those of the case of 3 bidders is another rationale for focusing on the symmetric equilibrium. In what follows, we will select this equilibrium as the continuation equilibrium. 7In the terminology of Cramton and Palfrey (1995), we allow for a form of ratiﬁability: there is updating of the outside option from the proposal. 8We assume that bidder 1 can commit to transfers, though not to bids. Transfers are paid in the ﬁrst stage, while bids are submitted in the second stage. 9Bidder 1 would not get the good for free if the seller sets a positive reserve price. We assume, for simplicity, that reserve prices are 0. However, this is not an innocuous normalization. With positive reserve prices, Claim 1 below is false; the symmetric equilibrium is to bid one’s private signal when this signal is above a cutoff, but this cutoff depends on beliefs. Hence, the equilibrium strategies in the continuation game, and not just the payoffs, depend on the updated beliefs.

5 function mapping her signal into transfers (or “bribes”). We will denote this function t¯ : [0, 1] R, and write t¯(s ) for the transfer that type s offers to bidder 2. ! 1 1

3.3 Simplifying bidder 2’s strategies

A strategy for bidder 2 in the ﬁrst stage of the game can be identiﬁed by a (measureable- valued) acceptance correspondence : R 2[0,1], where (t) is the (measurable) set of A ! A types that accept transfer t, while the types in the complement reject it. We have the following result.

Lemma 1. In any equilibrium, for any t R, the set (t) is of the form [0, s¯ (t)] for some 2 A 2 s¯ (t) [0, 1]. 2 2

Proof. Given that bidder 2 observes an offer of t, he will have some beliefs over the types of bidder 1 represented by F (s t, s ). Therefore, he will accept the transfer if 1 1| 2 and only if: s2 t [w(s2, s1) w(s1, s1)]dF1(s1 t, s2). ˆ0 | Since w is nondecreasing in its ﬁrst argument and the variables are afﬁliated, the right- hand side (RHS) is nondecreasing in s2. Thus, if the inequality holds for some s2, it holds for all s s . 20  2

Lemma 1 implies that a strategy for bidder 2 can be summarized by a cutoff function s¯ : R [0, 1] where, for any t, all s s¯ (t) accept t and all s > s¯ (t) reject it. 2 ! 2  2 2 2

3.4 (Weak perfect) Bayesian equilibrium

We can write bidder 1’s payoff if she is of type s1, offers transfer t, and bidder 2 follows a strategy summarized by cutoff sˆ2 as:

sˆ2 s1 U (s , sˆ , t)= [w(s , s ) t]dF(s s )+I(s > sˆ ) [w(s , s ) w(s , s )]dF(s s ). 1 1 2 ˆ 1 2 2 1 1 2 ˆ 1 2 2 2 2 1 0 | sˆ2 |

Similarly, we can write bidder 2’s payoff if he is of type s2, is offered transfer t, has beliefs over s represented by F ( t) and rejects the contract, as: 1 1 ·| s2 U2(s2, t)= [w(s2, s1) w(s1, s1)]dF1(s1 t). ˆ0 |

6 With the ﬁxed continuation equilibrium in the noncooperative auction, we will refer to (t¯( ), s¯ ( ), F ( )) as a (weak perfect) Bayesian equilibrium if: · 2 · 1 ·|·

(i) t¯(s ) arg max U (s , s¯ (t), t) for all s . 1 2 t 1 1 2 1 (ii) s¯ (t) arg max I(s s˜ )t + I(s > s˜ )U (s , t) for all s , t. 2 2 s˜2 2  2 2 2 2 2 2 (iii) for all t, beliefs F ( t) follow Bayes’ rule where possible. 1 ·|

If we consider U1(s1, sˆ2, t) as a function of sˆ2, we can plot bidder 1’s (expected) utility as a function of the mass of types of bidder 2 who accept the offer to collude. It can happen U1(s1, sˆ2, t) is actually decreasing in sˆ2. When this happens, it means that bidder 1 prefers fewer types of bidder 2 to accept his offer to collude (for certain ﬁxed values of s1 and t). Intuitively, this will happen when bidder 1 has a low estimate for the object’s value (s1 is low), and so less types of bidder 2 accepting means that bidder 1 has to pay the transfer t to fewer agents. When the object is not very valuable, bidder 1 may prefer to actually lose the object rather than to pay bidder 2 a transfer that is too high. So, U1(s1, sˆ2, t) is not necessarily increasing in sˆ2, the next result gives a characterization of this function that will be useful later.

10 Proposition. For any ﬁxed s1, t, the function U1(s1, sˆ2, t) is quasiconvex in sˆ2.

L Proof. We show that if U1 is increasing near some point sˆ2 , then it is also increasing L on (sˆ2 ,1], which in turn implies that U1 is quasiconvex in sˆ2. Fixing s1 and t, deﬁne a function g as

g(sˆ2)=U1(s1, sˆ2, t)

s1 sˆ2 s1 w(s1, s2)dF(s2 s1) tdF(s2 s1) w(s2, s2)dF(s2 s1), sˆ2 < s1 = 0 | 0 | sˆ2 | . ´ sˆ2 ´ ´ ( 0 [w(s1, s2) t]dF(s2 s1) sˆ2 s1 ´ | Note that g is everywhere continuous, and is everywhere differentiable, except per- haps sˆ2 = s1. The derivative of h is

[w(sˆ2, sˆ2) t] f (sˆ2 s1), sˆ2 < s1 g0(sˆ2)= | . ([w(s1, sˆ2) t] f (sˆ2 s1), sˆ2 s1 |

At sˆ2 = s1, since the derivative from the left is equal to the derivative from the right, g (s ) exists everywhere, and g (s )=[w(s , s ) t] f (s s ). Note that the sign of 0 1 0 1 1 1 1| 1 g ( ) is determined by the terms in brackets. Since w( , ) is increasing in both argu- 0 · · · ments, if g (sˆ ) 0 for some sˆ , then g (sˆ ) 0 for all sˆ sˆ as well. 0 2 2 0 20 20 2 10Recall that a function f is quasiconvex on a convex set S if f (lx +(1 l)y) max f (x), f (y)  { } for all x, y S and all l [0, 1]. 2 2 7 If the transfer t is prohibitively high, it is possible that U1(s1, sˆ2, t) is everywhere decreasing in sˆ2 (arg minsˆ2 U1(s1, sˆ2, t)=1), and bidder 1 may prefer all types of bidder 2 to reject. More interestingly, it may be that arg minsˆ2 U1(s1, sˆ2, t) is interior. The key implication of the proposition is that, ﬁxing s1 and t, if at some sˆ2 bidder 1 would prefer bidder 2 to use a lower cutoff, then he would prefer bidder 2 to use a cutoff of 0 rather than sˆ2. This is important because, in equilibrium, bidder 1 can always deviate to offering a transfer of 0, thereby inducing all types of bidder 2 to reject his offer. Thus, for any t offered in equilibrium, either s¯2(t)=0 or s¯2(t) is on the increasing part of U1.

3.5 Monotonic equilibria

We say that an equilibrium (t¯( ), s¯ ( ), F ( )) is monotonic or nondecreasing if both · 2 · 1 ·|· t¯ and s¯2 are nondecreasing. Under the next two assumptions, it turns out that all equilibria are nondecreasing.

s¯ (A1) For all s and s¯, the function s w(x, y)dF(y x) is nondecreasing in x. ´ | (A2) w(x, y) w(y, y) is nonincreasing in y for all y x. 

To understand (A1), a higher s1 has two opposing effects from the perspective of bidder 1: first, it increases the value of the object itself (w increases). However, because of affiliation, it also suggests that bidder 2’s signal is stochastically higher (F(s s ) decreases), which implies a lower chance of winning the auction. Assump- 2| 1 tion (A1) requires that the first effect outweigh the second. Assumption (A2) implies that a rise in bidder 2’s signal increases bidder 2’s bid in a noncooperative auction more than it increases bidder 1’s value for the object as long as bidder 1 has the higher signal (and vice-versa). Intuitively, both assumptions (A1) and (A2) require bidder i’s signal, si to influence his valuation for the object more than his opponent’s signal s i does.

Theorem 1 (Monotonic equilibria). Assume that both (A1) and (A2) hold. Then, in any equilibrium, both t¯ and s¯2 are nondecreasing on the equilibrium path.

Proof. See Lemmas 3 to 5 in the Appendix.

These conditions (A1) and (A2) are sufﬁcient for our results, but they seem more than necessary. Nonetheless, they are assumptions commonly featured in the literature. A sufﬁcient condition for (A1) to hold is that s1 and s2 are independent (possi- bly after conditioning on the common component z). Assumption (A2), on the other

8 hand, will hold both in the pure private value and pure common value models, or if the function w(x, y) has decreasing differences in (x, y).

Theorem 2 (Pooling at the top). In any equilibrium, there exists a type s such that t¯(s1)=t for some ﬁxed t for all s [s,1]. 1 2

Proof. Let (t¯, s¯ ) be an equilibrium. For any t in this equilibrium, let s (t)=inf s : 2 L { 1 t¯(s1)=t . Also, let g(t) be defined implicitly as the solution to the equation t = rej } rej U2 (g(t), t), where the function U2 is defined in Lemma 4 in the Appendix. Now, s¯ (t)=min g(t),1 . Note there is a subtle difference between s¯ (t) and g(t), in that 2 { } 2 we may have g(t) > 1. This occurs when all types s [0, 1] strictly prefer to accept 2 2 the offer t rather than reject, and so s¯2(t)=1. We first note that for any t > 0, we have g(t) > sL(t). To see this, consider a type sˆ2 = sL(t) who observes an offer of t > 0. rej Bayes’ rule implies that supportF(s sˆ , t) [s (t),1], and so U (sˆ , t)=0 for any 1| 2 ✓ L 2 2 consistent beliefs F(s sˆ , t), which in turn implies that g(t) > s (t). 1| 2 L Next we show that this implies there is pooling at the top. If t¯(s1)=0 for all s1, then there is obviously pooling at the top. Thus, assume that t¯(s1) > 0 for some s1, and that there is no pooling at the top. Then, by Theorem 1, there exists some # > 0 such that t¯(s ) is continuous on (1 #,1] and t¯(1 d) < t¯(1) for all d (0, #). Next, 1 2 note that g(t) is also continuous, and this, together with g(t¯(1)) > sL(t¯(1)) = 1 from the preceding paragraph, imply that g(t¯(1 d)) > 1 for all d > 0 sufficiently small. This means that s¯ (t¯(1 d)) = min g(t¯(1 d)),1 > 1 for all d > 0 sufficiently 2 { } small. Thus, type s = 1 can offer t¯(1 d), still get all types of bidder 2 to accept, and 1 pay a lower transfer (t¯(1 d) < t¯(1)), which is profitable deviation, and we have a contradiction.

The intuition behind this result is clear. If there is separation at the top, then bidder 2 will perfectly observe bidder 1’s type from her transfer when s1 = 1. After learning that bidder 1 has such a high signal, all types of bidder 2 will strictly prefer colluding than acting non-cooperatively. But this means that bidder 1 can then imitate type s = 1 # by lowering his offer a small amount e > 0 and still get bidder 2 to accept 1 with probability 1. She is thus paying less to bidder 2 while inducing the same action from him, which is clearly a proﬁtable deviation.

3.6 A continuous, partially separating equilibrium

We look for a partially separating equilibrium, where bidder 1’s strategy is continuous and perfectly separating on some interval [0, s¯1] while all s1 > s¯1 pool at some transfer t. In the separating region, bidder 2 correctly infers s1 and we can use local

9 IC (incentive compatibility) constraints to characterize t¯. In particular, bidder 1’s expected utility when she is of type s1 in the separating region and mimics type sˆ1 also in the separating region is:

s¯2(t¯(sˆ1)) U1(s1, s¯2(t¯(sˆ1)), t¯(sˆ1)) = [w(s1, s2) t¯(sˆ1)]dF(s2 s1). (1) ˆ0 |

Local IC requires that this function be maximized at sˆ1 = s1. Deﬁne h(s1, t) implicitly as the solution to:

t = w(h(s , t), s ) w(s , s ). 1 1 1 1

That is, h(s1, t) is the type of bidder 2 who is indifferent between accepting and rejecting t conditional on knowing s1. Assuming that w is continuously differentiable and using the implicit function theorem, we ﬁnd that:

∂h w (s , s )+w (s , s ) w (h(s , t), s ) = 1 1 1 2 1 1 2 1 1 , ∂s1 w1(h(s1, t), s1)

∂h 1 = . ∂t w1(h(s1, t), s1)

Considering t¯(s1) as a function of s1, we can write the total derivative of h with respect to s1 as

dh(s , t¯(s )) t¯ (s )+w (s , s )+w (s , s ) w (s¯ (t¯(s )), s ) 1 1 = 0 1 1 1 1 2 1 1 2 2 1 1 . ds1 w1(s¯2(t¯(s1)), s1)

In the separating region, we have s¯2(t¯(s1)) = h(s1, t¯(s1)). Local IC gives the following equation:

ds¯2(t¯(s1)) [w(s1, s¯2(t¯(s1))) t¯(s1)] f (s¯2(t¯(s1)) s1) t¯0(s1)F(s¯2(t¯(s1)) s1)=0. (2) | ds1 |

Plugging the expression for ds¯2/ds1 into equation (2), we get the following differential equation that characterizes t¯ in the separating region:

[w(s1, s¯2(t¯(s1))) t¯(s1)][w1(s1, s1)+w2(s1, s1) w2(s¯2(t¯(s1)), s1)] f (s¯2(t¯(s1)) s1) t¯0(s )= | . 1 w (s¯ (t¯(s )), s )F(s¯ (t¯(s )) s ) [w(s , s¯ (t¯(s ))) t¯(s )] f (s¯ (t¯(s )) s ) 1 2 1 1 2 1 | 1 1 2 1 1 2 1 | 1

With this derivation, we have the following result.

Theorem 3 (Differential characterization). For any s1, let h(s1,t) be the unique function that solves t = w(h(s , t), s ) w(s , s ). Suppose F( ) is log-concave. In any equilibrium 1 1 1 1 ·|·

10 in which t¯( ) is continuous and is not constant, then it is the unique solution to the following · differential equation with initial condition t¯(0)=0:

[w(s ,h(s ,t¯(s ))) t¯(s )][w (s ,s )+w (s ,s ) w (h(s ,t¯(s )),s )] f (h(s ,t¯(s )) s ) 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 | 1 ¯ w (h(s ,t¯(s )),s )F(h(s ,t¯(s )) s ) [w(s ,h(s ,t¯(s ))) t¯(s )] f (h(s ,t¯(s )) s ) , h(s1, t(s1)) < 1 t¯0(s1)= 1 1 1 1 1 1 | 1 1 1 1 1 1 1 | 1 (0, otherwise (3) ¯ ¯ Let s1⇤ be such that h(s1⇤, t(s1⇤)) = 1, and t⇤ = t(s1⇤). In this equilibrium,

1 h(t¯ (t), t), t < t⇤ s¯2(t)= . (1, t t⇤

4 Example: Pure common value

4.1 Equilibrium

Consider a pure interdependent-value example, with V(Y, S1, S2)=V(Y, S2, S1)= S1+S2 2 , with S1 and S2 distributed independently and uniformly on the interval [0, 1] x+y (the random variable Y is irrelevant for this example). This implies that w(x, y)= 2 . In the separating region, the differential equation characterizing t¯ becomes:

s1 2t(s ) s , s1 + 2t(s1) < 1 t¯0(s1)= 1 1 . (0, else

It is easy to show that the unique solution to this differential equation is:

1 t¯(s )=min s , . 1 1 3 ⇢

Similarly, if bidder 2 perfectly observes s1 from the offered transfer, the indifferent type will be s¯2(t)=2t + s1. Thus, the equilibrium strategy for bidder 2 is summarized by: s¯ (t)=min 3t,1 . 2 { }

11 4.2 Ex-ante payoffs

Conditional on s1, bidder 1’s interim-expected payoff is:

2 3s1 1 4 , s1 3 U˜ 1(s1) := U1(s1, s¯2(t¯(s1)), t¯(s1)) =  . ( s1 1 , s > 1/3 2 12 1 Integrating this over s , we ﬁnd that her ex-ante expected payoff 19 0.176. For 1 108 ⇡ bidder 2, his interim-expected proﬁt conditional on s2 is

s 2 1 3 s s 1 U˜ (s ) := E [U (s , t¯(s ))] = 2 1 ds + min s , ds . 2 2 2 2 1 1 s 1 1 ˆ0 2 ˆ 2 3 ✓ ◆ 3 ⇢ Integrating over s , his ex-ante expected payoff is 11 0.3056. The expected revenue 2 36 ⇡ accruing to the seller is 1 0.019. 54 ⇡

No-collusion benchmark

The ﬁrst obvious question to ask is how these numbers compare to the purely noncooperative game in which no collusion is possible. Both bidders are symmetric in this setting, so consider bidder 1:

1 s10 s10 s2 1 ds2ds10 = 0.08333. ˆ0 ˆ0 2 12 ⇡

1 The expected revenue to the seller is 3 . Clearly, with collusion, the seller is much worse off while both bidders are better off. However, bidder 2 captures a (signiﬁ- cantly) higher portion of the surplus than bidder 1. Thus, bidder 1 suffers a “ﬁrst- mover (relative) disadvantage.”

McAfee and McMillan (1992)

McAfee and McMillan (1992) provide the canonical “third-party” approach to modeling collusion. They focus on private values because, as they note, with common values, the optimal collusion mechanism with communication is trivial: the cartel al- locates the object according to some exogenous rule. Because reports do not affect the resulting allocation, no bidders’ types have any incentive to misreport. With common

12 values, this allocation rule is clearly efficient.11 A major assumption inherent in this result is that bidders commit to the cartel ex-ante. If bidders have private information when they decide upon collusion, this mechanism may not work. For example, a bidder with a higher signal for the object may prefer to enter the auction on her own, since she knows the object is valuable and desires a greater than 1/n chance of winning the object from the cartel. We model collusion at the interim stage. Computing the ex-ante expected surplus of the two agents when the good is allocated according to the flip of a fair coin, we find that they both receive utility of 1/4. While they find that all agents receive the same ex-ante surplus, we find a non- symmetric distribution. In particular, we find that bidder 1, the proposer, does worse than when the contract is designed before any private information is acquired, while bidder 2 does better. In addition, the McAfee and McMillan (1992) approach to collusion predicts that collusion always occurs and that the colluding bidders are able to extract the entire surplus. In our model, like in Es˝oand Schummer (2004), collusion may not occur in equilibrium and the seller is able to retain some positive profits.

Es˝oand Schummer (2004)

The private values model of Es˝oand Schummer (2004) is a special case of our model when Vi = Si for both bidders. The equilibrium we describe in Theorem 3 specializes to the same equilibrium that Es˝oand Schummer study. In this equilibrium, bidder 1’s payoff conditional on s1 is:

2 3s1 2 4 , s1 3 U˜ 1(s1)=  . (s 1 , s > 2 1 3 1 3

13 Integrating over s1, we ﬁnd that her ex ante expected payoff is 54 . For bidder 2, his expected payoff conditional on s2 is:

s 2 1 3 1 s U˜ (s )= (s s )ds + min , 1 ds . 2 2 2 1 1 s 1 ˆ0 ˆ 2 3 2 3 ⇢

11Communication is needed for the allocation to be efﬁcient if the seller has a reservation value or sets a reserve price; the cartel needs to establish if the object is worth purchasing. When the seller attaches no (use) value to the object or there is no reserve price, the efﬁcient allocation is to always allocate the object to some bidder. This allocation rule can (obviously) be implemented without any communication.

13 1 Integrating over s2 yields 3 . The seller’s expected proﬁt is

2 1 3 2 min s1, s2 ds2ds1 = . ˆ ˆ 3 { } 27 0 2 s1 The total ex-ante surplus if the good were allocated efﬁciently would be:

1 1 2 max s1, s2 ds2ds1 = . ˆ0 ˆ0 { } 3 Thus, we lose some surplus when bidder 1 has a lower value for the good but buys bidder 2 out of the auction. Again, both bidders do better than without collusion but bidder 2 does relatively better than bidder 1. We ﬁnd that bidder 2 does not gain quite as much in the private values case as he does in the common values case, with this extra surplus going mostly to the seller and the rest lost to inefﬁciency. Since the common value and private value models are not directly comparable (e.g., the total ex-ante surplus in the common value model is 1/2, while in the private value model it is 2/3), Table 1 below gives the proportion of the total surplus accruing to each bidder and the seller in each case.

5 Outside Competition

When the value functions to the bidders include some common value component, there is an additional rationale for information sharing between bidders that is not present when the cartel is all-inclusive. In the previous discussion, the actual content of bidder 2’s signal is irrelevant to bidder 1; all that matters is that bidder 2 stays out of the auction. To highlight the role of the information-sharing motivation for collusion we now introduce a third bidder to the model who is outside of the “cartel.” To highlight the

Bidder 1 Bidder 2 Seller Efﬁciency Loss Our Model 0.35 0.61 0.04 0 Our Model - No collusion 0.16 0.16 0.67 0 Private Values - No collusion 0.25 0.25 0.50 0 Es˝oand Schummer (2004) 0.36 0.50 0.11 0.03 McAfee and McMillan (1992) 0.50 0.50 0 0

Table 1: Comparison of ex-ante equilibrium payoffs against different benchmarks.

14 importance of communication, we specialize to a pure interdependent value setting. In this environment, the information of bidder 2 is now relevant to bidder 1 in updating her beliefs about the value of the object and refining her bid against the third bidder. This will allow us to study how competition from “outside the cartel” affects the distribution of surplus between the bidders and the seller. We now have 3 bidders, named 1, 2 and 3, and a seller; bidder 1 is again the “informed principal” who can approach agent 2 with a collusion proposal. Bidder 3 (another “he”) is an outsider who is unaware of the possibility of collusion between 1 and 2. For simplicity, we focus on the example from Section 4. The value of the object is now: S + S + S V(S , S , S )= 1 2 3 . 1 2 3 3 The rest of the game is as before. Define: x y w(x, y) := E [V (S , S , S ) S = x, max S , S = y] = + . 1 2 3 | 1 { 2 3} 3 2 The next proposition identifies equilibrium bidding behavior.

Proposition 2. Assume that, when bidder 2 accepts the transfer, he reports truthfully to bidder 1.12 Then, it is part of a (weak perfect) Bayesian equilibrium for all bidders to bid 5 b(si)=6 si when collusion fails and for bidder 1 to bid 5 5 b⇤(s , s )=min (s + s ) , 1 1 2 9 1 2 6 ⇢ following collusion with bidder 2.

Proof. Bidder 3 bids: 5 b(s )=w(s , s )= s 3 3 3 6 3 in the auction. Let the updated beliefs about the highest signal out of those of the other bidders be represented by some cdf G ( ) (bidder 1 will also update her beliefs i ·|C about s upon rejection of the contract). Outside options with respect to proposal 2 C are then: 5 si s u u (s ; ) := U s , s ; = i dG (u ). i i C i 6 i i C ˆ 3 i |C ✓ ◆ 0 ✓ ◆ 12As in section 3, bidder 2 has no reason to lie if he can commit to staying out of the auction. Inability to commit need not be a big problem with only two bidders; it is an equilibrium for bidder 1 to place a bid of 1 in the auction after colluding with bidder 2, effectively discouraging him from ‘sneaking back’ into the auction. Here, however, the problem is more serious. Bidder 1 cannot always discourage bidder 2 without taking a loss herself. Moreover, bidder 2 can manipulate a trusting bidder 1’s bid in the auction and learn about her signal from her offer. Of course, such bad incentives of bidder 2 break down bidder 1’s incentives to disclose information through her proposals in the ﬁrst place.

15 If the collusion agreement is accepted, bidder 1 learns bidder 2’s signal and faces only bidder 3 in the auction. Her expected payoff in the auction from submitting a bid of b R+, U (b, s , s ), is: 2 1 1 2 min 6 ,1 5 b s + s u 5 U (b, s , s ) : = { } 1 2 + u du 1 1 2 ˆ 3 3 6 0 ✓ ◆ min 6 ,1 5 b s + s u = { } 1 2 du. ˆ 3 2 0 ✓ ◆ 5 Now, for b < 6 ,

s1 + s2 3 6 U0 (b, s , s )= b 1 1 2 3 5 5 ✓ ◆ 18 U00(b, s , s )= < 0. 1 1 2 25 The unique best response is:

5 b⇤(s , s )= (s + s ) . 1 2 9 1 2

5 3 3 We have that b⇤(s1, s2) < 6 provided s1 + s2 < 2 . If s1 + s2 2 , then U1(b, s1, s2) 5 is strictly increasing in b up to 6 , at which point it becomes constant. Hence, a best response b⇤1 is: 5 5 b⇤(s , s )=min (s + s ) , , 1 1 2 9 1 2 6 ⇢ as claimed.

We next show that a result analogous to Lemma 1 holds in this setting for bidder 2.

Lemma 2. In any equilibrium, the set of types of bidder 2 who accept a transfer of t must be of the form [0, s¯2(t)].

Proof. Given that bidder 2 observes an offer of t, she will have some beliefs over the types of the highest signal besides his represented by G(u t). Then, he will accept the | transfer if and only if:

s2 s u s s2 u t 2 dG(u t)= 2 G(s t¯) dG(u t). ˆ 3 | 3 2| ˆ 3 | 0 ✓ ◆ 0

16 We show that the right-hand side is non-decreasing in s2. Integrating the second term by parts yields:

s s 2 u u s2 2 1 dG(u t)= G(u t) G(u t)du. ˆ0 3 | 3 | 0 ˆ0 3 | h i 1 s2 Hence, the right-hand side is 3 0 G(u t)du, which is clearly non-decreasing in s2. ´ |

Just as before, bidder 2’s strategy can be summarized by a cutoff such that all types below the cutoff accept the transfer t, and all types above reject. We again look for an equilibrium (t¯, s¯2) that is partially separating. In the separating region, we can write an equation analogous to Equation (1) for bidder 1’s utility if he is of true type s1 and he mimics type sˆ1 as:

s¯2(t¯(sˆ1)) ¯ ¯ U1(s1, s¯2(t(sˆ1)), sˆ1)= [U1⇤(s1, s2) t(sˆ1)]ds2. ˆ0

We can again use local IC to characterize t¯ in terms of a differential equation t¯0(s1)= g(s1, t¯(s1)) in the separating region. The equilibrium is qualitatively very similar to that found before. Low types of bidder 1 separate, up to the point where offering more guarantees all types of bidder 2 will accept, beyond which all types pool. We can also numerically calculate the equilibrium strategy for player 2, and the proportion of the total ex-ante surplus accruing to each of bidders 1, 2, and 3 and the seller. The results are shown in Tables 2 and 3. Several points of interest arise from these two tables. Comparing the three bidder model with and without collusion, we find that both bidders 1 and 2 are better off with collusion, and more of the gains go to the receiver (bidder 2) than the proposer (bidder 1). The benefits to the cartel come at the expense of the seller and of bidder 3, as expected. Next we compare the 2 bidder case to the 3 bidder case. Here, we note that with 2 bidders, collusion increased the profits of bidder 2 by 280% and those of bidder 1 by 118%. With three bidders, these increases drop to 150% for bidder 2 and 33% for bidder 1. While some of this loss is due to the presence of bidder 3, most of the lost

No Collusion Bidder 1 Bidder 2 Bidder 3 Seller 2 Bidders 0.16 0.16 - 0.67 3 Bidders 0.06 0.06 0.06 0.83

Table 2: Comparison of ex-ante equilibrium payoffs with 2 versus 3 bidders in the absence of the possibility of collusion.

17 With Collusion Bidder 1 Bidder 2 Bidder 3 Seller 2 Bidders 0.35 0.61 - 0.04 3 Bidders 0.08 0.15 0.05 0.72

Table 3: Comparison of ex-ante equilibrium payoffs with 2 versus 3 bidders when collusion is possible.

surplus goes to the seller instead. With 2 bidders, the presence of collusion reduced seller profits by 94%, while with some competition outside of the cartel, collusion among bidders 1 and 2 only decreases seller profits by about 13%. Some broad lessons can be drawn from this exercise. First, while collusion cer- tainly decreases seller profits, the presence of at least some competition outside of the cartel drastically decrease the effectiveness of the cartel. Second, the first-mover disadvantage prevails with outside competition. If we think of this game as embedded in some larger game in which the proposer is decided endogenously, it may be likely that neither bidder will propose, hoping instead to be the receiver. Thus, while there will be gains from colluding, waiting for the other bidder to take the first step might erode those gains.

6 Conclusion

In auctions in which collusion is illegal and collusion agreements are both designed and executed by the involved parties, the standard approach is inadequate. In this paper, we model collusion in a SPA with interdependent valuations and affiliated signals as a contract-design problem by an informed principal who runs the mechanism herself. We characterize equilibria in terms of a transfer function and an “acceptance cutoff” and provide sufficient conditions under which these two are non-decreasing on the equilibrium path. Moreover, show that there is always “pooling at the top” in the transfers and identify the differential equation that characterizes the transfer function on the separating region under an additional differentiability assumption, extending the results in Es˝oand Schummer (2004). It is a matter of ongoing work whether this equilibrium is the unique equilibrium that satisfies D1-type refinements. We find that collusion between the two bidders can severely harm the seller. How- ever, while both bidders are better off than not colluding, there is a (relative) first- mover disadvantage. If we think of this game as embedded in some larger game in which the proposer is decided endogenously, it may be likely that neither bidder will want to be the first to propose. An interesting extension that is left for future work is

18 to model the allocation of bargaining power in the collusion stage as a war attrition, where the loser makes the proposal. Moreover, the amount of surplus captured by the colluding bidders at the expense of the seller might be greatly reduced when “outside competition,” in the form of additional bidders outside the ring, is introduced. The generality of this feature of the example studied is another matter of ongoing work.

19 References

Aoyagi, M. (2003). Bid rotation and collusion in repeated auctions. Journal of Economic Theory 112:79–105.

Aoyagi, M. (2007). Efﬁcient collusion in repeated auctions with communication. Jour- nal of Economic Theory 134:61–92.

Bikhchandani, S. and Riley, J.G. (1991). Equilibria in open common value auctions. Journal of Economic Theory 53:101–130.

Cramton, P. and Palfrey, T. (1995). Ratiﬁable mechanisms: Learning from disagree- ment. Games and Economic Behavior 10.

Es˝o,P. and Schummer, J. (2004). Bribing and signaling in the second price auction. Games and Economic Behavior 47(2):299–324.

Hendricks, K., Porter, R. and Tan, G. (2008). Bidding rings and the winner’s curse. RAND Journal of Economics 39(4):1018–1041.

Maskin, E. and Tirole, J. (1990). The principal-agent relationship with an informed principal: The case of private values. Econometrica 58(2):379–409.

Maskin, E. and Tirole, J. (1992). The principal-agent relationship with an informed principal, ii: Common values. Econometrica 60(1):1–42.

McAfee, R.P. and McMillan, J. (1992). Bidding rings. American Economic Review 82(3):579–599.

Milgrom, P.R. (1981). Rational expectations, information acquisition, and compeitive bidding. Econometrica 49:921–943.

Milgrom, P.R. and Weber, R.J. (1982). A theory of auctions and competitive bidding. Econometrica 50(5):1089–1122.

Myerson, R. (1983). Mechanism design by an informed principal. Econometrica 51(6):1767–1797.

Mylovanov, T. and Kovac, E. (2012). Informed principal problems in generalized private values environments. Theoretical Economics Forthcoming.

Mylovanov, T. and Troeger, T. (2012). Mechanism design by an informed principal: the quasi-linear private-values case. Working Paper.

Quesada, L. (2005). Collusion as an informed principal problem. Working Paper.

20 A Proofs

Bidder 1’s expected payoff given she is of type s1, offers transfer t, and bidder 2 uses cutoff s˜2, is:

s˜2 s1 U (s , s˜ , t)= [w(s , s ) t]dF(s s )+I(s > s¯ ) [w(s , s ) w(s , s )]dF(s s ). 1 1 2 ˆ 1 2 2 1 1 2 ˆ 1 2 2 2 2 1 0 | s˜2 | (4)

Bidder 1’s equilibrium payoff when she is of type s1 and offers a transfer t is given by U1(s1, s¯2(t), t).

Lemma 3. In any equilibrium, s¯ (t) inf s : t¯(s )=t for all t > 0 in the range of t.¯ 2 { 1 1 }

Proof. Let s1 (t) denote the inﬁmum in the statement of the lemma. Assume that the opposite inequality is true and consider any s (s¯ (t), s (t)). The equilibrium strat- 20 2 2 1 egy instructs bidder 2 of type s20 to reject the offer of t. However, upon rejection, he learns that bidder 1 is of type at least s1 (t). Being bound to lose in the ensuing auction, accepting t is a proﬁtable deviation.

For the next lemma, we need some additional notation. For any transfer function 1 1 t¯ and any set T R+, let t¯ (T) be the preimage of T under t¯, t¯ (T) := s : t¯(s ) ✓ { 1 1 2 T . For any two such sets X, X R+, if x > x for all x X and all x X , we write } 0 ✓ 0 2 0 2 0 X0 >SSO X (X0 is strictly greater than X in the strong set order).

Lemma 4. Let (t¯, s¯2) represent an equilibrium. If t¯ is strictly decreasing on a neighbourhood of point sˆ1, then s¯2 must be strictly decreasing on a neighbourhood of t¯(sˆ1). Similarly, if there exists some tL, tH both in the range of t¯ such that tL < tH, but s¯2(tL) > s¯2(tH) (i.e., s¯2 is strictly decreasing at some points on the equilibrium path), then t¯ 1( t ) > t¯ 1( t ) { H} 6 SSO { L} (i.e., t¯ is strictly decreasing).

Proof. First, assume that s¯2(tL) > s¯2(tH) for some tH > tL both in the range of t¯, but t¯ 1( t ) > t¯ 1( t ). Let F(s sˆ , t) be the updated beliefs of bidder 2 over s , { H} SSO { L} 1| 2 1 conditional on his signal sˆ and the observed transfer t. t¯ 1( t ) > t¯ 1( t ) 2 { H} SSO { L} implies that F(s sˆ , t ) first-order stochastically dominates F(s sˆ , t ) for any sˆ . 1| 2 H 1| 2 L 2 Define a function Rej(sˆ , t)= sˆ2 [w(sˆ , s ) w(s , s )]dF(s sˆ , t) to be the ex- 2 0 2 1 1 1 1| 2 pected payoff to type sˆ2 of agent 2 from´ rejecting a transfer t (note that bidder 2 uses the updated beliefs F(s sˆ , t)). Now s¯ (t) is defined as the solution to t = Rej(s¯ (t), t). 1| 2 2 2 We have that Rej(sˆ , t ) Rej(sˆ , t ), by the fact that the integrand is decreasing and 2 H  2 L 21 F(s sˆ , t ) first-order stochastically dominates F(s sˆ , t ). Now, consider some type 1| 2 H 1| 2 L sˆ , and assume that sˆ accepts an offer of t , i.e., t Rej(sˆ , t ). This implies that: 2 2 L L 2 L t > t Rej(sˆ , t ) Rej(sˆ , t ) H L 2 L 2 H

or tH > Rej(sˆ2, tH), i.e., type sˆ2 also accepts an offer of tH. However, this is clearly a contradiction to the fact that s¯2(tH) < s¯2(tL). For the other direction, assume that t¯( ) is strictly decreasing at sˆ , but that s¯ ( ) is · 1 2 · weakly increasing around t¯(sˆ1). The case of separation is ruled out below using local IC constraints, and so we focus on the case of pooling only. First, consider pooling on both sides of sˆ1, so that and let tL := t¯(sˆ1) and tH := lim# 0 t¯(sˆ1 #). We have ! t > t . Now, assume that s¯ (t ) s¯ (t ). Equality can be immediately ruled out H L 2 L  2 H because obviously no type of bidder 1 would offer tH. and so here we only focus on pooling. First, consider pooling on both sides of sˆ1, i.e.,

tH s1 [sL, sˆ1) t¯(s1)= 2 (tL s1 [sˆ1, sH] 2

for some sL < sˆ1 < sH, and tL < tH. Towards a contradiction, assume that s¯2(tL) < 13 14 s¯2(tH), and notice that s¯2(tH) > s¯2(tL) > sˆ1. Thus, the second term in Equation (4) can be ignored for all types of bidder 1 in some interval [0, sˆ1 + d) for some d > 0. This ¯ ( ) means we can write bidder 1’s utility as U (s , s¯ (t), t)= s2 t [w(s , s ) t]dF(s s ). 1 1 2 0 1 2 2| 1 By assumption, we have U (s , s¯ (t ), t ) U (s , s¯ (t )´, t ) and U (sˆ + #, s¯ (t ), t ) 1 L 2 H H 1 L 2 L L 1 1 2 H H  U1(sˆ1 + #, s¯2(tL), tL). Now, the following string of inequalities is true:

s¯2(tH) w(sˆ1 + #, s2)dF(s2 sˆ1 + #) s¯2(t ) ´ L | s¯2(tH) w(sL, s2)dF(s2 sL) (assumption (A1)) s¯2(t ) ´ L | s¯2(tH) s¯2(tL) tHdF(s2 sL)+ [tH tL]dF(s2 sL) (IC for type sL) s¯2(t ) 0 ´ L | ´ | s¯2(tH) s¯2(tL) > tHdF(s2 sˆ1 + #)+ [tH tL]dF(s2 sˆ1 + #) (afﬁliation) s¯2(t ) 0 ´ L | ´ | s¯2(tH) w(sˆ1 + #, s2)dF(s2 sˆ1 + #) (IC for type sˆ1 + #) s¯2(t ) ´ L |

Putting together the ﬁrst and last expressions, we get a contradiction. The case of pooling only to the right of sˆ1 can be ruled out using the exact same argument.

13 Equality can be ruled out because then both tL and tH would induce the same response from bidder 2, and hence it is obvious that all agents would choose tL. 14Lemma 3 above only gives s¯ (t ) sˆ , but this inequality is actually strict because of pooling at 2 L 1 tL.

22 Pooling only to the left of sˆ1 requires a slight modiﬁcation. Assume that t¯(s1)=tH for all s (sˆ d , sˆ ), t¯(sˆ )=t , and t¯(s ) > t for all s (sˆ , sˆ + d ) and all 1 2 1 1 1 1 L 1 L 1 2 1 1 2 d1, d2 > 0 sufﬁciently small. For this case, we use essentially the same argument, only consider a type sˆ # for # (0, d ). Note by standard arguments, type sˆ must be 1 2 1 1 indifferent between offering tL and tH.

s¯2(tH) w(sˆ1 #, s2)dF(s2 sˆ1 #) s¯2(t ) ´ L | s¯2(tH) w(sˆ1, s2)dF(s2 sˆ1) (assumption (A1)) s¯2(t )  ´ L | s¯2(tH) s¯2(tL) = tHdF(s2 sˆ1)+ [tH tL]dF(s2 sˆ1) (IC for type sˆ1) s¯2(t ) 0 ´ L | ´ | s¯2(tH) s¯2(tL) < tHdF(s2 sˆ1 #)+ [tH tL]dF(s2 sˆ1 #) (afﬁliation) s¯2(t ) 0 ´ L | ´ | s¯2(tH) w(sˆ1 #, s2)dF(s2 sˆ1 #) (IC for type sˆ1 #) s¯2(t )  ´ L |

Once again, looking at the ﬁrst and last expressions, we see a contradiction.

Lemma 5. Given an equilibrium transfer function t,¯ assume that we can ﬁnd a point s M 2 (0, 1) such that, for some e > 0, t¯(s d) > t¯(s + d) for all d (0, e). Then: (i) there M M 2 cannot be pooling to the left of sM; i.e., there do not exist sL < sM and tL < tH such that

tH s1 [sL, sM) t¯(s1)= 2 . (tL s1 = sM

(ii) there cannot be pooling to the right of sM; i.e., there do not exist sM < s¯ and tL such that:

g(s1) s1 < sM t¯(s1)= (tL s1 [sM, s¯) 2 where g(s ) is some function, continuous on [s d, s ) for some d > 0 sufﬁciently small, 1 M M with the property that lims s g(s1) > tL. 1! M

Proof. Proof of (i). Assume that there exists an equilibrium with t¯ satisfying the property in the statement. Then, by Lemma 4, we know that s¯2(tH) < s¯2(tL). Using Equation (4), incentive compatibility for type sL requires

s¯2(tH) s¯2(tL) [w(sL, s2) tH]dF(s2 sL) [w(sL, s2) tL]dF(s2 sL). ˆ0 | ˆ0 |

23 s¯2(tL) w(sL,s2)dF(s2 sL) s¯2(tL) s¯2(tH) | This inequality implies that [w(sL, s2) tL]dF(s2 sL) < 0, or ´ < s¯2(t ) F(s¯2(t ) s ) F(s¯2(t ) s ) ´ H | L | L H | L tL. Note that this inequality can be written as:

ES S [w(sL, s2) s¯2(tH) s2 s¯2(tL)] < tL. 2| 1 |   However, we also know that:

ES S [w(sL, s2) 0 s2 s¯2(tH)] tH > tL. 2| 1 |  

Since w(sL, s2) is increasing in its 2nd argument, we have:

ES S [w(sL, s2) s¯2(tH) s2 s¯2(tL)] ES S [w(sL, s2) 0 s2 s¯2(tH)] tH > tL, 2| 1 |   2| 1 |   a contradiction. Proof of (ii). Choose some s < s such that t¯(s ) > t and s (s , s¯). Deﬁne L M L L H 2 M tH := t¯(sL) . Again by Lemma 4, we have s¯2(tH) < s¯2(tL). Now, IC for type sH requires

s¯2(tH) s¯2(tL) [w(sH, s2) tH]dF(s2 sH) [w(sH, s2) tL]dF(s2 sH). ˆ0 |  ˆ0 | This inequality implies that:

s¯2(tL) w(sH, s2)dF(s2 sH) s¯2(tH) | ´ > tL. F(s¯ (t ) s ) F(s¯ (t ) s ) 2 L | H 2 H | H We can write this as:

E[w(s , s ) s¯ (t ) s s¯ (t )] < t . H 2 | 2 H  2  2 L L Since w is nondecreasing in its ﬁrst argument, we have:

E[w(s , s ) 0 s s¯ (t )] E[w(s , s ) s¯ (t ) s s¯ (t )] < t < t . L 2 |  2  2 L  H 2 | 2 H  2  2 L L H The ﬁrst and last expressions give the desired contradiction.

Proof of Theorem 3 Existence. Recall the differential equation (3):

[w(s ,h(s ,t¯(s ))) t¯(s )][w (s ,s )+w (s ,s ) w (h(s ,t¯(s )),s )] f (h(s ,t¯(s )) s ) 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 | 1 ¯ w (h(s ,t¯(s )),s )F(h(s ,t¯(s )) s ) [w(s ,h(s ,t¯(s ))) t¯(s )] f (h(s ,t¯(s )) s ) , h(s1, t) < 1 t¯0(s1)= 1 1 1 1 1 1 | 1 1 1 1 1 1 1 | 1 . (0, else

It is possible to show that w (0, 0) > 0 and and f (0 s = 0) > 0 imply that 0 < t¯ (0) < 1 | 1 0

24 15 •. Now, the (local) inverse of t¯(s1), denoted S(t¯), is deﬁned by

S(0)=0

[t¯ w(S(t¯), h(S(t¯), t¯)] f (h(S(t¯), t¯) S(t¯)) + F(h(S(t¯), t), S(t¯))w1(h(S(t¯), t¯), S(t¯)) S0(t¯)= | . [w(S(t¯), h(S(t¯)) t] f (h(S(t¯)) S(t¯))[w (S(t¯), S(t¯)) + w (S(t¯), S(t¯)) w (h(S(t¯)), S(t¯))] | 1 2 2 1 By the Inverse Function Theorem, S0(0)= ¯ (0, •). Next, let t¯ be such that t0(0) 2 S0(t¯)=0. Then, w(S(t¯), h(S(t¯))) > t¯ and:

d F(h(S(t¯), t¯) S(t¯)) w1(h(S(t¯), t¯), S(t¯)) S00(t¯)= | dt¯ f (h(S(t¯), t¯)) S(t¯)) [w(S(t¯), h(S(t¯), t¯) t¯][w (S(t¯), S(t¯)) + w (S(t¯), S(t¯)) w (h(S(t¯), t¯), S(t¯))] ✓ | ◆ 1 2 2 F(h(S(t¯), t¯)) S(t¯)) w (h(S(t¯)), S(t¯)) + | 1 f (h(S(t¯), t¯)) S(t¯)) [w(S(t¯), h(S(t¯), t¯) t¯]2[w (S(t¯), S(t¯)) + w (S(t¯), S(t¯)) w (h(S(t¯), t¯), S(t¯))] | 1 2 2

d F(h(S(t¯),t¯) S(t¯)) | Now, dt¯ f (h(S(t¯),t¯)) S(t¯)) > 0 by log-concavity and | ⇣ ⌘ w (S(t¯), S(t¯)) + w (S(t¯), S(t¯)) w (h(S(t¯), t¯), S(t¯)) > 0 1 2 2 by assumption A2. Thus, S00(t¯) is positive. This implies that if t¯0(s1) is ever infinite, that s is only an inflection point of t¯( ). Thus, a equation (3) has a unique, continuous 1 · solution. To show that this is an equilibrium, first consider bidder 2. If bidder 2 observes 1 some t < t⇤, she believes bidder 1’s type is t¯ (t), and the obvious best response is 1 s¯2(t)=h(t¯ (t), t). If t = t⇤, bidder 2 updates her beliefs and places all weight on [s1⇤,1], in which case it is again easy to show that the best response is s¯2(t)=1. If t > t⇤, her beliefs can be anything, and the best response will depend on the beliefs. However, no matter what bidder 2 believes and what her response to an action t > t⇤ is, bidder 1 will never offer such a t, because she can always induce the same action from bidder 2 by offering a lower transfer (because t¯([0, t⇤]) = [0, 1]). So, consider bidder 1. As argued above, bidder 1 will never offer t > t⇤. Thus, we only need check that no type s1 wants to offer some t¯(sˆ1) for some sˆ1 = s1. First, d ∂w(s1,s26) define the function c(s1, s2, sˆ1)= [(w(s1, s2) t¯(sˆ1))f (s2 s1)] = f (s2 s1)+ ds1 | ∂s1 | ∂ f (s2 s1) (w(s1, s2) t¯(sˆ1)) | . Then, note that the envelope theorem gives ∂s1

s¯2(t¯(s1)) dp1(s1) = c(s1, s2, s1)ds2 ds1 ˆ0

Now, deﬁne a function f(s , sˆ )=U(s , s¯ (t¯(sˆ )), t¯(sˆ )) p(s ), and consider f (s , sˆ ): 1 1 1 2 1 1 1 s1 1 1 ¯ ¯ s¯2(t(sˆ1)) s¯2(t(s1)) ¯ 0 c(s1, s2, sˆ1)ds2 0 c(s1, s2, s1)ds2, s1 s¯2(t(sˆ1)) fs (s , sˆ )= ¯ ¯  1 1 1 ´ s¯2(t(sˆ1)) ´ s1 s¯2(t(s1)) ¯ ( 0 c(s1, s2, sˆ1)ds2 + s¯ (t¯(sˆ )) r(s1, s2)ds2 0 c(s1, s2, sˆ1)ds2, s1 > s¯2(t(sˆ1)) ´ ´ 2 1 ´

15The algebra was carried out with Mathematica. The formulas are cumbersome, and so are not printed here.

25 ∂w(s1,s2) ∂ f (s2 s1) where r(s1, s2)= f (s2 s1)+[w(s1, s2) w(s2, s2)] | . Now, since t¯( ) and ∂s1 | ∂s1 · s¯ ( ) are increasing, if s sˆ , then f (s , sˆ ) 0. 2 · 1  1 s1 1 1 ¯ s1 s¯2(t(s10 )) ¯ ¯ ¯ ¯ U1(s1, s¯2(t(s1)), t(s1)) = U1(0, s¯2(t(0)), t(0)) + c(s10 , s2)ds2 ds10 ˆ0 "ˆ0 #

d ∂w(s1,s2) ∂ f (s2 s1) where c(s1, s2)= [w(s1, s2) f (s2 s1)] = f (s2 s1)+w(s1, s2) | . Using ds1 | ∂s1 | ∂s1 this, we have

¯ s1 s¯2(t(s10 )) U (s , s¯ (t¯(s )), t¯(s )) U (sˆ , s¯ (t¯(sˆ )), t¯(sˆ )) = c(s , s )ds ds 1 1 2 1 1 1 1 2 1 1 ˆ ˆ 10 2 2 10 sˆ1 0 Switching the order of integration, the RHS becomes

s¯2(t¯(sˆ1)) s1 s¯2(t¯(s1)) s1 c(s10 , s2)ds10 ds2 + c(s10 , s2)ds10 ds2 ˆ ˆ ˆ ¯ ˆ 1 0 sˆ1 s¯2(t(sˆ1)) s1 (s2)

1 ¯ where s1 (s2) is the s1 such that s¯2(t(s1)) = s2. Since c(s10 , s2) is a derivative, the inner integral is easy to evaluate, and we get

s¯2(t¯(sˆ1)) [w(s1, s2) f (s2 s1) w(sˆ1, s2) f (s2 sˆ1)]ds2 (5) ˆ0 | | s¯2(t¯(s1)) + [w(s , s ) f (s s ) w(s 1(s ), s ))f (s s 1(s ))]ds ˆ 1 2 2 1 1 2 2 2 1 2 2 s¯2(t¯(sˆ1)) | |

Consider also the equation U (s , s¯ (t¯(sˆ )), t¯(sˆ )) U (sˆ , s¯ (t¯(sˆ )), sˆ ). This is: 1 1 2 1 1 1 1 2 1 1 ¯ s¯2(t(sˆ1)) ¯ ¯ ¯ 0 [(w(s1, s2) t(sˆ1))f (s2 s1) (w(sˆ1, s2) t(sˆ1))f (s2 sˆ1))ds2 s1 < s¯2(t(sˆ1)) ¯ | | ´ s¯2(t(sˆ1)) ¯ ¯ s1 ¯ ( 0 [(w(s1, s2) t(sˆ1))f (s2 s1) (w(sˆ1, s2) t(sˆ1))f (s2 sˆ1))ds2 + s¯ (t¯(sˆ ))[w(s1, s2) w(s2, s2)]dF(s2 s1) s1 s¯2(t(sˆ1)) ´ | | ´ 2 1 | Uniqueness. We ﬁrst show that in any continuous equilibrium, if there is pooling on some (non-degenerate) interval [s10 , s100], then s100 = 1. Assume not, so that s10 < ¯ ¯ s100 < 1 and let t00 = t(s100). Since the equilibrium is continuous, we have t( ) strictly ¯ · increasing on (s100, s100 + #) for some small # > 0, with lims s t(s1)=t00. Bidder 2’s 1# 100 optimal cutoff, s¯2(t00), is implicitly deﬁned as the solution to

s¯2(t00) t00 = [w(s¯ (t00), s ) w(s , s )]dF(s s¯ (t00)) ˆ 2 1 1 1 1| 2 s10 and is such that s¯ (t ) (h(s , t ), h(s , t )). Also, note that s < h(s , t ) < 1. Now, 2 00 2 10 00 100 00 100 100 00 consider a transfer t00 + d for some small d. Since t00 + d is in a separating region,

26 we have s¯ (t + d) > h(s , t ) s . Now, we calculate the change in utility from 2 00 100 00 100 offering t00 + d minus that from offering t00. There are two cases. When s100 > s¯2(t00), this expression is

s¯2(t00+d) s100 w(s00, s )dF(s s00)+ w(s , s )dF(s s00) ˆ 1 2 2| 1 ˆ 2 2 2| 1 s100 s¯2(t00) s¯2(t00+d) s¯2(t00+d) t dF(s s ) ddF(s s ) ˆ 00 2 100 ˆ 2 100 s¯2(t00) | 0 |

We also have limd 0 s¯2(t00 + d)=h(s100, t00) > s100 > s¯2(t00), and, by Lemma 7, t00 < ! w(s¯ (t ), s¯ (t )). So, as d 0, the last term becomes arbitrarily small, while the sum 2 00 2 00 ! of the first three terms approaches a number strictly greater than 0, meaning t00 + d is a profitable deviation for some d sufficiently small. A similar argument holds when s s¯ (t ). 100  2 00 Thus, there can be at most one pooling interval. Since we assume that the equilibrium is continuous but not constant, this means that there exists an s such that t¯( ) 10 · is strictly increasing and on [0, sˆ ] and is constant on (sˆ ,1] for some sˆ 1. We next 1 1 1  show that we must have t¯(0)=0. Assume that t¯(0)=t > 0. Since we know t¯( ) 0 · is strictly increasing at 0, bidder 2 will use cutoff h(0, t0), where w(h(0, t0),0)=t0. (0, ) Bidder 1’s payoff will be h t0 [w(0, s ) w(h(0, t ),0)]dF(s s ). However, the inte- 0 2 0 2| 1 grand is strictly negative´ for all s2 < h(0, t0), and so bidder 1 can deviate to offering 0 and be better off. Thus, t¯(0)=0; t¯0( ) satisfies equation (3). · ¯ Last, we must show that sˆ1 = s1⇤, where s1⇤ is defined implicitly by h(s1⇤, t(s1⇤)) = 1. Assume that sˆ < s . Then, h(sˆ , t¯(sˆ )) < 1. Consider a type sˆ e for some small e. 1 1⇤ 1 1 1 If type sˆ e deviates to offering t¯(sˆ ), his change in utility is 1 1 s¯ (t¯(sˆ )) s¯ (t¯(sˆ e)) 2 1 2 1 [w(sˆ1 e, s2) t¯(sˆ1)]dF(s2 sˆ1 e) [t¯(sˆ1) t¯(sˆ1 e)]dF(s2 sˆ1 e) ˆs¯ (t¯(sˆ e)) | ˆ0 | 2 1

Note that lime 0 t¯(sˆ1 e)=t¯(sˆ1), but lime 0 s¯2(t¯(sˆ1 e)) = h(sˆ1, t¯(sˆ1)) < s¯2(t¯(sˆ1)) < ! ! sˆ1. Thus, as e approaches 0, the second integral vanishes, while the ﬁrst converges to a strictly positive number, and so this devition is proﬁtable for all e small enough.

If, on the other hand, sˆ1 > s1⇤, then some type s1⇤ + e can deviate to offering t(s1⇤). ¯ ¯ Since s¯2(t(s1⇤ + e)) = s¯2(t(s1⇤)) = 1, but t(s1⇤) < t(s1⇤ + e), this is a proﬁtable deviation.

Lemma 6. Fix an equilibrium (t¯, s¯2) and an out of equilibrium transfer tˆ > t¯(s1). Deﬁne p1(s1)=U1(s1, s¯2(t¯(s1)), t¯(s1)) to be bidder 1’s equilibrium payoff. Assume that there exists a type s1 such that p1(s1) < U1(s1, 1, tˆ). Deﬁne s¯ (s1) to be the solution to

U1(s1, s¯ (s1), tˆ)=p1(s1) (6) whenever such a solution exists. Then, for all s1 such that s¯ (s1) exists and s1 < s¯ (s1), we have s¯ (s ) 0. 0 1 

27 Proof. That such a s¯ (s1) exists and is continuous is trivial. To shorten equations, deﬁne ∂w(s ,s ) ∂ f (s s ) the following functions: c(s , s )= 1 2 f (s s )+w(s , s ) 2| 1 and w˜ (s , s )= 1 2 ∂s1 2 1 1 2 ∂s1 1 2 | 2 ∂ f (s2 s1) ∂ F(s2 s1) w(s1, s2) w(s2, s2). Note that w( , ) positive and increasing and | = | · · ∂s1 ∂s1∂s2 0 (by increasing differences) implies that c(s , s ) 0. The envelope theorem gives 1 2

s¯2(t¯(s1)) ∂ f (s2 s1) p0 (s )= c(s , s ) t¯(s ) | ds + 1 1 ˆ 1 2 1 ∂s 2 0  1 max s1s¯2(t¯(s1)) { } ∂ f (s2 s1) w1(s1, s2) f (s2 s1)+w˜ (s1, s2) | ds2 ˆ ¯ | ∂s s¯2(t(s1))  1

If tˆ > t¯(s1), then it must be that s¯ (s1) > s¯2(t¯(s1)) (see footnote 16). Then, using the implicit function theorem on (6), we ﬁnd that

s¯ (s1) ¯ c(s1,s2)ds2+t(s1) s¯2(t(s1)) ¯ ´ ˆ , s1 s¯2(t(s1)) [w(s1,s¯ (s1)) t] f (s¯ (s1) s1)  8 s¯ (s ) s | ∂ f (s s ) 1 c(s ,s )ds 1 w(s ,s ) 2| 1 ds +t(s ) > s 1 2 2 s¯ (t¯(s )) 2 2 ∂s 2 1 s¯ 0(s1)=> 1 2 1 1 ¯ > ´ ´ ˆ , s¯2(t(s1)) < s1 s¯ (s1) > [w(s1,s¯ (s1)) t] f (s¯ (s1) s1)  > s¯ (s ) ∂ f (s s ) | < 1 ( ) 2| 1 + ( ) s¯ (t¯(s )) w s2,s2 ∂s ds2 t s1 ´ 2 1 1 , > s¯ ( ) [w(s¯ (s ),s¯ (s )) tˆ] f (s¯ (s ) s ) s1 s1 > 1 1 1 | 1 > > : ∂F(s¯ (s1) s1) ∂F(s¯2(t¯(s1)) s1) where t(s1)=tˆ | t¯(s1) | . Next, we want to show that this func- ∂s1 ∂s1 tion is everywhere negative. The fact that tˆ > t¯(s ) and F(s s ) having increasing 1 2| 1 differences (so that ∂F(s s )/∂s is increasing in s ) implies that t(s ) is negative. 2| 1 1 2 1 Lemma 7 shows that all of the denominators are positive. As for the numerators, recall that c(s , s ) 0. For the other terms, integration by parts gives 1 2 s¯ (s1) ∂F(s¯ (s ) s ) w(s , s )g(s )ds = w(s¯ (s ), s¯ (s )) 1 | 1 ˆ 2 2 2 2 1 1 s¯2(t¯(s1)) ∂s1 ∂F(s¯2(t¯(s1)) s1) ∂F(s2 s1) w(s¯2(t¯(s1)), s¯2(t¯(s1))) | | dw(s2, s2). ∂s1 ˆ ∂s1 Note that the last integral is negative, and so we can say

s¯ (s1) ∂F(s¯ (s ) s ) w(s , s )g(s )ds + w(s¯ (s ), s¯ (s )) 1 | 1 ˆ 2 2 2 2 1 1 s¯2(t¯(s1)) ∂s1 ∂F(s¯2(t¯(s1)) s1) w(s¯2(t¯(s1)), s¯2(t¯(s1))) | 0. ∂s1 

∂F(s¯ (s1) s1) ∂F(s¯2(t¯(s1)) s1) Finally, tˆ > t¯(s1), and | | (by increasing differences) ∂s1 ∂s1

28 It is worth noting that when the signals s1 and s2 are conditionally independent, s¯ 0(s1) is greatly simpliﬁed to:

max s¯ (s1),s1 { ¯ } w1(s1, s2)dF(s2) max s1,s¯2(t(s1)) s¯ 0(s1)= ´ { } . [w(min s , s¯ (s ) , s¯ (s )) tˆ] f (s¯ (s )) { 1 1 } 1 1

Lemma 7. Consider an equilibrium (t¯, s¯2) and an out of equilibrium transfer t.ˆ Then, for all s1: (i) w(s¯2(t¯(s1)), s¯2(t¯(s1))) > t¯(s1) (ii) w(s , s¯ (s )) t,ˆ if s¯ (s ) exists. 1 1 1 Proof. Assume the opposite inequality holds. Note that bidder 1 can always bid 0 and guarantee himself a payoff of U (s , 0, 0)= s1 [w(s , s ) w(s , s )]dF(s s ). So, 1 1 0 1 2 1 1 2| 1 individual rationality requires that U (s , s¯ (t¯(s ´)), t¯(s )) U (s , 0, 0), or 1 1 2 1 1 1 1 ¯ s¯2(t(s1)) ¯ 0 [w(s1, s2) t(s1)]dF(s2 s1)+ s1 | s1 I(s1 > s¯2(´t¯(s1))) [w(s1, s2) w(s2, s2)]dF(s2 s1) [w(s1, s2) w(s2, s2)]dF(s2 s1) s¯2(t¯(s1)) 0 ´ | ´ |

First, consider the case of s1 > s¯2(t¯(s1)). The above equation can be rewritten as

s¯2(t¯(s1)) s¯2(t¯(s1)) t¯(s1)dF(s2 s1) w(s2, s2)dF(s2 s1) ˆ0 |  ˆ0 |

If w(s¯2(t¯(s1)), s¯2(t¯(s1))) t¯(s1), then, since w is an increasing function, w(s2, s2) <  ¯ (¯( )) ¯ (¯( )) t¯(s ) for all s [0, s¯ (t¯(s ))], and hence s2 t s1 w(s , s )dF(s s ) < s2 t s1 t¯(s )dF(s s ), 1 2 2 2 1 0 2 2 2| 1 0 1 2| 1 which contradicts the equation above. ´ ´ When s1 < s¯2(t¯(s1)), bidder 1 only wins the object when the transfer is accepted. The highest possible value for the object when he wins is:

w(s1, s¯2(t¯(s1)) < w(s¯2(t¯(s1)), s¯2(t¯(s1)) < t¯(s1),

or t¯(s1) > w(s1, s¯2(t¯(s1)), which means that bidder 1 is paying more than the object is worth and is hence guaranteeing himself a negative payoff, which is clearly not individually rational. For the second statement, consider some out-of-equilibrium tˆ and corresponding s¯ (s ), and assume that w(s , s¯ (s )) < tˆ for some s . If s s¯ (s ), then we have 1 1 1 1 1  1 U (s , s¯ (s ), tˆ) < 0, while p (s ) 0, which is a contradiction to the deﬁnition of 1 1 1 1 1 s¯ (s1). Next, assume that s1 > s¯ (s1). There are two subcases. First, if tˆ > t¯(s1), then

29 16 s¯ (s1) > s¯2(t¯(s1)). Then, we can simplify equation (6) to:

s¯ (s1) w(s , s )dF(s s )=tFˆ (s¯ (s ) s ) t¯(s )F(s¯ (t¯(s )) s ) (7) ˆ 2 2 2 1 1 1 1 2 1 1 s¯2(t¯(s1)) | | | notice that the LHS is (strictly) negative, while the RHS is positive, which is a contradiction. When tˆ < t¯(s1), a similar argument to footnote 16 shows that s¯2(t¯(s1)) > s¯ (s1), and then we can derive an equation analogous to equation (7) to reach the desired contradiction.

16 To see this, note that U1 is (strictly) increasing in its second argument at s¯2(t¯(s1)) (otherwise, since U1 is quasiconvex in its second argument for any fixed s1, t, deviating to a transfer of 0 and inducing a cutoff of s¯2 = 0 would be a profitable deviation for bidder 1, contradicting the fact that (t¯, s¯ ) was an equilibrium). This implies that U (s , s¯ , tˆ) < U (s , s¯ , t¯(s )) U (s , s¯ (t¯(s )), t¯(s )) 2 1 1 1 1 1  1 1 2 1 1 for all 0 s¯ s¯ (t¯(s )), where the first inequality follows from tˆ > t¯(s ) and the second from   2 1 1 the quasiconvexity of U and the fact that U (s , s¯ (t¯(s )), t¯(s )) U (s , 0, 0) U (s , 0, t¯(s )) by 1 1 1 2 1 1 1 1 1 1 1 individual rationality. Thus, U1(s1, s¯ , tˆ) < p1(s1) for all s¯ s¯2(t¯(s1)), and so no such s¯ ’s can satisfy equation (6). 