An alternating-offers model of multilateral negotiations

Accepted Manuscript Title: An Alternating-Offers Model of Multilateral Negotiations Author: Charles J. Thomas PII: DOI: Reference:

S0167-2681(17)30308-6 https://doi.org/doi:10.1016/j.jebo.2017.11.004 JEBO 4186

To appear in:

Journal

Received date: Revised date: Accepted date:

9-1-2017 10-10-2017 9-11-2017

of

Economic

Behavior

&

Organization

Please cite this article as: Charles J. Thomas, An Alternating-Offers Model of Multilateral Negotiations, (2017), https://doi.org/10.1016/j.jebo.2017.11.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to download Manuscript: mln alternating offers (2017-10-09)a.pdf

An Alternating-Offers Model of Multilateral Negotiations∗

cr

ip t

Charles J. Thomas Economic Science Institute & Argyros School of Business and Economics Chapman University

Abstract

us

October 9, 2017

1

Introduction

te

d

M

an

I develop a model of the multilateral negotiations that are frequently observed when one party wishes to trade with one of several others offering potentially different amounts of surplus to be split. The model’s intuitively sensible equilibrium outcomes differ qualitatively from those in other models of these negotiations. I demonstrate one application of the model that provides empirical predictions about how the choice of transacting via negotiations or auctions is affected by factors including the number of trading partners, uncertainty when making the choice, and costly participation in the trading process. More generally the model provides a tractable foundation for analyzing strategic problems in settings featuring multilateral negotiations, including investment, product design, mergers, and hold-up.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Click here to view linked References

Bilateral negotiations play an important role in economic theory and practice, but negotiations often are multilateral in nature. For example, in a procurement setting a firm or government might negotiate to purchase from one of several suppliers whose products differ on dimensions such as quality or goodness-of-fit, such as Walgreen’s selection of AmerisourceBergen over Cardinal Health to provide drug-wholesaling services to the massive drugstore chain.1 Likewise, a takeover contest might involve multiple potential acquirers who differ in their synergies or opportunity costs from completing the transaction, such as the battle for control of Dell, Inc. amongst founder Michael Dell, Blackstone Group LP, and investor Carl Icahn.2 A high-end job candidate might have several ∗

Email: [email protected]. For their generosity in inviting me to be an Affiliated Research Scientist, I thank Chapman University’s Economic Science Institute & Argyros School of Business and Economics. This work was completed in part while I was a Visiting Associate Professor at Clemson University’s John E. Walker Department of Economics, whom I thank for their hospitality. Jeremy Bulow and Patrick Warren provided helpful comments, as did two anonymous referees. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Keywords: bargaining, auction, procurement, merger, hiring, investment JEL: C78, D44, D82 1 “Deal Transforms Global Pill Pipeline.” Wall Street Journal, Mar 20, 2013, B.1. 2 “Dell Buyout is a Fee-for-all - Banks Advising and Financing the Deal Could Reap More than $400 Million.” Wall Street Journal, Apr 05, 2013, C.1.

1 Page 1 of 42

te

d

M

an

us

cr

ip t

employers maneuvering for position, as occurred in early 2012 when the Indianapolis Colts released quarterback Peyton Manning to free agency.3 Lastly, a firm contemplating significant capital investment might have multiple state and local governments as eager suitors, as when Boeing built its 787 Dreamliner assembly line in South Carolina rather than in Washington or elsewhere.4 Such multilateral negotiations have received relatively little academic attention despite their empirical relevance, perhaps because their strategic complexity makes them difficult to model formally. They feature elements of bilateral negotiations and auctions, two exchange mechanisms that have been the subject of significant amounts of research.5 Of course, analyses of bilateral negotiations emphasize bilateral settings, with some exceptions described in Section 3 that require one party to abandon current negotiations before talking with a new potential trading partner, or that consider exogenously specified outside options rather than endogenously determined ones. Analyses of auctions focus on multilateral settings, but auctions lack the communication and interplay inherent to multilateral negotiations. In Section 2 I develop an infinite-horizon, alternating-offers model of multilateral negotiations that accommodates diverse settings such as those described earlier, but for concreteness I consider one buyer negotiating with several sellers. The buyer wants to make only one trade, and trade with different sellers can generate different amounts of surplus to be split. There is complete information about the available surpluses, and bargaining alternates between the buyer making simultaneous offers to all sellers and receiving simultaneous offers from all sellers. For settings in which one seller negotiates with several buyers, all of the model’s results can be applied simply by reversing the labels on the players’ roles. My model’s negotiation protocol differs from those in other approaches to modeling multilateral negotiations, and it leads to qualitatively different but intuitively sensible results. These differences are seen vividly by adding a less valuable seller to an initially bilateral setting. As I describe in Section 3, in models like Ray (2007) the new seller does not alter the terms of trade from those in the bilateral setting, no matter how close of a substitute the new seller is for the original seller. In models like Osborne and Rubinstein (1990, Ch. 9.3) the new seller dramatically benefits the buyer, no matter how poor of a substitute is the new seller. In contrast to those two approaches, my model’s predictions align with fundamental notions of competition: the new seller benefits the buyer if and only if the new seller is a sufficiently close substitute for the original seller. The basic model naturally extends Rubinstein’s (1982) bilateral negotiation model, and it illustrates how negotiated outcomes are affected by introducing additional sellers into a bilateral setting, by differences across sellers in terms of available surplus, and by the parties’ relative bargaining strengths. One unexpected finding is that the buyer can be better off when it becomes less patient in multilateral negotiations, which contrasts sharply with the effect of impatience in

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3

“Peyton Manning’s Long Game.” Sports Illustrated 116, no. 14: 50-55. “Seattle Frets as Boeing Looks South for Sites-Aircraft Maker’s Search for a New Dreamliner Assembly Plant Poses a Threat to Big Machinists Union and Washington State.” Wall Street Journal, Aug 11, 2009, A.4. 5 Kennan and Wilson (1993), Ausubel et al. (2002), and Serrano (2008) survey the bargaining literature, while McAfee and McMillan (1987) and Milgrom (1989) survey the auction literature. 4

2 Page 2 of 42

te

d

M

an

us

cr

ip t

bilateral negotiations. Being very impatient inclines the buyer to buy today in both bilateral and multilateral negotiations. While in bilateral negotiations extreme impatience allows the single seller to exploit the buyer’s aversion to waiting, in multilateral negotiations that same impatience provides the buyer with a credible commitment to buy today that forces each seller to compete vigorously so as not to lose the sale to a rival. In Section 4 I demonstrate one application of the model by using it to evaluate the buyer’s choice of conducting procurement via multilateral negotiations or an auction, which provides empirical predictions about the important issue of institutional choice. Comparing institutions has a long history in economics, from early research by Chamberlin (1948) and Smith (1962), to more recent research by Levin and Smith (1994), Bulow and Klemperer (1996, 2009), Bajari et al. (2009), and Athey et al. (2011). The application in Section 4 extends the basic model in various ways that make the institutional comparison non-trivial, because in the basic model the buyer weakly prefers multilateral negotiations. One extension models the available surpluses as realizations of random variables that are known when negotiations commence, but that are unknown when the procurement method is chosen. A seller learns its associated surplus if it chooses to incur an entry cost that reflects opportunity costs of its decision to compete, say to design prototypes, evaluate production costs, or assess its product’s fit with the buyer’s preferences. I find that the buyer tends to prefer negotiations when there are few sellers, when sellers’ products are distinct or their production costs differ greatly, and when sellers face sufficiently low entry costs. One striking finding when sellers incur entry costs is that the buyer’s preferred procurement mechanism can change multiple times within small ranges of those costs, which provides one explanation for the coexistence of auctions and multilateral negotiations across apparently similar buyers. This switching occurs because changing the sellers’ entry costs affects how the buyer strikes a balance between the procurement mechanism and the number of sellers to invite to participate. Sellers’ entry costs might differ across buyers in ways not obvious to an outside observer, say because of different degrees of complexity in determining buyers’ purchasing requirements. These differences can induce buyers to use different procurement mechanisms. More generally the model provides a tractable foundation for analyzing strategic problems in settings featuring multilateral negotiations, including investment, product design, mergers, holdup, dual-sourcing, entry, and collusion. Section 5 describes such research possibilities and provides concluding remarks. The Appendix contains all proofs, but in Section 2 I provide links to the relevant portions of the Appendix from the discussion of the negotiated payoffs, as a guide to readers interested in the technical details.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2

The Basic Model of Multilateral Negotiations

Consider a buyer negotiating to trade with one of N sellers. Trade between the buyer and seller i yields surplus Vi ≥ 0 for them to split, where V1 > 0 and V1 ≥ V2 ≥ · · · ≥ VN . The buyer and each 3 Page 3 of 42

te

d

M

an

us

cr

ip t

seller i have instantaneous rates of time preference rB > 0 and ri > 0. Unlike the Vi , the ranking of the ri across sellers is not tied to the index i. The game’s structure and parameters are common knowledge among all players, so there is complete information about the available surpluses and all players’ discount rates. Trade is conducted as follows, where time is measured in discrete periods t ∈ {0, 1, 2, . . .} that are of length ∆ > 0. In even-numbered period t ∈ {0, 2, 4, . . .} each seller i makes a proposal (si,t , Vi − si,t ) ∈ [0, Vi ]2 to the buyer that specifies the amount si,t of the surplus Vi that seller i demands for itself, with the remainder Vi − si,t that it offers to the buyer. The proposals are made simultaneously and are revealed publicly when received by the buyer, after which the buyer decides whether to accept one of the proposals. If the buyer accepts one of the proposals, then the negotiations conclude. If the buyer rejects all of the proposals, then in odd-numbered period t + 1 ∈ {1, 3, 5, . . .} the buyer makes proposals (bi,t+1 , Vi − bi,t+1 ) ∈ [0, Vi ]2 to each seller i that specify the amount bi,t+1 of the surplus Vi that the buyer demands for itself, with the remainder Vi −bi,t+1 that it offers to seller i. The proposals are made simultaneously and are revealed publicly when received by the sellers, after which the sellers simultaneously decide whether to accept their respective proposals. From the set of accepted proposals the buyer decides which transaction, if any, to consummate. If the buyer trades with a seller that accepted its proposal, then the negotiations conclude. If no transaction occurs, then play continues to the next period. A transaction between the buyer and seller i in even-numbered period t yields them respective payoffs (Vi − si,t ) e−rB t∆ and si,t e−ri t∆ , while a transaction in odd-numbered period t yields them respective payoffs bi,t e−rB t∆ and (Vi − bi,t ) e−ri t∆ . Losing sellers’ payoffs are 0. If a transaction never occurs, then each party’s payoff is 0. For notational convenience define δ k ≡ e−rk ∆ , where δ k ∈ (0, 1) is player k’s discount factor, for k ∈ {B, 1, 2, . . . , N }. Three issues must be clarified regarding the acceptance of proposals. First, I allow the buyer not to consummate any transaction even if one or more of its proposals were accepted. As will become apparent later, this assumption prevents the buyer from extracting the entire surplus V1 from seller 1 by committing to engage in unfavorable trades. Second, in some period the buyer might accept one of multiple proposals that give it the same payoff. If so, I assume that from those proposals the buyer accepts the one from the seller with the lowest index. Third, in some period several sellers might accept the buyer’s proposal. If trade occurs in such an instance, and if multiple proposals accepted by sellers give the buyer its highest payoff, I assume that from those proposals the buyer trades with the seller having the lowest index. The latter two tie-breaking assumptions simplify an issue in deriving equilibrium that is caused by an “openness” problem arising because the players’ allowable demands and offers are drawn from continuous sets. The preceding model extends Rubinstein’s (1982) bilateral negotiation model by letting one buyer negotiate simultaneously with multiple sellers. The assumptions about proposal acceptance and the proposals’ public nature keep the model close in spirit to Rubinstein’s model, while allowing for the multilateral aspect. The assumptions that the sellers make offers simultaneously and receive offers simultaneously attempt to capture two key features of the multilateral negotiations I consider:

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4 Page 4 of 42

M

an

us

cr

ip t

the actor on the “single-player” side of the negotiations can hold multiple offers, and it can hold an offer while seeking concessions. Such behavior is familiar from the academic job market, when a prospective hire with multiple offers attempts to play schools off one another to improve those offers along dimensions such as salary, teaching load, and summer support. Negotiations over each offer are not conducted in isolation, but rather in light of the other offers the prospective recruit holds and is likewise trying to improve. The players’ strategies consist of proposals or accept/reject decisions at every decision point, all of which can depend on all prior moves that led to a specific point in the game. I solve for the game’s pure strategy subgame perfect Nash equilibria (SPNE), and for simplicity I focus on stationary SPNE outcomes: SPNE outcomes supported by strategies for which each player k receives the payoff π bo k in all subgames that begin with the buyer making offers, and receives the so payoff π k in all subgames that begin with sellers making offers.6 I characterize this game’s stationary SPNE outcomes by modifying the approach pioneered by Shaked and Sutton (1984) for solving Rubinstein’s (1982) bilateral negotiation model. Their method uses four variables. The first two are π B and π B , the infimum and supremum of the set of the buyer’s SPNE payoffs in subgames beginning with offers from the buyer. The second two variables are π 1 and π 1 , the infimum and supremum of the set of seller 1’s SPNE payoffs in subgames beginning with offers from the seller. Shaked and Sutton (1984) show that the following four constraints must hold.

d

π B ≤ V1 − δ 1 π 1

π 1 ≥ V1 − δ B π B

π 1 ≤ V1 − δ B π B

te

π B ≥ V1 − δ 1 π 1

Roughly speaking, the first and third constraints tell a negotiator, “When you are making offers, you need not give too much.” The second and fourth tell a negotiator, “When you are making offers, you cannot take too much.” Manipulating these constraints reveals that

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

πB = πB =

³

1−δ 1 1−δ1 δ B

´

V1 and π 1 = π 1 =

³

1−δB 1−δ 1 δ B

´

V1 .

Having established the SPNE payoffs are unique, one can show there is a unique SPNE. The stationary SPNE outcomes with multilateral negotiations are derived by recognizing that versions of the preceding constraints continue to hold, except in particular circumstances in which the credible prospect of trading with another seller requires modifying one or more of the constraints. In some cases the modifications involve introducing a tighter constraint, and in others involve replacing a constraint with a looser one. While these modifications significantly complicate the analysis, Theorem 1 reveals that trade in any subgame occurs between the buyer and seller 1 in the subgame’s initial period. 6

Characterizing all SPNE outcomes has proven to be difficult, in part because of the apparent necessity of allowing the buyer to retract accepted offers. I discuss this and related issues after Theorem 1.

5 Page 5 of 42

Theorem 1 In subgames that begin with the buyer making offers, in all stationary SPNE outcomes the payoffs for the buyer and seller 1 are

π bo B =

³

1−δ 1 1−δ1 δ B

V2 δ 2B

´

V1 and π bo 1 = δ1

³

1−δ B 1−δ 1 δB

and π bo 1 = V1 − and π bo 1 =0

π bo B = V1

and π bo 1 =0

h ³ ´ ´ 1−δ 1 V1 if V2 ∈ 0, δ 2B 1−δ V1 1 δB h ³ ´ i 1−δ 1 2 if V2 ∈ δ 2B 1−δ , δ V V 1 1 B 1 δB ¢ ¡ 2 if V2 ∈ δ B V1 , δ B V1 if V2 ∈ [δ B V1 , V1 ] .

cr

π bo B = V1

V2 δ 2B

´

ip t

π bo B =

π so B =

V2 δB

³

1−δ 1 1−δ1 δ B

´

³

h ³ ´ ´ 1−δ 1 V1 if V2 ∈ 0, δ 2B 1−δ V1 1 δB h ³ ´ i 1−δ 1 2 = V1 − δVB2 if V2 ∈ δ 2B 1−δ , δ V V 1 B 1 1 δB ¢ ¡ 2 = (1 − δ B ) V1 if V2 ∈ δ B V1 , δ B V1

V1 and π so 1 = and π so 1

1−δ B 1−δ 1 δB

´

an

π so B = δB

us

In subgames that begin with sellers making offers, in all stationary SPNE outcomes the payoffs for the buyer and seller 1 are

and π so 1

π so B = V2

and π so 1 = V1 − V2

M

π so B = δ B V1

if V2 ∈ [δ B V1 , V1 ] .

te

d

The unique stationary SPNE payoffs in multilateral negotiations are more complex than the unique SPNE payoffs in bilateral negotiations, but the derivations of both share the fundamental concept that a party formulating a proposal must consider its counterpart’s next-best alternative to accepting that proposal. In bilateral negotiations the counterpart’s next-best alternative is to incur delay and make a proposal next period, which leads to a player receiving an equilibrium offer equal to its net present value of letting play continue to the next period. Multilateral negotiations feature the additional influence of the buyer’s threat to trade with another seller, which in some cases the buyer uses to dramatic effect. Theorem 1 reveals that the surpluses from sellers 3, .., N are irrelevant to the negotiated outcome, which reflects the role of next-best alternatives because all players recognize sellers 1 and 2 as the buyer’s two best trading partners. If the buyer ever contemplated trading with any seller other than seller 1, then the buyer and seller 1 can find a mutually beneficial trade because trading with seller 1 yields the largest surplus to be split. Likewise, if the buyer’s next-best alternative to trading with seller 1 is trading with another seller rather than letting the negotiations continue, then seller 2 is the relevant threat to seller 1 because trade with seller 2 yields the next-highest surplus to seller 1’s. The interplay between V1 and V2 affects the players’ negotiating behavior. Seller 2 competes

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

all-out for the buyer’s business, both when making and receiving offers, because it expects to be outdone by seller 1; seller 2 will offer the entire surplus V2 to the buyer, and it will accept any amount of surplus the buyer offers.7 The buyer and seller 1 account for seller 2’s incentives when 7 This behavior also explains why seller 2’s patience is irrelevant to the negotiated outcome, because its behavior is based solely on its payoff in the current period.

6 Page 6 of 42

te

d

M

an

us

cr

ip t

they make offers or decide to accept or reject offers, provided that the buyer prefers trading with seller 2 versus letting the negotiations continue. One might question the inclusion of sellers 3, ..., N in the model given their irrelevance to the negotiated outcome, but there are at least two reasons for doing so. First, their irrelevance could not be assessed if they were excluded from the model. For example, without sellers 3, ..., N one might wonder if their presence would lead to a recursive structure in the negotiated payoffs, with the payoff from trading with seller 1 affected by the payoff from trading with seller 2, the payoff from trading with seller 2 affected by the payoff from trading with seller 3, and so on. Second, the number of sellers affects the values of the highest and second-highest surpluses in some of the extensions analyzed later, which is relevant for applications of the model in which sellers make strategic decisions that affect the sizes of the surpluses from trading with them. Theorem 1’s characterization of the buyer’s and seller 1’s stationary SPNE payoffs demonstrates that seller 2 affects the negotiations when V2 is sufficiently high, a determination that depends on the relationship among V1 , V2 , δ 1 , and δ B . Seller 2 affects the negotiations if V1 and V2 are sufficiently close, because in such cases sellers 1 and 2 are sufficiently close competitors in terms of what surplus they can offer the buyer. Seller 2 also affects the negotiations if δ 1 is sufficiently high or δ B is sufficiently low, because in such cases the buyer has a relatively weak bargaining position and would obtain a small portion of V1 if seller 2 were not present. Hence, the surplus available from seller 2 is a credible alternative to trading with seller 1. It is instructive ³ to focus³ on the´impact ´ of adding sellers to a bilateral setting, which reveals that 1−δ 1 2 for low values of V2 < δ B 1−δ1 δB V1 the stationary SPNE payoffs equal those from Rubinstein’s model of bilateral negotiations. To see why, suppose the buyer makes offers and that V2 is strictly 8 less than the buyer’s net present value of letting play continue to the next period (V2 < δ B π so B ). For such values of V2 , trading with seller 2 is not a credible threat in the current period, nor in the subsequent period when sellers make offers (because V2 < π so B , too). Sellers 2, ..., N therefore are strategically irrelevant when the surpluses available from them are so low, so the negotiations between the buyer and seller 1 proceed as if those other sellers were not present. Figure 1 illustrates this range of V2 by plotting the buyer’s SPNE payoffs in subgames beginning with offers from the buyer (panel (a)), and in subgames beginning with offers from the sellers (panel (b)). As V2 increases from 0, for a range of V2 the buyer’s SPNE payoffs are fixed at their levels from bilateral negotiations.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 1 Here ¡ ¢ For high values of V2 > δ 2B V1 the stationary SPNE outcome features the buyer making offers that extract the entire surplus V1 from seller 1. To see why, suppose the buyer makes offers and V2 strictly exceeds the buyer’s net present value of letting play continue to the next period 9 (V2 > δ B π so When V2 is so high it is credible for the buyer to trade with seller 2 if seller 1 B ). 8 9

Lemma 5 provides the initial analysis of this case, and it is then applied in Lemmas 7 and 8. Lemma 4 provides the initial analysis of this case, and it is then applied in Lemmas 9-12.

7 Page 7 of 42

us

cr

ip t

rejects the buyer’s offer, by demanding V2 − > δ B π so B that seller 2 will accept. Hence, it is as if the buyer can make a take-it-or-leave-it offer to seller 1, which enables the extraction of the entire surplus V1 . For such values of V2 , Figure 1 (b) shows the SPNE outcome varies based on what seller 1 must offer when sellers make offers. When δ B V1 > V2 , the buyer’s payoff is δ B V1 : seller 1 offers the buyer just enough not to reject all offers and get a payoff of V1 next period, and trade with seller 2 is not a credible threat. When δ B V1 < V2 , the buyer’s payoff is V2 : seller 1 offers the buyer just enough not to trade with seller 2, which is a credible threat rather than rejecting all offers and getting a payoff of V1 next period. Finally, for intermediate values of V2 the stationary SPNE outcome reflects the remaining case in which V2 equals the buyer’s net present value of letting play continue to the next period 10 The condition V = δ π so that is relevant when the buyer makes offers actually (V2 = δ B π so 2 B B B ). ³ ´ V2 pins down what the buyer’s payoff must be when the sellers make offers π so = B δ B . Moreover,

M

an

when the sellers make offers seller 1 is not constrained by seller 2, because δVB2 > V2 , but instead follows the common equilibrium feature in bilateral negotiation models of offering the buyer just enough to induce trade. Specifically, when the sellers make offers, the buyer’s payoff equals the net ¡ ¢ bo present value of what it expects to get next period π so B = δ B³π B . This ´ equilibrium requirement V2 bo pins down what the buyer must receive when it makes offers π B = δ2 . Interestingly, the buyer B

te

d

does not follow the same approach when it makes offers: seller 1’s payoff V1 − δV22 is less than the B net present value of what it expects to get next period. Instead, seller 1’s payoff is constrained by the requirement that the buyer not prefer to trade today with seller 2, which is equivalent to the net present value of the buyer’s payoff from waiting until the next period (V2 versus getting δVB2 next period). At this point it is worth returning to the assumption that the buyer is not committed to trade if at least one of its proposals is accepted. If the buyer were so committed, then with its proposals the buyer always could extract V1 from seller 1. To see why, imagine the buyer offered > 0 to sellers 1 and 2. In any equilibrium, seller 2 accepts whenever seller 1 rejects, which forces seller 1 to accept rather than receive a payoff of 0. Letting → 0 completes the argument, which illustrates that such commitment effectively allows the buyer to always make seller 1 a take-it-or-leave-it offer. The problem is that the buyer might prefer not to trade with seller 2, if by doing so the buyer’s payoff V2 − is less than what the buyer would receive if the negotiations continued to the next period. In the high range of V2 described earlier (V2 > δ B π so the buyer B ), for sufficiently small actually prefers trading today with seller 2 rather than letting the negotiations continue, so there is no sense in which the buyer has committed to a trade it prefers to reject.11 The assumption that the buyer makes offers to multiple sellers also affects the surplus extraction

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

10

Lemma 6 provides the initial analysis of this case, and it is then applied in Lemmas 13 and 14. While imposing commitment is problematic, it is important to note that offer-retraction creates another problem. In bilateral negotiations Muthoo (1999, Ch. 7.3) shows that letting players retract accepted proposals makes any split of the surplus and any finite delay feasible as a SPNE. The same outcome seems likely in my model, including SPNE in which the buyer receives less than V2 . I avoid this problem by assuming future payoffs are invariant to current play, whereas such conditional moves are crucial to Muthoo’s result. Reconciling the problems of commitment and offer-retraction seems like a valuable contribution. 11

8 Page 8 of 42

ip t

just discussed. If the buyer can make an offer only to one seller at a time, then clearly in that period there is no other seller with whom the buyer can credibly threaten to trade if the first seller rejects the buyer’s offer. Therefore, the buyer cannot fully extract V1 from seller 1. Figure 2 reveals that Theorem 1’s SPNE payoffs have some unusual properties with respect to discounting, relative to the conventional wisdom derived from bilateral negotiation models. As a function of the discount factor δ B , the figure plots the buyer’s stationary SPNE payoff in multilateral negotiations (solid line) and in bilateral negotiations (dashed line).

cr

Figure 2 Here

te

d

M

an

us

Panel (a) considers subgames that begin with offers by the buyer. When the buyer is very impatient it extracts the entire surplus V1 from seller 1, because the buyer is so impatient it can credibly commit to trade this period with seller 2 if seller 1 rejects the buyer’s offer.12 This corresponds to the condition V2 > δ B π so For intermediate values of δ B the B described earlier. buyer does worse when it becomes more patient, which contrasts with conventional wisdom. This corresponds to the condition V2 = δ B π so B . Finally, for high δ B the buyer’s payoff is the same in multilateral and bilateral negotiations, because the buyer is sufficiently patient that it can protect its interests without the threat of trading with seller 2. This corresponds to the condition V2 < δ B π so B. Note that the buyer’s payoff at the beginning of this range of high δ B is strictly less than its payoff for very low δ B . Only as δ B approaches 1 does the buyer’s payoff return to the same level as when δ B was small. Panel (b) considers subgames that begin with offers by the sellers. For low δ B seller 1 expects the buyer to extract the entire surplus V1 next period (when V2 > δ B π so B ), so to get a positive payoff seller 1 must offer at least δ B V1 . However, if V2 > δ B V1 , then the threat from seller 2 requires seller 1 to offer V2 . Hence, for low δ B the buyer’s payoff is V2 , and eventually switches to δ B V1 as δ B increases. Once again, for intermediate δ B the buyer does worse as it becomes more patient (when V2 = δ B π so B ), and for high δ B the buyer’s payoff is the same in multilateral and bilateral negotiations (when V2 < δ B π so B ). Corollary 1 evaluates the stationary SPNE outcomes from Theorem 1 as the time between offers ∆ → 0. Muthoo (1999, Ch. 3.2) suggests this limiting case is the appropriate one to consider, because a party making a counteroffer has incentives to do so quickly to reduce its cost of delay. The limiting values of the payoffs as ∆ → 0 also do not depend on who makes the initial offers, so there is no first-mover advantage in the limit. Letting ∆ → 0 simplifies matters by eliminating two of the relevant ranges for V2 from Theorem 1, and the stationary SPNE payoffs have an intuitively appealing form.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

12 As mentioned earlier, if the buyer were required to trade according to its best accepted offer, then the buyer could always extract V1 when making offers: Seller 2 accepts any offer giving it a strictly positive payoff whenever seller 1 rejects, so seller 1 cannot reject an extremely unfavorable offer.

9 Page 9 of 42

Corollary 1 As the time between offers ∆ → 0, in all stationary SPNE outcomes in all subgames, the payoffs for the buyer and seller 1 approach h ³ h ³ ´ i ´ i rB r1 ∗ = min V − V , π ∗B = max V2 , r1 +r V and π 1 1 2 1 r1 +rB V1 . B

ip t

Players’ stationary SPNE payoffs in a subgame do not depend on who makes offers.

te

d

M

an

us

cr

Corollary 1 fully characterizes the buyer’s and seller 1’s payoffs in all stationary SPNE outcomes as the time between offers ∆ → 0, with the division of the maximum surplus V1 depending on whether seller 2 constrains the negotiations. If V2 is large enough to constrain the negotiations, then the buyer’s payoff is higher than it would be if seller 2 were not present, and seller 1’s payoff is lower. This is direct evidence of the benefit to the buyer of adding another seller to an initially bilateral setting. If V2 is too small to constrain the negotiations, then the buyer’s and seller 1’s payoffs equal those in Rubinstein’s model. Seller 2’s impact is analogous to the Outside Option Principle found in models of bilateral negotiations with exogenously-specified outside options, such as Binmore et al. (1989). Section 3 discusses this relationship further. How players’ stationary SPNE payoffs vary with the discount rates also depends on whether seller 2 constrains the negotiations. If V2 is large enough to constrain the negotiations, then changes in the buyer’s and seller 1’s discount rates do not affect the players’ payoffs. For example, one special case of seller 2 constraining the negotiations is when V1 = V2 , in which case the buyer gets the entire surplus and the discount rates are irrelevant. If V2 is too small to constrain the negotiations, then standard comparative statics emerge: as the buyer becomes more patient it does strictly better in the negotiations and seller 1 does strictly worse, while the reverse holds as seller 1 becomes more patient. In Section 4 I use the payoffs π ∗B and π ∗1 to evaluate the buyer’s choice of conducting procurement via multilateral negotiations or an auction. The limiting case seems appropriate given the points raised just prior to Corollary 1, and the unique stationary SPNE outcome is a natural one to consider. Before proceeding to that analysis, I place my model in context.

3

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Relationship to Prior Research

My model of multilateral negotiations fits into a broader literature that extends game-theoretic models of bilateral negotiations to analyze negotiations involving three or more parties. In this section I provide context for my model by describing representative complete-information models from this broader literature. I begin with models of general multi-party negotiations, to illustrate the variety of scenarios and issues that have been considered. Next I compare my model’s structure and predictions to those from alternative models of multilateral negotiations, and from bilateral negotiation models in which one player has an exogenously-specified outside option. The latter models can be viewed as another way to model multilateral negotiations.

10 Page 10 of 42

3.1

Models of General Multi-Party Negotiations

te

d

M

an

us

cr

ip t

Despite the interest in and importance of bilateral negotiations, researchers recognized that many interesting negotiation scenarios involve three or more players. As a result, several game-theoretic models have been developed to analyze a variety of multi-party negotiations. Krishna and Serrano (1996) analyze multiple players bargaining to split one joint surplus, which is one natural extension of two parties negotiating to allocate a single surplus. Negotiations begin with player 1 offering publicly-observed portions of the surplus to the other players, who make simultaneous decisions whether to accept or reject. Any players who accept receive immediately the portion they are offered (from the offering player) and exit the negotiations. If not all players accept, then in the next period a remaining player offers publicly-observed portions of the remaining surplus to the other remaining players. Play continues in this fashion, potentially indefinitely, until all players have reached an agreement. The authors show that this game with “exit” by players who accept an offer has a unique SPNE, which contrasts with the multiplicity of SPNE found in related models that require unanimous acceptance for any negotiated payoffs to be awarded. Osborne and Rubinstein (1990, Ch. 3.13) discuss models of the latter type and provide references to other early analyses of this negotiation scenario, while Herings, et al. (2017) provide a recent treatment of such models. Penta (2011) considers an exchange economy with an arbitrary number of players who each have strictly positive endowments of each of an arbitrary number of goods. Players can differ in their endowments, discount factors, and utility functions. Each period a player is chosen to propose prices and player-specific “maximum trading constraints” that limit each player’s quantity demanded of each good, the other players accept or reject in a specified order, and agreement requires unanimous acceptance. Upon agreement the non-proposing players demand or supply quantities of each good to create their best affordable consumption bundle, and the proposer is the residual claimant who clears the market. Within this broad class of bargaining protocols, the author shows that stationary SPNE implement Walrasian allocations if the players are sufficiently patient. Manea (2011) and Abreu and Manea (2012a,b) analyze the effect of network structure on negotiated outcomes when only connected agents can trade. This approach refines the exchange approach in Penta (2011) by only allowing certain trades to occur. Players share a common discount factor, and all pairs of connected agents can jointly produce an identical surplus, but these assumptions can be relaxed at the price of a more complex analysis. Each period a randomly selected player in a randomly selected pairwise link makes a proposal to its linked partner. If agreement is reached, both players leave the network and play continues next period. Manea (2011) considers a stationary environment in which the departing players are replaced by identical players, while Abreu and Manea (2012a,b) consider a non-stationary environment in which the departing players are not replaced. If agreement is not reached, then play moves to the next period, beginning with another random draw of a link and a proposer. Equilibrium outcomes depend not only a player’s number of links in the network, but also on the nature of the links of

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

11 Page 11 of 42

3.2

te

d

M

an

us

cr

ip t

that player’s linked partners. Specific results differ across the models, but in general the authors show that each player’s equilibrium payoff is uniquely determined for every discount factor, and they characterize the players’ equilibrium payoffs as the players become extremely patient. Ray (2007) provides an overview of research that considers coalition formation in multi-party negotiations in which a range of diverse coalitional structures all can create value. In principle each player can engage in profitable trades simultaneously with multiple other players. Moreover, there can be externalities, in the sense that the value of a particular coalition can depend on the coalitional structure that emerges amongst other players. Each period a specified player proposes a coalition and associated payoffs to the coalition’s members, those members sequentially choose whether to accept or reject the proposal, and agreement requires unanimous acceptance. Within this framework one can characterize SPNE payoffs for arbitrary coalitional values. In terms of payoff structure, the general formulation of coalitional values in the approach described by Ray (2007) can accommodate all of the preceding ones, except for the stationary structure of Manea (2011) in which players who reach agreement are replaced by identical players. The payoff structure from models like Krishna and Serrano (1996) corresponds to allowing only the grand coalition of all players to generate value; from exchange models like Penta (2011) corresponds to having payoffs from single-player and multi-player coalitions depend on the coalition members’ endowments; and from network models like Manea (2011) and Manea and Abreu (2012a,b) corresponds to allowing only coalitions of linked players to generate value. Of course, differences across models in the extensive-form of the bargaining protocols can lead to dramatically different predictions for outcomes such as payoffs, the division of gains from trade, the efficiency of negotiated agreements, and the time to reach agreement.

Alternative Models of Multilateral Negotiations

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Earlier research considers various extensions of bilateral negotiation models to address the idea of one party negotiating to trade with one of several others. The following representative examples use models with complete information to ask questions similar to mine, and I compare their structure and predictions with the model described in Theorem 1.13 The two modeling approaches closest to mine are described in Osborne and Rubinstein (1990) and Ray (2007). Our three approaches use different bargaining protocols that lead to qualitatively distinct outcomes, but my model’s outcomes seem the most intuitively appealing. Osborne and Rubinstein (1990, Ch. 9.3) model a buyer negotiating to trade with one of two sellers offering surpluses V1 and V2 ∈ (0, V1 ] to be split.14 Play alternates between simultaneous offers from the buyer and to the buyer. Seller 1 responds to the buyer’s offer before seller 2 does, 13

Other papers that share some similarities include Reinganum and Daughety (1991, 1992), Muthoo (1995), Chatterjee and Dutta (1998), and Marx and Shaffer (2010). Two papers that consider multilateral negotiations with incomplete information are McAdams and Schwarz (2007) and Thomas (2011). See Thomas (2011) for references to additional papers with incomplete information that consider some aspects of the multilateral negotiation problem analyzed there. 14 Players have the same discount factor, but it might be easy to generalize the model in this respect.

12 Page 12 of 42

us

cr

ip t

and the buyer is required to trade with the first seller to accept its offer. The buyer is required to demand the same payoff from both sellers, a curious restriction that dramatically affects the negotiated outcome. Without that restriction, one can show that the buyer’s offers extract the entire surplus V1 from seller 1, precisely because the buyer must trade if at least one seller accepts (even if the buyer would prefer not to trade). In the limit as the time between offers ∆ → 0, the buyer obtains the entire surplus V1 in every subgame, regardless of who makes offers. This result holds even if V2 is arbitrarily small, while in my model seller 2 is strategically irrelevant when it is such a poor substitute for seller 1. While assumptions need not match real-world protocols for a model to be useful, one must be wary if such assumptions harm the party on whom they are imposed. As Binmore (1985) states when introducing the model underlying the Osborne and Rubinstein model:

an

(p. 270): Of course, if a particular bargaining model imposes constraints on a player’s behavior that he would prefer to violate and no mechanism exists in the situation one is trying to model that would prevent such violations, then little insight can be expected from the model.

M

(p. 283): The only firm principle would seem to be that one cannot expect players to submit to constraints that limit their payoffs unless there is some mechanism that forces the constraints on them.

te

d

Hendon and Tranaes (1991) also model a buyer negotiating to trade with one of two sellers offering surpluses V1 and V2 ∈ (0, V1 ] to be split, but each period the buyer is randomly matched with one seller and a randomly chosen player makes a single offer. The authors find an equilibrium of the game only for certain parameter values, but more importantly the random assignment and lack of simultaneous competition enormously impact the negotiated outcome. For example, if the sellers are identical and the common discount factor is very small, then the buyer’s payoff is very small if its matched seller gets to make the offer. Likewise, if the sellers differ, then the negotiated outcome can be inefficient because the buyer might trade with seller 2 rather than with seller 1. The approach in Ray (2007) can accommodate one party that wants to trade with one of several others, but the negotiation protocol avoids the key features of the multilateral negotiations I consider: holding multiple offers, and the ability to hold an offer while seeking concessions. Specifically, the buyer can make only one offer in a period in which it makes offers, and likewise can receive only one offer in a period in which sellers make offers. Using Ray’s protocol, a buyer negotiating to trade with one of two sellers with available surpluses V1 and V2 ∈ (0, V1 ] obtains a payoff equal to its payoff in bilateral negotiations with seller 1.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Introducing seller 2 to an initially bilateral setting has no impact on the negotiated outcome, even if seller 2’s available surplus is identical to seller 1’s. In my model, seller 2 matters if and only if it is a sufficiently close competitor to seller 1. The reason for Ray’s result is that only one offer can exist at a time. For example, the buyer can make an offer to seller 2 only after rejecting seller 1’s offer. If seller 2 turns down that offer, 13

Page 13 of 42

te

d

M

an

us

cr

ip t

then seller 2 makes an offer to the buyer. Only upon rejecting seller 2’s offer can the buyer return to seller 1. Consequently, there is no way for the buyer to leverage seller 2’s existence to obtain concessions from seller 1, when either making or receiving an offer. Other models with endogenous outside options assume the sellers are identical, and it is not obvious how to solve them with asymmetric surpluses or discount rates. In contrast, my model has straightforward solutions in such asymmetric settings. Moreover, my model’s predicted outcomes qualitatively differ from those models’, in the only comparable case of identical sellers: In the models described below, the buyer in most or all cases receives less than the entire surplus, whereas in my model the buyer receives the entire surplus due to the fierce competition among the sellers.15 This important difference suggests similar qualitative differences will emerge if one could solve asymmetric versions of the models. Shaked and Sutton (1984) model bilateral negotiations with alternating offers. The buyer bargains with one seller at a time, but can switch to another identical seller after a commonly known length of time. Offers from one seller are void upon switching to another seller, and after switching the buyer makes the first offer. The authors find that the presence of another seller constitutes a credible threat that permits the buyer to obtain more surplus than if switching were impossible. In equilibrium the buyer does not receive the entire surplus, even as the time between offers ∆ → 0, but the amount the buyer gets increases as the switching delay shrinks. As the switching delay approaches the length of a bargaining period, the buyer’s payoff approaches the entire surplus, because in that limiting case the buyer, after receiving an offer, can instantaneously make an offer to another seller.16 Essentially, the model introduces a form of simultaneity that occurs only occasionally. Vincent (1992) considers simultaneous negotiations by adding an identical seller to a bilateral negotiation setting in which only the sellers make offers. He finds that the buyer might not receive the entire surplus, due to the sellers’ ability to support collusive equilibria through the prospect of potentially an infinite number of bargaining periods. My model allows both the buyer and the sellers to make offers, which crucially enables a seller to deviate profitably from collusive behavior by accepting an offer that ends the game. Manea’s (2011) analysis of the effect of network structure on negotiated outcomes describes a special case with one buyer connected to N identical sellers. Because successful traders are replaced in the network, a seller excluded from the current trade is not excluded from a future trade. Like Hendon and Tranaes (1991), the model assumes that each period a randomly chosen player makes a single offer. Once again the random matching and the restriction to one offer at a time affect the negotiated outcome. The sellers have positive payoffs that decrease as the number of sellers increases, because a seller who fails to trade when matched has a longer expected waiting time to be matched again.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

15

An exception is Corominas-Bosch (2004), but she uses the model from Osborne and Rubinstein (1990, Ch. 9.3). With asymmetric sellers, in this limiting case it seems likely that model has the same unappealing outcome as that for the model of Osborne and Rubinstein (1990, Ch. 9.3): the buyer extracts the entire surplus from the superior trading partner, even if the surplus from the inferior trading partner is arbitrarily small. 16

14 Page 14 of 42

3.3

Bilateral Negotiation Models with Exogenous One-Sided Outside Options

M

an

us

cr

ip t

Bilateral negotiation models with exogenous outside options can be interpreted as allowing for trade with different sellers to generate different amounts of surplus to be split, if the buyer’s outside option gives a payoff V2 that differs from the surplus V1 available from trading with the sole strategic seller. While considering exogenous outside options is not necessarily worse from a modeling perspective geared toward prediction, there are at least two reasons to prefer my model of endogenous outside options. First, my model illustrates how the outside option is determined, that the payoff from the outside option represents the entire surplus available from that alternative trade, and that the patience of the alternative trading partner is irrelevant. Second, applications like those in Sections 4 and 5 make sense only if the outside options result from decisions by strategic actors. Binmore et al. (1989) modify Rubinstein’s bilateral negotiation model so that the buyer can take up a commonly known and exogenously specified outside option only when it rejects the seller’s offer, while Muthoo (1999, Ch. 5.6) assumes the buyer can take up its outside option only after its offer is rejected. Shaked (1994) allows the buyer to take up its outside option after either player rejects, but in each period Nature randomly determines which player makes an offer. All three models’ equilibrium payoffs share similarities with those in Theorem 1 if the payoff from the exogenous outside option equals V2 . In general, the outside option matters when it is large, but

4

te

d

not when it is small. In another vein, Chatterjee and Lee (1998) model bilateral negotiations with complete information in which the buyer can hold an offer from one seller while it incurs a cost to acquire an offer from another seller. Unlike in my model, the competing offer is drawn from a probability distribution rather than being the outcome of strategic interaction.

Multilateral Negotiations Versus Auctions

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

In practice the buyer chooses how to conduct procurement, which is one reason economists emphasize understanding the performance of different exchange mechanisms.17 In this section I use Corollary 1’s unique stationary SPNE payoffs to compare two commonly-used procurement mechanisms: multilateral negotiations and auctions. Bulow and Klemperer (1996) analyze a buyer’s choice between negotiations and auctions in a setting with homogeneous products and incomplete information about sellers’ costs. They illustrate circumstances in which a buyer prefers a simple second-price auction with N +1 sellers to an optimal mechanism with N sellers. The authors do not analyze a specific negotiation protocol, but let the optimal mechanism provide an upper bound on the buyer’s payoff from negotiating. Their results suggest that attracting more sellers can be more important than increasing the buyer’s bargaining 17 For example, consider how the work by Chamberlin (1948) and Smith (1962) spurred research on the importance of institutions in determining economic outcomes.

15 Page 15 of 42

M

an

us

cr

ip t

strength.18,19 Prompted partly by observing multilateral negotiations and auctions being used by similar buyers in the same market, Thomas and Wilson (2002, 2005, 2014) use experiments to compare both exchange mechanisms. In settings with incomplete information they consider first-price and second-price auctions, homogeneous and differentiated products, and negotiations that differ in the buyer’s ability to provide a seller with verifiable information about rival sellers’ offers. The authors find that the mechanisms’ outcomes are similar, and in some cases are statistically indistinguishable. They conducted experiments because no formally-solved models existed that considered the sort of negotiations that interested them. I evaluate the buyer’s choice by extending Section 2’s basic model in ways that make the institutional comparison nontrivial; it will be evident that in the basic model the buyer weakly prefers multilateral negotiations. This section’s goal is to provide some simple extensions with theoretical and empirical insights, and that also suggest more complex or comprehensive extensions that might merit further study. In particular, this analysis might shed light for those wishing to explore the relationship between auctions and multilateral negotiations in settings with incomplete information. For example, the experiments by Thomas and Wilson (2002, 2005, 2014) found that relationship was affected by the number of sellers, in part because the number of sellers affected the expected gap between V1 and V2 .

te

d

Three phases comprise the extended game: In the surplus-determination phase Nature determines the Vi draws according to a commonly known probability distribution. The planning phase consists of three choices that occur in a specific order: First, the buyer chooses publicly between conducting multilateral negotiations or a first-price auction. Second, the buyer chooses publicly which of the N sellers to invite to participate. Third, the invited sellers make simultaneous and public decisions whether to participate. In the procurement phase the participating sellers compete via either the multilateral negotiations from Section 2, or a first-price auction in which each seller i makes a simultaneous price offer pi ∈ [0, Vi ] to the buyer. The buyer accepts the offer giving the highest payoff, Vi − pi , with ties handled as in the multilateral negotiations. Kim and Che (2004) establish that the buyer’s payoff in the first-price auction is the second-highest surplus from the set of participating sellers, and that it is the same in a Dutch, English, or second-price auction.20 The procurement phase occurs last in the extensions that follow, but either the surplus-determination or the planning phase can occur first. Those two phases’ order determines whether the players’ planning choices occur after or before they know the available surpluses.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

18 I do not address this issue in my analysis, but it is straightforward to create examples with the opposite conclusion; this suggests the importance of incomplete information in making such comparisons. 19 More recently, Bulow and Klemperer (2009) consider this choice when the mechanisms involve costly entry but differ in their timing. Their auctions involve simultaneous entry decisions followed by price competition. Their negotiations involve sequential entry by sellers, in which later sellers enter only if trade with earlier sellers is insufficiently attractive. 20 I assume the buyer does not have the commitment power to conduct an optimal auction, which in this setting would have the buyer impose a reserve price of 0 for seller 1. If the buyer attempted such a move without the necessary commitment power, then seller 1 and others presumably would make counteroffers, which would effectively lead to multilateral negotiations.

16 Page 16 of 42

4.1

Institutional Choice: Fixed Vi

=

π ∗B

−C

mln

∙ µ = max V2 ,

r1 r1 + rB

¶

¸

V1 − C mln ,

cr

π mln B

ip t

Suppose the surplus-determination phase precedes the planning phase and the buyer incurs an incremental cost of conducting multilateral negotiations versus an auction, C mln ∈ R. Because participation is costless, I assume the buyer invites all sellers and that all sellers participate. The only substantive difference from the basic model is the cost C mln , which is positive or negative depending on which mechanism costs more to conduct.21 From Corollary 1 the buyer’s payoff in multilateral negotiations is

us

while its payoff in a first-price auction is π fBpa = V2 . The following result is straightforward.

∙ µ < max 0,

r1 r1 + rB

¶

¸

V1 − V2 .

M

C

mln

an

Proposition 1 If the surplus-determination phase precedes the planning phase, then the buyer strictly prefers multilateral negotiations if and only if

d

The buyer’s relative preference for multilateral negotiations intensifies as C mln decreases, as V1 increases or V2 decreases, and as rrB1 decreases.

te

This simple extension of the basic model illustrates what factors incline the buyer to use multilateral negotiations. Figure 3 shows the buyer’s payoff in both institutions as a function of the ratio of discount rates, rrB1 , for some positive incremental cost of conducting multilateral negotiations, C mln . Each panel shows a parameter change that increases the range of rrB1 for which the buyer prefers multilateral negotiations. Panel (a) shows that decreasing C mln has the obvious effect of increasing multilateral negotiations’ attractiveness, because they become relatively cheaper to conduct. Panel (b) shows that increasing V1 increases the buyer’s payoff from multilateral negotiations while leaving unchanged its payoff from a first-price auction. Increasing V1 in multilateral negotiations increases the amount of surplus to be split between the buyer and seller 1, but in the auction has no effect because the buyer’s payoff is V2 . Panel (c) shows that decreasing V2 lowers the buyer’s payoff in both institutions, but to different extents. In the auction decreasing V2 reduces the buyer’s payoff one-for-one, because the buyer’s payoff is V2 . In multilateral negotiations decreasing V2 leaves the buyer’s payoff unchanged in the original instances in which seller 2 is irrelevant to the negotiated outcome, decreases the buyer’s payoff less than one-for-one in a range of cases in which seller 2 initially constrained the negotiations, and otherwise decreases the buyer’s payoff one-for-one. The net effect makes multilateral negotiations more attractive for a larger range of relative discount rates, rrB1 .

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

21

One also could include costs incurred by the buyer for each seller that participates. In Subsection 4.3 I consider costs incurred by each seller that participates.

17 Page 17 of 42

Figure 3 Here Finally, each panel of Figure 3 illustrates that multilateral negotiations are more attractive as decreases, which is a crucial element of the buyer’s choice between negotiations and an auction. Reducing rrB1 increases the buyer’s bargaining strength relative to seller 1, and by conducting an auction the buyer forgoes any bargaining advantage it might have.

4.2

ip t

rB r1

Institutional Choice: Random Vi and Costless Participation

d

M

an

us

cr

Now suppose the planning phase precedes the surplus-determination phase. The available surpluses are unknown when the buyer chooses the procurement mechanism and which sellers to invite, and when the sellers make participation decisions. For simplicity I assume the Vi are independently and identically drawn from U [0, 1]. I also assume ri = rS for each seller i, because otherwise the details of the institutional choice depend in a straightforward but tedious way on the ri and the realized Vi . Finally, I assume the buyer invites all sellers and that all sellers participate, because participation is costless. The buyer’s payoff in multilateral negotiations depends on the highest and second-highest realized Vi , while its payoff in a first-price auction depends on the second-highest realized Vi . To calculate the buyer’s expected payoffs one therefore must consider distributions of the highest and second-highest order statistics of N draws from U [0, 1]. Denoting those order statistics by V(1) and V(2) , the distribution of V(1) is

te

¡ ¢ G1 (v) ≡ Pr V(1) ≤ v = v N ,

the distribution of V(2) conditional on the value of V(1) is

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

¡ ¢ vN−1 G2 (v | w) ≡ Pr V(2) ≤ v | V(1) = w = N −1 , w

and the unconditional distribution of V(2) is

¡ ¢ G2 (v) ≡ Pr V(2) ≤ v = v N + N (1 − v)v N−1 .

Corollary 1 gives the buyer’s payoff for any particular realization of the Vi , so the buyer’s expected payoff in multilateral negotiations is π mln B

=

∙Z

0

=

1 ½Z v(1) 0

N −1+

³

∙ µ max v(2) ,

rS rS +rB

N +1

´N

rS rS + rB

¶

¸

v(1) G02

¾ ¸ ¡ ¢ ¡ ¢ 0 v(2) | v(1) dv(2) G1 v(1) dv(1) − C mln

− C mln .

18 Page 18 of 42

Similarly, the buyer’s expected payoff in a first-price auction is π fBpa

=

Z

1

0

¡ ¢ N −1 . v(2) G02 v(2) dv(2) = N +1

ip t

Not counting the incremental cost C mln , notice that the buyer’s expected negotiated payoff equals its expected auction payoff, plus an extra term that reflects the likelihood that the realizations of V(1) and V(2) are such that seller 2 is irrelevant to the negotiated outcome.

rS rS +rB

´N

N +1

.

us

C mln <

³

cr

Proposition 2 If the planning phase precedes the surplus-determination phase and the Vi are iid draws from U [0, 1], then the buyer strictly prefers multilateral negotiations if and only if

an

The buyer’s relative preference for multilateral negotiations intensifies as C mln decreases, as N decreases, and as rrBS decreases.

te

d

M

Proposition 2 illustrates that the buyer tends to prefer multilateral negotiations for low values of C mln , N, and rrBS . Decreasing C mln has the obvious effect of making multilateral negotiations relatively cheaper to conduct. Decreasing N increases the expected difference between the highest and second-highest surpluses, which Corollary 1 demonstrates makes it more likely that negotiations outperform an auction. Likewise, decreasing rrBS strengthens the buyer’s bargaining position, an advantage it forgoes by conducting an auction. One also can assess how the variance of the Vi draws affects the buyer’s institutional choice, say by considering mean-preserving changes in the distribution. A formal analysis is messy, even with the Vi uniformly distributed, but the following informal argument suggests that increasing the variance increases the relative attractiveness of multilateral negotiations. £1 ¤ £ 1¤ 1 Suppose the V are iid draws from U − θ, + θ for θ ∈ 0, 2 . If θ is small enough that i 2 2 ³ ´¡ ¢ rS 1 1 2 − θ ≥ rS +rB 2 + θ , then seller 2 constrains the negotiations for any realizations of the Vi . ¡ ¡ ¢ ¢ That is, even for the highest possible V(1) = 12 + θ and the lowest possible V(2) = 12 − θ , V(2) exceeds the buyer’s share of V(1) from bilateral negotiations. Consequently, the buyer’s expected payoff is the same with multilateral negotiations or an auction, not counting any differential cost of conducting negotiations.22 This informal analysis suggests negotiations are more likely to be used when sellers are more likely to differ in their available surplus, such as when the buyer’s tastes tend to be quite distinct for different sellers’ products, or when there is significant cost variability across sellers. Multilateral negotiations outperform auctions in such cases because the buyer’s bargaining ability strongly influences the negotiated outcome, and that ability is wasted by using an auction.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

22 The informal analysis is messy for larger values of θ, because in calculating the buyer’s expected payoff one must account for whether the realizations of the Vi are such that seller 2 constrains the negotiations.

19 Page 19 of 42

4.3

Institutional Choice: Random Vi and Costly Participation

³

rS rS +rB

´N

M

π mln S (N ) =

1−

an

us

cr

ip t

Finally, consider extending the preceding subsection’s analysis by having sellers incur a cost of participating before they learn the Vi draws, C entry ∈ R+ . This cost can reflect opportunity costs of designing prototypes, evaluating production costs, or assessing the product’s fit with the buyer’s preferences. I assume C mln = 0 to focus attention on the uncertainty when decisions are made. Costly entry has been considered in other oligopoly models. For example, costly entry is examined in a Bertrand setting by Lang and Rosenthal (1991), in a Cournot setting by Dixit and Shapiro (1986), and in an auction setting by Levin and Smith (1994). A common finding is that the buyer might benefit from decreasing the size of the pool of potential entrants, which in my extended model amounts to inviting fewer than N sellers to participate. As a first step I consider the sellers’ participation decisions when the buyer invites all N sellers to participate, for each institution x ∈ {mln, fpa}. Let π xS (k) denote a seller’s expected payoff from participating in institution x when a total of k sellers participate. If C entry < π xS (N ), then each seller participates with probability 1 because it expects a positive net payoff even if all of its rivals participate. With the Vi drawn from U [0, 1],

N (N + 1)

and π fSpa (N ) =

1 . N (N + 1)

te

d

f pa Because π mln S (N) < π S (N ), in multilateral negotiations sellers stop participating with probability 1 at a lower value of C entry than in an auction. If C entry > π xS (1), then each seller participates with probability 0 because it expects a negative net payoff even if no rivals participate. With the Vi drawn from U [0, 1],

µ

rB rS + rB

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

π mln S (1)

=

¶µ ¶ 1 2

1 and π fSpa (1) = . 2

f pa Because π mln S (1) < π S (1), in multilateral negotiations sellers stop participating entirely at a lower value of C entry than in an auction. For later reference note that these values of C entry are invariant to the number of invited sellers, because they are determined solely from a seller’s expected payoff when it is the only participating seller. Finally, if π xS (N ) ≤ C entry ≤ π xS (1), then I consider strategies such that each seller participates with probability ρx ∈ (0, 1). The entry probability depends on the number of invited sellers and the value of C entry .23 The equilibrium value of ρx equates a seller’s expected payoff from participating and not participating. Specifically, ρx solves

# ! "N −1 Ã X N −1 x k x n−1−k x π S (k + 1) − C entry = 0. (ρ ) (1 − ρ ) k k=0 23 While there exist asymmetric equilibria in which a subset of invited sellers participate with probability 0, I focus on symmetric equilibria because of the strategic environment’s symmetry.

20 Page 20 of 42

Once the entry probability is derived, the buyer’s expected payoff is π xB

=

N X k=0

Ã

N k

!

(ρx )k (1 − ρx )n−1−k π xB (k) ,

an

Figure 4 Here

us

cr

ip t

where π xB (k) denotes the buyer’s expected payoff in institution x when k sellers participate. Even with the simple expressions for π xB (k) that precede Proposition 2 (substituting k for N as needed), the equilibrium values of ρx and π xB do not have convenient analytic expressions. However, ρx can be solved numerically, from which π xB is easily calculated. Figure 4 reports π mln and π fBpa as a function of C entry , with panels (a)-(c) featuring different B numbers of invited sellers. Each panel shows π mln for three values of rrBS , while π fBpa is invariant B to the discount rates.

te

d

M

The first insight from Figure 4 is that the buyer strictly prefers multilateral negotiations for low entry costs, and strictly prefers a first-price auction for high entry costs. From Proposition 2 the buyer prefers multilateral negotiations over an auction for any specific number of participants when C mln = 0, and both institutions’ efficiency implies that the sellers by contrast prefer an auction. For sufficiently low entry costs, all invited sellers participate in both institutions. Therefore, the buyer prefers multilateral negotiations. As the entry cost increases, sellers in multilateral negotiations participate less frequently than they do in an auction. The decreased participation harms the buyer directly in the negotiations because with fewer sellers the expected value of the highest surplus declines. Eventually this decline swamps the buyer’s advantage from its bargaining entry is such that no sellers enter with negotiations, ability. This effect is abundantly clear when C ³ ´ entry < π f pa (1) . but some sellers might enter with an auction π mln (1) < C S S Finding that the buyer’s preferred procurement method depends on entry costs illustrates an important reason for considering costly entry. Levin and Smith (1994) found no such effects in their analysis of costly entry with different auction formats, because the formats they considered all were revenue-equivalent for any particular number of actual entrants. Hence, the sellers’ entry probabilities were the same across institutions. In contrast, the fact that payoffs in multilateral negotiations and a first-price auction differ for any specific number of participating sellers causes the buyer’s preference to depend on the level of C entry . The second insight from Figure 4 is that the value of C entry at which the buyer’s institutional choice changes increases as the buyer’s relative bargaining strength diminishes. Extreme values of rB rB entry > 0, rS illustrate this point clearly. As rS → 0 the buyer prefers a first-price auction for any C because the buyer’s overwhelming bargaining ability ensures that no sellers participate if the buyer uses multilateral negotiations. As rrBS → ∞ multilateral negotiations and an auction become more similar for any specific number of participating sellers. Hence, the sellers’ participation probabilities also get more similar, and the value of C entry increases at which the buyer’s preferred choice changes.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

21 Page 21 of 42

an

us

cr

ip t

Careful examination of Figure 4 reveals that the buyer’s expected payoff is higher with fewer invited sellers, for some values of C entry . This result is consistent with findings mentioned earlier from Lang and Rosenthal (1991) and Levin and Smith (1994), and it stems from the interplay between the number of invited sellers and the invited sellers’ probability of participating. One implication of this finding is that the buyer’s ability to restrict the number of sellers it invites might eliminate the change in the buyer’s institutional choice that was highlighted above. If C entry is such that the buyer prefers an auction to multilateral negotiations for a certain number of invited sellers, then the buyer might still prefer multilateral negotiations after suitably reducing the number of invited sellers. Figure 5 illustrates the buyer’s expected payoff when it optimally chooses how many sellers to invite, restricting attention to a setting in which there are N = 10 sellers available. Each panel reflects a different ratio of the discount rates, rrBS . Each line in Figure 5 is the upper envelope from plotting the buyer’s expected payoff for each number of invited sellers for a particular institution, such as the three reported in Figure 4. Figure 5 Here

te

d

M

The first insight from Figure 5 is that, even optimizing over the number of invited sellers, the buyer prefers multilateral negotiations for sufficiently low values of C entry , and prefers a first-price auction for sufficiently high values of C entry . For very low values of C entry , all invited sellers participate even if all sellers are invited. In this case the buyer invites all ³sellers and prefers ´ f pa multilateral negotiations, as shown in Proposition 2. For values of C entry ∈ π mln (1), π (1) , S S no sellers participate in multilateral negotiations, but participate with positive probability in a first-price auction. Consequently, for such values of C entry the buyer prefers a first-price auction to multilateral negotiations. The second insight from Figure 5 is that the buyer’s institutional choice can change multiple times as C entry increases through intermediate values. For example, in panel (b) the buyer’s optimal choice is to invite 2 sellers and use multilateral negotiations when C entry = 0.14, to invite 2 sellers and use a first-price auction when C entry = 0.17, and to invite 1 seller and use multilateral negotiations when C entry = 0.23. Once C entry > 0.25 the buyer switches back to inviting 2 sellers and using a first-price auction. Finally, one could assess how the effects of costly entry are influenced by the variance of the Vi draws. An informal analysis similar to that in Subsection 4.2 suggests the two institutions are equivalent if the variance of the Vi draws is sufficiently low. Hence, entry costs would not affect the buyer’s institutional choice.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

5

Conclusions and Future Research

This paper develops a model of multilateral negotiations with complete information by adding players to one side of Rubinstein’s classic infinite-horizon, alternating-offers model of bilateral 22 Page 22 of 42

te

d

M

an

us

cr

ip t

negotiations, to help formalize negotiations that frequently occur in areas such as procurement, corporate takeovers, high-end job markets, and large-scale investment decisions. For concreteness I consider a buyer negotiating to purchase from one of several sellers offering potentially different amounts of surplus to be split, and the model’s unique stationary SPNE outcome involves the buyer trading immediately with the surplus-maximizing seller. I show that the presence of additional sellers in certain instances enables the buyer to receive higher payoffs than it otherwise would. In other instances the additional sellers are irrelevant, because the surpluses available from trading with them are too low for the buyer to credibly threaten to trade with them rather than with seller 1. In sharp contrast to bilateral negotiations, with multilateral negotiations the buyer’s payoff can be highest when it is extremely impatient, because its strong inclination to avoid delay forces each seller to compete vigorously so as not to lose the sale. This unexpected feature vanishes in the limit as the time between offers goes to 0, and the stationary SPNE outcome has an intuitively appealing structure that is consonant with the SPNE outcome from bilateral negotiations: The prospect of trade with seller 2 constrains the multilateral negotiations if and only if V2 exceeds what the buyer would obtain in bilateral negotiations with seller 1. I demonstrate one of the model’s many applications by analyzing the buyer’s choice to conduct procurement via multilateral negotiations or an auction. As a general matter the buyer prefers multilateral negotiations when there are few sellers, when the sellers’ products are distinct or their production costs differ greatly, and when the buyer is relatively patient. In such cases the buyer’s bargaining ability significantly affects the negotiations, and the buyer forgoes that ability by using an auction. However, the buyer prefers an auction if it is sufficiently costly for sellers to participate in the procurement process, because sellers’ anticipated low payoffs in multilateral negotiations make them less likely to participate than if the buyer used an auction. These differences in the buyer’s institutional preference are maintained if the buyer can strategically limit the number of sellers it invites to participate, and a new one emerges: For intermediate entry costs the buyer can prefer either institution, and the preferred choice can change multiple times as the entry cost increases. The analysis of the buyer’s choice helps explain why multilateral negotiations and auctions are used by similar buyers, an observation that spurred the experimental research by Thomas and Wilson (2002, 2005, 2014). In some instances the institutions might coexist because each gives the buyer the same expected payoff, such as when there is little variation in the surpluses available from different sellers. In other instances buyers might have distinct preferences that depend on factors including variation in the cost of sellers’ participation across buyers (say because of complexity in determining a seller’s fit with the buyer’s preferences), differences in the buyers’ bargaining ability relative to the sellers’, or differences in the buyers’ costs of conducting multilateral negotiations. The model illustrates a setting in which multilateral negotiations can be evaluated straightforwardly, and the following examples give a sense of the variety of ways in which it can be applied. One could evaluate horizontal mergers in markets where strategic interaction among firms involves

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

23 Page 23 of 42

6

an

us

cr

ip t

multilateral negotiations. The basic model is simple enough that one could include merger-specific efficiencies or changes in bargaining positions, changes in product offerings from the merging firms or their rivals, or entry by new firms. Likewise, one could consider collusion amongst sellers who repeatedly compete, where each negotiation is the stage game of a repeated game. The model’s emphasis on surplus allows consideration of cost differences across sellers, or of horizontal and vertical product differentiation. One also could consider incentives regarding dual-sourcing, sellers’ investments to reduce costs or improve their products’ goodness-of-fit with the buyer’s preferences, the buyer’s design of its purchasing requirements, or joint decisions between the buyer and each seller on relationship-specific investments. Finally, one could use the model as a starting point for designing experiments. In addition to comparing experimental outcomes to the model’s predictions regarding the available surpluses, the parties’ discount rates, or the number of sellers, one also can assess whether a structured model of multilateral negotiations can reasonably explain the outcomes of more-realistic unstructured negotiations.

Appendix

te

d

M

This appendix contains proofs of Theorem 1, Corollary 1, and Propositions 1 and 2. It begins with several lemmas that are used in the proof of Theorem 1. I will frequently refer to SPNE strategies by σ ≡ {σ B , σ 1 , σ 2 , ..., σ N }, where σ k denotes player bo k’s strategy in the infinite-horizon game, for k ∈ {B, 1, . . . , N }. Let π so k and π k respectively denote player k’s stationary SPNE payoffs in subgames that begin with offers from the sellers and from the buyer, for some SPNE σ. Lemma 1 In subgames beginning with offers from the sellers, all SPNE featuring stationary outcomes involve trade between the buyer and seller 1 in the subgame’s initial period.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Proof of Lemma 1: In an arbitrary subgame beginning with offers from the sellers, consider a bo so bo SPNE σ that supports the stationary payoffs π so k and π k . Note that π B ≥ δ B π B , because the buyer always can let play continue to the next period by rejecting all offers in the subgame’s initial period. I first show that σ involves trade between the buyer and seller 1. The proof involves three cases. Case 1: Suppose σ involves trade between the buyer and a seller k 6= 1 for which Vk = V1 . Relabel the sellers so that trade involves seller 1. Case 2: Suppose σ involves trade between the buyer and a seller k for which Vk < V1 . By so following σ, π so B ≤ Vk < V1 and π 1 = 0. In the subgame’s initial period, I can show that seller 1 can profitably deviate from σ by offering the buyer V1 − > π so B for some > 0. First, the buyer strictly prefers accepting seller 1’s offer, because the payoff V1 − the buyer receives by accepting strictly exceeds both its payoff from accepting seller k’s offer (which is at most π so B , depending on whether σ specified a trade in the 24 Page 24 of 42

te

d

M

an

us

cr

ip t

subgame’s initial period), and its payoff from rejecting all offers and letting play continue to the so next period (which is δ B π bo B ≤ π B ). Second, seller 1 strictly prefers offering V1 − , because the payoff seller 1 receives by doing strictly exceeds its payoff from following σ (which is 0). The existence of this profitable deviation implies σ cannot involve the buyer trading with a seller k for which Vk < V1 . so bo bo Case 3: Suppose σ involves no trade in any period. By following σ, π so B = π 1 = π B = π 1 = 0. In the subgame’s initial period, I can show that seller 1 can profitably deviate from σ by offering the buyer V1 − > 0 for some > 0. First, the buyer strictly prefers accepting seller 1’s offer, because the payoff V1 − the buyer receives by accepting strictly exceeds both its payoff from accepting any other seller’s offer, and its payoff from rejecting all offers and letting play continue to the next period (both of which must be 0, else the buyer’s payoff could not be 0 according to σ). Second, seller 1 strictly prefers offering V1 − , because the payoff seller 1 receives by doing so strictly exceeds its payoff from following σ (which is 0). The existence of this profitable deviation implies σ cannot involve no trade in any period. Cases 1-3 exhaust all possibilities other than trading with seller 1, and so in all subgames beginning with offers from the sellers, all SPNE featuring stationary outcomes involve trade with seller 1. I next show that σ involves trade in the subgame’s initial period. Suppose not. Because of so discounting, this delay in trading implies π so 1 + π B < V1 . In the subgame’s initial period, I can show that seller 1 can profitably deviate from σ by offering the buyer π so B + for some > 0 such so so that π 1 +π B + < V1 . First, the buyer strictly prefers accepting seller 1’s offer, because the payoff π so B + the buyer receives by accepting strictly exceeds both its payoff from accepting any other seller’s offer (which is at most π so B ), and its payoff from rejecting all offers and letting play continue so so to the next period (which is δ B π bo B ≤ π B ). Second, seller 1 strictly prefers offering π B + , because the payoff V1 − (π so B + ) seller 1 receives by doing so strictly exceeds its payoff from following σ so (which is π 1 < V1 − (π so B + )). The existence of this profitable deviation implies σ cannot involve trade not occurring in the subgame’s initial period. ¥

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Lemma 2 In subgames beginning with offers from the buyer, all SPNE featuring stationary outcomes involve trade between the buyer and seller 1 in the subgame’s initial period. Proof of Lemma 2: In an arbitrary subgame beginning with offers from the buyer, consider a so bo so SPNE σ that supports the stationary payoffs π bo k and π k . Note that π B ≥ δ B π B , because the buyer always can let play continue to the next period by refusing to trade with any seller who accepted the buyer’s proposal in the subgame’s initial period. I first show that σ involves trade between the buyer and seller 1. The proof involves three cases. Case 1: Suppose σ involves trade between the buyer and a seller k 6= 1 for which Vk = V1 . Relabel the sellers so that trade involves seller 1.

25 Page 25 of 42

te

d

M

an

us

cr

ip t

Case 2: Suppose σ involves trade between the buyer and a seller k for which Vk < V1 . Trade must occur in the subgame’s initial period, because by Lemma 1 trade involves the buyer and seller bo 1 if play continues to the subgame’s next period. By following σ, π bo B ≤ Vk < V1 , and π 1 = 0. so If π bo B > δ B π B , then in the subgame’s initial period I can show that the buyer can profitably deviate from σ by offering Vj to each seller j 6= 1, k , and offering > 0 to sellers 1 and k such that so bo V1 − > Vk − ≥ π bo B − > δ B π B , and V1 − > π B . First consider the buyer’s decision to trade after the sellers make their accept/reject decisions. If seller k accepts and seller 1 rejects, then the buyer strictly prefers trading with seller k because the payoff Vk − the buyer receives by doing so strictly exceeds both its payoff from trading with any other seller j 6= 1, k (which is 0), and its payoff from letting play continue to the next period (which is δ B π so B ). If seller 1 accepts, then the buyer strictly prefers trading with seller 1 because the payoff V1 − the buyer receives by doing so strictly exceeds both its payoff from trading with any other seller (which is at most Vk − ), and its payoff from letting play continue to the next period (which is δ B π so B ). Next consider the sellers’ accept/reject decisions after receiving the deviating offers. In any equilibrium seller k accepts if seller 1 rejects, because seller k anticipates trading with the buyer and receiving a payoff that strictly exceeds its payoff from rejecting (which is 0). Consequently, in any equilibrium seller 1 must accept the buyer’s deviating offer, because it anticipates trading with the buyer and receiving a payoff that strictly exceeds its payoff from rejecting (which is 0). Finally, making the specified deviating offers gives the buyer a strictly higher payoff than from following σ. The existence of this profitable deviation implies σ cannot involve the buyer trading with a seller k for which Vk < V1 . so If π bo B = δ B π B , then in the subgame’s initial period I can show that the buyer can profitably deviate from σ by offering Vj to each seller j 6= 1 , and offering δ 1 (V1 − π so B ) + < V1 to seller 1 for some > 0. First consider the buyer’s decision to trade after the sellers make their accept/reject decisions. If seller 1 accepts, then the buyer strictly prefers trading with seller 1 because the payoff V1 − δ 1 (V1 − π so B ) − the buyer receives by doing strictly exceeds its payoff from trading with any other seller (which is 0), and its payoff from letting play continue to the next period. To see the latter point, by trading with seller 1 the buyer’s payoff can be written as (1 − δ 1 ) V1 + δ 1 π so B − . There are three cases to consider.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

bo • If π so B = V1 , then by trading in the current period the buyer’s payoff is V1 − > δ B V1 = π B for sufficiently small > 0. Thus, the buyer prefers trading in the current period.

• If 0 < π so B < V1 , then by trading in the current period the buyer’s payoff strictly exceeds π bo so πB − = δB − > π bo B , for sufficiently small > 0. Thus, the buyer prefers trading in the B current period. • If 0 = π so , then π bo = 0, and by trading in the current period the buyer’s payoff is (1 − δ 1 ) V1 − ¡B bo ¢ B > 0 = π B for sufficiently small > 0. Thus, the buyer prefers trading in the current period. Next consider the sellers’ accept/reject decisions. If the buyer offers seller 1 δ 1 (V1 − π so B)+ , then seller 1 strictly prefers accepting. By accepting seller 1 anticipates trading with the buyer 26 Page 26 of 42

te

d

M

an

us

cr

ip t

this period and getting a payoff of δ 1 (V1 − π so B ) + , while by rejecting seller 1 expects at best a so strictly lower payoff δ 1 (V1 − π B ). Therefore, making the specified offers gives the buyer a strictly higher payoff than from following σ. The existence of this profitable deviation implies σ cannot involve the buyer trading with a seller k for which Vk < V1 . Case 3: Suppose σ involves no trade in any period. Once play reaches the subgame’s second period (in which sellers make offers to the buyer), by Lemma 1 trade will occur, which is a contradiction. Cases 1-3 exhaust all possibilities other than trading with seller 1, and so in all subgames beginning with offers from the buyer, all SPNE featuring stationary outcomes involve trade with seller 1. I next show that σ involves trade in the subgame’s initial period. Suppose not. Because of bo discounting, this delay in trading implies π bo 1 + π B < V1 . In the subgame’s initial period, I can show that the buyer can profitably deviate from σ by offering seller 1 π bo 1 + for some > 0 such bo bo that π 1 + π B + < V1 , while making offers to the remaining sellers according to σ. First, the buyer strictly prefers trading with seller 1 if seller 1 accepts this offer, because the buyer’s payoff ¡ ¢ V1 − π bo from doing so strictly exceeds its payoff from trading with any other seller (which 1 + bo is at most π B ), and its payoff from rejecting all offers and letting play continue to the next period bo (which is δ B π so B ≤ π B ). Second, seller 1 strictly prefers accepting the buyer’s offer, because its payoff π bo 1 + from doing so strictly exceeds its payoff from rejecting the offer (which is at most bo π 1 ). Therefore, the buyer strictly prefers offering π bo 1 + to seller 1, because doing so gives the buyer a strictly higher payoff than from following σ. The existence of this profitable deviation implies σ cannot involve trade not occurring in the subgame’s initial period. ¥ £ ¤ bo Lemma 3 In all stationary SPNE outcomes, π so 1 = V1 − max V2 , δ B π B .

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Proof of Lemma 3: In an arbitrary subgame beginning with offers from the sellers, consider a bo SPNE σ that supports the stationary payoffs π so k and π k . ¤ £ bo Suppose π so 1 < V1 − max V2 , δ B π B . In the subgame’s initial period, I can show that seller 1 ¤ £ bo can profitably deviate from σ by demanding π so 1 + < V1 − max V2 , δ B π B , for some > 0. The buyer strictly prefers accepting this demand, because its payoff from doing so is V1 − (π so 1 + ) > £ ¤ £ ¤ bo max V2 , δ B π bo B , where max V2 , δ B π B is the highest payoff the buyer can get by rejecting seller 1’s demand (either at most V2 by trading this period with another seller, or δ B π bo B by rejecting all offers and letting play continue to the next period). Given the buyer’s anticipated behavior, seller so 1’s payoff from demanding π so 1 + strictly exceeds its payoff from following σ (which is π 1 ). The ¤ £ bo existence of this profitable deviation implies that π so 1 ≥ V1 − max V2 , δ B π B . ¤ £ bo Now suppose π so 1 > V1 − max V2 , δ B π B . In the subgame’s initial period, by Lemma 1 the ¤ £ bo bo buyer’s payoff according to σ is V1 − π so 1 < max V2 , δ B π B . If δ B π B ≥ V2 , then the buyer strictly prefers rejecting all offers and letting play continue to the next period, because the buyer’s payoff bo δ B π bo B from doing so strictly exceeds its payoff from following σ. If V2 > δ B π B , then seller 2 strictly ¤ £ bo > 0. The buyer prefers deviating from σ by offering V2 − > max V1 − π so 1 , δ B π B , for some 27

Page 27 of 42

ip t

strictly prefers accepting seller 2’s offer, because the buyer’s payoff V2 − from doing so strictly exceeds its payoff both from accepting seller 1’s offer (which is V1 − π so 1 ), and its payoff from rejecting all offers and letting play continue to the next period (which is δ B π bo B ). Given the buyer’s anticipated behavior, seller 2’s payoff from offering V2 − strictly exceeds its payoff from following ¤ £ bo σ (which is 0). The existence of these profitable deviations implies π so 1 ≤ V1 − max V2 , δ B π B . ¤ £ bo The arguments in the two preceding paragraphs imply π so 1 = V1 − max V2 , δ B π B . ¥

cr

bo Lemma 4 In all stationary SPNE outcomes, if V2 > δ B (V1 − π so 1 ), then π B = V1 .

te

d

M

an

us

Proof of Lemma 4: In an arbitrary subgame beginning with offers from the buyer, consider a so so SPNE σ that supports the stationary payoffs π bo k and π k , and for which V2 > δ B (V1 − π 1 ). Suppose π bo B < V1 . In the subgame’s initial period, I can show that the buyer can profitably so deviate from σ by demanding V1 − 2 > π bo B from seller 1 and V2 − > δ B (V1 − π 1 ) from seller 2, for some > 0, and demanding 0 from all other sellers. First consider the buyer’s decision whether to trade with any seller that has accepted the buyer’s proposal. If seller 1 accepts the buyer’s offer, then the buyer strictly prefers trading this period with seller 1, because the buyer’s payoff V1 − 2 from doing so strictly exceeds the buyer’s payoff from trading this period with any other seller (which is at most V2 − ), and its payoff from letting play continue to the next period (which is δ B (V1 − π so 1 )). If seller 2 accepts the buyer’s offer and seller 1 rejects, then the buyer strictly prefers trading this period with seller 2, because the buyer’s payoff V2 − from doing so strictly exceeds its payoff from trading this period with any other seller (which is 0), and its payoff from letting play continue to the next period (which is δ B (V1 − π so 1 )). Now consider the sellers’ decisions to accept or reject the buyer’s proposals. If seller 1 rejects the buyer’s offer, then seller 2 strictly prefers accepting to rejecting the buyer’s offer: the buyer will trade this period with seller 2, as noted in the preceding paragraph. Seller 2’s payoff in this case is > 0, which strictly exceeds its payoff from rejecting the buyer’s offer (which is 0). Because of seller 2’s incentives, if seller 1 rejects the buyer’s offer, then seller 1’s payoff is 0. If seller 1 accepts the buyer’s offer, then seller 1’s payoff is 2 > 0. Hence, seller 1 accepts the buyer’s offer 2 . Finally, given the sellers’ anticipated behavior the buyer strictly prefers making these deviating offers, because the buyer’s payoff V1 − 2 from doing strictly exceeds its payoff from following σ. The existence of this profitable deviation implies that it cannot be that π bo B < V1 , according to σ, bo so it must be the case that π B = V1 . ¥

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

bo so Lemma 5 In all stationary SPNE outcomes, if V2 < δ B (V1 − π so 1 ), then π B = V1 − δ 1 π 1 .

Proof of Lemma 5: In an arbitrary subgame beginning with offers from the buyer, consider a so so SPNE σ that supports the stationary payoffs π bo k and π k , and for which V2 < δ B (V1 − π 1 ). so Suppose π bo In the subgame’s initial period, I can show that the buyer can B < V1 − δ 1 π 1 . bo profitably deviate from σ by demanding π bo B + from seller 1, for some > 0 such that π B + < 28 Page 28 of 42

d

M

an

us

cr

ip t

V1 − δ 1 π so 1 . First consider the buyer’s decision to trade after the sellers make their accept/reject decisions. If seller 1 accepts, then the buyer strictly prefers trading with seller 1 this period, because the buyer’s payoff π bo B + from doing so strictly exceeds its payoff from trading with another seller or letting play continue to next period (both of which are at most π bo B ). Next consider the sellers’ accept/reject decisions after observing the deviating offer. Seller 1 strictly prefers accepting the ¡ ¢ offer, because seller 1’s payoff V1 − π bo from doing so strictly exceeds its payoff from rejecting B + so (which is at most δ 1 π 1 ). Given seller 1’s anticipated behavior, the buyer strictly prefers deviating from σ in this fashion, because the buyer’s payoff π bo B + doing so strictly exceeds its payoff from so following σ. The existence of this profitable deviation implies that π bo B ≥ V1 − δ 1 π 1 . so Now suppose π bo B > V1 − δ 1 π 1 . By following σ, according to Lemma 2 trade takes place in bo the subgame’s initial period, which implies seller 1’s payoff is π bo 1 = V1 − π B . If seller 1 deviates from σ by rejecting the buyer’s equilibrium offer, and if the buyer does not trade with any other bo seller in the subgame’s initial period, then seller 1’s payoff is δ 1 π so 1 > π 1 . If seller 1 rejects the buyer’s offer, then the buyer will not trade with any other seller in the subgame’s initial period. The highest payoff the buyer can get is V2 by trading with seller 2, but by waiting until the next Therefore, it is profitable for seller 1 to period the buyer gets a payoff of δ B (V1 − π so 1 ) > V2 . reject the buyer’s equilibrium offer in the subgame’s initial period, which contradicts σ as a SPNE. so Therefore, it must be the case that π bo B ≤ V1 − δ 1 π 1 . so The arguments in the two preceding paragraphs imply π bo B = V1 − δ 1 π 1 . ¥ V2 δB .

te

so Lemma 6 In all stationary SPNE outcomes, if V2 = δ B (V1 − π so 1 ), then π 1 = V1 −

Proof of Lemma 6: Follows from simple algebra. ¥ With the preceding results, the stationary equilibrium payoffs can be derived for each of four cases.

6.1

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Case 1: V2 < δB (V1 − π so 1 )

Lemma 7 If V2 < δ B (V1 − π so 1 ), then π bo B =

³

1−δ 1 1−δ 1 δ B

´

V1

π bo 1 = δ1

³

1−δB 1−δ 1 δB

´

π so B = δB

V1

³

1−δ1 1−δ 1 δB

´

V1

π so 1 =

³

1−δ B 1−δ1 δ B

´

V1 .

¤ £ bo Proof of Lemma 7: By Lemma 3, π so 1 = V1 − max V2 , δ B π B . Substituting the expression for ¤ £ bo π so 1 into the condition for Lemma 7 requires V2 < δ B max V2 , δ B π B , so for Lemma 7’s condition ¤ £ bo to hold requires δ B π bo B = max V2 , δ B π B . so so bo By Lemma 5, π bo B = V1 − δ 1 π 1 . Substituting the expression for π 1 into the expression for π B yields ¶ µ ³ ´ 1 − δ1 bo bo bo V1 . π B = V1 − δ 1 V1 − δ B π B ⇐⇒ π B = 1 − δ1δB ³ ´ 1−δB so yields π so = into the expression for π Substituting the derived value for π bo 1 1 B 1−δ 1 δ B V1 . By 29 Page 29 of 42

³

1−δ B 1−δ1 δ B

´

V1 and π so B = δB

2 Lemma 8 If V2 < δ B (V1 − π so 1 ), then V2 < δ B

³

³

1−δ 1 1−δ1 δ B

1−δ1 1−δ 1 δ B

´

so Proof of Lemma 8: If V2 < δ B (V1 − π so 1 ), then π 1 =

´

V1 . These are the desired results.

V1 . ³

1−δ B 1−δ1 δ B

bo Case 2: V2 > δB (V1 − π so 1 ) and V2 ≥ δ B π B

bo Lemma 9 If V2 > δ B (V1 − π so 1 ) and V2 ≥ δ B π B , then

π bo 1 =0

π so B = V2

π so 1 = V1 − V2 .

an

π bo B = V1

us

6.2

V1 by Lemma 7. Substituting ´ 1−δ 1 1−δ1 δ B V1 , which is the desired

cr

that expression into the condition for Lemma 8 yields V2 < δ 2B result. ¥

³

´

ip t

Lemmas 1 and 2, π bo 1 = δ1 ¥

M

¤ £ bo Proof of Lemma 9: By Lemma 3, π so 1 = V1 −max V2 , δ B π B , so the second condition for Lemma bo bo so 9 implies π so 1 = V1 − V2 . By Lemma 4, π B = V1 . By Lemmas 1 and 2, π 1 = 0 and π B = V2 . These are the desired results. ¥

d

bo Lemma 10 If V2 > δ B (V1 − π so 1 ) and V2 ≥ δ B π B , then V2 ≥ δ B V1 .

te

bo Proof of Lemma 10: If V2 > δ B (V1 − π so 1 ) and V2 ≥ δ B π B , then substituting the stationary SPNE payoffs derived in Lemma 9 into the conditions for Lemma 10 yields V2 > δ B V2 and V2 ≥ δ B V1 . The first constraint holds trivially, so the relevant constraint is the second one. This is the desired result. ¥

6.3

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

bo Case 3: V2 > δB (V1 − π so 1 ) and V2 < δ B π B

bo Lemma 11 If V2 > δ B (V1 − π so 1 ) and V2 < δ B π B , then

π bo B = V1

π bo 1 =0

π so B = δ B V1

π so 1 = (1 − δ B ) V1 .

¤ £ so bo Proof of Lemma 11: By Lemma 4, π bo B = V1 . By Lemma 3, π 1 = V1 − max V2 , δ B π B , so bo the second condition for Lemma 11 implies π so 1 = (1 − δ B ) V1 . By Lemmas 1 and 2, π 1 = 0 and π so B = δ B V1 . These are the desired results. ¥ 2 bo Lemma 12 If V2 > δ B (V1 − π so 1 ) and V2 < δ B π B , then δ B V1 < V2 < δ B V1 . bo Proof of Lemma 12: If V2 > δ B (V1 − π so 1 ) and V2 < δ B π B , then substituting the stationary SPNE payoffs derived in Lemma 11 into the conditions for Lemma 12 yields V2 > δ 2B V1 and V2 < δ B V1 . These are the desired results. ¥

30 Page 30 of 42

6.4

Case 4: V2 = δB (V1 − π so 1 )

Lemma 13 If V2 = δ B (V1 − π so 1 ), then π bo B =

V2 δ 2B

π bo 1 = V1 −

V2 δ 2B

π so B =

V2 δB

π so 1 = V1 −

V2 δB .

and π so B =

V2 δB .

These are the desired results. ¥

2 Lemma 14 If V2 = δ B (V1 − π so 1 ), then δ B

³

1−δ1 1−δ 1 δB

´

B

cr

V2 δ2B

V1 ≤ V2 ≤ δ 2B V1 .

us

π bo 1 = V1 −

ip t

¤ £ V2 so bo Proof of Lemma 13: By Lemma 6, π so 1 = V1 − δ B . By Lemma 3, π 1 = V1 − max V2 , δ B π B , ¤ £ V2 bo so substituting the value of π so Given that 1 derived in Lemma 6 yields δB = max V2 , δ B π B . ¤ £ V2 bo bo bo δ B < 1, it must be that δ B π B = max V2 , δ B π B . Therefore, π B = δ2 . By Lemmas 1 and 2,

d

M

an

Proof of Lemma 14: If V2 = δ B (V1 − π so 1 ), then the stationary SPNE payoffs derived in Lemma V2 V2 so bo 13 are π 1 = V1 − δB and π B = δ2 . From the first expression, the restriction that π so 1 ≥ 0 B implies that the proposed solution’s validity requires V2 ≤ δ B V1 . From the second expression, the 2 restriction that π bo B ≤ V1 implies that the proposed solution’s validity requires V2 ≤ δ B V1 . The second constraint is tighter than is the first, so the second constraint is the relevant of the two constraints. so Another restriction is that π bo If not, then the buyer would be offering seller 1 1 ≤ δ1 π1 . strictly more than necessary to induce seller 1 to accept the buyer’s offer. Given the values of the stationary SPNE payoffs derived in Lemma 13, the proposed solution’s validity requires

te

µ ¶ µ ¶ V2 V2 1 − δ1 2 V1 − 2 ≤ δ 1 V1 − ⇐⇒ δ B V1 ≤ V2 . δB 1 − δ1 δB δB

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Combining ³ ´the two parameter restrictions implies that the proposed solution’s validity requires 1−δ1 2 δ B 1−δ1 δB V1 ≤ V2 ≤ δ 2B V1 , which is the desired result. ¥ The preceding Lemmas now can be used to prove Theorem 1. Proof of Theorem h h considers ³ 1: The ´ ´proof separately ³ ´four sets i of possible values for V2 . 1−δ 1 1−δ1 2 2 If V2 ∈ 0, δ B 1−δ1 δB V1 , then V2 ∈ / δ B 1−δ1 δB V1 , V1 . The contrapositives of Lemmas 10 and 12 imply it is not the case that V2 > δ B (V1 − π so 1 ), while the contrapositive of Lemma so 14 implies that it is not the case that V2 = δ B (V1 − π 1 ). Hence, it must be the case that so V2 < δ B (V1 − by iLemma 7 the hpayoffs³ must be 1. h π 1³). Therefore, ´ ´ as´specified in Theorem ¤ ¡ 2 1−δ 1 1−δ 1 2 2 2 If V2 ∈ δ B 1−δ1 δB V1 , δ B V1 , then V2 ∈ / 0, δ B 1−δ1 δB V1 and V2 ∈ / δ B V1 , V1 . The contrapositive of Lemma 8 implies that it is not the case that V2 < δ B (V1 − π so 1 ), while the contrapositives of Lemmas 10 and 12 imply it is not the case that V2 > δ B (V1 − π so 1 ). Hence, it so must be the case that V2 = δ B (V1 − π 1 ). Therefore, by Lemma 13 the payoffs must be as specified in Theorem 1. ³ ´ ´ ´ i h h ³ ¢ ¡ 1−δ 1 1−δ1 2 2 V , V V , and V2 ∈ If V2 ∈ δ 2B V1 , δ B V1 , then V2 ∈ / 0, δ 2B 1−δ ∈ / δ , δ V / 1 2 1 1 B 1−δ 1 δ B B 1 δB so [δ B V1 , V1 ]. The contrapositive of Lemma 8 implies that it is not the case that V2 < δ B (V1 − π 1 ), 31 Page 31 of 42

cr

ip t

the contrapositive of Lemma 14 implies that it is not the case that V2 = δ B (V1 − π so 1 ), while the contrapositive of Lemma 10 implies that it is not the case that both V2 > δ B (V1 − π so 1 ) and bo so bo V2 ≥ δ B π B hold. Hence, it must be the case that V2 > δ B (V1 − π 1 ) and V2 < δ B π B . Therefore, by Lemma 11 the payoffs must be as specified in Theorem 1. If V2 ∈ [δ B V1 , V1 ], then V2 ∈ / [0, δ B V1 ). The contrapositive of Lemma 8 implies that it is not so the case that V2 < δ B (V1 − π 1 ), the contrapositive of Lemma 14 implies that it is not the case that V2 = δ B (V1 − π so 1 ), while the contrapositive of Lemma 12 implies that it is not the case that bo so both V2 > δ B (V1 − π so 1 ) and V2 < δ B π B hold. Hence, it must be the case that V2 > δ B (V1 − π 1 ) and V2 ≥ δ B π bo B . Therefore, by Lemma 9 the payoffs must be as specified in Theorem 1. ¥

π bo B = V1 π bo B = V1 and −rB ∆ π so B =e

π so B =

V2 e−rB ∆

³

h ³ ´ ´ −r1 ∆ V1 if V2 ∈ 0, e−2rB ∆ 1−e1−e V1 −r1 ∆ e−rB ∆ ³ ´ i h −r1 ∆ −2rB ∆ V V if V2 ∈ e−2rB ∆ 1−e1−e , e 1 1 −r1 ∆ e−rB ∆ ¢ ¡ −2r ∆ if V2 ∈ e B V1 , e−rB ∆ V1 £ ¤ if V2 ∈ e−rB ∆ V1 , V1

1−e−r1 ∆ 1−e−r1 ∆ e−rB ∆

−rB ∆ V π so 1 B =e

π so B = V2

an

V2 e−2rB ∆

´

´

M

1−e−r1 ∆ 1−e−r1 ∆ e−rB ∆

h ³ ´ ´ −r1 ∆ V1 if V2 ∈ 0, e−2rB ∆ 1−e1−e V1 −r1 ∆ e−rB ∆ h ³ ´ i −r1 ∆ −2rB ∆ V if V2 ∈ e−2rB ∆ 1−e1−e , e V 1 1 −r1 ∆ e−rB ∆ ¢ ¡ −2r ∆ if V2 ∈ e B V1 , e−rB ∆ V1 ¤ £ if V2 ∈ e−rB ∆ V1 , V1 .

d

π bo B =

³

te

π bo B =

us

Proof of Corollary 1: Recalling that the discount factor δ k is defined as δ k ≡ e−rk ∆ , the buyer’s payoffs from Theorem 1 can be written as

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

so By Lemmas 1 and 2 one can derive the associated values for π bo 1 and π 1 . ∆ −→ 0 requires using L’Hopital’s rule, which gives the desired result. ¥

6.5

Taking the limit as

Proofs of Propositions 1 and 2

Proof of Proposition 1: The buyer strictly prefers multilateral negotiations if and only if its payoff from multilateral negotiations strictly exceeds its payoff from a first-price auction. That condition amounts to ¶ ¸ ∙ µ r1 f pa mln π B > π B ⇐⇒ max V2 , V1 − C mln > V2 . r1 + rB The latter expression is equivalent to ∙ µ C mln < max 0,

r1 r1 + rB

¶

¸ V1 − V2 ,

(1)

which is the desired result. 32 Page 32 of 42

f pa The buyer’s relative preference for multilateral negotiations intensifies as π mln B − π B increases. That condition corresponds to relaxing (1), which is accomplished by decreasing C mln , increasing V1 , decreasing V2 , or decreasing rrB1 . ¥

N +1

³

´N

rS rS +rB

− C mln >

N +1

,

(2)

an

C mln <

us

The latter expression is equivalent to rS rS +rB

N −1 . N +1

cr

π mln > π fBpa ⇐⇒ B

³

´N

N −1+

ip t

Proof of Proposition 2: The buyer strictly prefers multilateral negotiations if and only if its expected payoff from multilateral negotiations strictly exceeds its expected payoff from a first-price auction. That condition amounts to

References

d

M

which is the desired result. f pa The buyer’s relative preference for multilateral negotiations intensifies as π mln B − π B increases. That condition corresponds to relaxing (2), which is accomplished by decreasing C mln , decreasing N , or decreasing rrBS . ¥

te

[1] Abreu, D., and Manea, M. (2012a) “Markov Equilibria in a Model of Bargaining in Networks,” Games and Economic Behavior 75, 1-16. [2] Abreu, D., and Manea, M. (2012b) “Bargaining and Efficiency in Networks,” Journal of Economic Theory 147, 43-70.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[3] Athey, S., Levin, J., and Seira, E. (2011) “Comparing Open and Sealed Bid Auctions: Evidence from Timber Auctions,” Quarterly Journal of Economics 126, 207-257. [4] Ausubel, L., Cramton, P., and Deneckere, R. (2002) “Bargaining with Incomplete Information,” in Aumann, R., and Hart, S. (eds.) Handbook of Game Theory 3 (Amsterdam: Elsevier Science B.V.). [5] Bajari, P., McMillan, R., and Tadelis, S. (2009) “Auctions Versus Negotiations in Procurement: An Empirical Analysis,” Journal of Law, Economics, and Organizations 25(2), 372-399. [6] Binmore, K. (1985) “Bargaining and Coalitions,” in Roth, A. Game-Theoretic Models of Bargaining (Cambridge: Cambridge University Press).

(ed.)

[7] Binmore, K., Shaked, A., and Sutton, J. (1989) “An Outside Option Experiment,” Quarterly Journal of Economics 104, 753-770.

33 Page 33 of 42

[8] Bulow, J., and Klemperer, P. (1996) “Auctions Versus Negotiations,” American Economic Review 86(1), 180-194. [9] Bulow, J., and Klemperer, P. (2009) “Why Do Sellers (Usually) Prefer Auctions?,” American Economic Review 99(4), 1544-1575.

ip t

[10] Chamberlin, E. (1948) “An Experimental Imperfect Market,” Journal of Political Economy 56, 95-108.

cr

[11] Chatterjee, K., and Dutta, B. (1998) “Rubinstein Auctions: On Competition for Bargaining Partners,” Games and Economic Behavior 23(2), 119-145.

us

[12] Chatterjee, K., and Lee, C. (1998) “Bargaining and Search with Incomplete Information About Outside Options,” Games and Economic Behavior 22(2), 203-237.

an

[13] Corominas-Bosch, M. (2004) “Bargaining in a Network of Buyers and Seller,” Journal of Economic Theory 115, 35-77.

M

[14] Dixit, A., and Shapiro, C. (1986) “Entry Dynamics with Mixed Strategies,” in Thomas, L.G. (ed.) The Economics of Strategic Planning (Lexington: Lexington Books).

d

[15] Hendon, E., and Tranaes, T. (1991) “Sequential Bargaining in a Market with One Seller and Two Different Buyers,” Games and Economic Behavior 3(4), 453-466.

te

[16] Herings, P.J.J., Meshalkin, A., and Predtetchinski, A. (2017) “A One-Period Memory Folk Theorem for Multilateral Bargaining Games,” Games and Economic Behavior 103, 185-198. [17] Kennan, J., and Wilson, R. (1993) “Bargaining with Private Information,” Journal of Economic Literature 31, 45-104.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[18] Kim, J., and Che, Y.-K. (2004) “Asymmetric Information About Rivals’ Types in Standard Auctions,” Games and Economic Behavior 46, 383-397. [19] Krishna, V., and Serrano, R. (1996) “Multilateral Bargaining,” Review of Economic Studies 63(1), 61-80. [20] Lang, K., and Rosenthal, R. (1991) “The Contractors’ Game,” RAND Journal of Economics 22(3), 329—338. [21] Levin, D., and Smith, J. (1994) “Equilibrium in Auctions with Entry,” American Economic Review 84(3), 585—599. [22] Manea, M. (2011) “Bargaining in Stationary Networks,” American Economic Review 101(5), 2042-2080. [23] Marx, L., and Shaffer, G. (2010) “Break-up Fees and Bargaining Power in Sequential Contracting,” International Journal of Industrial Organization 28, 451-463. 34 Page 34 of 42

[24] McAdams, D., and Schwarz, M. (2007) “Credible Sales Mechanisms and Intermediaries,” American Economic Review 97(1), 260-276. [25] McAfee, R.P., and McMillan, J. (1987) “Auctions and Bidding,” Journal of Economic Literature 25, 699-738.

ip t

[26] McAfee, R.P., and Vincent, D. (1997) “Sequentially Optimal Auctions,” Games and Economic Behavior 18(2), 246-276.

cr

[27] Milgrom, P. (1989) “Auctions and Bidding: A Primer,” Journal of Economic Perspectives 3, 3-22.

us

[28] Muthoo, A. (1995) “On the Strategic Role of Outside Options in Bilateral Bargaining,” Operations Research 43(2), 292-297.

an

[29] Muthoo, A. (1999) Bargaining Theory with Applications (Cambridge: Cambridge University Press). Academic

M

[30] Osborne, M., and Rubinstein, A. (1990) Bargaining and Markets (San Diego: Press).

d

[31] Penta, A. (2011) “Multilateral Bargaining and Walrasian Equilibrium,” Journal of Mathematical Economics 47, 417-424.

te

[32] Ray, D. (2007) A Game-Theoretic Perspective on Coalition Formation (Oxford: Oxford University Press). [33] Reinganum, J., and Daughety, A. (1991) “Endogenous Availability in Search Equilibrium,” RAND Journal of Economics 22(2), 287-306.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[34] Reinganum, J., and Daughety, A. (1992) “Search Equilibrium with Endogenous Recall,” RAND Journal of Economics 23(2), 184-202. [35] Rubinstein, A. (1982) “Perfect Equilibrium in a Bargaining Model,” Econometrica 50(1), 97109. [36] Serrano, R. (2008) “Bargaining,” in S. Durlauf and L. Blume The New Palgrave Dictionary of Economics, 2nd edition (London: McMillan).

(eds.)

[37] Shaked, A. (1994) “Opting Out: Bazaars Versus Hi Tech Markets,” Investigaciones Economicas 18(3), 421-432. [38] Shaked, A., and Sutton, J. (1984) “Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model,” Econometrica 52, 1351-1364. [39] Smith, V. (1962) “An Experimental Study of Competitive Market Behavior,” Journal of Political Economy 70(2), 111-137. 35 Page 35 of 42

[40] Thomas, C.J. (2011) “Equilibrium Behavior in a Model of Multilateral Negotiations,” working paper, Clemson University. [41] Thomas, C.J., and Wilson, B.J. (2002) “A Comparison of Auctions and Multilateral Negotiations,” RAND Journal of Economics 33(1), 140-155.

ip t

[42] Thomas, C.J., and Wilson, B.J. (2005) “Verifiable Offers and the Relationship Between Auctions and Multilateral Negotiations,” Economic Journal 115(506), 1016-1031.

cr

[43] Thomas, C.J., and Wilson, B.J. (2014) “Horizontal Product Differentiation in Auctions and Multilateral Negotiations,” Economica 81, 768-787.

te

d

M

an

us

[44] Vincent, D. (1992) “Modelling Competitive Behavior,” RAND Journal of Economics 23(4), 590-599.

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

36 Page 36 of 42

ߨ஻௕௢ 1

ip t

0.8 0.6

us

cr

0.4

0.2

0.4

an

0.2

0.6

0.8

ܸଶ 1

M

(a) Buyer’s stationary SPNE payoffs when buyer makes offers

d

ߨ஻௦௢ 1

0.6

te

0.8

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.4 0.2

0.2

0.4

0.6

0.8

1

ܸଶ

(b) Buyer’s stationary SPNE payoffs when sellers make offers

Figure 1. Buyer’s Stationary Payoffs in Multilateral Negotiations, as a Function of V2. Panel (a) shows the buyer’s SPNE payoffs as a function of V2, in subgames beginning with offers from the buyer. Panel (b) shows the buyer’s SPNE payoffs as a function of V2, in subgames beginning with offers from the sellers. These figures are calculated using V1 = 1, δ1 = 0.8, δB = 0.8. Dashed line is 45○-line.

Page 37 of 42

ߨ஻௕௢

1

ip t

0.8 0.6

0.2

0.4

0.6

an

0.2

us

cr

0.4

0.8

1

ߜ஻

1

ߜ஻

1

d

ߨ஻௦௢

M

(a) Buyer’s stationary SPNE payoffs when buyer makes offers

0.6

te

0.8

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.4 0.2

0.2

0.4

0.6

0.8

(b) Buyer’s stationary SPNE payoffs when sellers make offers

Figure 2. Buyer’s Stationary Payoffs in Multilateral Negotiations, as a Function of δB. Panel (a) shows the buyer’s SPNE payoffs as a function of δB, in subgames beginning with offers from the buyer. Panel (b) shows the buyer’s SPNE payoffs as a function of δB, in subgames beginning with offers from the sellers. These figures are calculated using V1 = 1 , V2 = 0.6, δ1 = 0.8. Solid line is stationary SPNE payoff, and dashed line is SPNE payoff from bilateral negotiations.

Page 38 of 42

πB V1 − C% mln

π Bmln

π% Bmln

π Bfpa = π% Bfpa

ip t

V1 − C mln

V2

cr

rB r1

us

(a) Decreasing Cmln

V%1 − C mln

π Bmln

π% Bmln

π Bfpa = π% Bfpa

M

V1 − C mln

an

πB

d

V2

te

rB r1

(b) Increasing V1

πB

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

V1 − C mln

π Bmln

π% Bmln

π Bfpa

π% Bfpa

V2 V%2

(c) Decreasing V2

rB r1

Figure 3. Comparative Statics of Buyer’s Payoffs in Multilateral Negotiations and a First-Price Auction. Each panel shows the buyer’s payoffs with initial parameters and with one changed parameter (changes denoted by a tilde). Panel (a) decreases Cmln, panel (b) increases V1, and panel (c) decreases V2. Each change intensifies the buyer’s relative preference for multilateral negotiations over a first-price auction.

Page 39 of 42

πB 1

π Bmln

(

rB rS

= 0.25

)

π Bmln

(

rB rS

)

=1

π Bmln

(

rB rS

=4

)

0.8 0.6

ip t

π Bfpa

0.4

cr

0.2

C entry

0 0.4

πB 1

0.6

(a) N = 2

π Bmln

(

rB rS

= 0.25

)

π Bmln

0.8 0.6

rB rS

)

=1

1

π Bmln

(

rB rS

=4

)

M

π Bfpa

(

0.8

us

0.2

an

0

0.4

d

0.2

0

πB

1

C entry

te

0

0.2

0.4 (b) N = 4

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

π Bmln

(

rB rS

= 0.25

)

π Bmln

(

rB rS

0.6

)

=1

0.8

π Bmln

(

rB rS

1

=4

)

0.8 0.6

π Bfpa

0.4 0.2

C entry

0 0

0.2

0.4

0.6

0.8

1

(c) N = 6 Figure 4. Buyer’s Expected Payoffs in Multilateral Negotiations and a First-Price Auction, with Fixed Number of Sellers. Each panel shows the buyer’s payoffs for a particular value of N, and with different ratios of discount rates rB/rS. N = 2 in panel (a), N = 4 in panel (b), and N = 6 in panel (c). Multilateral negotiation payoffs are dashed lines, first-price auction payoff is solid line.

Page 40 of 42

πB 1 0.8

π Bmln

π Bfpa

ip t

0.6 0.4

cr

0.2

C entry

0

0.2

0.4 0.6 (a) rB/rS = 0.25

πB

π Bmln

0.8

π Bfpa

M

0.6

1

an

1

0.8

us

0

0.4

d

0.2

0

πB

C entry

te

0

0.2

Ac ce p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.4 0.6 (b) rB/rS = 1

0.8

1

1

0.8

π Bmln

π Bfpa

0.6 0.4 0.2

C entry

0 0

0.2

0.4 0.6 (c) rB/rS = 4

0.8

1

Figure 5. Buyer’s Expected Payoffs in Multilateral Negotiations and a First-Price Auction, with Optimal Number of Sellers. Each panel shows the buyer’s payoffs for a particular ratio of discount factors, and with optimal number of invited sellers. rB/rS = 0.25 in panel (a), rB/rS = 1 in panel (b), and rB/rS = 4 in panel (c). Multilateral negotiation payoff is dashed line, first-price auction payoff is solid line.

Page 41 of 42

*Highlights (for review)

A model of multilateral negotiations is developed. The model’s predictions are intuitively sensible and tractable. The model’s predictions differ qualitatively from those in other models.

Ac ce p

te

d

M

an

us

cr

ip t

One application of the model is shown: choosing between auctions and negotiations.

Page 42 of 42