Optimal Sales Mechanism with Outside Options and Withdrawal Rights∗ Dongkyu Chang† September 28, 2017
Abstract
This paper studies the optimal design of sales mechanisms when a buyer can break off the negotiation for an outside option at any time. The main results indicate that the profit-maximizing mechanism reduces price over time, and thus a set of buyer types delay purchasing the good. Moreover, to prevent the buyer from breaking off the negotiation, the profit-maximizing mechanism also features an upfront payment, which is compensated later by an additional price discount. The seller can implement the profit-maximizing mechanism by offering a menu of European call options. (JEL C78, D82, D86) Keywords: commitment, delay, mechanism, outside option, sales negotiation, option contract, upfront payment, withdrawal right
∗
This paper grew out of my doctoral dissertation submitted to Yale University. I am indebted to my advisors, Eduardo Faingold, Johannes Hörner, and Larry Samuelson. I also benefited from invaluable discussion with Yeon-Koo Che, Ilwoo Hwang, Kyungmin (Teddy) Kim, Jong Jae Lee, Aniko Öry, Jiwoong Shin, and the audiences at various seminars and conferences. The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 21506616). † Department of Economics and Finance, City University of Hong Kong, 83 Tat Chee Ave., Kowloon Tong, Hong Kong,
[email protected]
Table of Contents 1 Introduction 1.1
Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3
2 Model
4
2.1
Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3 Two-type Example 4 Characterization of Optimal Mechanism
7 12
4.1
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.2
Relaxed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.3
Optimal Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5 Option Contracts
19
5.1
Delay of Transaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5.2
Upfront Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
6 Comparative Statics Analysis
21
6.1
Autarky Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
6.2
Benefit From Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
7 Extensions and Discussions
26
7.1
Time-varying Outside Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
7.2
Interdependent Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
7.3
Negotiation Without Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
8 Conclusion
30
Bibliography
31
Appendix A Proofs
33
A.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
A.2 Proof of Lemma 2 and Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
A.3 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
A.4 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
A.5 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
A.6 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
A.7 Proof of Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
A.8 Proof of Proposition 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
1
Introduction
In sales negotiations, parties often face a choice between haggling and promptly breaking off the negotiation. There is clearly a trade-off. They can haggle in search of a delayed agreement. However, it is sometimes better for both parties to immediately break off the negotiation and opt for outside options. This trade-off is resolved in various manners. For instance, a delayed agreement is normal in real estate sales negotiations, whereas retailers typically adopt a fixed-price mechanism with no room for haggling. Our aim is to address the following questions: what makes negotiating parties choose a delay rather than an immediate breakdown of the negotiation, or vice versa? How does the negotiating parties’ effort to avoid breakdown or delay affect the dynamics of the offer, the division of the surplus, and other features of the negotiation process and outcome? Suppose that a buyer and a seller negotiate the terms of trade for a single indivisible good. The buyer (informed party) privately learns her type prior to the negotiation, which determines both the value of the good and the value of her outside option. The seller (uninformed party) can choose and commit to any mechanism, thus allowing him to choose a time-dependent price path, arrange auxiliary transfers between two parties that may depend on the buyer’s reports, break off the negotiation at any time, and so on.1 Sales negotiations with the seller’s full commitment power have long been studied in the economics literature.2 The novelty of our model is twofold. First, the buyer’s outside option randomly arrives during the negotiation, and more importantly, the value of the outside option depends on her type (type-dependent outside option). Second, the buyer can withdraw from the negotiation table at any time during the negotiation (withdrawal rights), in which case she can still enjoy the outside option. Implicitly, it is impossible or prohibitively costly to sign any contract that abrogates the withdrawal right. The main results of the paper identify when and how a delay occurs in profit-maximizing (hereafter, optimal) mechanisms rather than an immediate outcome. The seller strictly prefers delaying to immediately breaking off the negotiation if (i) the value of the outside option is positively related to the buyer’s value of the good, and (ii) outside options are highly dispersed among buyer types.3 Otherwise, it is optimal for the seller to commit to a fixed-price mechanism, such that the buyer would immediately purchase the seller’s good or break off the negotiation.4 The occurrence of a delay in optimal mechanisms shows that the the no-haggling result (Stokey, 1
Board and Pycia (2014), Hwang (2016), and Hwang and Li (2017) study the same problem but without the seller’s commitment power. 2 For example, Stokey (1979) and Riley and Zeckhauser (1983) study a model identical to ours model except that the informed party’s outside option is type-independent and normalized to zero. Samuelson (1984) considers the case in which the uninformed party’s reservation price also depends on the informed party’s private type (interdependent values). 3 The dispersion of outside options is measured by the elasticity of the value of the outside option with respect to the value of the seller’s good across buyer types. Outside options are said to be more dispersed (less dispersed) if the elasticity is greater than one (less than one). 4 We treat the negotiating parties breaking off the negotiation as another manner of resolving the bargaining situation. Hence, we will assume that no delay occurs if the negotiating parties either trade the seller’s good or break off the negotiation at time zero.
1
1979; Harris and Raviv, 1981; Riley and Zeckhauser, 1983; Samuelson, 1984; Manelli and Vincent, 2007) does not hold if outside options are heterogeneous (type-dependent) across different buyer types. The comparative statics with respect to outside options also provides an explanation for why bargaining delays remain the norm in markets for real estate, automobiles, inputs for manufacturing, and so forth; buyers in these markets have different outside options owing to their financial status and/or information, and thus the seller can make a higher profit by delaying. In contrast, a fixed-price mechanism is optimal for typical retail transactions in which buyers have similar outside options. To see why delay can be beneficial for the seller, observe first that the outside option remains available for the buyer even after the negotiation breaks down. Because the value of the outside option is type-dependent and unknown to the seller, any possibility of breakdown concedes information rent to buyer types with a valuable outside option. This means that, all else being equal, the seller can extract more profit by delaying the negotiation instead of breaking it off right away. This intuition can be better understood through a two-type example in which the buyer may have either high valuation (high type) or low valuation (low type) of the seller’s good. For simple exposition, suppose that the value of the high type’s outside option is positive while the low type’s outside option is worth zero, which means that the ratio of the high type’s outside option to the low type’s is infinite. In other words, the dispersion of outside options prevails over the dispersion of valuations. Furthermore, suppose that the outside option is always available to both types during the negotiation, again only for simple exposition. Observe first that a fixed-price mechanism is optimal among all one-shot mechanisms in which the negotiation situation is resolved at time zero for certain. By definition of the one-shot mechanism, the negotiation breaks down and the buyer exercises her outside option if a one-shot mechanism fails to achieve an immediate agreement. Hence, the problem of solving for the optimal one-shot mechanism is mathematically equivalent to the static screening problem with no outside options, except that now the difference between the value of the good and the value of the outside option constitutes the reservation price of the buyer. The optimality of a fixed-price mechanism among one-shot mechanisms then follows from the no-haggling result. To see how a delay generates more profit, suppose that the seller alternatively offers the following two options. First, the buyer can trade immediately at a high price that is acceptable to the high type only. Second, she can trade at a lower price, but it is available only after a delay. Faced with these two options, the high type would never opt for the second option if the delay is sufficiently long, as long as her outside option generates a higher payoff than the second option. In contrast, the low type would always choose the second option regardless of how long the delay is. By offering an option to delay, the seller can therefore extract more profit without conceding any further information rent to either type.5 Notably, the delay can never be replaced by any other form of screening scheme. Particularly, the seller would ultimately earn a strictly lower profit if the delay of the second option were to replace a lottery that dictates trade and breakdown with positive probabilities. A mechanism with a lottery necessarily allows a higher information rent for the high type, because she can still enjoy the outside 5
Here, the information rent refers to the difference between the buyer’s expected payoff from participating in a mechanism and the value of her outside option.
2
option even if the lottery dictates a negotiation breakdown. In turn, a higher information rent causes a lower expected profit for the seller. In addition to a delay in the transaction, the optimal mechanisms also feature the buyer’s upfront payment. The buyer would be compensated for the upfront payment by the corresponding price discount when she purchases the good in the future. However, if she withdraws from the table without trading, the buyer would lose the money paid as the upfront payment. Hence, the upfront payment scheme prevents negotiation breakdown from occurring during the negotiation, which results in less information rent to the buyer and more profit to the seller.6 The upfront payment for the price discount in the future makes the optimal mechanisms resemble call option contracts. Indeed, the seller can achieve the optimal profit level by offering a menu of European call options, which is widely used in financial markets as well as other production contracts. Moreover, if there are multiple optimal mechanisms, all of them feature the option-like structure with a delay in transaction (maturity date in the future) and an upfront payment (premimum). The strict optimality of option-like contracts stands in contrast to the case without outside options, in which a posting-price mechanism (with a time-varying price path) is optimal with or without the seller’s commitment power (Skreta, 2006). The result also contrasts with the revenue management problem in which committing to a posting-price mechanism is often sufficient to achieve the optimal profit level; see, for example, Hörner and Samuelson (2011, Section IV.C.2) and Board and Skrzypacz (2016). The paper’s findings also have an implication for bargaining theory. To incentivize some buyer types to delay, the optimal mechanisms decrease the price over time. Interestingly, in our environment, a declining price path requires more commitment power for the seller than does a fixed price. In a recent paper, Board and Pycia (2014) consider the bargaining problem without the seller committing to any mechanism but otherwise identical to our model. They show that the seller insists on a fixed price indefinitely in an (essentially) unique equilibrium. The seller could earn more profit by decreasing the price over time, but this strategy violates sequential rationality. This observation contrasts with the prediction of the Coase conjecture and hence provides a new perspective on the role of commitment in bargaining (or equivalently, a durable-good monopoly). Ever since Coase (1972), the bargaining literature has emphasized the seller’s inability to commit to a fixed price and its adverse effect on his profit.7 However, combined with the observation of Board and Pycia, our results demonstrate that the Coase conjecture is overturned when buyers have typedependent outside options and withdrawal rights; it is the seller’s inability to commit to a declining price path that harms the seller without sufficient commitment power.
1.1
Related Literature
This paper is related to the literature regarding mechanism design with type-dependent participation constraints. Lewis and Sappington (1989), Jullien (2000), and Rochet and Stole (2002), among others, 6
A similar upfront scheme, followed by (type-dependent) price discounts, is often observed in consumer lending contracts (Einav, Jenkins, and Levin, 2012), collateral contracts (Geanakoplos, 2003), and also various call option contracts. 7 In bargaining without outside options, whether values are private or interdependent, a fixed-price mechanism is the optimal mechanism (Riley and Zeckhauser, 1983; Samuelson, 1984). However, Gul, Sonnenschein, and Wilson (1986) and Deneckere and Liang (2006) show, in different cases, that the equilibrium outcome exhibits a declining price path when the seller has no commitment power, which results in the profit level being short of that under the optimal mechanism.
3
discuss type-dependent participation constraints in the context of nonlinear pricing. This paper differs from those papers in two respects. First, they impose no constraint on the quantity that the seller may sell to a buyer, whereas in this paper, the two parties are restricted to trade only one unit of an indivisible good.8 Second, and more importantly, these papers focus on the case without withdrawal rights, meaning that the buyer cannot opt for an outside option once she has participated in a mechanism. Recently, Krähmer and Strausz (2015) discuss the implications of the buyer’s withdrawal right in the context of sequential screening. Contrary to our model, however, they suppose that the buyer’s payoff upon withdrawal is independent of the buyer’s type. While this paper focuses on the case in which two parties trade a single good, other authors, for example, Thanassoulis (2004), Manelli and Vincent (2006, 2007), Pycia (2006), Pavlov (2011), Hart and Reny (2015), and Rochet and Thanassoulis (2016), have shown that the no-haggling result may fail when parties negotiate the terms of trade of multiple goods at the same time. It is also known that the no-haggling result may not hold when different buyer types have different budget constraints (Che and Gale, 2000) or different risk attitudes (Laffont and Martimort, 2002, Section 2.13). The noncooperative bargaining game literature has also identified various sources of delay in negotiations. Different from our model, however, most papers in this literature focus on the case in which the seller (uninformed party) cannot commit to any sophisticated mechanism. Fuchs and Skrzypacz (2010), Hwang (2016), and Hwang and Li (2017) show that the interplay between outside options and asymmetric information can result in a delay in equilibrium when the seller has no commitment power. Among many others, Evans (1989), Vincent (1989), and Deneckere and Liang (2006) show that interdependent values (the lemon problem) can result in a delay. Abreu and Gul (2000), Compte and Jehiel (2002), and Atakan and Ekmekci (2014), among many others, analyze delays in bargaining with reputational concerns.
2
Model
2.1
Environment
A seller and a buyer negotiate the terms of trade for an indivisible good in continuous time t ≥ 0. The buyer’s valuation of the good v(θ) ∈ [v, v¯] ⊂ R+ is constant over time, and it depends on her type θ, ¯ which is drawn from a compact set Θ ⊂ R+ . The realization of θ is the buyer’s private information. The seller has a prior over Θ with a cumulative distribution function F and differentiable density function f such that f (θ) > 0 for any θ ∈ Θ. The seller’s reservation price is normalized to zero.9 The buyer has an outside option, the value of which w(θ) ∈ [w, w] ⊂ R+ also depends on θ. In the baseline model, we assume that w(θ) is constant over time.10 The outside option and the seller’s good are mutually exclusive, so that the outside option is no longer worth once the two parties trade, and vice versa. The availability of the outside option Zt ∈ {0, 1} at t is also private information of the buyer, where Zt = 1 iff the outside option is available. Formally, Z ≡ (Zt )t≥0 is a càdlàg process 8
This constraint makes the existing techniques in the literature inapplicable, which forces us to develop and use an alternative approach, which is presented in Section 4. 9 The case with interdependent values is discussed in Section 7.2. 10 All main results are robust to the case where the outside option stochastically changes over time. See Section 7.1.
4
defined on a filtered probability space (Ω, F, {FtZ }t≥0 , P).11 We assume that Z0 is a constant random variable to avoid technical issues due to multi-dimensional signals. The negotiation is mediated by a mechanism that is introduced in the next subsection. For the present, we first describe the possible bargaining outcomes and the final payoffs in each case. There are two ways for the negotiation to conclude. First, the negotiation ends if the two parties trade the seller’s good at t ≥ 0, in which case the buyer and the seller obtain e−rt v(θ) − p and p, respectively, as their final payoffs, where p ∈ R is the total discounted net transfer from the buyer to the seller, and r > 0 is the common discount rate. Parties can also break off the negotiation without trade (withdrawal from the negotiation) at any time t ≥ 0. The seller’s continuation payoff is 0 in this case. The buyer can still enjoy the outside option, and hence her expected continuation payoff is t φt (θ; Z t ) := E e−r(γt (Z )−t) |Z t w(θ) where γt (Z t ) is the conditional random variable defined by γt (Z t ) := inf{s ≥ t : Zs = 1}|Z t .
(1)
Therefore, the final payoffs of the buyer and the seller are e−rt φt (θ; Z t ) − p and p, respectively, where p is again the total discounted net transfer. If the negotiation continues forever, the final payoffs of the seller and the buyer are equal to the total discounted net transfers. Independent of the negotiation process, each buyer type θ ∈ Θ can guarantee the expected payoff h i φ(θ) ≡ φ0 (θ; Z0 ) = E e−rγ0 (Z0 ) w(θ) by exercising her outside option as soon as it becomes available. We will call φ(θ) the autarky payoff of a buyer type θ ∈ Θ, and u(θ) := v(θ) − φ(θ) will be referred to as the net-valuation. Let U := {u(θ) : θ ∈ Θ} denote the support of net-valuations. In some examples, we focus on the case that Zt = 0 but the outside option arrives during the negotiation according to a Poisson process with rate λ > 0, and then remains available until the end of the negotiation. In this case, ( t
φt (θ; Z ) = and u(θ) = v(θ) −
2.2 2.2.1
λ λ+r w(θ)
λ λ+r w(θ)
w(θ)
if Zt = 0 if Zt = 1
for any Z t , t and θ.
Mechanisms Direct Mechanisms
The negotiation is mediated by a direct mechanism that recommends actions and transfers at each time based on the buyer’s message. The seller can commit to any mechanism, and as it is standard, he can also bind himself to obey any recommendation. Denote by mt the buyer’s message at t ≥ 0, 11 We will use the notation such as {Xt , t ≥ 0}, (Xt )t∈[0,∞) , (Xt )t≥0 , and X to denote the same stochastic process. Also, for any stochastic process X and t ≥ 0, X t ≡ (Xs )s∈[0,t] , X t− ≡ (Xs )s∈[0,t) , and X ∞ ≡ (Xs )s∈[0,∞) .
5
where m0 ∈ Θ indicates the buyer’s type, and mt ∈ {0, 1} indicates the realization of Zt for t > 0.12 Let mt ≡ (ms )s∈[0,t] generically denote a history of messages up to t ≥ 0, and let m ≡ (ms )s∈[0,∞) be a full history. Let M be the set of all full histories of messages. A direct mechanism µ is characterized as a controlled stochastic process µ = (A, P ) ≡ {(At (m), Pt (m))t≥0 ∈ ∆({−1, 0, 1}[0,∞) × R[0,∞) ) : m ∈ M } whose sample path represents the recommendation at each t conditional on m ∈ M being sent. (At (m), Pt (m)) must depend only on mt , and hence we use the notation (At (mt ), Pt (mt )) interchangeably with (At (m), Pt (m)) with no risky of confusion. At (mt ) = 1 if parties are recommended to trade at t, At (mt ) = −1 if they are recommended to break off the negotiation, and At (mt ) = 0 otherwise. Pt (mt ) ∈ R is the recommended (undiscounted) total transfer up to t, meaning that R P0 (mt ) + s∈(0,t] e−rs dPs (mt ) is the total discounted transfer up to t.13 The timeline of the mechanism is as follows: At each t ≥ 0, the buyer sends mt and then, the recommendation (At (mt ), Pt (mt )) is announced. At (mt ) and Pt (mt ) are implemented at once if the buyer accepts the recommendation. Two parties break off the negotiation if the buyer vetoes the recommendation (At (mt ), Pt (mt )), in which case none of previous actions or transfers is undone. 2.2.2
Incentive-compatibility
The buyer’s problem is to choose which messages to send (communication strategy) and when to veto the mechanism (veto strategy) after each history. A communication strategy is denoted by ν = (νt )t≥0 where νt prescribes the message at t conditional on the history so far. Let N be the set b b t (ν)}t≥0 be the filtration generated of all communication strategies. For each ν ∈ N let F(ν) ≡ {F b b by (At (ν), Pt (ν), Zt )t≥0 , and let T (F(ν)) be the collection of all F(ν)-stopping times. Then, a veto b strategy, which we will usually denote by ψ, is a stopping time in ∪ν∈N T (F(ν)) that indicates when b the buyer vetoes. (ν, ψ) is called feasible iff ψ ∈ T (F(ν)), and let Σ be the set of all feasible strategies. We are now ready to formally state the buyer’s problem. For any communication strategy ν ∈ N, τ µ (ν) := inf{t ≥ 0 : At (ν) = 1} and σ µ (ν) := inf{t ≥ 0 : At (ν) = −1} as the first times at which µ recommends each non-null action when the buyer plays ν.14 Additionally, µ
e−rτ (ν) v(θ) if τ µ (ν) < σ µ (ν) ∧ ψ µ µ µ b (θ; ν, ψ) := e−rσ (ν) φσµ (ν) (θ; Z σ (ν) ) if σ µ (ν) < τ µ (ν) ∧ ψ e−rψ φψ (θ; Z ψ ) otherwise
for any (ν, ψ) ∈ Σ and θ ∈ Θ. Then, each buyer type θ’s expected payoff in µ is
µ
V (θ) := max (ν,ψ)∈Σ
E b (θ; ν, ψ) − 1{ψ>0} · P0 (ν) − µ
12
Z
−rs
e
dPs (ν) ,
(2)
s∈(0,ψ)
Recall that Z0 is assumed to be a constant random variable; hence, the buyer does not need to reveal Z0 at t = 0. We also impose usual conditions on (A, P ). Among other ones, we assume that it always has a càdlàg path. 14 We use the convention that inf ∅ = ∞ and e∞ = 0. 13
6
where 1 denotes the indicator function. A communication strategy ν of a buyer type θ is said to be truthful if ν0 = θ and (νt )t∈(0,∞) = (Zt )t∈(0,∞) almost surely, and a veto strategy ψ is said to be obedient if ψ = ∞ almost surely. A direct mechanism µ = (A, P ) is called incentive-compatible iff the truthful and obedient strategy solves (2) for any θ. Denote by MIC the set all incentive-compatible mechanisms. Note that, without loss, we may assume that any mechanism in MIC recommends each buyer type a non-null action only at one point of time. Define ΠS (µ) as the expected payoff of the seller in mechanism µ = (A, P ) ∈ MIC : Z ΠS (µ) = θ∈Θ
"
E P0 (θ, Z ∞ ) +
Z
# e−rs dPs (θ, Z ∞ ) dF (θ)
s∈(0,∞)
where Z ∞ ≡ (Zt )t∈[0,∞) stands for a typical realization of Z. Finally, a mechanism µ ∈ MIC is called ¯ S := supµ˜∈MIC ΠS (˜ profit-maximizing mechanism or optimal mechanism if ΠS (µ) = Π µ). 2.2.3
Further Definitions and Notation
Neither a trade nor a transfer occurs after t = inf{˜ s : A(θ, Z ∞ ) 6= 1 and P (θ, Z ∞ ) = Pt (θ, Z ∞ ) ∀s ≥ s˜} when faced with type θ; hence, we define the expected delay of transaction, when faced with θ, by h i δ(θ; µ) := E inf{t : As (θ, Z ∞ ) ∈ {0, −1} and Ps (θ, Z ∞ ) = Pt (θ, Z ∞ ) for any s ≥ t} and average expected delay by
Z
δ(θ; µ) dF (θ)
δ(µ) := θ∈Θ
for any µ = (A, P ) ∈ MIC . We call µ ∈ MIC a one-shot mechanism if δ(µ) = 0 and the optimal oneshot mechanism if it achieves the highest profit among all one-shot mechanisms. A one-shot mechanism is called a fixed-price mechanism (committing to p) iff all buyer types either trade at price p or break off the negotiation immediately. Finally, µ = (A, P ) ∈ MIC is called posting-price mechanism iff there exist a stochastic process (pt )t∈[0,∞) and a collection of random times (t(θ))θ∈Θ such that, for any θ ∈ Θ,15 At (θ, Z ∞ ) = 1{t=t(θ)} and Pt (θ, Z ∞ ) = pt(θ) · 1{t ≥ t(θ)} almost surely. In other words, the outcome of µ can be implemented by committing to (possibly stochastic) price path (pt )t∈[0,∞) and letting the buyer choose to trade at any time t ≥ 0 by paying pt to the seller. We will call (pt )t∈[0,∞) and (t(θ))θ∈Θ the price path and the trading time associated to µ, respectively.
3
Two-type Example
Suppose that the buyer is of the high type (θH ) or the low type (θL ) with probabilities q(θH ) and q(θL ) = 1 − q(θH ) respectively, where v(θH ) > v(θL ) > 0 and u(θH ) = v(θH ) − φ(θH ) > u(θL ) = v(θL ) − φ(θL ) > 0. 15
t(θ) = ∞ at a state of the world in which the buyer type θ does not purchase the good from the seller.
7
That is, the high type has both higher valuation of the good and higher net-valuation. Also, in order to focus on the case of the greatest interest,16 suppose q(θH ) is sufficiently large so that q(θH ) > max
v(θL ) v(θL ) − φ(θL ) , v(θH ) v(θH ) − φ(θH )
> 0.
(3)
Finally, for simple exposition, also suppose that the buyer’s outside option is always available during the negotiation; that is, P{Zt = 1 for any t ≥ 0} = 1, and hence φ(θ) = w(θ) for both types. The last assumption will be dropped later at the end of this section. For the benchmark, first suppose that the seller can only commit to a one-shot mechanism in which all actions and transfers occur at t = 0. Any one-shot mechanism is identified by x(θ) and p(θ), the probability that each θ trades and her payment at t = 0. The buyer would immediately take the outside option if no transaction occurs, and hence, a one-shot mechanism is incentive-compatible iff v(θ)x(θ) + w(θ)(1 − x(θ)) − p(θ) ≥ max{v(θ)x(θ0 ) + w(θ)(1 − x(θ0 )) − p(θ0 ), w(θ)} ∀θ, θ0 ∈ Θ. (4) Then, we can identify an optimal one-shot mechanism by solving the following program: Πone-shot = S
q(θH )p(θH ) + q(θL )p(θL )
sup
(5)
(x(θ),p(θ))θ∈{θH ,θL }
subject to u(θ)x(θ) − p(θ) ≥ max{u(θ)x(θ0 ) − p(θ0 ), 0} ∀θ, θ0 ∈ {θH , θL } where the constraint is a rearrangement of (4). This is equivalent to the static screening problem, except that v(θ) is replaced by u(θ) in the incentive-compatibility constraints. Hence, the no-haggling result (Samuelson, 1984, Proposition 1) implies that the optimal one-shot mechanism always commits to a single fixed price pone-shot = u(θH ) that is accepted at t = 0 if θ = θH ; otherwise, the buyer instantly exercises the outside option. PROPOSITION 1. The fixed-price mechanism that commits to pone-shot = u(θH ) = v(θH ) − φ(θH ) constitutes an optimal one-shot mechanism. The next question is whether any other mechanism can generate profit strictly higher than Πone-shot S by delaying. The answer to this question hinges on the relative dispersion of w compared to the dispersion of v, which can be measured by the (discrete) elasticity of w with respect to v: ∞ if w(θH ) > w(θL ) = 0 w,v := 0 if w(θH ) = w(θL ) = 0 (w(θH )−w(θL ))/w(θL ) otherwise. (v(θH )−v(θL ))/v(θL )
To see the role of the elasticity of w(θ) in sales negotiations, observe first that a buyer of type θ never chooses to trade at any time after ( t(θ) := 16
1 r
v(θ) log w(θ)
if w(θ) > 0 if w(θ) = 0,
∞
We show in Appendix A.3 that a fixed-price mechanism is optimal whenever (3) fails.
8
because e−rt (v(θ) − p) < w(θ) for any p ≥ 0 and t > t(θ). That is, type-dependent outside options induce type-dependent deadlines in bargaining before which the negotiation situation has to be resolved. The significance of w,v lies in the following observation: w,v ≥ 1
⇐⇒
t(θH ) ≤ t(θL ).
That is, the high type has a tighter deadline iff w(θ) is relatively more dispersed than v(θ). Now suppose that w,v ∈ [1, ∞) or, equivalently, t(θH ) ≤ t(θL ). This condition holds, for example, if w(θ) = βv(θ) for both buyer types, where β ∈ (0, 1). Any offer made by the seller at t ∈ [t(θH ), t(θL )] does not affect the incentive constraint for the high type. The seller can exploit this slackness in the incentive constraint to inter-temporally discriminate between the two buyer types, without conceding any information rent. Specifically, consider the following mechanism. • The seller commits to the deterministic price path ( pt =
pone-shot for t < t† p† for t ≥ t†
(6)
†
where t† ∈ [t(θH ), t(θL )] and p† < v(θL ) − ert w(θL ). • The buyer chooses whether to purchase the good (by paying pt ), exercise her outside option, or delay at any t ≥ 0. It is clearly optimal for the high type and the low type to purchase the good at t = 0 and t = t† respectively. Note that this mechanism earns a positive profit even when faced with the low type, contrary to the fixed-price mechanism that generates zero profit from the low type. The mechanism , which shows that there exists at least one can therefore earn a (weakly) higher profit than Πone-shot S optimal mechanism that involve a positive duration of delay. There are generally multiple optimal mechanisms when w,v ≥ 1. One of them is the posting-price mechanism with price path ( pt =
for 0 ≤ t <
u(θH ) v(θL )w(θH )−v(θH )w(θL ) w(θH )−w(θL )
for t ≥
1 r
1 r
v(θH )−v(θL ) log w(θ H )−w(θL )
v(θH )−v(θL ) log w(θ H )−w(θL )
(7)
and deterministic trading time
t(θ) :=
0 v(θ ) − v(θL ) H 1r log w(θH ) − w(θL )
if θ = θH if θ = θL .
The seller’s expected profit is u(θH )q(θH ) + (1 − q(θH ))
v(θL )w(θH ) − v(θH )w(θL ) v(θH ) − v(θL )
9
(8)
𝑝𝑡 𝑢(!𝐻 )
𝑣(!𝐿 )"(!𝐻 ) − 𝑣(!𝐻 )"(!𝐿 ) "(!𝐻 ) − "(!𝐿 )
0
1 𝑣(!𝐻 ) − 𝑣(!𝐿 ) log 𝑟 "(!𝐻 ) − "(!𝐿 )
𝑡
Figure 1: Optimal price path when w,v > 1
if w,v ≥ 1.17 If we further assume that w,v > 1, the profit (8) is which is weakly larger than Πone-shot S , and hence, all optimal mechanisms necessarily involve a positive delay. strictly larger than Πone-shot S The above argument exploits the slackness in incentive constraints due to t(θH ) ≤ t(θL ), which is no longer applies when w,v ∈ (−∞, 1). In the extreme case that w(θH ) = w(θL ) > 0 and w,v = 0, for instance, the no-haggling result applies, and hence, the fixed-price mechanism is optimal. In the appendix, we show that all optimal mechanisms are necessarily a fixed-price mechanism, and hence no delay occurs whenever w,v ∈ (−∞, 1). PROPOSITION 2. If w,v ≥ 1, there exists at least one optimal mechanism that involves a delay. Moreover, if w,v > 1, all optimal mechanisms involve a delay. Proof. See Appendix A.3. One may wonder whether the delay could be replaced by a stochastic one-shot mechanism, the outcomes of which are resolved at t = 0 for sure (and hence no delay in transaction occurs). Intertemporal price discrimination is indeed equivalent to stochastic price discrimination without typedependent outside options and withdrawal rights. However, such equivalence breaks down in our model, and the delay cannot be replaced by any stochastic offer. To see this, it actually suffices to note that the domain of (x(θ), p(θ))θ∈Θ in (5) covers all stochastic one-shot mechanisms. The intuition behind this observation can be explained as follows. In our model, the buyer can use her outside option as insurance against negotiation breakdowns. Therefore, other things being equal, the buyer’s information rent (in particular, information rent for a type with a better outside option) tends to increase as the probability of breakdown increases.18 Compared to (7), therefore, any nondeterministic one-shot mechanism that induces the negotiation to break off with positive probability would result in more information rent to the buyer and a strictly lower profit to the seller. 17 18
We use the convention 10 = ∞ and log(∞) = ∞. The information rent refers to the gap between the autarky payoff and the payoff from participating in a mechanism.
10
Now we turn to the case in which the outside option may not be available at some t ≥ 0. For concreteness, suppose that Zt = 0 but the outside option arrives during the negotiation according to a Poisson process with rate λ > 0. Propositions 1 and 2 still remain true in this case without any modification. In particular, a delay has to occur in all optimal mechanisms whenever w,v > 1. However, it turns out that a posting-price mechanism like (7) is not optimal anymore, and the seller actually has to additionally arrange an upfront payment scheme to achieve the optimal profit level. For example, consider the following mechanism which is optimal if w,v > 1.19 • The buyer sends a message m0 ∈ {θH , θL }, and then makes the upfront payment ( α(m0 ) =
if m0 = θH if m0 = θL
0 r λ+r w(θL )
(9)
prior to the negotiation. There is no further non-trivial communication afterward. • The two parties break off the negotiation immediately if the buyer declines to pay the upfront payment. Otherwise, the mechanism posts a price v(θH )−v(θL ) u(θH ) for 0 ≤ t < 1r log λ+r λ w(θH )−w(θ L) pt = v(θL )w(θH )−v(θH )w(θL ) for t ≥ 1 log λ+r v(θH )−v(θL ) r
w(θH )−w(θL )
λ w(θH )−w(θL )
at each time t ≥ 0. If the buyer chooses to purchase the good at time t, she immediately pays pt (m0 ) = pt − ert α(m0 )
(10)
to the seller, where m0 denotes the buyer’s initial message. • Faced with this mechanism, it is optimal for both types to truthfully reveal hertype prior to the v(θH )−v(θL ) negotiation, and then the high type and the low type trade at t = 0 and t = 1r log λ+r λ w(θH )−w(θL ) respectively. It is suboptimal for both types to exercise the outside option. This mechanism is similar to the posting-price mechanism with the price path (7). Notable distinctions are, however, the type-dependent upfront payment α(m0 ) and the corresponding price discount, (9) and (10), respectively. The role of the upfront payment and the price discount is to prevent the buyer from opting out during the negotiation. Indeed, without them, the low type would exercise the outside option if it arrives at t < 1r log λ+r λ , which results in a lower profit level to the seller. More generally, without an upfront payment, the break-off necessarily occurs in any mechanism such that (i) the low type’s individual rationality constraint binds, and (ii) a delay of transaction occurs.20 As discussed above, a possibility of breakdown generally results in more information rent to the high type, and hence any mechanism without an upfront payment is never optimal whenever w,v > 1. In particular, all posting-price mechanisms are necessarily suboptimal. 19
A proof can be found in Appendix A.3. That is to say, the break-off necessarily occurs with positive probability in any mechanism µ ∈ MIC such that δ(µ) > 0 and V µ (θL ) = φ(θL ). 20
11
PROPOSITION 3. Suppose w,v > 1. (i) If the outside option is always available through the negotiation, then there exists a posting-price ¯ S as the seller’s expected profit. mechanism that generates Π (ii) If the outside option arrives to the buyer according to a Poisson process with rate λ > 0, any ¯ S. posting-price mechanism (without an upfront payment) can never achieve Π Proof. See Appendix A.4. Overall, the no-haggling result fails in two ways whenever w,v > 1. First, a delay occurs in all optimal mechanisms. Also, if we assume that the outside option arrives at a Poisson rate λ > 0, any posting-price mechanism is suboptimal. These two observations will be generalized in the next section to continuous-types.
4
Characterization of Optimal Mechanism
4.1
Assumptions
From now on, suppose that Θ is a closed interval (continuous-type space). We may assume without loss that v(θ) = θ, so that u(θ) = θ −φ(θ) for all θ ∈ Θ. Throughout our analysis, we will also maintain the following assumptions. ASSUMPTION 1. Θ = [θ, θ] ⊂ [0, ∞). u(θ) ≥ 0 for any θ ∈ [θ, θ]. ASSUMPTION 2. Both
f (θ) 1−F (θ)
and θf (θ) + F (θ) are strictly increasing at θ ∈ (θ, θ).
ASSUMPTION 3. w(θ) is convex, continuous, and piecewise twice differentiable. The first assumption can be made without loss because any buyer type with a negative net-valuation is necessarily excluded from any optimal mechanism. The second assumption, which is one of the standard simplifying assumptions used in the literature, excludes bunching from the optimal mechanisms. The convexity of w(θ) may sound restrictive, but it actually arises in various contexts as in Example 1 below. Also, note that Assumption 3 does not exclude the case in which θ and w(θ) are negatively related over a subinterval of [θ, θ]. EXAMPLE 1. Suppose that the buyer can switch to an outside seller at any time during the negotiation with the incumbent seller, which serves as her outside option. Each θ ∈ Θ assigns a value αθ + β ≥ 0 to the outside seller’s good, where α, β ∈ R. The value of the outside option w(θ) = max{αθ + β − p, 0} is convex in θ, if the outside seller commits to a single price p ≥ 0. The convexity also arises in any incentive-compatible mechanism that the outside seller may use. The elasticity of φ with respect to θ plays a key role in the following discussion. Formally, ( φ (θ) :=
θφ0 (θ− ) φ(θ)
if φ(θ) > 0
0
if φ(θ) = 0 12
∀θ ∈ Θ,
where φ0 (θ− ) stands for the left-derivative of φ. Note that φ (θ) measures the local relative dispersion of φ and θ. φ (θ) ≤ 1 if and only if θ ≤ θ† , where ( θ† :=
θ if φ (θ) ≤ 1 for all θ inf {θ ∈ Θ : φ (θ) > 1} otherwise.
(11)
Also, define θ(p) := min{θ ∈ Θ : u(θ) ≥ p},
θ(p) := max{θ ∈ Θ : u(θ) ≥ p},
and U(p) := [θ(p), θ(p)] = {θ ∈ Θ : u(θ) ≥ p} and T(p) := [θ† , θ(p)].
(12)
for any p ≥ 0.
4.2
Relaxed Problem
For any µ = (A, P ) ∈ MIC and θ ∈ Θ define the (expected) discounted trading volume and the total (expected) discounted transfer by "
h i µ ∞ x (θ) := E e−rτ (θ,Z ) |θ
and p (θ) := E P0 (θ, Z ) +
µ
∞
µ
Z e s∈(0,∞)
−rs
# dPs (θ, Z ) θ , ∞
respectively. Also, ( µ
y (θ) :=
1 φ(θ)
E e−rσ
µ (θ,Z ∞ )
φσµ (θ,Z ∞ ) (θ; Z σ
µ (θ,Z ∞ )
)|θ
if w(θ) > 0 otherwise.
0
By definition, xµ (θ) + y µ (θ) ∈ [0, 1] for any θ ∈ Θ. LEMMA 1. For any µ ∈ MIC , there exists always another µ ˜ ∈ MIC such that ΠS (µ) = ΠS (˜ µ) and ˜ + y µ˜ (θ)φ(θ) ˜ ˜ xµ˜ (θ)θ + y µ˜ (θ)φ(θ) − pµ˜ (θ) ≥ max{φ(θ), xµ˜ (θ)θ − pµ˜ (θ)} for any θ, θ˜ ∈ Θ. Proof. In Appendix A.1. ¯ S: The last lemma leads to the following relaxed problem which provides an upper bound for Π Z ΠR := max p(θ)dF (θ) (x(θ),y(θ),p(θ))θ∈Θ
(R)
θ∈Θ
subject to x(θ)θ + y(θ)φ(θ) − p(θ) ≥ φ(θ) ˜ + y(θ)φ(θ) ˜ ˜ x(θ)θ + y(θ)φ(θ) − p(θ) ≥ x(θ)θ − p(θ) 0 ≤ x(θ) ≤ 1 − y(θ) ≤ 1 and p(θ) ∈ R 13
∀θ ∈ Θ,
(13)
∀θ, θ˜ ∈ Θ,
(14)
∀θ ∈ Θ.
(15)
LEMMA 2. A solution to (R) exists, and ¯ S ≤ ΠR = max Π
"Z
p∈U
U(p)
Z
p dF (θ) +
# θφ (θ ) − φ(θ) dF (θ) . 0
T(p)
−
(16)
Proof. See Appendix A.2. The proof can be found in the appendix. Here, we briefly discuss the technical aspects of the proof. Note that (R) is mathematically equivalent to the static screening problem with two distinctive features: two-dimensional allocation rule (x(θ), y(θ)) and a type-dependent individual-rationality constraint (13). Two technical issues arise due to these features. First, as the allocation rule is multi-dimensional, the monotonicity of the allocation rule does not guarantee the incentive-compatibility. Second, the type-dependent individual-rationality constraint (13) makes it difficult to tell a priori for which types constraint (13) binds.21 Applying the envelope theorem (Milgrom and Segal, 2002), we can transform (R) into the following form: Z 1 − F (θ) 1 − F (θ) x(θ) − φ(θ) − φ0 (θ− ) (1 − y(θ)) dF (θ) + π(θ) − φ(θ) max θ− f (θ) f (θ) (x(·),y(·),p(·)) Θ subject to ˜ + y(θ)φ(θ) ˜ ˜ x(θ)θ + y(θ)φ(θ) − p(θ) ≥ x(θ)θ − p(θ) Z θ x(s) − φ0 (s− )(1 − y(s)) ds ≥ 0 π(θ) − φ(θ) = π(θ) − φ(θ) +
∀θ, θ˜ ∈ Θ,
(17)
∀θ ∈ Θ,
(18)
∀θ ∈ Θ.
(19)
θ
0 ≤ y(θ) ≤ 1 − x(θ) ≤ 1
where π(θ) := θx(θ) + φ(θ) − p(θ) for any θ ∈ Θ. If we further drop (17), the program degenerates into a SCLP (Separated Continuous Linear Program) for which we could obtain a closed-form solution. It turns out that the solution for the SCLP problem satisfies (17) all the time, and hence it also solves (R). The details can be found in the appendix.
4.3
Optimal Mechanism
Now we are ready to solve for a (globally) optimal mechanism. First, fix p∗ = arg max p∈U
"Z U(p)
p dF (θ) +
Z T(p)
# θφ0 (θ− ) − φ(θ) dF (θ) .
(20)
If the optimization problem (20) has multiple maximizers, take one among them as p∗ so that U(p) ⊂ U(p∗ ) for any other arg maximum p.22 With this choice of p∗ , consider the following mechanism (which 21
Jullien (2000, Section 5.4) analyzes a mechanism design problem closely related to (R) in the context of nonlinear pricing for a divisible good, and provides a technique to cope with the type-dependent individual-rationality constraint. But unfortunately, (R) fails two assumptions, Homogeneity and CVU (Jullien, 2000, p. 10 and p. 23, respectively), which are required for his analysis. These conditions fail due to the buyer’s having only a unit demand for the good. 22 ∗ p is well-defined because U(p) is monotone in p (when sets are ordered by set inclusion).
14
generalizes the optimal mechanism for the two-type example in the last section). • At the beginning of the negotiation, the buyer sends a message m0 ∈ Θ, and then makes an upfront payment ( w(m0 ) − φ(m0 ) if m0 ∈ T(p∗ ) α(m0 ) = (21) 0 otherwise. • Parties immediately break off the negotiation if the buyer declines to pay the upfront payment. Otherwise, the buyer obtains the right to purchase the good at any t ≥ 0 by additionally paying p(t, m0 ) = p∗t − ert α(m0 ) where p∗t
=
if 0 ≤ t <
p∗
(φ0 )−1 e−rt − ert φ((φ0 )−1 e−rt ) if
1 r
1 r
log φ0 (θ1∗− )
(22)
log φ0 (θ1∗− ) ≤ t < ∞
and θ∗ := θ∗ (p∗ ) = min{θ ∈ Θ : u(θ) ≥ p∗ }. The buyer can also exercise her outside option at any t ≥ 0 when the outside option is available. Suppose for now that Z0 = 1. In this case, γ0 (Z0 ) = 0, and hence the amount of the upfront payment α(m0 ) is zero for any m0 ∈ Θ, and p(t, m0 ) = p∗t for all t ≥ 0 and m0 ∈ Θ. That is, the mechanism features no upfront payment, and all buyer types face the same price path, and hence, the mechanism degenerates to a posting-price mechanism. To derive the trading time for each buyer type, note that it is optimal for the buyer to exercise the outside option at t = 0 if and only if max e−rt (θ − p∗t ) ≤ w(θ) = φ(θ), t≥0
(23)
and otherwise she will never exercise the outside option. Denote the optimal purchase time of a buyer of type θ by −rt ∗ inf arg max e (θ − pt ) if maxt≥0 e−rt (θ − p∗t ) ≥ φ(θ) t≥0 t(θ) := ∞ if e−rt (θ − p∗ ) < φ(θ) for all t ≥ 0. t
A straightforward calculation would show that 0 1 1 t(θ) = r log φ0 (θ− ) ∞ and V˜ (θ) := e−rt (θ − p∗t ) t=t(θ)
if θ ∈ U(p∗ ) if θ ∈ T(p∗ ) otherwise,
∗ ∗ θ − p if θ ∈ U(p ) = φ(θ) if θ ∈ T(p∗ ) 0 otherwise
15
(24)
(25)
for any θ.23 Hence, the above mechanism is equivalent to the posting-price mechanism with the price path (22) and the trading time (24). This posting-price mechanism generates Z U(p∗ )
p∗ dF (θ) +
Z T(p∗ )
e−rt(θ) p∗t(θ) dF (θ) =
Z U(p∗ )
p∗dF (θ) +
Z T(p∗ )
θφ0 (θ− ) − φ(θ) dF (θ)
(26)
as the seller’s expected profit, and hence the above mechanism is optimal by Lemma 2. Now suppose that γ0 (Z0 ) > 0, so that α(m0 ) is positive for some m0 ∈ Θ. If a buyer type θ sends m0 in the pre-negotiation communication and then purchases the good at time t, her final payoff is U (m0 , t) = −α(m0 ) + e−rt θ − (p∗t − ert α(m0 )) = e−rt [θ − p∗t ] .
(27)
Note that U (m0 , t) does not depend on m0 , and hence we may assume that all buyer types would send m0 = θ in the pre-negotiation communication. Moreover, U (m0 , t) coincides with the left-hand side of (23). Hence, each buyer type’s expected payoff from trading with the seller is
V˜ (θ) = e−rt (θ − p∗t ) t=t(θ)
∗ ∗ θ − p if θ ∈ U(p ) = φ(θ) if θ ∈ T(p∗ ) 0 otherwise
where t(θ) is as specified in (24). Now consider a buyer type θ ∈ U(p∗ ). V˜ (θ) ≥ φ(θ) and t(θ) = 0 for such a buyer type, and hence, she would trade with the seller at t = 0. The total payment to the seller is α(θ) + p(t, θ) = α(θ) + (p∗ − α(θ)) = p∗ . Next, consider a buyer type θ ∈ T(p∗ ). This buyer type has no reason to break off the negotiation at t = 0 because V˜ (θ) = φ(θ). What if the outside option arrives at t ∈ [0, t(θ)]? If she waits until her optimal trade time t(θ) and then purchases the good, she can guarantee e−r(t(θ)−t) (θ − p(t, θ)) = e−r(t(θ)−t) (θ − p∗t(θ) + ert(θ) α(θ)) ≥ e−rt(θ) (θ − p∗t(θ) + ert(θ) α(θ)) = φ(θ) + α(θ) = w(θ). Hence, this buyer type never exercises the outside option, and instead, trades with the seller at t = t(θ). Finally, consider a buyer type θ such that θ 6∈ U(p∗ ) ∪ T(p∗ ). Clearly, V˜ (θ) ≤ φ(θ), and hence this type would rather break off the negotiation immediately. As a result, the expected profit of the seller is again equal to (26). We denote the above mechanism, combined with the buyer’s optimal decision faced with this mechanism as described above, by µ∗ in the remainder of the paper. µ∗ can also be written using the 23
Again, we use the convention (φ0 )−1 (e−rt ) = inf{θ ∈ Θ : φ0 (θ− ) ≥ e−rt } for any t ≥ 0,
16
1 0
= ∞, and log(∞) = ∞.
notation developed in Section 2 as follows.
At (m) =
n 1 t=
∗ 1{t = 0} o − 1{t n > 0} o if m0 ∈ U(p ) 1 1 1 1 if m0 ∈ T(p∗ ) r log φ0 (m− ) − 1 t > r log φ0 (m− ) 0
Pt (m) =
0
−1
otherwise.
p∗ · 1{t = 0}
α(m0 ) · 1{t = 0} +
m0 φ0 (m− 0 )−φ(m0 )−α(m0 ) φ0 (m− 0 )
if m0 ∈ U(p∗ ) n ·1 t=
1 r
1 log φ0 (m − )
o
0
0
if m0 ∈ T(p∗ ) otherwise.
The next proposition immediately follows from the discussion so far. PROPOSITION 4. µ∗ is an optimal mechanism. The optimal mechanism µ∗ has several notable features departing from the prediction of the nohaggling result. First, the transaction is delayed faced with buyer types in T(p∗ ). The expected duration of delay is given by ∗
Z
t(θ) dF (θ) = P{θ ∈ T(p )} · E ∗
δ(µ ) = θ∈T(p∗ )
1 1 ∗ log 0 − θ ∈ T(p ) , r φ (θ )
which is positive if and only if T(p∗ ) is of positive (Lebesgue) measure. Second, the optimal mechanism µ∗ commits to the declining price path. In the simplest case that Z0 = 1, the price path coincides with (22), which is deterministic and decreases in t. Third, if Z0 = 0, buyer types in T(p∗ ) are asked to pay upfront payments α(θ) > 0, which are compensated by the price discount later on. These feature help to prevent the buyer from breaking off the negotiation, which, compared to other mechanisms, results in less information rent to the buyer and higher profit to the seller. The intuition is basically identical to the discussion in Section 3, and hence we omit discussing it again. EXAMPLE 2 (Uniform-Quadratic Case). Consider a uniform-quadratic environment such that θ is drawn from the uniform distribution over [0, 1], and w(θ) = θ2 for all θ ∈ [0, 1]. Suppose that the outside option arrives to the buyer at a Poisson rate λ ∈ (0, ∞], where λ = ∞ indicates the case such that Zt = 1 for all t ≥ 0. Define ( λ λ+r if λ < ∞ B := 1 if λ = ∞ so that φ(θ) = Bw(θ). Suppose that B is sufficiently close to 1 (or equivalently, λ is sufficiently large), so that the first-order condition alone pins down the solutions of all optimization problems below. 1 First, the optimal one-shot mechanism is the fixed-price mechanism that commits to hpone-shot = 6B i. This price is accepted instantly by a type θ if and only if θ ∈ [θ(pone-shot ), θ(pone-shot )] =
and thus, the trading volume in the fixed-price mechanism is z one-shot = optimal mechanism, define p∗ by the arg maximum of "Z max p∈U
U(p)
p dF (θ) +
Z T(p)
√
3 3B .
√ √ 3− 3 3+ 3 6B , 6B
To characterize the
# " √ 3 # √ 1 − 1 − 4Bp 1 − 4Bp θφ (θ) − φ(θ) dF (θ) = max p + , p∈U B 24B 2 0
17
,
pt*
zt
p*
1
z∞
p one-shot
one-shot
z
p**
_
t 10
0
z0
t
20
_
t 10
0
20
t
Figure 2: Graph for Example 2. Θ = [0, 1], F (θ) = θ, w(θ) = φ(θ) = θ2 , B = 0.95, and r = 0.1.
or equivalently,
√ 10 − 2 p = 49B ∗
Also,
∗
T(p ) = 0,
=⇒
1−
√
1 − 4Bp∗ . 2B
α(θ) = (1 − B)θ2 · 1{θ∈T(p∗ )} ,
and ( p(t, θ) =
p∗t
rt
− e α(θ) =
p∗ − ert (1 − B)θ2 · 1{θ∈T(p∗ )} 1 −rt 4B e
if 0 ≤ t <
− ert (1 − B)θ2 · 1{θ∈T(p∗ )} if
1 r
1 r
1 log 1−√1−4Bp ∗
1 log 1−√1−4Bp ∗ ≤ t < ∞
for all buyer types θ ∈ Θ. The optimal purchase time of the buyer is √ i ∗ ), 1] = 1+ 1−4Bp∗ , 1 ∞ if θ ∈ (θ(p 2B h √ i √ ∗ 1+ 1−4Bp∗ ∗ ∗ t(θ) = 0 if θ ∈ U(p ) = [θ(p ), θ(p∗ )] = 1− 1−4Bp , 2B 2B h √ 1− 1−4Bp∗ ∗ ∗ 1 log 1 , r 2Bθ if θ ∈ T(p ) = [0, θ(p )) = 0, 2B
(28)
(29)
and the expected delay is √
Z 0
1−
1−4Bp∗ 2B
t(θ)dF (θ) =
1−
√
i p 1 − 4Bp∗ h 1 − log 1 − 1 − 4Bp∗ > 0. 2Br
Note that α(θ) → 0 as λ → ∞, and hence µ∗ converges to the posting-price mechanism associated with the price path (p∗t )t≥0 . However, the expected delay δ(µ∗ ) does not shrink to zero as λ → ∞.
18
The (undiscounted) cumulative trading volume up to time t ≥ 0 is Z zt :=
( dF (θ) =
θ:t(θ)≤t √
√
1−4Bp∗ B √ 1+ 1−4Bp∗ 1 −rt − 2B e 2B
if 0 ≤ t ≤
1 r
1 log 1−√1−4Bp ∗
otherwise
∗
as t → ∞. µ∗ suppresses the breakdown of the negotiation which converges to z∞ = 1+ 1−4Bp 2B with buyer types θ ∈ T(p∗ ) by promising to offer lower prices in the future. This results in a lower probability of negotiation breakdown than the fixed-price mechanism, which explains the gap between z∞ and z one-shot in the right panel of Figure 2.
5
Option Contracts
With the delay of transaction and upfront payments, µ∗ features an option-contract-like structure. Indeed, µ∗ is equivalent in terms of the outcome to a menu of European call options. Suppose that, based on the buyer’s message θ ∈ Θ at t = 0, the seller offers the European call option with premium α(θ) and maturity time t(θ) as specified in (21) and (24) respectively, and the strike price
p(θ) =
w(θ) φ0 (θ− )
if θ ∈ U(p∗ ) if θ ∈ T(p∗ )
0
otherwise.
p∗ θ−
The buyer can maximize her expected payoff by truthfully revealing her type, and hence the menu of European call options implement the outcome of µ∗ . Note that the seller can benefit from the optionlike structure mainly because it is effective at minimizing the possibility of negotiation break-off. One can show that, if the seller could force the buyer to sign a contract that abrogates her withdrawal right and outside option, the seller can maximize his profit simply by randomizing over fixed-price mechanisms. The model generally admits multiple (globally) optimal mechanisms. However, we will argue in the following subsections that the option-like structure is not specific to µ∗ . More precisely, we will show that all optimal mechanisms feature the option-like structure a delay of transaction (maturity date in the future) and an upfront payment (premium) whenever the Lebesgue measure of T(p∗ ) is positive. The European call option is one of the simplest ones among all such option-like optimal mechanisms.
5.1
Delay of Transaction
We begin by identifying the optimal one-shot mechanism for the benchmark. The same argument as for Proposition 1 shows that a fixed-price mechanism is optimal among all one-shot mechanisms. PROPOSITION 5. The fixed-price mechanism that commits to the price Z one-shot p ∈ arg max p dF (θ) p∈U
constitutes an optimal one-shot mechanism. 19
U(p)
(30)
From now on, let µ∗one−shot denote the optimal one-shot mechanism characterized by the last proposition. If the optimization problem in (30) has multiple maximizers, we again take one among them as pone−shot so that U(p) ⊂ U(pone−shot ) for any other arg maximum p. Note that ΠS (µ∗one−shot ) =
Z U(pone−shot )
pone−shot dF (θ).
One natural question is whether there is any optimal mechanism, other than µ∗ , which involves no delay. Proposition 5 also helps us to answer this question. By comparing the optimization problem (30) with the relaxed problem (R), it is clear that δ(µ) > 0 for all optimal mechanisms if and only if "Z max
Z p d F (θ) +
p∈U
U(p)
# Z 0 − θφ (θ ) − φ(θ) d F (θ) > max p∈U
T(p)
U(p)
p dF (θ).
Note that θφ0 (θ) − φ(θ) > 0 for all θ ∈ ∪p≥0 T(p), and hence the inequality always holds at least weakly. If the inequality holds strictly, all one-shot mechanisms are strictly suboptimal. This is the case if and only if the (Lebesgue) measure of T(p∗ ) is positive. Recall that this is also the necessary and sufficient condition for δ(µ∗ ) > 0, and hence we obtain the following corollary. COROLLARY 1. The following statements are equivalent: (i) The Lebesgue measure of T(p∗ ) is positive. (ii) δ(µ∗ ) > 0. (iii) δ(µ) > 0 for any optimal mechanism µ. Statement (i) in the last corollary holds in many natural examples. For example, this condition always holds whenever w(·) is strictly convex and θ = 0.
5.2
Upfront Payments
Now we turn to the upfront payment scheme of the optimal mechanism µ∗ . Our purpose is to show that a similar upfront payment scheme is present if Z0 = 0 (not only in µ∗ ) in all optimal mechanisms. We first define upfront payment scheme formally. Define each buyer type’s upfront payment and the average upfront payment in a mechanism µ = (A, P ) as follows: Φ(θ; µ) := E
"
1{0<τ µ (θ,Z ∞ )∧σ(θ,Z ∞ )} · P0 (θ, Z ∞ ) + Z
Z 0
Φ(θ; µ) dF (θ).
Φ(µ) :=
# e−rs dPs (θ, Z ∞ ) θ , (31)
θ∈Θ
Φ(θ; µ) is the expected discounted transfer from the buyer to the seller before the negotiation concludes. Note that s = τ µ (θ, Z ∞ ) ∧ σ(θ, Z ∞ ) is excluded from the integration region in (31). We say that a mechanism features an upfront payment scheme if Φ(µ) > 0.
20
To formally state our result, we also need the following condition. φ(θ) < w(θ) and ∃¯ > 0 such that
P{γ0 (Z0 ) ∈ (0, )} > 0 ∀ ∈ (0, ¯).
(U )
Recall that γ0 (Z0 ) denotes the earliest time at which the outside option becomes available to the buyer. Hence, this condition is satisfied, for example, if Z0 = 0 but the outside option arrives to the buyer at a Poisson rate λ > 0, and it is trivially failed if Z0 = 1 with probability 1 (because φ(θ) = w(θ) in this case). PROPOSITION 6. Suppose that Condition U holds and the Lebesgue measure of T(p∗ ) is positive. Then, for any optimal mechanism µ, (i) Φ(µ) ≥ Φ(µ∗ ) > 0, and (ii) Φ(θ; µ) ≥ Φ(θ; µ∗ ) for almost every θ ∈ Θ. Proof. See Appendix A.5. Part (i) of the proposition states that all optimal mechanisms feature an upfront payment scheme. Part (ii) states that, compared to any other optimal mechanism, µ∗ features minimal upfront payments for almost every buyer types. This property makes µ∗ more attractive compared to other optimal mechanisms, especially when the buyer is possibly unable to pay a large amount of upfront payments due to a liquidity constraint in the early stage of the interaction. Combining the last proposition with Corollary 1, we obtain the following corollary. COROLLARY 2. Suppose that Condition U holds and the Lebesgue measure of T(p∗ ) is positive. Then, for any optimal mechanism µ, δ(µ) > 0 and Φ(µ) > 0. Summing up, all the optimal mechanisms have the option-like structure with an upfront payment (premium) and delay of transaction (maturity date in the future) whenever the measure of T(p∗ ) is positive.
6 6.1
Comparative Statics Analysis Autarky Payoffs
Corollary 1 provides the necessary and sufficient condition for a delay being present in all optimal mechanisms, which also appears in Proposition 6 and Corollary 2. In this subsection, we will show that this condition holds more likely in an environment with more dispersed autarky payoffs among buyer types in terms of φ . We will identify each environment by its type distribution F and autarky payoff function φ.24 Define E as the set of all environments that satisfy Assumptions 1–3. Without loss, we assume that F 24 We also need to specify Z = (Zt )t≥0 for a complete description of the model. However, recall that it is independent of Z whether δ(µ∗ ) > 0 (see Corollary 1), and hence we do not specify Z.
21
has a unit length of support in any environment in E. For any E = (F, φ) ∈ E, define Z QE := θ∈supp(F )
1{φ (θ)>1} dF (θ)
as the fraction of types at which autarky payoff is locally elastic, where supp(F ) is the support of F . Furthermore, let µ∗E be the optimal mechanism (as characterized in Section 4.3) for environment E. We first focus on a subset of E such that φ0 (θ− ) ∈ (−∞, 1] for all θ ∈ Θ, and thus, the net-valuation u(θ) = θ − φ(θ) increases in θ. We will call such an environment monotone. Other environments will be discussed subsequently. For any type distribution F with a unit length of support, define E|F := {(E 0 , F 0 ) ∈ E : F 0 = F } 0 0 0 0 − Em |F := {(E , F ) ∈ E : F = F and φ (θ ) ∈ (−∞, 1] for any θ ∈ supp(F )}
where the superscript m stands for monotone. ∗ PROPOSITION 7. Fix a type distribution F such that Em |F is nonempty. There exists a cutoff Q ∈ [0, 1] ∗ such that δ(µ∗E ) > 0 for E ∈ Em |F if and only if QE > Q .
Proof. See Appendix A.6. Note that whether a delay occurs or not in an environment in Em |F is completely pinned down by the fraction of types at which autarky payoff is locally elastic. The intuition for this proposition is basically identical to one for the two-type example in Section 3. Recall that a delay occurs in the θ are decreasing in θ. two-type example iff the type-dependent deadlines in bargaining t(θ) := 1r log φ(θ) Any offer made after t(θ) is irrelevant for the incentive constraint of type θ, and such slackness of the incentive constraint allows the seller to generate more profit by delaying the transaction rather than 1−φ (θ) < 0 iff φ (θ) > 1, breaking off the negotiation. Assuming the differentiability of t(θ), dt(θ) dθ = rθ which is consistent with our observation that a delay is more likely to occur in an environment with a large QE . m COROLLARY 3. Fix a type distribution F such that Em |F is nonempty, and consider E1 = (F, φ1 ) ∈ E|F such that δ(µ∗E1 ) > 0. Then δ(µ∗E2 ) > 0 for any E2 = (F, φ2 ) ∈ Em |F such that φ1 (θ) and φ2 (θ) have the same sign and |φ2 (θ)| ≥ |φ1 (θ)| for all θ ∈ supp(F ).
We can inarguably say that E2 = (F, φ2 ) has more dispersed autarky payoffs than E1 = (F, φ1 ) in the last corollary. The corollary tells us that if the optimal mechanism involves in one environment, so it also does in another environment with more dispersed autarky payoffs. This is an immediate implication of Proposition 7, and hence we omit its proof. EXAMPLE 3. Consider Ex = (Fx , φx ) such that Fx is an uniform distribution over [x, x + 1] and φx (θ) =
β (θ − x)2 + x ∀θ ∈ [x, x + 1] 2
22
Π S (µ* ) - ΠS (µ*one-shot ) ___________________ ΠS (µ* one-shot ) 0.10
x=0
0.08 0.06 0.04
x = 0.05
0.02
x = 0.1 0
β* (0)
0.5
β* (0.05)
β* (0.1)
1
1.5
2
β
Figure 3: Graph for Example 3
where x ≥ 0 and 0 ≤ β ≤ 2. The elasticity of the autarky payoff is given by ( φx (θ) =
2βθ(θ−x) β(θ−x)2 +2x
if θ ∈ (x, x + 1] if θ = x
0
and it is nonnegative and increasing in β for any θ. Suppose that β ∈ [0, 1], so that Ex is monotone. One can show that δ(µ∗Ex ) > 0 if and only if QEx > Q∗ (x) := (1 + x)/2
⇐⇒ β ∗ (x) :=
8x < β ≤ 1, (1 + 3x)(1 − x)
consistent with Proposition 7. A graphical illustration of this observation can be found in Figure 3. ΠS (µ∗ )−ΠS (µ∗one−shot ) Note that the optimal mechanism involves a positive expected delay iff > 0. ΠS (µ∗ ) one−shot
We now turn our attention to non-monotonic environments for which Proposition 7 cannot apply. For example, see the dashed part of the lowest curve (x = 0.1) in Figure 3. It shows that a delay eventually disappears with a sufficiently higher level of β, seemingly contradictory to Proposition 7. But this observation is actually not surprising, considering that β affects not only the dispersion of autarky payoffs, but also from trading (net-valuation). Note that the net-valuation the total net-gain β 2 u(θ) = θ − φx (θ) = θ − 2 (θ − x) + x decreases in β for any θ. This means that the seller’s potential gain from price discrimination also decreases in β. Accordingly, the gain from delaying the transaction also disappears when β is sufficiently large, even though the dispersion of autarky payoffs is large. To isolate the effect of a change in autarky payoffs from the effect of a change in net-valuations, we restrict attention to a subclass of E such that net-valuations are identically distributed. For any E = (F, φ) ∈ E let θE (q) := inf{θ ∈ Θ : q ≤ F (θ)} and uE (q) := θE (q) − φ(θE (q)) be a buyer type at quantile q ∈ [0, 1] of F and her net-valuation, respectively. Fix an environment 23
ΠS(µ* )- ΠS (µ*one-shot ) _____________________ ΠS (µ*one-shot ) 0.10
β=1.25
0.08 0.06
β=1.50
0.04 0.02
β=1.75 0
0
0.05
0.1
0.15
0.2
x
Figure 4: Graph for Example 4
E = (F, φ) ∈ E, and define ˜ ∈ E : u ˜ (q) = uE (q) ∀q ∈ [0, 1]} E∗E := {E E as the set of all environments with the same net-valuation at each quantile of the type distribution. Restricting attention to E∗E , we can obtain a partial generalization of Proposition 7. PROPOSITION 8. Fix an environment E ∈ E. There are cutoffs Q∗H and Q∗L such that ˜ ∈ E∗ E E
and
QE˜ > Q∗H
=⇒
δ(µ∗E˜ ) > 0
(32)
˜ ∈ E∗E E
and
QE˜ ≤ Q∗L
=⇒
δ(µ∗E˜ ) = 0.
(33)
Proof. See Appendix A.7. EXAMPLE 4. Consider again the environments discussed in Example 3. Fix β ∈ [0, 2] and x ≥ 0, and note that β θEx (q) = x + q, and uEx (q) = θEx (q) − φx (θEx (q)) = q − q 2 2 for each quantile q ∈ [0, 1]. uEx (q) is independent of x, and hence Ex ∈ E∗E0 for any x ≥ 0. Moreover, φx (θEx (q)) =
θEx (q)φ0x (θEx (q)) 2βq(q + x) = φx (θEx (q)) βq 2 + 2x
∀q ∈ [0, 1], x ≥ 0,
n o q and hende QEx = max 1 + x − x2 + 2x , 0 decreases in x. Also, there is a cutoff x∗ (β) > 0 such β that δ(µ∗Ex ) > 0 (or equivalently, ΠS (µ∗ ) > ΠS (µ∗one−shot )) if and only if x < x∗ (β).
24
6.2
Benefit From Delay
In this subsection we will compare µ∗ and µ∗one−shot to identify the exact benefit from the delay of transaction. The delay in µ∗ has two effects on welfare. First, it induces low buyer types to wait rather than break off the negotiation immediately. As a result, additional trading opportunities are created for low buyer types, and hence, more surplus is generated from the negotiation process. Second, as discussed at the end of Section 3, the seller generally can concede less information rent to the buyer, particularly when faced with high buyer types, by avoiding negotiation breakdown. The next proposition confirms this intuition formally. For any µ ∈ MIC define the informational rent for a buyer type θ and net surplus generated from a buyer type θ by "
R(θ; µ) := V (θ) − φ(θ) and S(θ; µ) := R(θ; µ) + E P0 (θ, Z ) + ∞
µ
Z
# −rs
e
∞
dPs (θ, Z )
s∈(0,∞)
respectively. Additionally, define the trading region T R(µ) by the set of types that trade with the seller in µ with positive probability. Then, T R(µ∗one-shot ) = U(pone-shot ) = [θ(pone-shot ), θ(pone-shot )] and T R(µ∗ ) = U(p∗ ) ∪ T(p∗ ) where p∗ is the price offered by the seller in µ∗ at t = 0. PROPOSITION 9.
(i) p∗ ≥ pone-shot .
(ii) 0 = S(θ; µ∗one−shot ) ≤ S(θ; µ∗ ) and 0 = R(θ; µ∗ ) ≤ R(θ; µ∗one−shot ) for almost every θ ∈ T R(µ∗ ) ∩ [θ, θ(pone-shot )]. (iii) R(θ; µ∗ ) ≤ R(θ; µ∗one−shot ) for almost every θ ∈ [θ(pone-shot ), θ] ∩ T R(µ∗ ). Proof. In Appendix A.8. p∗ ≥ pone-shot follows from the standard monotone comparative statics argument (Vives, 2000, Theorem 2.3), and it echoes the intuition discussed at the beginning of this subsection. That is, µ∗ can concede less information rent than µ∗one−shot , and thus, the seller can charge a higher price at t = 0. The other two observations also confirm the same intuition. µ∗ creates the additional trade opportunity with θ ∈ [θ, θ(pone−shot )] ∩ T R(µ∗ ) who were excluded from µ∗one-shot . This additional trading opportunity generates additional surplus, and moreover, the seller could exploit all the net surplus without any rent for these buyer types (R(θ; µ∗ ) = 0 for all these types). The seller also concedes less rent for types in [θ(pone−shot ), θ] ∩ T R(µ∗ ). All these types could trade at t = 0 in µ∗one−shot ; hence, no additional net surplus is generated from them. However, µ∗ can extract more rent from these types. One remaining question is whether the optimal mechanism µ∗ generates a higher social surplus compared to µ∗one−shot . µ∗ indeed has both surplus-enhancing and surplus-deteriorating features, compared to µ∗one−shot . For example, the market penetration is higher in µ∗ , which is generally beneficial for the social surplus. However, some buyer types delay their purchase in µ∗ , while they could trade im-
25
* S(𝜃;μone-shot )
* * S(𝜃;μone-shot ) - R(𝜃;μone-shot ) S(𝜃;μ* )
R(𝜃;μ*one-shot )
S(𝜃;μ*) - R(𝜃;μ* )
R(𝜃;μ* )
0 𝜃† =0
max TR(𝜃;μ* )
𝜃_ (pone-shot )
_
0
𝜃
𝜃 †=0
𝜃=1
𝜃_ (pone-shot )
max TR(𝜃;μ* )
_
𝜃
𝜃=1
Figure 5: Comparison between µ∗ and µ∗one−shot (Θ = [0, 1], F (θ) = θ, φ(θ) = θ2 )
mediately in µ∗one−shot , and such delays tend to deteriorate the social surplus. Which effect dominates has to be determined on a case-by-case basis.
7 7.1
Extensions and Discussions Time-varying Outside Option
All main results presented in the previous sections are robust to the case in which the value of the outside option stochastically evolves over time. Specifically, suppose that the buyer’s continuation payoff from breaking off the negotiation at t ≥ 0 is given by Zt w(θ) ≥ 0, where (Zt )t≥0 is now an arbitrary càdlàg stochastic process that satisfies the assumptions presented below. Let {FtZ }t≥0 and T ({FtZ }t≥0 ) denote the filtration generated by Z and the collection of {FtZ }t≥0 -stopping times, respectively. ASSUMPTION 4. Z0 is a constant random variable, and sup β∈T ({FtZ }t≥0 )
h
i
E e−rβ Zβ |Z t < ∞ almost surely for any t ≥ 0.
ASSUMPTION 5. There exists ¯ > 0 such that ( ) h i −rβ P sup E e Zβ < sup {Zt : t ∈ (0, )} > 0 ∀ ∈ (0, ¯). β∈T ({FtZ }t≥0 )
Note that the baseline model presented in Section 2 is the special case such that supp(Zt ) ⊂ {0, 1} for
26
any t ≥ 0. Define for any t ≥ 0, θ ∈ Θ, and Z t φt (θ; Z t ) :=
h
sup β∈T ({FtZ }t≥0 )
i
E e−r(β−t) Zβ |Z t w(θ)
and let φ(θ) = φ0 (θ; Z0 ) continue to denote the autarky payoff for each θ ∈ Θ. We may use all other notations we have developed so far without any change. Then, one can easily show that all main results in the previous sections remain true with the time-varying outside option. Most importantly, the following proposition, which is analogous to Proposition 4 and Corollary 2, holds. PROPOSITION 10. Suppose that the buyer’s continuation payoff from breaking off the negotiation at t ≥ 0 is Zt w(θ), and also suppose that Assumptions 1–5 hold. Then, whenever the Lebesgue measure of T(p∗ ) is positive, (i) δ(µ) > 0 and Φ(µ) > 0 for any optimal mechanism µ, and ¯ S by offering a menu of European call options. (ii) the seller can achieve the optimal profit level Π We can prove this proposition in essentially the identical way as we proved Proposition 4 and Corollary 2, and hence the proof is omitted here.
7.2
Interdependent Values
Thus far, we have assumed that the buyer’s private type θ does not directly affect the seller’s payoff. That is to say, we have assumed that values are private. However, this does not hold for some applications of interest. For example, the seller’s value of the good (true opportunity cost) may depend on information only available to the buyer in the real estate market. It is beyond the scope of this paper to study a fully general model with interdependent values. However, we can still show that a delay may occur in optimal mechanisms when values are interdependent. Suppose that the seller’s value of the good is c(θ) ≥ 0 which depends on the buyer’s private information θ ∈ Θ. The buyer’s payoff function remains the same as in the private value case. Then the seller’s objective is to find µ = (A, P ) ∈ MIC that maximizes Z ΠS (µ) = θ∈Θ
"
E P0 (θ, Z ∞ ) +
Z
−rτ µ (θ,Z ∞ )
e−rs dPs (θ, Z ∞ ) − e
s∈(0,∞)
# c(θ) θ dF (θ).
We assume that c(·) is differentiable, and we also impose the following assumptions.25 ASSUMPTION A. Θ = [θ, θ] ⊂ [0, ∞). u(θ) − c(θ) ≥ 0 for any θ ∈ [θ, θ]. ASSUMPTION B. Both
f (θ) 1−F (θ)
and (θ − c(θ))f (θ) + F (θ) are strictly increasing at θ ∈ (θ, θ).
ASSUMPTION C. w(θ) is convex, continuous, and piecewise twice-differentiable. Note that Assumptions A–C degenerate to Assumptions 1–3 if c(θ) = 0 for all θ ∈ Θ. To ensure the tractability of the model, we need two additional assumptions. 25 All results in this subsection remain valid even if, in addition to Assumptions A–E, we allow the value of the outside option to be time-varying in the sense of Section 7.1.
27
ASSUMPTION D. u(θ) − c(θ) is increasing at any θ ∈ Θ. ASSUMPTION E. θf 0 (θ) + f (θ) ≥ 0 for any θ ∈ Θ. Assumptions D and E are monotonicity assumptions which require that u(θ) − c(θ) (net-gain from trading) and θf (θ) increases in θ ∈ Θ, respectively. These two assumptions, together with Assumption C, allow us to avoid bunching. Finally, we generalize the definition (12) as follows: U(p) := {θ ∈ Θ : θ − φ(θ) ≥ p}, T(p) := θ ∈ Θ : (θ − c(θ))φ0 (θ− ) > φ(θ), φ0 (θ− ) < 1, and θ − φ(θ) < p .
(34) (35)
Similar to the private value case, it is optimal among all one-shot mechanisms to commit to a fixed price Z pone-shot ∈ arg max p∈U
U(p)
(p − c(θ)) dF (θ).
(36)
The proof is similar to the proof of Proposition 1 of Samuelson (1984), and hence it is omitted here. Additionally, we can define and solve the relaxed problem similarly to the private value case. LEMMA 3. Under Assumptions A–E, ¯S ≤ Π
"Z max p∈{θ−φ(θ):θ∈Θ}
U(p)
[p − c(θ)] dF (θ) +
Z T(p)
# (θ − c(θ))φ0 (θ− ) − φ(θ) dF (θ) .
(37)
Proof. See Appendix A.2 With interdependent values, the following direct mechanism achieves the upper bound (37). Let θ ∈ Θ denote the type that the buyer reports to the mechanism at t = 0. • If θ ∈ U(p∗∗ ), the two parties trade immediately in exchange for the buyer’s payment p∗∗ to the seller, where p∗∗ = arg max p≥0
"Z U(p)
p dF (θ) +
Z T(p)
# (θ − c(θ))φ0 (θ− ) − φ(θ) dF (θ)
similar to the private value case. If there are multiple arg maxima in the above optimization problem, choose the smallest arg maximum as p∗∗ . • If θ ∈ T(p∗∗ ), the buyer pays an upfront payment α(θ) = w(θ) − φ(θ) at t = 0. The two parties delay until t(θ) = p(θ) = θ −
φ(θ)+α(θ) φ0 (θ− )
1 r
log φ0 (θ1 − ) and then trade with net-transfer
from the buyer to the seller.
• Otherwise, no trade occurs, and the buyer breaks of the negotiation at t = 0.
28
Denote this mechanism by µ∗∗ . Note that µ∗∗ coincides with µ∗ (Section 4.3) if c(·) = 0 for all θ. µ∗∗ also features a positive duration of (expected) delay (whenever T(p∗∗ ) is nonempty) and upfront payment scheme (whenever w(θ) 6= φ(θ) for some buyer types). PROPOSITION 11. Suppose that values are interdependent, and Assumptions A–E hold. Then, µ∗∗ is optimal. Corollary 2 also holds with no modification. Some comparative statics results in Section 6 may not extend to the interdependent value case, as now the benefit from delay depends on not only φ(·) but also on c(·). However, Proposition 9 still remains valid if we now let µ∗one-shot denote the optimal one-shot mechanism that commits to (36), and redefine surplus generated from a buyer type θ by h i µ ∞ µ ∞ µ ∞ S(θ; µ) := E e−rτ (θ,Z ) (θ − c(θ)) + e−rσ (θ,Z ) φσµ (θ,Z ∞ ) (θ; Z σ (θ,Z ) ) − φ(θ) for any µ = (A, P ) ∈ MIC .
7.3
Negotiation Without Commitment
In this subsection, we will compare the optimal mechanism with the bargaining outcome when the seller cannot commit to any mechanism. Specifically, suppose that the seller posts a price pt at each t ∈ T (∆) := {(k − 1)∆ : k ∈ N} where ∆ > 0 is a small positive real number. The seller can commit to the price offered at t ∈ T (∆) until t + ∆. That is, ps = pt for any t ≤ s < t + ∆ where t ∈ T (∆). At any point in time t ≥ 0, the buyer can accept the standing offer pt , exercise her outside option, or delay. The negotiation ends when the buyer accepts a seller’s offer or exercises her outside option. This bargaining protocol induces an extensive-form game which we will simply call bargaining game without commitment. Note that the seller’s commitment power indeed vanishes as ∆ → 0. We will study perfect Bayesian equilibria (PBEs) for this game.26 A PBE is called essentially unique if all other equilibria lead to the same payoff profile. We assume that Θ is a compact subset of Rn for some n ∈ N, and c(·) can be any (measurable) function from Θ to R+ . For simplicity, we will restrict our attention to the case in which Zt = 1 for any t ≥ 0. With private values, this bargaining game has already been studied by Board and Pycia (2014). The following proposition is a slight generalization of their main result (Proposition 1) to interdependent values. The proof is almost identical to the proof of Proposition 1 in Board and Pycia (2014), and is omitted here. PROPOSITION 12. Suppose that w = inf{w(θ) : θ ∈ Θ} > 0 and Zt = 1 for any t ≥ 0. There exists a PBE in which all buyer types trade with the seller or exercise their outside options at t = 0. This PBE is essentially unique. This proposition shows that no delay occurs in the essentially unique equilibrium for the case without the seller’s having commitment. In equilibrium, the seller offers Z ? p ∈ arg max (p − c(θ)) dF (θ) p≥0
θ∈{θ:u(θ)≥p}
26
See Definition 8.2 in Fudenberg and Tirole (1991) for the definition of PBEs. Fudenberg and Tirole in fact define PBEs for finite games with incomplete information only, but their generalization to this setting is straightforward, and hence omitted here.
29
at any time t ∈ T (∆). A buyer of type θ accepts p? at t = 0 if and only if her u(θ) ≥ p? , and otherwise she immediately exercises the outside option. Note that the seller’s equilibrium expected profit is equal to the profit from the optimal one-shot mechanism. Hence, the equilibrium profit is strictly lower than he could earn with full commitment power. Why does the seller not reduce his price offer after his first price is rejected, despite that he understands that it could increase his expected profit? Roughly speaking, a declining price path can be justified in equilibrium only if the seller’s belief about the buyer’s net-valuation becomes pessimistic over time. However, all else being equal, buyer types with low net-valuation tend to exercise the outside option earlier than ones with high net-valuation, which makes the seller’s belief rather more (at least weakly) optimistic over time; hence, a declining price path is hardly credible in equilibrium. This observation shows that the role of commitment is reversed with type-dependent outside option. The bargaining literature has showed that, without outside options, the seller can benefit from committing to a single price, while he cannot help revising his offer over time without commitment.27 With the outside option, conversely, the seller can benefit by committing to revising his offers over time if he has full commitment power, but he has no choice but to stick to a single take-it-or-leave-it price if he has no commitment power. This observation also has an implication for bargaining strategies in real-world practices. While additional commitment power is always (at least weakly) beneficial, full commitment power is extremely costly to obtain in practice. As a result, practical bargaining strategies must be able to advise priorities in choosing which commitment devices to be invested. The observation in this section shows that an obsession with a “take-it-or-leave-it” strategy can be suboptimal in negotiations in which one’s opponent has outside options, contrary to the conventional wisdom.
8
Conclusion
This paper considered sales negotiations in which the buyer can opt out for a type-dependent outside option during the negotiation. The outside option has a stark effect on the outcome. Most notably, delays of transaction and upfront payment schemes arise in all optimal mechanisms due to the seller’s motive to prevent negotiation breakdown. Although this paper frames the interaction between the two parties as a sales negotiation, the model of ours can easily be translated to other situations to explain delays in business-to-business negotiations, upfront payment (down payment) in the consumer loan service, among others. Finally, we provide suggestions for further extensions of the model. The outside option was exogenously given in this paper, and its origin was not explicitly modeled. It would also be interesting for future research to endogenize the buyer’s outside option. Another possibility would be to enrich the model by incorporating the seller’s moves to reduce the value of the counterparty’s outside option or the buyer’s investment in her outside option before and/or during the negotiation. The majority 27
Gul, Sonnenschein, and Wilson (1986) and Deneckere and Liang (2006) demonstrated that, when values are private and interdependent respectively, the price declines over time and delay occurs when the seller lacks commitment power. On the other hand, Samuelson (1984) demonstrated previously that the seller-optimal mechanism always commits to a single price. The same observation was made by Sobel (1991) for the durable-good monopoly problem with a constant incoming flow of new buyers to the market.
30
of business discussions about bargaining center on such tactics; hence, incorporating strategic moves could improve our understanding of real-world sales problems.
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Hwang, I., and F. Li (2017): “Transparency of Outside Options in Bargaining,” Journal of Economic Theory, 167, 116–147. Jullien, B. (2000): “Participation Constraints in Adverse Selection Models,” Journal of Economic Theory, 47, 1–47. Krähmer, D., and R. Strausz (2015): “Optimal Sales Contracts with Withdrawal Rights,” Review of Economic Studies, 82, 762–790. Laffont, J.-J., and D. Martimort (2002): The Theory of Incentives: The Principal-Agent Model. Princeton University Press. Lewis, T. R., and D. E. M. Sappington (1989): “Countervailing Incentives in Agency Problems,” Journal of Economic Theory, 49(2), 294–313. Manelli, A. M., and D. R. Vincent (2006): “Bundling as an Optimal Selling Mechanism for a Multiple-good Monopolist,” Journal of Economic Theory, 127(1), 1–35. (2007): “Multidimensional Mechanism Design: Revenue Maximization and the Multiple-good Monopoly,” Journal of Economic Theory, 137(1), 153–185. Mangasarian, O. L. (1979): “Unlqueness of Solutlon In Linear Programming,” Linear Algebra and its Applications, 25, 151–162. Milgrom, P., and I. Segal (2002): “Envelope Theorems for Arbitrary Choice Sets,” Econometrica, 70(2), 583–601. Pavlov, G. (2011): “Optimal Mechanism for Selling Two Goods,” The B.E. Journal of Theoretical Economics, 11(1), Article 3. Pullan, M. C. (1995): “Forms of Optimal Solutions for Separated Continuous Linear Programs,” SIAM Journal on Control and Optimization, 33(6), 1952. Pycia, M. (2006): “Stochastic vs Deterministic Mechanisms in Multidimensional Screening,” mimeo, University of California, Los Angeles. Riley, J., and R. Zeckhauser (1983): “Optimal Selling Strategies: When To Haggle, When To Hold Firm,” Quarterly Journal of Economics, 98(2), 267–289. Rochet, J.-C., and L. A. Stole (2002): “Nonlinear Pricing with Random Participation,” The Review of Economic Studies, 69(1), 277–311. Rochet, J.-c., and J. Thanassoulis (2016): “Stochastic Bundling and Multiproduct Intertemporal Price Discrimination,” mimeo. Samuelson, W. (1984): “Bargaining Under Asymmetric Information,” Econometrica, 52(4), 995– 1005. Skreta, V. (2006): “Sequentially Optimal Mechanisms,” Review of Economic Studies, 73(4), 1085– 1111. Sobel, J. (1991): “Durable Goods Monopoly with Entry of New Consumers,” Econometrica, 59(5), 1455–1485. Stokey, N. L. (1979): “Intertemporal Price Discrimination,” Quarterly Journal of Economics, 93(3), 355–371. Thanassoulis, J. (2004): “Haggling over Substitutes,” Journal of Economic Theory, 117(2), 217–245. Vincent, D. R. (1989): “Bargaining with Common Values,” Journal of Economic Theory, 48(1), 47–62. Vives, X. (2000): “Oligopoly Pricing: Old Ideas and New Tools,” MIT Press. 32
Appendix A A.1
Proofs
Proof of Lemma 1
Note that we may focus on incentive-compatible mechanisms such that
P{Zσµ (θ,Z ∞ ) = 1} = 1 ∀θ ∈ Θ.
(A.1)
Otherwise, without affecting the incentive-compatibility or the seller’s profit, the mechanism always can delay its recommendation that the two parties break-off the negotiation until the outside option becomes available. For all such mechanisms, h i h i µ ∞ µ ∞ µ ∞ E e−rσ (θ,Z ) φσµ (θ,Z ∞ ) (θ; Z σ (θ,Z ) )|θ = E e−rσ (θ,Z ) |θ w(θ) and hence,
( µ
y (θ) =
E e−rσ
µ (θ,Z ∞ )
|θ /E e−rγ0 (Z0 ) if w(θ) > 0 0 otherwise
where γ0 (Z0 ) is the time that the outside option is available for the first time; see (1). Note that V µ (θ) = θxµ (θ) + φ(θ)y µ (θ) − pµ (θ). by the definitions of xµ (θ), y µ (θ), and pµ (θ). Next, consider the case in which a buyer of type θ deviates ˜ such that (i) ν˜0 = θ˜ and νt > 0 is truthful from the truthful-obedient strategy and instead plays (˜ ν, ψ), for any t > 0; and (ii) ψ˜ is obedient. The expected payoff from the deviation is i h ˜ ∞) ˜ ∞) σ µ (θ,Z ˜ := xµ (θ, ˜ Z ∞ )θ + E e−rσµ (θ,Z ˜ V µ (θ|θ) φσµ (θ,Z )|θ − pµ (θ) (A.2) ˜ ∞ ) (θ; Z which is never larger than V µ (θ) by the incentive-compatibility of µ. However, due to (A.1), it is optimal for the buyer type θ to exercise her outside option immediately after separation: i h i h µ ˜ ∞ ˜ ∞) ˜ ∞) σ µ (θ,Z −rσ µ (θ,Z w(θ). E e−rσ (θ,Z ) φσµ (θ,Z (θ; Z (θ))|θ = E e ˜ ∞) Consequently, h i ˜ ∞) ˜ = xµ (θ)θ ˜ + E e−rσµ (θ,Z ˜ = xµ (θ)θ ˜ + y µ (θ)φ(θ) ˜ ˜ V µ (θ) ≥ V µ (θ|θ) w(θ) − pµ (θ) − pµ (θ) for any θ, θ˜ ∈ Θ. Finally, the individual rationality also requires that the buyer’s expected payoff from µ to never be strictly lower than her autarky payoff: V µ (θ) = xµ (θ)θ + y µ (θ)φ(θ) − pµ (θ) ≥ φ(θ) ∀θ ∈ Θ.
A.2
Proof of Lemma 2 and Lemma 3
We prove Lemmas 2 and 3 together in this subsection. Values are said to be private if c(θ) = 0 for all θ; otherwise, values are interdependent.
33
A.2.1
Preliminary Observations
First, define ( θˆ :=
if 1 − c0 (θ) − φ0 (θ− ) ≥ 0 for all θ < θ otherwise
θ inf{θ ∈ Θ : 1 −
c0 (θ)
φ0 (θ− )
−
< 0}
and ρ(θ; x, y) := x − φ0 (θ− )(1 − y) ˆ ˆ (θ) (θ) 0 (θ − ) F (θ)−F θ − c(θ) − F (θ)−F x − φ(θ) − φ (1 − y) if θ < θˆ f (θ) f (θ) χ(θ; x, y) := (θ) (θ) x − φ(θ) − φ0 (θ− ) 1−F (1 − y) otherwise θ − c(θ) − 1−F f (θ) f (θ)
(A.3) (A.4)
for any θ ∈ Θ and x, y ∈ [0, 1]. Consider any (x(θ), y(θ), p(θ)) that satisfies all the constraints of (R), and πB (θ) := θx(θ) + φ(θ)y(θ) − p(θ) ∀θ ∈ [θ, θ]. We will often use the following notation: π B = πB (θ),
π B = πB (θ),
ˆ and π ˆB = πB (θ).
By the envelope theorem, Z θˆ ˆ ˆ πB (θ) − φ(θ) − ρ(s; x(s), y(s))ds if θ ≤ θˆ πB (θ) − φ(θ) = Zθ θ ˆ ˆ πB (θ) − φ(θ) + ρ(s; x(s), y(s))ds otherwise.
(A.5)
θˆ
Integrating by parts, Z
θˆ
(p(θ) − c(θ)x(θ)) dF (θ) =
θ
Z
θ
θˆ
Z
θˆ
ˆ χ(θ; x(θ), y(θ))dF (θ) + F (θ)(φ(θ) − πB (θ)),
θ
(p(θ) − c(θ)x(θ)) dF (θ) =
Z
θ
θˆ
ˆ ˆ − πB (θ)). ˆ χ(θ; x(θ), y(θ))dF (θ) + (1 − F (θ))(φ( θ)
For now, suppose that θˆ = θ. Define for any (x, y) : Θ → [0, 1]2 and π B ≥ 0 θˆ
Z b Π(x, y, π B ) :=
(p(θ) − c(θ)x(θ)) dF (θ)
θ
Z
θ
=
ˆ ˆ − πB (θ)) ˆ + F (θ)(φ(θ) ˆ χ(θ; x(θ), y(θ))dF (θ) + (1 − F (θ))(φ( θ) − πB (θ))
θ
Z
θ
χ(θ; x, y)dF (θ) + φ(θ) − π B .
= θ
34
(A.6)
Additionally, for any π B ∈ U := {θ − φ(θ) : θ ∈ Θ} Π0 (π B ) :=
max
(x(·),y(·),π B )∈[0,1]Θ ×[0,1]Θ ×U
(R0∗ )
b Π(x, y, π B )
subject to Z π B − φ(θ) = π B − φ(θ) −
θ
(A.7)
ρ(s; x(s), y(s))ds θ
Z
θ
π B − φ(θ) −
ˆ θ] ρ(s; x(s), y(s))ds ≥ 0 ∀θ ∈ [θ,
(A.8)
θ
(A.9)
0 ≤ y(θ) ≤ 1 − x(θ) ≤ 1 ∀θ ∈ [θ, θ],
where constraints (A.7) and (A.8) come from (A.5). Then, the maximized objective function ΠR of (R) is bounded from above by maxπB ≥φ(θ) Π0 (π B ). Note that (A.7) and (A.8) jointly imply the following inequality: Z θ π B − φ(θ) + ρ(s; x(s), y(s))ds ≥ 0 ∀θ ∈ Θ. (A.10) θ
LEMMA A.1. Suppose that θˆ = θ. Moreover, suppose that Assumptions 1–3 hold if values are private, or suppose that Assumptions A–E hold if values are interdependent. Then, there is a solution for (R0∗ ) such that (x∗ (θ), y ∗ (θ)) ∈ {(1, 0), (φ0 (θ), 0), (0, 0)} if (θ − c(θ))φ0 (θ− ) ≥ φ(θ) (x∗ (θ), y ∗ (θ))
∈ {(1, 0), (0, 1), (0, 0)}
(A.11)
otherwise.
for almost every θ ∈ Θ. Proof. Problem (R0∗ ) is a special case of separated continuous linear programs (SCLPs) whose theoretical properties have been studied by Anderson, Nash, and Perold (1983) and Pullan (1995), among others. The existence of a solution is guaranteed by Theorem 1 in Anderson, Nash, and Perold (1983). Let (x∗ , y ∗ , π ∗B ) be a solution for (R0∗ ). (x∗ (θ), y ∗ (θ)) necessarily solves the following auxiliary optimization problem: max χ(θ; x, y) s.t. 0 ≤ y ≤ 1 − x ≤ 1 and x − φ0 (θ− )(1 − y) = 0 x,y
(A.12)
for almost every θ ∈ Θ for which (A.8) binds. This problem admits a solution such that ( (φ0 (θ), 0) if (θ − c(θ))φ0 (θ− ) ≥ φ(θ) (x(θ), y(θ)) = (0, 1) otherwise. On the other hand, applying Theorems 1 and 2 in Anderson, Nash, and Perold (1983), there is a solution such that (x∗ (θ), y ∗ (θ)) ∈ {(0, 0), (1, 0), (0, 1)} for any θ ∈ Θ for which (A.8) does not bind. The following lemma summarizes some properties of (R0∗ ) that will be frequently used later.
35
LEMMA A.2. Suppose that θˆ = θ. Also, suppose that Assumptions 1–3 hold if values are private, or suppose that Assumptions A–E hold if values are interdependent. Then, there is a solution (x∗ , y ∗ , π ∗B ) of (R0∗ ) such that (i) (x∗ (θ), y ∗ (θ)) ∈ {(1, 0), (0, 1)} for almost every θ such that φ0 (θ− ) ≤ 0; Z (ii) If π B − φ(θ) +
θ˜
ρ(s; x(s), y(s))ds = 0, (x∗ (θ), y ∗ (θ)) ∈ {(1, 0), (0, 1), (φ0 (θ− ), 0)} for almost
θ
˜ every θ ≥ θ; (iii) π ∗B = φ(θ). Proof. Suppose that for a contradiction to (i), for any solution (x∗ , y ∗ , π ∗B ) of (R0∗ ) such that (A.13) holds, there is a closed set L0 ⊂ θ ∈ Θ : (x∗ (θ), y ∗ (θ)) = (0, 0) and φ0 (θ− ) ≤ 0 of a positive measure. For ease of exposition we consider the case in which L0 is a closed interval in the form of [a, b], although all the arguments below work for any closed set. By the intermediate value theorem, there must be h ∈ L0 such that Z a
h
1 (−φ (θ ))dθ = b − h+ 1 − F (h) 0
−
Z
b
c(θ)f (θ)dθ h
Consider (xI , y I , π IB ) such that π IB = π ∗B and (1, 0) if θ ∈ [h, b] I I (x (θ), y (θ)) := (0, 1) if θ ∈ [a, h) (x∗ (θ), y ∗ (θ)) otherwise which satisfy all the constraints of (R0∗ ). However, Z hh Z bh i i I I I ∗ ∗ ∗ b b χ(θ; 1, 0) − χ(θ; 0, 0) f (θ)dθ + χ(θ; 0, 1) − χ(θ; 0, 0) f (θ)dθ Π(x , y , π B )−Π(x , y , π B ) = h a Z bh Z hh i i = (θ − c(θ))f (θ) + F (θ)−1 dθ + φ(θ)f (θ) + φ0 (θ− )(F (θ)−1) dθ h a Z b = bF (b) − hF (h) + φ(h)F (h) − φ(a)F (a)−(b − h) − φ(h) + φ(a) − c(θ)f (θ)dθ h Z b > (b − h)(F (h) − 1) + (φ(h) − φ(a))(F (h) − 1)− c(θ)f (θ)dθ h " # Rb Z h 0 − h c(θ)f (θ)dθ = (F (h) − 1) b − h + φ (θ )dθ+ θ =0 1 − F (h) a which is impossible because (x∗ , y ∗ , π ∗B ) solves (R0∗ ) by the hypothesis. Next, suppose for a contradiction to (ii) that there is a closed set n o H0 ⊂ θ ≥ θ˜ : (x∗ (θ), y ∗ (θ)) = (0, 0) and φ0 (θ− ) > 0 36
R θ˜ of a positive measure, where π B − φ(θ) + θ ρ(s; x(s), y(s))ds = 0. Again, for ease of exposition, we consider the case that H0 is a closed interval in the form of [a, b], although all the arguments below work for any closed set. We first claim that there is another closed set A := [h, d] ⊂ Θ such that θ˜ ≤ h < d < a and (x∗ (θ), y ∗ (θ)) = (1, 0) for any θ ∈ A. Otherwise, because φ(θ; x(θ), y(θ)) = 0 for any θ ∈ Θ such that (x∗ (θ), y ∗ (θ)) 6∈ {(1, 0), (0, 0)}, b
Z
b
Z ρ(s; x(s), y(s))ds ≤
π B − φ(θ) +
Z ρ(s; 0, 0)ds =
θ
b
(−φ0 (θ− ))ds < 0
a
a
which contradicts (A.10). Additionally, without loss, we may assume that `(H0 ) = b−a = `(A) = d−h. Now (˜ x, y˜) such that ( (x∗ (θ), y ∗ (θ)) if θ 6∈ A ∪ H0 (˜ x(θ), y˜(θ)) = (1/2, 0) if θ ∈ A ∪ H0 . (˜ x, y˜, φ(θ)) satisfies (A.7)–(A.9). Moreover, b x, y˜, φ(θ)) − Π(x b ∗, y∗, π∗ ) Π(˜ B Z Z 1 − F (θ) 1 1 − F (θ) 1 θ − c(θ) − dF (θ) + θ − c(θ) − dF (θ) =− 2 θ∈A f (θ) 2 θ∈H0 f (θ) Z Z 1 1 =− ((θ − c(θ))f (θ) − 1 + F (θ)) dθ + ((θ − c(θ))f (θ) − 1 + F (θ)) dθ 2 θ∈A 2 θ∈H0 Z 1 1 1 − F (θ) ≥ − + θ − c(θ) − dF (θ) = 0 2 2 f (θ) θ∈H0 which contradicts the hypothesis. Here, the final inequality is due to Assumption 2. Finally we show π ∗B = φ(θ). Suppose not. Then, ( θ > θII := sup θ ∈ Θ : πB (θ) − φ(θ) −
Z
θ
Z ρ(θ; 1, 0)dθ −
θ
Now define
( II
II
(x (θ), y (θ)) :=
θ
) ρ(θ; x∗ (θ), y ∗ (θ))ds = 0 .
θ
(1, 0) if θ ≥ θII (x∗ (θ), y ∗ (θ)) otherwise.
II πB := φ(θ) + π B − φ(θ) −
Z
θ
ρ(s; xII (s), y II (s))ds.
θ II ) satisfies all the constraints of (R∗ ) and π II = φ(θ). We claim that Then, (xII , y II , πB 0 B
b II , y II , π II ) − Π(x b ∗, y∗, π∗ ) = Π(x B B
Z
θ
θII θ
Z =
θII
∗ χ(θ; xII (θ), y II (θ)) − χ(θ; x∗ (θ), y ∗ (θ)) dF (θ) − (π II B − πB ) χ(θ; xII (θ), y II (θ)) − χ(θ; x∗ (θ), y ∗ (θ)) dF (θ) Z
θ
− θII
II [ρ(θ; x∗ (θ), y ∗ (θ), π ∗B ) − ρ(θ; xII (θ), y II (θ), πB )]dθ
37
is positive, which contradicts our hypothesis. Note that, as (θ − c(θ))f (θ) + F (θ) increases in θ (Assumptions 2 and B), [χ(θ; 1, 0) − χ(θ; 0, 0)]f (θ) = (θ − c(θ))f (θ) − 1 + F (θ) ≥ −1 = ρ(θ; 0, 0) − ρ(θ; 1, 0) for any θ ∈ Θ with the inequality holding strictly for a positive measure of θ. Similarly, for any θ [χ(θ; 1, 0) − χ(θ; φ0 (θ− ), 0)]f (θ) =(1 − φ0 (θ))((θ − c(θ))f (θ) − 1 + F (θ)) ≥ρ(θ; φ0 (θ− ), 0) − ρ(θ; 1, 0) = − (1 − φ0 (θ− )) and [χ(θ; 1, 0) − χ(θ; 0, 1)]f (θ) =(θ − c(θ) − φ(θ))f (θ) − (1 − φ0 (θ− ))(1 − F (θ)) ≥ρ(θ; 0, 1) − ρ(θ; 1, 0) = − (1 − φ0 (θ− )). b II , y II , π II ) − Π(x b ∗ , y ∗ , π ∗ ) is positive. Hence, Π(x B B A.2.2
Proof for Lemma 3
We first restate Lemma 3 using the notation developed in the appendix. LEMMA A.3. Suppose that values are interdependent, π B ∈ U, (x∗ , y ∗ , π ∗B ) such that π ∗B = φ(θ) and (1, 0) ∗ ∗ 0 (x (θ), y (θ)) = (φ (θ− ), 0) (0, 1)
and that Assumptions A–E hold. For any
if θ ∈ U(θ − π B ) if θ ∈ T(θ − π B ) otherwise
solves (R0∗ ). Proof. First of all, note that θˆ = θ by Assumption D. Hence, Lemmas A.1 and A.2 hold, which will be frequently used throughout the proof. Step I: Choose an arbitrary solution (x∗ , y ∗ , π ∗B ) for (R0∗ ). By Lemma A.2, we may assume that (x∗ (θ), y ∗ (θ)) ∈ {(1, 0), (φ0 (θ− ), 0)} if (θ − c(θ))φ0 (θ− ) ≥ φ(θ) . (x∗ (θ), y ∗ (θ)) ∈ {(1, 0), (0, 1)} otherwise. Consider the following constrained optimization problem: Z Z max e(θ)dθ subject to I∈C(Θ) θ∈I
d(θ)dθ = 0
θ∈I
where C(Θ) is the set of all closed subsets of Θ = [θ, θ], and ( (χ(θ; 1, 0) − χ(θ; φ0 (θ− ), 0))f (θ) if (θ − c(θ))φ0 (θ− ) ≥ φ(θ) e(θ) = (χ(θ; 1, 0) − χ(θ; 0, 1))f (θ) if (θ − c(θ))φ0 (θ− ) < φ(θ), 38
(A.13)
(R1∗ )
and
( d(θ) =
ρ(θ; 1, 0) − ρ(θ; φ0 (θ− ), 0) if (θ − c(θ))φ0 (θ− ) ≥ φ(θ) ρ(θ; 1, 0) − ρ(θ; 0, 1) if (θ − c(θ))φ0 (θ− ) < φ(θ).
Let I ∗ be a solution to (R1∗ ). Then, by Lemma A.2, (1, 0) if θ ∈ I ∗ ∗ ∗ 0 − ∗ (x (θ), y (θ)) = (φ (θ ), 0) if θ 6∈ I and (θ − c(θ))φ0 (θ− ) ≥ φ(θ) (0, 1) if θ 6∈ I ∗ and (θ − c(θ))φ0 (θ− ) < φ(θ) solves (R0∗ ). First, any θ such that φ0 (θ− ) ≥ 1 has to be in I ∗ , because d(θ) = 0 and e(θ) ≥ 0. Step II: e(θ)/d(θ) is increasing at any θ such that φ0 (θ− ) < 1. To illustrate, note that, over any interval such that φ0 (θ− ) < 1 and (θ − c(θ))φ0 (θ− ) ≥ φ(θ), e(θ)/d(θ) = (θ − c(θ))f (θ) + F (θ) − 1 is increasing by Assumption B. Now consider θ such that (θ − c(θ))φ0 (θ− ) < φ(θ). Over any interval that f (θ) is e(θ) increasing, d(θ) = θ−c(θ)−φ(θ) f (θ) − (1 − F (θ)) is increasing in θ. Now, consider an interval over which 1−φ0 (θ− ) f (θ) is decreasing. Whenever both φ0 (θ) and φ00 (θ) exist (which is the case except for finite points), " # d e(θ) 1 (θ − c(θ) − φ(θ))(1 − φ0 (θ))f 0 (θ) + 2(1 − φ0 (θ))2 f (θ) = dθ d(θ) (1 − φ0 (θ))2 −c0 (θ)(1 − φ0 (θ))f (θ) + φ00 (θ)(θ − c(θ) − φ(θ))f (θ) ≥ (θ − c(θ))f 0 (θ) + 2f (θ) −
c0 (θ) φ00 (θ)(θ − c(θ) − φ(θ))f (θ) f (θ) + 1 − φ0 (θ) (1 − φ0 (θ))2 φ00 (θ)(θ−c(θ)−φ(θ))f (θ) is nonnegative, (1−φ0 (θ))2 ≥ 0 whenever f 0 (θ) ≤ 0. If c0 (θ) ≤ 0,
where the inequality is due to (θ − c(θ))φ0 (θ) < φ(θ). Note that 0
c (θ) and hence, it suffices to show (θ − c(θ))f 0 (θ) + 2f (θ) − 1−φ 0 (θ) f (θ)
(θ − c(θ))f 0 (θ) + 2f (θ) −
c0 (θ) f (θ) ≥ (θ − c(θ))f 0 (θ) + 2f (θ) − c0 (θ)f (θ) 1 − φ0 (θ) d = (θ − c(θ))f (θ) + F (θ) ≥ 0. dθ
On the other hand, if c0 (θ) > 0, (θ − c(θ))f 0 (θ) + 2f (θ) −
c0 (θ) d 0 f (θ) ≥ θf (θ) + 2f (θ) − f (θ) = θf (θ) ≥ 0. 1 − φ0 (θ) dθ
(A.14)
Finally, suppose there is θ? at which the sign of φ0 (θ− )(θ − c(θ)) − φ(θ) is reversed at θ = θ? . First, suppose there is > 0 such that φ0 (θ− )(θ−c(θ)) < φ(θ) for all θ ∈ (θ? −, θ? ], and φ0 (θ− )(θ−c(θ)) ≥ φ(θ) for all θ ∈ (θ? , θ? + ). For any θ ∈ (θ? , θ? + ) and θ˜ ∈ (θ? − , θ? ], ˜ ˜ − φ(θ) ˜ e(θ) e(θ) θ˜ − c(θ) ˜ − F (θ) ˜ − = (θ − c(θ))f (θ) + F (θ) − f (θ) ˜ d(θ) d(θ) 1 − φ0 (θ˜− ) ˜ ˜ ˜ ˜ (θ) ˜ + F (θ) ˜ − θ − c(θ) − φ(θ) f (θ) ˜ − F (θ) ˜ ≥ (θ˜ − c(θ))f 1 − φ0 (θ˜− ) h i ˜ φ(θ) ˜ − (θ˜ − c(θ))φ ˜ 0 (θ˜− ) /(1 − φ0 (θ˜− )) > 0, = f (θ) and hence e(θ)/d(θ) is increasing over (θ? − , θ? + ). Finally, suppose that there is > 0 such that 39
φ0 (θ− )(θ − c(θ)) > φ(θ) for all θ ∈ (θ? − , θ? ], and φ0 (θ− )(θ − c(θ)) ≤ φ(θ) for all θ ∈ (θ? , θ? + ). Due to the convexity φ0 (θ− )(θ − c(θ)) − φ(θ) must be continuous at θ = θ? .28 Then, the above argument shows that e(θ)/d(θ) is increasing around θ? . e(θ) Step III: Finally, the monotonicity of d(θ) implies that there is a solution to (R1∗ ) in the form of [θ0 , θ] 0 ∗ for some θ ∈ Θ. The constraint of (R1 ) should be binding; hence,
Z
θ
(1 − φ0 (θ− ))dθ = 0
=⇒
θ0 − φ(θ0 ) = θ − φ(θ)
θ0
which completes the proof. The only remaining question is how to choose π ¯B optimally. It is evident that the objective function of (R0∗ ) is decreasing in π B , which is also decreasing in π B ; see constraint (A.7). Hence, it is clearly optimal to choose π B = φ(θ), the minimum value that satisfies constraints of (R0∗ ). In turn, this implies that "Z # Z ∗ 0 − ΠR ≤ max Π0 (π B ) = (p − c(θ)) dF (θ) + (θ − c(θ))φ (θ ) − φ(θ) d F (θ) U(p∗ )
π B ≥φ(θ)
T(p∗ )
where p∗ = max{θ1 , θ2 } − φ(max{θ1 , θ2 }) θ1 := inf{θ ∈ Θ : (θ − c(θ))f (θ) + F (θ) − 1 ≥ 0} 0 − 1 − F (θ) θ2 := inf θ ∈ Θ : θ − c(θ) − φ(θ) ≥ (1 − φ (θ )) f (θ) One can easily show that this upper bound for ΠR is exactly achieved with (1, 0) if θ ∈ U(p∗ ) (x(θ), y(θ)) = (φ0 (θ− ), 0) if θ ∈ T(p∗ ) (0, 1) otherwise and
Z p(θ) = x(θ)θ + y(θ)φ(θ) − φ(θ) −
θ
ρ(s; x(s), y(s))ds, θ
which completes the proof. A.2.3
Proof for Lemma 2
It is straightforward to see that the proof for Lemma A.3 also proves Lemma 2 for the special case such that θˆ = θ.29 Hence, suppose that θˆ < θ without loss. One can extend the optimization problem (R0∗ ) to obtain an upper bound for the maximized objective function ΠR of (R). However, that upper bound is not tight. To obtain the right upper bound, we need to impose additional constraints. Suppose that φ0 (θ− )(θ − c(θ)) − φ(θ) must jump up at θ = θ? if it is discontinuous at the point, which is impossible given our choice of θ? . 29 When values are private and θˆ = θ, Assumptions A–D and Assumptions 1–3 are equivalent with each other. In the proof of Lemma A.3, we make use of Assumption E only when we prove (A.14), which is redundant if values are private. 28
40
(x(θ), y(θ), p(θ))θ∈Θ is a solution for (R). Again, define πB (θ) := x(θ)θ + y(θ)φ(θ) − p(θ). This solution must satisfy conditions (13) and (14). Additionally, from (A.5), ˆ − φ(θ) ˆ + πB (θ) − φ(θ) = πB (θ)
Z
θ
ρ(s; x(s), y(s))ds θˆ
(A.15)
ˆ Now, suppose that y(θ0 ) = 1 for some θ0 ≥ θ. ˆ (13) requires for any θ ≥ θ. πB (θ) = x(θ)θ + y(θ)φ(θ) − p(θ) ≥ φ(θ) − p(θ0 ) = φ(θ) + πB (θ0 ) − φ(θ0 ) for any θ0 ≥ θ; hence, (A.15) must be increasing at any θ ≥ θ0 . However, for almost every θ ≥ θ0 , ρ(θ0 ; x, y) = x − φ0 (θ0 )(1 − y) is strictly smaller than 0 unless y = 1; hence, y(θ0 ) = 1 for some θ0 ≥ θˆ
=⇒
y(θ) = 1 ∀θ ≥ θ0 .
(A.16)
Motivated by this observation, define ˜ := e Π(x, y, π B , π ˆB , θ)
Z
θ
ˆ ˆ −π ˆ θ) ˆB ), χ(θ; x(θ), y(θ))dF (θ) + F (θ)(φ(θ) − π B ) + (1 − F (θ))(φ(
θ
and consider the following optimization problem for any > 0: Π∗ () :=
max
Θ )2 ×Θ3 ˜ (x,y,π B ,ˆ πB ,θ)∈([0,1]
ˆ θ) ˜ e Π(x, y, πB (θ), πB (θ),
(R2∗ )
subject to ˆ − π B − φ(θ) = π ˆB − φ(θ)
Z
θˆ
ρ(s; x(s), y(s))ds
(A.17)
θ
ˆ − π ˆB − φ(θ)
Z
θˆ
ˆ ρ(s; x(s), y(s))ds ≥ 0 ∀θ ∈ [θ, θ)
(A.18)
ˆ θ] ρ(s; x(s), y(s))ds ≥ 0 ∀θ ∈ [θ,
(A.19)
θ
ˆ + π ˆB − φ(θ)
Z
θ
θˆ
0 ≤ y(θ) ≤ 1 − x(θ) ≤ 1 ∀θ ∈ [θ, θ] ˆ θ] ˜ y(θ) ≤ 1 − /θ ∀θ ∈ [θ, ∀θ > θ˜
y(θ) = 1 θ˜ ≥ θˆ
(A.20) (A.21) (A.22) (A.23)
where the last three constraints are motivated by the observation (A.16). Then, ΠR ≤ lim sup→0 Π∗ (). LEMMA A.4. Suppose that values are private, and that Assumptions 1–3 hold. Then, ! Z Z Π∗ () = max p dF (θ) + θφ0 (θ− ) − φ(θ) dF (θ) . p∈{θ−φ(θ):θ∈Θ}
θ∈U(p)
θ∈T(p)
for any sufficiently small > 0. 41
∗ ,θ ˜∗ ) is a solution to (R∗ ). Proof. Fix a small positive number > 0 and suppose that (x∗ , y ∗ , π ∗B , π ˆB 2 Define ∗ ˆ : θ − φ(θ) ≥ p? }. p? = θˆ − π ˆB and θ? := inf{θ ∈ [θ, θ]
Note that the choice of π B and (x(θ), y(θ)) for θ ≤ θˆ affects constraints (A.17)-(A.23) for θ > θˆ only through π ˆB . Hence, by Lemma A.3, we may assume without loss that (1, 0) if θ ≤ θˆ and θ ∈ U(p? ) ∗ ∗ 0 − (x (θ), y (θ)) = (φ (θ ), 0) if θ ≤ θˆ and θ ∈ T(p? ) ˆ (0, 1) for any other θ ≤ θ. Now, we argue that the lemma immediately follows if ˆ θ˜∗ ). x∗ (θ) + y ∗ (θ) = 1 ∀θ ∈ (θ, (A.24) i h (θ) is Suppose that (A.24) is the case, and note that χ(θ; x, 1 − x) = x (θ − φ(θ)) − (1 − φ0 (θ− )) 1−F f (θ) ˆ Hence, weakly increasing in x for any θ ≥ θ. ˆ θ˜∗ ] and, x∗ (θ) = 1 for any θ ∈ [θ, θ˜∗ = θ or θ˜∗ − p? = φ(θ˜∗ ) This shows that the equation in the statement of the lemma is true. ˆ θ˜∗ ) : x∗ (θ) + y ∗ (θ) < 1} is of positive Suppose, for contradiction of (A.24), that E := {θ ∈ (θ, measure. We may assume without loss that θf (θ) − 1 + F (θ) < 0 for any θ ∈ E.30 Additionally, as ˆ so that ρ(θ; x, y) < 0 for any θ > θˆ whenever x + y < 1, p? must be strictly smaller than θˆ − φ(θ), ∗ > φ(θ). ˆ Now take a small positive number η > 0 such that 0 < η < `(E), and define π ˆB ( p‡ := inf
p ≥ p? :
)
Z ρ(θ; 1, 0)dθ ≤ η ˆ ? ≤θ−φ(θ)≤p} {θ∈[θ,θ]:p
( θ‡ := inf
θ ≥ θˆ :
Z
h i ρ(θ; 1 − y ∗ (θ), y ∗ (θ)) − ρ(θ; x∗ (θ), y ∗ (θ)) dθ ≤ η
) .
θ∈E∩[θ,θ]
Consider (1 − y ∗ (θ), y ∗ (θ)) if θ ∈ E ∩ [θ‡ , θ] ˆ and θφ0 (θ− ) ≥ φ(θ) (φ0 (θ− ), 0) if θ − φ(θ) ∈ [p? , p‡ ], θ ≤ θ, (x‡ (θ), y ‡ (θ)) = ˆ and θφ0 (θ− ) < φ(θ) (0, 1) if θ − φ(θ) ∈ [p? , p‡ ], θ ≤ θ, ∗ ∗ (x (θ), y (θ)) otherwise. 30
For any θ ≥ θˆ such that θf (θ) − 1 + F (θ) ≥ 0 h i χ(θ; 1 − y ∗ (θ), y ∗ (θ))f (θ) − χ(θ; x∗ (θ), y ∗ (θ))f (θ) = θf (θ) − 1 + F (θ) (1 − y ∗ (θ) − x∗ (θ)) ≥ 0 ρ(θ; 1 − y ∗ (θ), y ∗ (θ))f (θ) − ρ(θ; x∗ (θ), y ∗ (θ))f (θ) = (1 − y ∗ (θ) − x∗ (θ)) ≥ 0.
Hence, we can weakly increase the objective function without interfering with the constraints by increasing x(θ) for those buyer types up to 1 − y ∗ (θ).
42
and
‡ ∗ ∗ π ˆB =π ˆB − (p‡ − p∗ ) = π ˆB − η,
π ‡B = φ(θ),
θ‡ = θ˜∗ .
‡ ˜‡ ˆB , θ ) satisfies all constraints for (R2∗ ). Moreover, note that Then (x‡ , y ‡ , π ‡B , π
Z h i ‡ ‡ ∗ ∗ ρ(θ; x (θ), y (θ)) − ρ(θ; x (θ), y (θ)) dθ =
Z η=
(1 − y ∗ (θ) − x∗ (θ))dθ
θ∈E∩[θ‡ ,θ]
θ∈E∩[θ‡ ,θ]
and Z η= ρ(θ; x∗ (θ), y ∗ (θ)) − ρ(θ; x‡ (θ), y ‡ (θ)) dθ ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p Z (1 − φ0 (θ− ))dθ. = ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p
∗ ˜∗ e := Π(x e ∗, y∗, π∗ , π e ‡ ‡ ‡ ˆ ‡ , θ˜‡ ) ≥ 0. First, we may assume Now, we claim that ∆Π B ˆB , θ ) − Π(x , y , π B , π B that (by taking a sufficiently small η) either
θ? φ0 (θ? ) ≥ φ(θ? ) for almost any θ
such that θ − φ(θ) ∈ [p? , p‡ ] or
(A.25)
θ? φ0 (θ? ) ≤ φ(θ? ) for almost any θ
such that θ − φ(θ) ∈ [p? , p‡ ]
(A.26)
Suppose that (A.25). Then, Z ˆ + e = (1 − F (θ))η ∆Π
h i χ(θ; 1 − y ∗ (θ), y ∗ (θ)) − χ(θ; x∗ (θ), y ∗ (θ)) dF (θ) θ∈E∩[θ‡ ,θ] Z h i + χ(θ; φ0 (θ− ), 0) − χ(θ; 1, 0) dF (θ) ? ,p‡ ] ˆ θ≤θ:θ−φ(θ)∈[p Z h i ˆ + = (1 − F (θ))η (1 − y ∗ (θ) − x∗ (θ)) θf (θ) − 1 + F (θ) dθ θ∈E∩[θ‡ ,θ] Z h i ˆ + F (θ) dθ − (1 − φ0 (θ− )) θf (θ) − F (θ) ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p
On other hand, if (A.26) is the case, Z h i ˆ + e = (1 − F (θ))η χ(θ; 1 − y ∗ (θ), y ∗ (θ)) − χ(θ; x∗ (θ), y ∗ (θ)) dF (θ) ∆Π θ∈E∩[θ‡ ,θ] Z h i + χ(θ; 0, 1) − χ(θ; 1, 0) dF (θ) ? ,p‡ ] ˆ θ≤θ:θ−φ(θ)∈[p Z h i ˆ + = (1 − F (θ))η (1 − y ∗ (θ) − x∗ (θ)) θf (θ) − 1 + F (θ) dθ θ∈E∩[θ‡ ,θ] Z (θ − φ(θ))f (θ) 0 − ˆ − (1 − φ (θ )) − (F (θ) − F (θ)) dθ 1 − φ0 (θ− ) ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p Z h i ˆ + ≥ (1 − F (θ))η (1 − y ∗ (θ) − x∗ (θ)) θf (θ) − 1 + F (θ) dθ θ∈E∩[θ‡ ,θ] Z h i ˆ + F (θ) dθ. − (1 − φ0 (θ− )) θf (θ) − F (θ) ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p
43
In both cases, Z
ˆ + e ≥ (1 − F (θ))η ∆Π
θ∈E∩[θ‡ ,θ]
h i (1 − y ∗ (θ) − x∗ (θ)) θf (θ) − 1 + F (θ) dθ
Z − ? ,p‡ ]} ˆ θ∈{θ≤θ:θ−φ(θ)∈[p
≥η
min
h
h i ˆ + F (θ) dθ (1 − φ0 (θ− )) θf (θ) − F (θ)
i
ˆ + F (θ) − θf (θ) − F (θ)
θ∈E∩[θ‡ ,θ]
max
? ,p‡ ] ˆ θ∈{θ≤θ:θ−φ(θ)∈[p }
h i ˆ + F (θ) θf (θ) − F (θ)
!
≥0 where the inequalities are due to the monotonicity of θf (θ) + F (θ). Now, we are ready to prove Lemma 2. Let p∗ be a maximizer of the maximization problem in the statement of Lemma A.4. Define ∗ if θ − φ(θ) ≥ p∗ (1, 0, p ) (x(θ), y(θ), p(θ)) = (φ0 (θ− ), 0, θφ0 (θ− ) − φ(θ)) if θ − φ(θ) < p∗ , φ0 (θ− ) ≤ 1, and θφ0 (θ− ) ≥ φ(θ) (0, 1, 0) otherwise. Then, it is straightforward to show that (x(θ), y(θ), p(θ))θ∈Θ satisfies constraints (13), (14), and (15); hence, Z p(θ) dθ ≤ ΠR . θ∈Θ
On the other hand, one can easily verify that Z p(θ) dθ = lim sup Π∗ () ≥ ΠR . →0
θ∈Θ
In sum, Z ΠR =
A.3
Z
max p∈{θ−φ(θ):θ∈Θ}
! 0
−
θφ (θ ) − φ(θ) d F (θ) .
p d F (θ) + ˆ θ∈[θ† ,θ]∩{θ:θ−φ(θ)
θ∈{θ:θ−φ(θ)≥p}
Proof of Proposition 2
Throughout this proof, θH refers to the type with a larger net-valuation. That is to say, u(θH ) ≥ u(θL ). Contrary to our discussion in the main text, however, we will allow v(θH ) ≤ v(θL ). Similar to the case with continuous types (see Section 4.2), we can define the relaxed problem for the two-type example as follows: b ΠR := max q(θH )p(θH ) + q(θL )p(θL ) (R) (x(θ),y(θ),p(θ))θ∈{θH ,θL }
subject to v(θ)x(θ) + φ(θ)y(θ) − p(θ) ≥ v(θ)x(θ0 ) + φ(θ)y(θ0 ) − p(θ0 ) ∀θ, θ0 ∈ {θH , θL }
(A.27)
v(θ)x(θ) + φ(θ)y(θ) − p(θ) ≥ φ(θ) ∀θ ∈ {θH , θL }
(A.28)
0 ≤ y(θ) ≤ 1 − x(θ) ≤ 1 ∀θ ∈ {θH , θL }
(A.29)
44
Any (x, y, p) = (x∗ (θ), y ∗ (θ), p∗ (θ)))θ∈{θH ,θL } that satisfies constraints (A.27)–(A.29) will be called b is a bounded linear programming problem, and hence, the existence of a feasible. Note that (R) ¯ S ≤ ΠR ; hence, it suffices to show that solution is guaranteed. Clearly, Π ( φ(θH )−φ(θL ) q(θH )v(θH ) L) H )−v(θH )φ(θL ) u(θH )q(θH ) + q(θL ) v(θL )φ(θ if φ(θ v(θH )−v(θL ) v(θL ) ≤ v(θH )−v(θL ) < 1 < v(θL ) (A.30) ΠR = max{q(θH )u(θH ), u(θL )} otherwise. to prove Proposition 2. Note that it also proves that a fixed-price mechanism is optimal whenever condition (3) fails (see footnote 16). LEMMA A.5. Suppose that v(θ) > φ(θ) ≥ 0 for both θ = θH and θL , and that v(θL ) ≥ v(θH ). Then, ΠR = max{q(θH )u(θH ), u(θL )}. Proof. First, note that if v(θL ) ≥ v(θH ), v(θH ) − φ(θH ) is necessarily strictly larger than v(θL ) − φ(θL ). b We may assume the following: Consider any solution (x∗ , y ∗ , p∗ ) of (R). y ∗ (θH ) = 0 and x∗ (θL ) + y ∗ (θL ) = 1.
(A.31)
Otherwise, consider x ˜(θH ) = x∗ (θH ) + y ∗ (θH ), y˜(θH ) = 0, p˜(θH ) = p∗ (θH ) + y ∗ (θH )(v(θH ) − φ(θH )), x ˜(θL ) = 1 − y ∗ (θ), y˜(θL ) = y ∗ (θL ), p˜(θL ) = p∗ (θL ) + v(θL )(1 − y ∗ (θ) − x∗ (θL )). One can easily show that (˜ x, y˜, p˜) is feasible, but q(θH )˜ p(θH )+q(θL )˜ p(θL ) ≥ q(θH )p∗ (θH )+q(θL )p∗ (θL ). Claim 1: x∗ (θH ) = 1. Suppose, for contradiction, that x∗ (θH ) < 1. First, constraint (A.28) implies p(θL ) ≤ v(θL )x∗ (θL ) + φ(θL )y ∗ (θL ) − φ(θL ) = (v(θL ) − φ(θL ))x∗ (θL )
(A.32)
where the equality is due to (A.31). On the other hand, constraint (A.27) necessarily binds for θ = θL and θ0 = θH because otherwise we could increase the objective function by increasing both x∗ (θH ) and p∗ (θH ) by ∈ (0, 1 − x∗ (θH )) and v(θH ), respectively. Hence, v(θL )x∗ (θL ) + φ(θL )(1 − x∗ (θL )) − p∗ (θL ) = v(θL )x∗ (θH ) − p∗ (θH ), which implies that p∗ (θH ) = v(θL )(x∗ (θH ) − x∗ (θL )) − φ(θL )(1 − x∗ (θL )) + p∗ (θL ) < (v(θL ) − φ(θL ))(1 − x(θL )) + p∗ (θL ) ≤ v(θL ) − φ(θL ) where the last inequality is due to (A.32). However, q(θH )p∗ (θH ) + q(θL )p∗ (θL ) < q(θH )(v(θL ) − φ(θL )) + q(θL )x∗ (θL )(v(θL ) − φ(θL )) ≤ v(θL ) − φ(θL ), 45
(A.33)
and hence, (x∗ , y ∗ , p∗ ) is strictly dominated by x(θ) = 1,
y(θ) = 0 p(θ) = v(θL ) − φ(θL ) ∀θ ∈ {θH , θL },
b hence, Claim 1 must be true. contradicting the hypothesis that (x∗ , y ∗ , p∗ ) solves (R); Claim 2: v(θL )x∗ (θL ) + φ(θL )y ∗ (θL ) − p∗ (θL ) = φ(θL ). Suppose not. First note that if constraint (A.28) is slack for θL , it has to bind for θH so that p∗ (θH ) = v(θH ) − φ(θH ). But then, the right-hand side of (A.27) for θ = θL and θ0 = θL becomes v(θL ) − (v(θH ) − φ(θH )) which is strictly smaller than φ(θL ). Hence, both (A.28) for θ = θL and (A.27) for (θ, θ0 ) = (θL , θH ) are slack for (x∗ , y ∗ , p∗ ). However, in this case, (x∗ , y ∗ , p∗ ) is clearly suboptimal; hence, Claim 2 must be true. By (A.31) and the last two claims, we can simplify constraint (A.27) as v(θH ) − p(θH ) ≥ v(θH )x(θL ) − φ(θH )(1 − x(θL )) − u(θL )x(θL ) φ(θL ) ≥ v(θL ) − p(θH ) and constraint (A.28) as v(θH ) − p(θH ) ≥ φ(θH ) v(θL )x(θL ) + φ(θL )(1 − x(θL )) − p(θL ) = φ(θL ) It is straightforward to see that all the constraints are jointly satisfied if and only if u(θL ) ≤ p(θH ) ≤ u(θH )(1 − x(θL )) + u(θL )x(θL ).
(A.34)
Hence, ΠR =
max x(θL )∈[0,1]
n o q(θH )u(θH )(1 − x(θL )) + u(θL )x(θL ) = max{u(θH )q(θH ), u(θL )}.
Now, we turn to the case v(θH ) > v(θL ). LEMMA A.6. Suppose that v(θ) > φ(θ) ≥ 0 for both θ = θH and θL and that v(θH ) > v(θL ). For any b solution (x, y, p) of (R), (i) x(θH ) = 1, (ii) constraint (A.28) binds for θ = θL , and (iii) constraint (A.27) binds for (θ, θ0 ) = (θH , θL ). 46
b and consider (˜ Proof of (i). Fix any (x, y, p) that is feasible for (R), x, y˜, p˜) such that x ˜(θH ) = 1, y˜(θH ) = 0, p˜(θH ) = p(θH ) + v(θH )(1 − x(θ)) − φ(θH )y(θH ), x ˜(θL ) = x(θL ), y˜(θL ) = y(θL ), p˜(θL ) = p(θL ).
(A.35)
b is larger at (˜ One can confirm that (˜ x, y˜, p˜) is also feasible, and the objective function of (R) x, y˜, p˜) b than at (x, y, p). This shows that x(θH ) = 1 for any solution of (R). Proof of (ii). We can prove that (A.28) binds for θ = θL by the same argument for Claim 2 in Lemma A.5. b such that x∗ (θH ) = x∗ (θ∗ ) = 1. Proof of (iii). Suppose that there is a solution (x∗ , y ∗ , p∗ ) of (R) L ∗ ∗ ∗ ∗ ∗ b only if p (θH ) = p (θL ) = v(θL ) − φ(θL ), and in this case, (A.27) binds (x , y , p ) could solve (R) for θ = θH and θ0 = θL . Now, suppose, for contradiction, that v(θH ) − p∗ (θH ) > v(θH )x∗ (θL ) + φ(θH )y ∗ (θL ) − p∗ (θL ) and x∗ (θL ) < 1. However, the objective function is smaller at (x∗ , y ∗ , p∗ ) than (˜ x, y˜, p˜) such that x ˜(θH ) = x∗ (θH ),
y˜(θH ) = y ∗ (θH ),
x ˜(θL ) = + (1 − )x∗ (θL ),
p˜(θH ) = p∗ (θH ),
y˜(θL ) = (1 − )y ∗ (θL ),
and p˜(θL ) = p∗ (θL ) + (1 − x∗ (θL ))v(θL ) + y ∗ (θL )φ(θL ) b as long as > 0 is sufficiently small. which satisfies all the constraints for (R) In the last lemma we may replace constraints (A.27)–(A.29) with u(θL ) ≤ p(θH ) ≤ u(θH )
(A.36)
p(θH ) = u(θL ) + (v(θH ) − v(θL ))(1 − x(θL )) − (φ(θH ) − φ(θL ))y(θL )
(A.37)
p(θL ) = v(θL )x(θL ) − φ(θL )(1 − y(θL )).
(A.38)
x(θH ) = 1,
0 ≤ x(θL ) ≤ 1 − y(θL ) ≤ 1.
(A.39)
b with (A.37), (A.38), and (A.39), respectively, we can see Substituting p(θH ), p(θL ), and x(θH ) in (R) b must solve that any solution (x(θL ), y(θL ), p(θH )) of (R) max (x(θL ),y(θL ))
[v(θL )x(θL ) − (1 − y(θL ))φ(θL )] + q(θH ) [v(θH )(1 − x(θL )) − φ(θH )y(θL )]
subject to u(θL ) ≤ u(θL ) + (v(θH ) − v(θL ))(1 − x(θL )) − (φ(θH ) − φ(θL ))y(θL ) ≤ u(θH )
(A.40)
0 ≤ x(θL ) ≤ 1 − y(θL ) ≤ 1.
(A.41)
LEMMA A.7. Constraints (A.40) and (A.41) are jointly satisfied if and only if φ(θH ) − φ(θL ) ≥x . (x(θL ), y(θL )) ∈ Ψ := (x, y) : 0 ≤ x ≤ 1 − y ≤ 1 and (1 − y) v(θH ) − v(θL ) 47
Proof. The second inequality in (A.40) u(θL ) + (v(θH ) − v(θL ))(1 − x(θL )) − (φ(θH ) − φ(θL ))y(θL ) ≤ u(θH ) H )−φ(θL ) is satisfied if and only if (1 − y(θL )) φ(θ v(θH )−v(θL ) ≥ x(θL ). Hence, the necessary part of the lemma is immediate. To prove sufficiency, note that the first inequality (A.40) holds for any (x(θL ), y(θL )) such that 0 ≤ x(θL ) ≤ 1 − y(θL ) ≤ 1; hence, it is redundant.
The solution (x(θ L ), y(θL )) must occur at extreme points of the set Ψ, which are (0, 1), (0, 1), φ(θH )−φ(θL ) and v(θH )−v(θL ) , 0 . Comparing the objective function evaluated for all these three candidates shows (A.30).
A.4
Proof of Proposition 3
The statement (i) follows the optimality of the posting-price mechanism with the price path (7). To b is prove (ii), note that, whenever w,v > 1, a solution for (R) x(θH ) = 1, x(θL ) =
y(θH ) = 0,
φ(θH ) − φ(θL ) , v(θH ) − v(θL )
(A.42)
p(θH ) = u(θ) y(θL ) = 0,
p(θL ) =
φ(θL )v(θH ) − φ(θH )v(θL ) . v(θH ) − w(θL )
Note that this solution does not depend on qH ∈ (0, 1), and hence it remains solve the following linear programming for any ∈ R sufficiently close to 0 in R space. ΠR :=
max (x(θ),y(θ),p(θ))θ∈{θH ,θL }
(q(θH ) + )p(θH ) + (1 − q(θH ) − )p(θL )
subject to v(θ)x(θ) + φ(θ)y(θ) − p(θ) ≥ v(θ)x(θ0 ) + φ(θ)y(θ0 ) − p(θ0 ) ∀θ, θ0 ∈ {θH , θL } v(θ)x(θ) + φ(θ)y(θ) − p(θ) ≥ φ(θ) ∀θ ∈ {θH , θL } 0 ≤ y(θ) ≤ 1 − x(θ) ≤ 1 ∀θ ∈ {θH , θL } b As an implication, Then, by Theorem 1 in Mangasarian (1979), (A.42) is a unique solution of (R). V µ (θL ) = φ(θL ) =
λ w(θL ), λ+r
xµ (θL ) =
φ(θH ) − φ(θL ) < 1, v(θH ) − v(θL )
and y µ (θL ) = 0
(A.43)
for any optimal mechanism µ. Finally, we argue that any posting-price mechanism cannot satisfy (A.43). Suppose there is a posting-price mechanism with (A.43), and let t(θL ) the trading time of the low type in this mechanism. Because x(θL ) < 1, there is > 0 such that the following event occurs with positive probability µ: (a) The outside option arrives before t = /2 (that is Zt = 1 at t = /2). (b) No action and no payment is made at or before t = between the seller and the low type. In particular, the low type does purchase the good before t = .
48
(c) The low type’s continuation payoff from continuing the negotiation reaches below w(θL ) at some point of time between t = /2 and t = . Note that (a) is due to the property of Poisson arrival processes, and (b) follows xµ (θL ) < 1 and the λ definition of posting-price mechanism. Finally, (c) comes from V µ (θL ) = λ+r w(θL ) < w(θL ). Note that if the above event occurs, the low type would optimally exercise the outside option at or before t = , and hence y µ (θL ) cannot be zero. Contradiction. This shows that any posting-price mechanism is suboptimal, and the same argument also shows that any optimal mechanism necessarily involves an upfront scheme for the low type.
A.5
Proof of Proposition 6
Choose an arbitrary optimal mechanism µ = (A, P ). Φ(θ; µ∗ ) > 0 if and only if θ ∈ T(p∗ ), and hence we only need to show that Φ(θ; µ) ≥ Φ(θ; µ∗ ) for almost every θ ∈ T(p∗ ). Step 1: We first claim that y µ (θ) = 0 for all θ ∈ T(p∗ ). Suppose for contradiction that y µ (θ) > 0 for any θ ∈ K, where K ⊂ T(p∗ ) and has a positive Lebesgue measure. Now consider a new mechanism µ ˜ ∗ as follows: If the buyer reveals her type as θ 6∈ T(p ), the seller offers µ to the buyer. Otherwise, the seller offers µ with probability xµ (θ) or µ∗ with probability 1 − xµ (θ) respectively. One can easily show that this mechanism µ ˜ is incentive-compatible, and moreover, generates strictly more profit than µ. Step 2: V µ (θ) = φ(θ) for any θ ∈ T(p∗ ). Otherwise, by the linearity of the seller’s problem, there must be θ? such that xµ (θ) = 1 for any θ ∈ [θ? , θ] ∩ T(p∗ ). But this is impossible due to our choice of p∗ such that U(p) ⊂ U(p∗ ) for any other arg maximum p for (20). Step 3: Now we are ready to prove that Φ(θ; µ) ≥ Φ(θ; µ∗ ) > 0 for any θ ∈ T(p∗ ). Fix θ ∈ T(p∗ ). We first show that Φ(θ; µ) > 0. Suppose not for contradiction. Because xµ (θ) ∈ (0, 1) and y µ (θ) = 0, we may assume that the buyer purchases the good at t(θ) = 1r log xµ1(θ) by paying p(θ) = pµ (θ)/xµ (θ). Now pick > 0 such that h i E e−r(t(θ)−s) (θ − p(θ)) < w(θ) ∀s ∈ [0, ) Because of Condition U , we may assume that the outside option arrives before t = with a positive probability. In this case, however, because the buyer’s continuation payoff from delaying until t(θ) is strictly less than w(θ), the buyer would exercise the outside option. This contradicts the fact that y µ (θ) = 0. Finally, we have to show that Φ(θ; µ) ≥ Φ(θ; µ∗ ) = w(θ) − φ(θ) > 0. But w(θ) − φ(θ) is clearly the minimum upfront payment for a buyer type θ such that V µ (θ) = φ(θ) and xµ (θ) ∈ (0, 1).
A.6
Proof of Proposition 7
Combining Lemma 2, Proposition 5, and Proposition 4, one can show that ΠS (µ∗ ) > ΠS (µ∗one-shot ) if and only if T(p∗ ) is nonempty, which is equivalent to θ(p∗ ) > θ† . Under the assumption of the proposition, this inequality is in turn equivalent to QE = 1 − F (θ† ) > 1 − F (θ(p∗ )) so that the proposition holds true with Q∗ = 1 − F (θ(p∗ )). 49
A.7
Proof of Proposition 8
Assumption 3 (convexity of w) implies that uE (q) is either decreasing in q or single-peaked. µ∗ = µ∗one−shot for all environments in E∗E whenever uE (q) is decreasing in q (Proposition 4), in which case Proposition 8 is vacuously true. Hence, we may assume that uE (q) is singled-peaked without loss. For any p ∈ U = {u(θ) : θ ∈ Θ}, let µone−shot (p) be the one-shot mechanism that commits to a single price p indefinitely. µone−shot (p) yields the same profit in any environment in E∗E , and let ΠE (p) denote this profit level. The maximum theorem guarantees that ΠE (p) is continuous in p ∈ U. Define p∗one-shot := min p ∈ U : ΠE (p) ≥ ΠE (p0 ) ∀p0 ∈ U
and Q∗H := inf{q ∈ [0, 1] : uE (q) = p∗one-shot }
˜ ∈ E∗ such that Q ˜ > Q∗ , and consider the following mechanism. Now, fix any E E H E • Let m0 ∈ Θ denote a type that the buyer reports to the mechanism at t = 0. If m0 ∈ U(p∗one-shot ), the two parties trade immediately in exchange for the buyer’s payment p∗one-shot . • If m0 ∈ T(p∗one-shot ), the buyer pays an upfront payment w to the seller at t = 0, where w = 1 sup{w(θ) : θ ∈ Θ}. The two parties delay until t(θ) = 1r log φ0 (m − , and then trade the good with ) 0
the net-transfer pt = p(θ) = m0 −
φ(m0 )+w φ0 (m− 0 )
to the seller.
• No trade occurs if m0 6∈ U(p∗one-shot ) ∪ T(p∗one-shot ), and the buyer exercises the outside option at t = γ0 (Z0 ). One can easily show that this mechanism is incentive-compatible, and generates more profit than ΠE (p∗one-shot ), which in turn implies that the expected profit generated in µ∗ exceeds ΠE (p∗one-shot ). This completes the proof for the statement (32). Note that (33) is trivial; for example, the statement is true with Q∗L = 0.
A.8
Proof of Proposition 9
R(θ; µ∗ ) = R(θ; µ∗one−shot ) = 0 and S(θ; µ∗ ) = S(θ; µ∗one−shot ) = 0 for any θ ∈ [θ, min(θ† , θ(pone−shot ))], and hence, we may focus on θ ≥ min(θ† , θ(pone−shot )). Step I: Note that p∗0 , the price offered at time t = 0 in µ∗ is ( p∗0
"Z
= min arg max p∈U
U(p)
p dF (θ) +
Z
#) θφ (θ ) − φ(θ) dF (θ) , 0
T(p)
−
where U = {θ − φ(θ) : θ ∈ Θ}. Additionally, p∗one−shot , the price committed by µ∗one−shot is ( p∗one−shot = min arg max p∈U
"Z U(p)
#) p dF (θ)
Because θφ0 (θ− ) ≥ φ(θ) for any θ ∈ ∪p≥0 T(p), d dp
"Z U(p)
p dF (θ) +
Z T(p)
# "Z # d θφ0 (θ− ) − φ(θ) dF (θ) ≥ p dF (θ) dp U(p)
50
for any p ∈ U, and hence, p∗0 ≥ p∗one−shot . This also implies ∗ θ† ≤ max T(p∗0 ) ≤ θone−shot ≤ min U(p∗0 ).
∗ ∗ Step II: p∗0 ≥ p∗one−shot implies [θ† , θone−shot ] ∩ T R(µ∗ ) ⊂ T(p∗0 ). All types in (θ† , θone−shot ) ∩ T R(µ∗ ) are completely excluded from µ∗one−shot while they trade with the seller in µ∗ , and hence, ∗ 0 = S(θ; µ∗one−shot ) ≤ S(θ; µ∗ ) for almost every [θ† , θone−shot ] ∩ T R(µ∗ ). ∗ Moreover, from the construction of µ∗ , all these buyer types in (θ† , θone−shot ) ∩ T R(µ∗ ) obtain
V
µ∗
−rt
(θ) = e
θ−
e−rt p∗t
t= r1 log
1 φ0 (θ − )
= φ0 (θ− )θ − (φ0 (θ− )θ − φ(θ)) = φ(θ),
hence, ∗ R(θ; µ∗ ) = 0 for almost every [θ† , θone−shot ] ∩ T R(µ∗ ). ∗ Step III: p∗0 ≥ p∗one−shot implies max T R(µ∗ ) ≤ max T R(µ∗one−shot ), and hence, [θone−shot , θ]∩T R(µ∗ ) ⊂ T R(µ∗one−shot ). Note that any buyer type θ ∈ T R(µ∗one−shot ) trades with the seller t = 0 with probability 1, and therefore S(θ; µ∗one−shot ) ≥ S(θ; µ) for any µ ∈ MIC . In particular, ∗ S(θ; µ∗ ) ≤ S(θ; µ∗one−shot ) for almost every θ ∈ [θone−shot , θ] ∩ T R(µ∗ ). ∗ , θ] ∩ T R(µ∗ ) ∩ U(p∗0 ), If θ ∈ [θone−shot
R(θ; µ∗ ) = θ − p∗0 − φ(θ) ≤ θ − p∗one−shot − φ(θ) = R(θ; µ∗one−shot ) ∗ , θ] ∩ T R(µ∗ ) ∩ T(p∗0 ), If θ ∈ [θone−shot
R(θ; µ∗ ) = 0 ≤ θ − p∗one−shot − φ(θ) = R(θ; µ∗one−shot ).
51
