The Efficiency of Bargaining with Many Items

Transcription

1 The Efficiency of Bargaining with Many Items Matthew O. Jackson, Hugo F. Sonnenschein, and Yiqing Xing March 2015 Abstract For an important class of alternating-offer bargaining problems with significant two-sided asymmetric information, we demonstrate that all sequential equilibria are efficient or approximately efficient. We focus on the case in which the bargaining problem contains many aspects, the value of each of which is private information, but which aggregate to an overall surplus from trade that is approximately known. The results may help explain why so many situations with significant asymmetric information exhibit little departure from first best efficiency (e.g., time lost to labor negotiations is negligible world-wide). Keywords: Bargaining, Exchange, Trade, Multiple Objects, Linking, Contract Theory, Mechanism Design, Implementation, Bayesian Equilibrium, Efficiency, Sequential Equilibrium, Trembling. JEL Classification Numbers: A13, C72, C78, D47, D82. Department of Economics, Stanford University, Stanford, California USA, and Department of Economics, University of Chicago, Chicago, Illinois, Jackson is also an external faculty member at the Santa Fe Institute and a fellow of CIFAR. s: jacksonm@stanford.edu, hfsonnen@uchicago.edu and xingyq@stanford.edu. Financial support from the NSF under grant SES as well as ARO MURI award W911NF We thank seminar participants at the University of Chicago, Princeton University, Stanford University and the Decentralization conference, for helpful comments. Electronic copy available at:

2 1 Introduction It is not curious that strikes exist, but it is curious that they are so rare. According to Kennan (2005), in the United States between 1948 and 2005, idleness due to strikes never exceeded one half of one percent of total working days in any year. 1 In fact, since 1990 average lost time has been about 20 minutes per year per worker in the U.S.; and, even in a more strike-prone country like Spain, the number is less than 1/3 of a day per worker per year (again, according to Kennan (2005)). The puzzle here has its roots in the two seminal papers in bargaining: Rubinstein (1982) and Myerson and Satterthwaite (1983). On the one hand, if there is complete information about the value of a potential bargain and any impatience among bargainers, then Rubinstein showed that in a canonical and descriptively attractive alternating offer model of bargaining there is always immediate agreement and efficiency in a unique equilibrium outcome. While that is encouraging, it is far from the explanation of the data quoted above, as it is likely that there is some private information in many settings in which firms and workers bargain over labor contracts, and in bargaining settings more generally. In that light, Myerson and Satterthwaite s important result comes to bear which shows that with two-sided uncertainty about the value of trade, it can be that all equilibria result in substantial inefficiency. 2 In this paper we provide an explanation for why it can be that bargaining with substantial two-sided uncertainty can still systematically result in efficient outcomes. We analyze settings in which there may be several or many aspects to the bargain, 3 each of which may have significant uncertainty, but on average aggregate to a fairly predictable surplus. For example, there may be substantial uncertainty to how much workers value a health package, time off for child care, pension provisions, sick days, flexible schedule, safety rules, disability insurance, grievance provisions, wages, and so forth; and similarly how management views the relative costs of each of these items. If each item was bargained over in isolation, following 1 Kennan notes similar numbers for Canada, where he mentions that lost work time was about a third of a day per year per worker. Kennan also states that Although the data are not readily available for a broad sample of developed countries, the pattern described above seems quite general: days lost due to strikes amount to only a fraction of a day per worker per annum, on average, exceeding one day only in a few exceptional years. 2 See, e.g. Kennan and Wilson (1993), Ausubel, Cramton and Deneckere (2001) for surveys on bargaining with private information. Later results include Deneckere and Liang (2006) for the case of interdependent information; and Feinberg and Skrzypacz (2005) for the case of higher-order uncertainty. 3 For the literature on exchanges with multiple objects: Armstrong (1999) considers monopolistic price discrimination problems with multiple products, and Sen (2000) and Inderst (2003) consider bargaining over both price and quantity (of homogeneous goods). All those models have one-sided private information about buyers valuation. In contrast, our paper consider the case where agents on both sides hold private information on heterogeneous products, so incentive issues potentially are substantial on both sides and become impediment to efficiency. 1 Electronic copy available at:

3 Myerson and Satterthwaite (1983) there would be substantial losses in efficiency. However, in aggregate, both the workers and the firm may have a good idea of the overall cost or value that an equilibrium total compensation package should have. As we prove, when bargaining over many aspects, or many items, at once, it is possible to reach efficient outcomes (in all equilibria) even though this would not happen if the aspects or items were each bargained over in complete isolation. Thus, the settings we examine live between the extremes considered by Rubinstein (1982) and Myerson and Satterthwaite (1983): each item has substantial uncertainty, but the overall surplus is (nearly) known. A priori, it is not clear which of the extreme reasonings should apply, if either. We show that which logic applies depends on the properties of the bargaining protocol: If the protocol allows the agents to bargain over more than one item at a time then efficiency ensues and for all equilibria, while if the protocol has agents bargaining over the items independently, then none of equilibria are inefficient, even though offers could be coordinated across items. 4 Thus, the contrast across bargaining protocols is quite stark. The critical insight is that to achieve efficiency the responder cannot select some items and not others at various prices, but must face coordinated prices across items. A reaction to our results might be that we already knew the answer to one of the questions posed here, namely that efficiency is possible. In particular, in an earlier work (Jackson and Sonnenschein (2007)), we showed that in settings in which collective decisions over many objects could be linked together, there exist mechanisms that result in (approximately) efficient outcomes. There is an important difference however. Those familiar with that paper will recall that it takes a mechanism design approach to the problem of efficient allocation and makes substantial use of a mechanism designer or mediator who is well-informed regarding the nature of the Bayesian uncertainty. Here we consider a more ambitious conclusion in which the mechanism is not designed, but is a natural bargaining protocol. We examine the circumstances under which bargaining competition necessarily leads to an efficient allocation of resources (think of the relative infrequency of strikes and efficient contractual arrangements) when there are many items to be bargained over with significant uncertainty regarding their valuations, but relatively good knowledge regarding the overall surplus. We consider extensions of the Rubinstein-Stahl bargaining protocol to the case where the bargain has many items. We refer to these extensions as alternating-offer bargaining protocols. As in Rubinstein-Stahl, the bargaining agents have similar strategy spaces and they make offers in turn. Offers involve demands of surplus together with constraints on what is made available to the recipient if she accepts the offer. Bargaining ends when the process of offers, counter-offers, and eventual agreement followed by transfers, has run its course. In the 4 If offers cannot be coordinated across items, then the Myerson-Satterthwaite result applies directly and in many cases guarantees significant departure from efficiency. See subsection 4.2 for more discussion. 2

4 spirit of the word bargaining we consider protocols that impose relatively little and are not customized to account for the individuals who may meet to bargain or their particular uncertainty. We have in mind the essence of the dialogue in a labor-management negotiation. Although the alternating-offer bargaining protocols that we consider can be viewed as mechanisms, 5 they describe interactions between agents who come together to bargain and are not a priori restricted. In particular, the protocols allow us to investigate when it is that the competitive forces of bargaining, which arise naturally in a world where agents can claim to have any valuations and ask for any share of a hypothetical surplus, will result in efficiency. Our modeling leads to the conclusion that for the important class of problems under consideration the forces of bargaining competition are sufficient for efficiency. Furthermore, since our description of bargaining competition does not vary with the structure of uncertainty in the world, we provide a new angle on the robust mechanisms literature. In particular, the bargaining protocols that we examine do not depend on any information about the distribution of agents types or the possible surplus, hence can work across many different environments. 6 Some very natural bargaining protocols provide for a sort of universal implementation, in that they result in efficient outcomes in all equilibria for a general class of settings. 2 Multi-Object Bargaining We first introduce the multi-object bargaining setting. 2.1 Multiple Objects and Preferences A multi-object bargaining problem consists of: a deal with n objects (or aspects, items), 5 There is an extensive mechanism design literature characterizing the feasibility of efficient allocations. See, e.g., Jackson (2003) and Segal and Whinston (2012) and the references therein for static case; and Skrzypacz and Toikka (2014) for dynamic case. 6 This follows in the broader spirit of Wilson s (1987) criticisms of mechanisms that depend on agents (higher-order) beliefs. Satterthwaite, Williams, and Zachariadis (2014) also view such mechanisms as impractical as [the agents ] beliefs are not a datum that is practically available for defining economic institutions (p.249). In contrast, robust and detail-free have mainly been used in literature to refer to resolving the more explicit aspect of the Wilson s critique, namely the assumption of common knowledge among agents: e.g., see Bergemann and Morris (2005) and Roughgarden and Talgam-Cohen (2013). The exception is Matsushima (2008) who used detail-free with a meaning more similar to ours in an auction environment. To avoid confusions in terminologies we use universality to capture the feature that a protocol/mechanism is not defined upon the knowledge of the prior distribution. 3

5 two agents, {s, b}, the seller s and the buyer b, finite valuation or type spaces Θ i IR, i {s, b}, with Θ = (Θ s ) n (Θ b ) n, and a probability distribution described by a density function f over types Θ. We let f i denote the marginal of f on Θ i. We assume throughout that there exist (θ s, θ b ) Θ s Θ b such that θ b > θ s, so that there are, at least potentially, nontrivial gains from trade. In the beginning of period 0, the types are drawn according to the density f and agents observe their own types only. The realized types are denoted θ i = (θ i1,..., θ ik,... θ in ), with θ ik being agent i s value for aspect/object k. Bargaining occurs at t = 0, 1, 2,... The generality of f allows for correlated values and also allows for different distributions over various classes of objects (say some big, some small). We discuss conditions on f under which efficient bargaining holds with our main results. The bargainers are considered to be engaged in making a single joint decision or deal that generalizes Rubinstein-Stahl in that it has several aspects, while in Rubinstein-Stahl, modified to allow for Bayesian uncertainty, surplus is divided between a buyer and a seller when the good is transferred and dollars are paid. Here two firms may be bargaining on a contract which includes specification of a service to be provided, or a good to be produced, and this could include many aspects which need to be specified, which include the time to production, various specs of the object (e.g., its weight, performance, durability), the quality of the object, penalties for failure to deliver. Or for example it could be a contract between a faculty member and a university, specifying a teaching load, a sabbatical policy, research funding, summer support, a salary, etc. In another interpretation there is a buyer and a seller with the seller being able to deliver any of the n objects to the buyer. The allocation problem is then which objects will be transferred and how much money will pass from the buyer to the seller. 7 With this noted, we use the terminology of objects or items, with the reader keeping in mind that this all applies to multiple aspects of a single exchange. 2.2 Payoffs Time advances in discrete periods and is discounted according to δ i (0, 1), i {s, b}. We also assume that the agents payoffs across objects are additively separable, but this is not essential to the analysis (see subsection 4.3 for more discussions). 7 The case in which some objects are initially held by one agent and other objects by the other could be easily accommodated, but we keep the notation uncluttered. 4

6 If X t is the set of objects traded at date t, and p t is a value (total price) paid from the buyer to the seller in period t, then the seller s net (realized) utility from trade as seen from time 0 is ) (p t k Xt θ sk U 0 s = t δ t s and the utility to the buyer is U 0 b = t δ t b (( ) p t ) k X t θ bk. Note that this specification allows different objects to trade at different times (via the X t s). A special case is in which trade of all goods must occur in only one period (so for instance, disability insurance might or might not be included and then the price would implicitly include how much of it would be paid by the employer). The discounting can be interpreted in at least two (standard) ways. 1) Production occurs at time t, so the seller incurs a cost θ sk for item k at time t, and the buyer also gets the payoff from consuming at t. This interpretation also applies to the example of strikes between a firm and a union, where the costs and benefits are held until the employment relationship is restored (, i.e. the agreement is reached in the bargaining, at t). 2) The seller holds the items each of which generates a flow payoff in every period up to period t, when she forgoes the future flow payoffs for those traded items, i.e. θ sk is the time-t value of flow payoffs the seller could get from item k. 2.3 Benchmark Protocols Rubinstein-Stahl Bargaining with Many Objects: A Minimalist Extension As an important benchmark, we specify a natural extension of the Rubinstein-Stahl bargaining protocol to multiple objects. One of the agents, say the seller, announces a string of her valuations θ s = ( θ s1,..., θ sn ) and demands a surplus v s V, from some compact set V of possible total values. The other agent, say the buyer, accepts or rejects. If accepted, the buyer picks a subset of items to buy, X {1,..., n}, X, and delivers a net payoff equal to v s based on the seller s announced values: so, the total price given to the seller by the buyer is equal to v s + θ sk. k X 5

7 In the case in which it is the buyer who has announced θ b = ( θ b1,..., θ bn ) and demanded a net payoff v b IR +, then the seller picks the X and the price paid by the buyer is θ bk v b. k X The game ends. If rejected we start again with the roles of the agents reversed (and one period of discounting ensues) The Item-by-Item Bargaining Protocol Another natural benchmark protocol is one in which agents simultaneously bargain over the n objects, but the bargaining is independent across items - so each item is bargained upon via its own price and sequence of offers and counter-offers. It is still possible that agents may tie the bargaining together across items, but only via their equilibrium actions, as otherwise it is simply n separate bargaining processes that are conducted in parallel. One of the agents, say the seller, announces a string of prices for items p s = (p s1,..., p sn ). The other agent, say the buyer, can accept any subset of the items, which are then traded at those prices. If any objects are untraded, we start again with the roles of the agents reversed (and one period of discounting ensues). The buyer then announces a price for each item that still remains untraded. The seller can accept any subset of those remaining items, which are then traded at the buyer s suggested prices. If any objects are untraded, we start again with the roles reversed. We continue in this manner indefinitely or until all goods are traded. Agents obtain net utility discounted to period 0 utiles for the value of any trades relative to the period in which a particular object was traded. We note that in some cases, with bargaining over a contract with many aspects, this sort of bargaining may raise issues of feasibility and interpretation if agents cannot consume until all of the objects are agreed upon. For instance, an employment contract would have to specify wages, pension plan, hours, holidays, etc., and employment might not be feasible until all of the facets were agreed upon. In that case, it might be that only the previous 6

8 protocol would be feasible and this item-by-item protocol might not. In contrast, a consumer thinking about buying several carpets, might bargain with a seller on a item-by-item basis as in this protocol, or for a set of carpets as in the previous protocol A Combinatorial Bargaining Protocol One could also move to the opposite extreme: instead of bargaining item-by-item, people could bargain in ways that allow them to consider trading all possible subsets. Agents alternate in offering. One of the agents, say the seller, names a total price for each possible subset of the objects. Formally, an offer is a mapping p : 2 n R +. The other agent, say the buyer, accepts or rejects. If accepted, the buyer picks items to trade, X {1,..., n}, and the buyer pays the price p(x) to the seller. If rejected we start again with the roles of the agents reversed (and one period of discounting ensues). The richness in message space of the combinatorial protocol is a disadvantage in practice: the size of an offer is 2 n, exploding at exponential rate as n grows. For instance, with n = 20 each offer needs to specify k prices, which is not very realistic in practice, where agents tend to use reduced forms like the one introduced in Section where a much smaller message spaces suffices to convey the most important information. In Section we discuss another natural protocol with small message spaces A Frequency Protocol as a Reduced-Form Protocol To go to another extreme, we examine a protocol in which agents use minimal announcements (of frequencies of types and demanded values), until an agreement is reached and then announce specific values for actual objects in a second stage. This fits with many settings in which people bargain over basic terms, and then after they have reached a tentative agreement then they fill in the details. Consider the following frequency protocol constituting of two phases: Phase 1: The offerer, say the seller quotes a frequency distribution φ n s Φ n s Θ n s, 8 and a v s V. 8 Φ n i is the set of possible frequencies (with n items) on Θ n s. Notice that the quoted frequency φ n i may different from i s true frequency. 7

9 The recipient accepts or rejects. If accepted we move to Phase 2. If rejected we begin Phase 1 again with the roles of the agents reversed (and one period of discounting ensues). Phase 2: (We do this as if the seller made the offer and the buyer accepted in Phase 1, there is an obvious change for the other case) The seller announces θ s n Θ n s that has a frequency distribution φ n s. Either the buyer picks a subset of items to buy X {1,..., n}, and makes a payment of at least v s + θ k X sk n to the seller for those objects; or the buyer says No and there is no trade. The game ends. Note that all of the protocols introduced in Sections are universal, in the sense that the protocols are defined independently of the prior distribution f of the players valuations. This also applies to the frequency protocol as the feasible frequencies that one can announce do not depend on the prior distribution. In contrast, the linking mechanisms introduced in Jackson and Sonnenschein (2007) were defined relative to the distribution of types (see Section 2.3.6). Universal protocols have some advantages in practice as they can become traditions or customs for how to bargain and apply across many settings, rather than requiring adjustment to fit details of the setting A General Definition of an Alternating-Offer Bargaining Protocol With these various examples of alternating-offer bargaining protocols in hand, we provide a definition of what we mean in general by an alternating-offer bargaining protocol when there are n objects. Given a finite set A, let Π(A) denote the set of all partitions of A, with a generic partition denoted π. One of the agents, i (the offerer), announces from a finite set of possible announcements ( offers ) A 0 i, with a generic offer denoted a 0 i. The other agent, j (the recipient), observes some information about the offer I(a 0 i ) which is the element of a partition π 0 Π(A 0 i ) that contains a 0 i. That agent then responds from a set A 0 j(i(a 0 i )), with a generic response denoted by a 0 j. 8

10 As a function of a 0 i, a 0 j some goods (possibly the empty set), denoted X(a 0 i, a 0 j) {1,..., n}, are traded, and some price is paid from the buyer to the seller, and those goods are consumed. The set of goods traded and the price are commonly observable. One period of discounting ensues. Inductively, in period t, agent i(t) makes offers from a finite set A i(t) (h t 1 ) which could depend on h t 1 (a 0 i(0), a0 j(0),..., at 1 i(t 1), at 1 j(t 1) ), the full history of bargaining through the last period 9 The other agent j(t) reacts from a set A j(t) (h t 1, I(a t i(t) )), where I(at i(t) ) is the element of a partition π t Π(A t i) that contains a t i and is the information that j observes about i s latest offer. As a function of h t = (h t 1, a t i(t), at j(t) ), some of the remaining goods are traded and consumed. This continues as long as there are goods remaining to be traded. The information observed, I(a t i(t) ), is shaped by the protocol: I(at i(t) ) = {at i(t) } if the offerer s action is perfectly observed by the responder, I(a t i(t) ) = At i(t) if the offerer s action is not observable. Non-observability allows us to consider simultaneous announcement protocols within the same framework. The frequency protocol can be fit into this framework by having the offerer announce both the frequency, demand, and type profile simultaneously, but then the responder only sees the frequency and not the type profile. To see how our first protocol from Section fits into this definition, note that it is easy let it be such that once any goods are traded, then the remaining announcements are all irrelevant and no further trade is possible and so the game effectively ends. The item-by-item protocol is one in which the set of items changes over time, but otherwise the mechanism looks stationary in terms of the announcements A Linking Mechanism This protocol is based on the mechanisms from Jackson and Sonnenschein (2007). Agents simultaneously announce θ i Θ n i with the restriction that their announced 9 For instance, only some objects might remain, or a protocol might not allow an agent to change prices in certain directions, etc. This specification also allows for various rules for choosing the proposer - an easy extension is to allow for random recognition of i(t). 9

11 frequency of valuations must match the empirical distribution f i ; 10 i.e., for each θ Θ i. {k θ ik = θ} = f i (θ) The outcome is efficient trade at some pre-determined prices relative to the announced θ b, θ s. So, trade of a particular object occurs if and only if θ b > θ s. In the case in which the distribution of the types of each agent is known and exchangeable, any(!) function p(θ si, θ bi ) will actually result in an incentive compatible mechanism (including, for instance, the Rubinstein prices) - which follows from Theorem 1 in Jackson and Sonnenschein (2007). Notice this mechanism does not have universality since the knowledge of the prior distribution is essential to design the mechanism. 3 Challenges in Multi-Object Bargaining In order to illustrate some of the challenges and main ideas, we consider a particular example. In the example, the seller s valuations lie in Θ s = {0, 8}, and the buyer s valuations lie in Θ b = {2, 10}. To start, consider an f such that the marginal distribution to any particular object k is such that it puts equal likelihood on each combination of (θ sk, θ bk ) {(0, 2), (0, 10), (8, 2), (8, 10)}. If the bargaining were just on one object, then we know that there is no Bayesian incentive compatible, Pareto efficient, and ex post individually rational direct mechanism for this problem, and thus there is no alternating offer bargaining protocol with the aforementioned properties. This follows in a general form from Myerson and Satterthwaite (1983), but can be established directly for this setting. Now consider bargaining with multiple objects, and for illustration purpose suppose the agents have valuations independently drawn across objects. As the number of objects becomes large, although uncertainty about the total surplus may vanish, it is still unclear whether we should expect that the Rubinstein-Stahl extension from Section or the item-by-item protocol from Section will lead to nearly efficient outcomes. There are two forces that push in different directions regarding whether we get efficiency as we have many objects for trade: 10 For ease of exposition, we suppose that this is possible. See Jackson and Sonnenschein (2007) for details of how to extend the mechanism in the case in which the integer counts of an announced distribution cannot match up with the expected frequencies exactly. 10

12 There is less uncertainty about overall/per item surplus. There is more uncertainty about which items to trade and at which prices. Thus, there are several issues here with such alternating-offer bargaining protocols: (i) Does reducing uncertainty about overall surplus lead to nearly efficient trade? and (ii) When there are multiple objects can the issue of deciding over which objects to trade at which prices be solved? and (iii) How do the answers to these questions depend on whether we are talking about some versus all equilibria? First note that the challenge is due to the combination of incentives and individual rationality constraints that arise with such bargaining protocols: they naturally imply (interim) individual rationality, since agents bargain (and need to accept an offer) after they observe their own valuations. 11 A successful elicitation of trade-relevant information needs to subsidize an agent differently enough depending on the agent s valuations; on the other hand, interim individual rationality pins down how much an agent should receive with her worst possible type. The two together determine the least amount of share each agent should receive, and efficiency is impossible when the total expected surplus is not big enough to deliver both agents their shares. Let us discuss some of these challenges in turn, beginning with the limiting case in which overall surplus is commonly known. For that case, we show that all sequential equilibria of the the protocol from Section lead to immediate and efficient exchange, despite the substantial uncertainty about which objects to trade. Moreover, we show that the same is true for a general class of bargaining protocols. 4 Multi-Object Bargaining with Known Surplus We focus first on a case in which the surplus is commonly known, but which objects should trade and at which prices is not known. We think of this as the natural limit of worlds in which there are a large number of aspects. (One might think of a parallel between 11 It is important to note that eliminating individual rationality allows for an efficient and (dominant strategy!) incentive compatible mechanism for this problem: The price is always 5, and the agents announce their types and trade occurs unless the announced types are (8,2) in which case no trade occurs. It is easily seen that this mechanism has truth as a dominant strategy and it is efficient, but it violates the individual rationality of the agents who sometimes ends up with negative values. In fact, the mechanism fails to be interim individually rational for either agent even simply conditional upon type (a type 8 seller only ever gets a price of 5, and a type 2 buyer sometimes pays 5 but never less) - although it is ex ante individually rational. (See Gresik 1991 for results on the extension of Myerson and Satterthwaite s results to more general distributions and for interim individual rationality, and noting the possibility of finding ex ante individually rational mechanisms for some distributions.) 11

13 the assumption of commonly-known surplus with only a finite number of objects and the assumption that finite numbers of agents in Walrasian economies take a common price vector as given - concentrating on the insights of how such economies work and then later checking that the results still hold when one drops the assumption and looks at limits of sequences of economies.) We discuss the extensions to the case of a nearly known surplus in Section 5. In particular, a bargaining problem (n, Θ, f) (as defined above) has a known surplus S > 0 if there exists S > 0 such that S(θ s, θ b ) 1 n for every (θ s, θ b ) for which f(θ s, θ b ) > 0. k=1,...,n max(θ bk θ sk, 0) = S Thus, a bargaining problem with known surplus is such that agents know what the total gains from trade are, but not necessarily which objects should be traded. A first question is whether in this case one ends up with results that are closer to Rubinstein s conclusion of efficiency since overall surplus is known, or Myerson and Satterthwaite s inefficiency results since there can still be arbitrary uncertainty about whether it is efficient to trade any particular object. 4.1 An Efficiency Result We first consider the natural extension of the Rubinstein-Stahl protocol that we introduced in Subsection As we will see in Theorem 1, despite the substantial uncertainty about the value of any given object, all sequential equilibria of that protocol lead to to immediate and efficient trade and a unique division of the total surplus. Theorem 1 If a bargaining problem (n, Θ, f) has a known surplus S > 0, then in all sequential equilibria under the protocol introduced in subsection 2.3.1: the agreement is reached immediately, the full surplus is realized, and agents expected net payoffs from trade are their Rubinstein shares, i.e. (1 δ b) S 1 δ b δ s seller, and δ b(1 δ s) S 1 δ b δ s for the buyer. for the All proofs appear in the appendix. We emphasize that this protocol is universal in that the above result above holds for exactly the same protocol for any f with a known surplus. This distinguishes the result from a mechanism-design result in which the mechanism is tailored to the f. Thus, this is not only distinguished because we are taking a positive perspective (examining a mechanism which 12

14 seems natural in terms of how people actually bargain) as opposed to a normative one (using direct mechanisms to prove that efficiency is possible if the designer has sufficient knowledge), but also because the protocol in question is both simple and directly adapts with the environment. The intuition behind Theorem 1 is as follows. If there were any inefficiency in the anticipated equilibrium path, then since the agents know the potential surplus and can make demands for shares of that total surplus, there is an offer that they each know makes them strictly better off if it is immediately accepted. The existence of such an offer rules out inefficient equilibria. The argument for the precise Rubinstein shares is based on an extension of that by Shaked and Sutton (1984). 4.2 The Bargaining Protocol Matters One may conjecture that with known overall surplus, the above result about efficient trade would extend to most any bargaining protocol. This is not the case. The uncertainty about specific matchups can distort the agents incentives and lead to substantial delay and inefficiency in protocols in which agents cannot make offers that require trading on all objects at once. To illustrate this point, we consider the item-by-item bargaining protocol in which agents bargain on objects separately but simultaneously as defined in Section Under this protocol, not only do there exist equilibria with substantial inefficiency, but in many situations substantial inefficiency occurs in all sequential equilibria. This is shown in the following example. Example 1 Consider agents bargaining over n = 2 objects under the item-by-item bargaining protocol. The seller has type (θ s1, θ s2 ) = (0, 0), and there are two equally likely type profiles for the buyer: (θ b1, θ b2 ) = (2, 10) or (10, 2). In all sequential equilibria at most one of the two objects is traded in the initial period. The proof behind the claim in the example appears in the Appendix. Out of situations with uncertainty, this example is such that it should be easiest to reach efficiency since it is commonly known that all objects should trade and there is only onesided uncertainty about whether trade should occur. 12 Nevertheless, even in such a simple environment, none of the sequential equilibria are efficient with item-by-item bargaining. This result is in contrast to Theorem 1, in which all sequential sequential equilibria are efficient under the protocol from Section For reviews of single-object bargaining with one-sided uncertainty, see, e.g. Fudenberg, Levine and Tirole (1985) and Ausubel, Cramton, and Deneckere (2002). 13

15 The inefficiency in Example 1 derives from the seller s incentives to screen the buyer s type to try to obtain a high price for the more valuable item. Such incentives are mitigated when agents can bargain on items overall, as agents have better knowledge about the overall surplus. Instead, in protocols that entail separate bargaining, even though agents can coordinate their actions across objects they cannot take advantage of the greater knowledge that they have about the overall values of trade by offers that involve overall surplus, and the ensuing selection problems are not easily overcome. In the above example, the inefficiency is necessarily large when discounting is substantial, but that leaves open the possibility that, for δ very close to 1, approximate efficiency could also be achieved via the item-by-item protocol. Indeed, extending results from Ausubel and Deneckere (1989) to this multi-object problem, it becomes clear there exist a large set of equilibria only some of which would be approximately efficient in this example. Nonetheless, this example illustrates that the item-by-item protocol performs much differently from protocols in which objects are bargained over together, and that substantial inefficiencies are possible and sometimes ubiquitous when items are bargained over separately, but completely precluded when objects are bargained over simultaneously. 4.3 A General Feature for a Protocol to Provide Efficiency The bargaining protocol allows the agents to bargain over the surplus and discover which objects to trade in an integrated manner that takes advantage of their knowledge of the overall surplus, while item-by-item bargaining does not. There is a general sense in which any protocol that has such integration leads to full efficiency (in all equilibria), as we now formalize. Consider a bargaining problem (n, Θ, f) with a known surplus S. Definition 1 (Share-demanding offers and protocols) A protocol includes a sharev demanding offer in some period t, for some i(t), θ i(t) Θ n i h t 1 if: 13 there exists a i A i(t) (h t 1 ) such that and v [0, S] and after a history for every a j A j(t) (h t 1, I(a t i(t) )) either there is no trade or the realized surplus for i(t) in the current period is at least v, and for any θ j(t) for which f(θ i(t), θ j(t) ) > 0: there exists a j A j(t) (h t 1, I(a t i(t) )) for which the realized surplus in the current period for i(t) is v and the realized surplus in the current period is S v for θ j(t). 13 Payoffs expressed here are not-discounted; i.e., they are evaluated in the current period. 14

16 A bargaining protocol includes share-demanding offers for some set V [0, S], if at any point of the protocol through which no trade has yet occurred, the current offerer i(t) has a share-v demanding offer for each v V and type θ i(t) for which f i(t) (θ i(t) ) > 0. When there is no ambiguity, we say a bargaining protocol includes share-demanding offers if it includes share-demanding offers for [0, S]. We say that a bargaining protocol is fully alternating if the offerer s action is always observable to the recipient, i.e. I(a t i(t) ) = {at i(t) }. Most of the protocols that we have discussed are fully alternating. An exception is the frequency protocol as it involves subperiods so that only frequency announcements are observable in the first phase. Nonetheless, the result below extends to it, but requires a special proof. Since that proof is a special case of Theorem 3 we omit it. Theorem 2 If a bargaining problem with n objects has a known surplus S > 0 and the protocol includes share-demanding offers and is fully alternating, then in all sequential equilibria: the agreement is reached immediately, the full surplus is realized, and the agents expected payoffs equal to their Rubinstein shares; i.e., (1 δ b) S 1 δ b δ s and δ b(1 δ s) S 1 δ b δ s for the buyer. for the seller, The protocol introduced in includes share-demanding offers: at any point of the game and for any share v, a share-v demanding offer is (θ i, v), i.e. the current offerer lists the valuations truthfully, and demands a total share of v. Such an offer, once accepted, gives the offerer exactly payoff of v regardless of the recipient s choice of the objects, and gives the recipient S v if the recipient chooses to trade all objects with values higher than the costs. On the other hand, the item-by-item bargaining protocol does not include share-demanding offers: the offerer s payoff depends on which objects the recipient accepts to trade - and the offerer cannot ask for an overall surplus that must be taken as a whole rather than in part. This creates a tension between the offerer s incentives and the realization of full surplus. For instance, in Example 1 the seller is able to demand a value of v = 6, for instance by asking for a price of 3 on each object. However, the buyer would rather just accept the price of 3 on the more valuable object, thus realizing a total surplus of 7, instead of accepting both and only getting a surplus of 6. This would only lead to a surplus of 3 for the seller. Besides the protocol 2.3.1, many other protocols include share-demanding offers. For example, the combinatorial protocol (from Section 2.3.3) includes share-demanding offers. 15

17 In addition, note that any offer available in the protocol introduced in Subsection has an equivalent offer in the combinatorial protocol. 14 As a result, the combinatorial protocol can be viewed as a richer protocol in terms of the message space. Moreover, that protocol has the advantage of allowing for a general payoff structure. In particular, the agents payoffs can be non-additively separable across objects, but quasi-linear in money; i.e. v s = p u s (X, θ s ), v b = u b (X, θ s ) p, S(θ s, θ b ) = max X (u b (X, θ b ) u s (X, θ s )), where θ i Θ n i is agent i s type. Again time is discounted according to δ i (0, 1). Notice that the above are total payoffs and surplus. The frequency protocol (from Section 2.3.4) also includes share demanding offers, and even though it is not a fully alternating protocol an argument similar to that underlying Theorem 2 shows that its equilibria also result in efficient outcomes. 15 The simplicity of frequency protocol s strategy space is also helpful when we turn to settings with incomplete information about the surplus. Again we note that the protocols just mentioned all satisfy our universality condition; i.e., the protocols work for any distribution and surplus. 4.4 Reopened Bargaining for Not-Yet-Traded Objects Our efficiency results discussed above are robust to reopening the bargaining over the notyet-traded objects. In particular: In the protocol 2.3.1, after the recipient accepts an offer, selects which items to trade, and pays a total price (that delivers the surplus demanded by the offerer), the agents may continue to bargain over the remaining objects, with the last recipient being the first offerer in the continuation - with the same protocol used on the remaining items. 14 Any offer (e.g. from the seller) (θ s, v s ) in the protocol has an equivalent offer p(x) = k X θ sk + v s, X in the combinatorial protocol. 15 This holds when type distributions are exchangeable; a sufficient condition for which is that the valuations are i.i.d. across items and independent across players. Then in all exchangeable sequential equilibria, where agent adopts the same strategy (in Phase 1) for types that have the same frequency, Phase 2 has a unique equilibrium: the recipient chooses those items with positive value-cost gaps, provided the sum of those gaps exceeds the payoff demanded by the offerer; and as a best response, the offerer s lists the valuations as truthful as possible - in particular, the offerer is truthful when the truthful frequency is announced in Phase 1 (which is the case in the share demanding offers). 16

18 In the combinatorial protocol 2.3.3, after the recipient accepts an offer, picks a subset and pays the price for that subset, the agents may continue to bargain over the remaining objects, with the last recipient being the first offerer in the continuation according to the same combinatorial protocol. In both cases, the agents receive the payoffs from the objects traded (and the price paid) at the time when the objects are traded. This implies that different discounting applies to objects traded in different periods. Both protocols with reopening still include share-demanding offers. Take the protocol as an example: being truthful and demanding a total surplus of nv always guarantees nv to self once the offer is accepted, regardless of which objects are not picked up and how they are traded when the bargaining reopens; on the other hand, when the recipient accepts such an offer, the best strategy is to choose to trade all objects with the buyer s values higher than the seller s costs so that no objects with positive surplus are left not traded. Therefore, as a corollary to Theorem 2, it follows that the efficiency of all sequential equilibria is robust to the opportunity of reopening the bargaining. The opportunity to reopen exchange complicates analyses of many settings, from Walrasian exchange, to auctions, to contracting. Here, the robustness comes from the sharedemanding offers and the knowledge of the overall surplus. When there is substantial uncertainty, the potential to reopen discussions distorts incentives for screening, which are completely circumvented with the share-demanding offers. 5 Multi-Object Bargaining with a Nearly-Known Surplus Our analysis with a known total surplus illustrates the idea that with known surplus, sharedemanding offers provide for efficient bargaining and are really essential for such results. The ability to bargain over a full bundle, means that the known surplus dominates over the screening of particular items. Thus, we end up with a sort of Rubinstein result, rather than a Myerson-Satterthwaite result, with many items and uncertainty over each item but a known surplus overall. The exact knowledge of the full surplus is a expository device, as we can imagine that with large numbers of objects agents will have a good idea of the total gains from trade, but still have substantial uncertainty about which precise set of objects should trade. Thus, it is useful to verify that there is not a substantial discontinuity between having the total surplus being nearly-known versus exactly-known. And, given that all equilibria are efficient in the limit, it is enough to look for upper-hemi continuity. 17

19 In this section we explore that continuity. There are several technical difficulties with establishing upper-hemi continuity. The first is that incomplete information game theory is still not well-understood in the case of continua of types and actions - as measurability issues and other issues of updating beliefs conditional on atomless events is cumbersome (e.g., sequential equilibria have only even been recently defined for such settings, and existence is still a tricky problem). This prompts us to work with finite environments and protocols - so that measurability issues are avoided. The second difficulty is that in order to have the total surplus be nearly-known it makes sense to work large numbers of objects - so that one can appeal to laws of large numbers. Unfortunately, that also means that the action space in our previously discussed bargaining protocols explodes exponentially. This leads to challenges in characterizing how beliefs must evolve in equilibria. The third difficulty is that the updating of beliefs is problematic even in very simple incomplete information games - and upper-hemi continuity can require some restrictions. To handle these issues we work with four different approaches: We analyze the frequency protocol (which limits the size of the strategies and makes it easier to identify beliefs); We work with a fixed number of objects with uncertainty that converges to full knowledge, and have agents tremble so that beliefs are tied down; We work with large number of objects but place bounds on the rate at which agents update beliefs over time; and We analyze other protocols in which the strategy space satisfies a size restriction, but still allows for value-demanding offers. So, in this section we begin by discussing the setting and illustrating some of the technical issues via some simple examples. We then turn to each of the four approaches mentioned above. 5.1 Multi-Object Bargaining with Converging Surplus We index things by a sequence m, to allow (but not require) the rate at which uncertainty converges not to depend directly on the number of objects. objects. A sequence of bargaining problems with priors f m converging to S > 0 if S m p S, as m 18 (Θ nm s The m-th economy has n m Θ nm b ) have surpluses

20 where S m is the per-item surplus in the m-th problem; i.e., a random variable S(θ s, θ b ) 1 n m k=1,...,n m (θ bk θ sk ) + with types (θ s, θ b ) drawn according to f m. This embodies the idea that there can be substantial uncertainty about which objects should trade and at which prices. In addition, the above allows for correlations of valuations across objects and between agents. It also allows for heterogeneities across objects. 5.2 A Technical Challenge: Failure of Upper-Hemicontiunity of Sequential Equilibria at the Limit of Certainty We first illustrate the challenge that is substantial but is of a technical nature: sequential equilibria often fail a fundamental upper hemi-continuity condition. We view this as a shortcoming of sequential equilibrium and the current tool-box of game theory. Moreover, this is not solved by standard existing refinements. A game with arbitrarily small (Bayesian) uncertainty is very different from its counterpart with certainty, in the sense that some sequences of sequential equilibria of the former game have no limit in the set of sequential equilibria (subgame-perfect equilibria) of the limit game with certainty. This happens because the the notion of sequential equilibria allows for a lot of freedom in off-path beliefs, and as a result lots of outcomes can be supported as part of a sequential equilibrium by extreme off-path beliefs, and leads to a failure of basic conditions like upper hemicontinuity of the equilibrium correspondence (which generally holds for Bayesian equilibrium: see, e.g. Jackson, Simon, Swinkels and Zame (2002), but fails for sequential equilibria). This challenge is not specific to our multi-object bargaining games. It applies to many simple games, and in particular, upper hemi-continuity even fails in simple single-object Rubinstein bargaining with the most basic forms of uncertainty. 16 Consider alternating-offer bargaining with one object and one-sided uncertainty (let δ s = δ b = δ = 0.8): the buyer s value is commonly known as 10, and the seller s cost is either 0 or 8, so that it is commonly known that the agents should always trade immediately to get efficiency. In addition, suppose the seller s cost has increasing probability on 8 along the sequence. One may conjecture that all sequential equilibria in this game converge to 16 The problem that we are pointing out here is endemic: the example still works with perturbations in the payoffs and/or how the small uncertainty is introduced, as it is freedom in specifying beliefs that cause problems, and not exact indifferences (which lead to lower hemi-continuity problems). Thus, there is a fundamental sort of discontinuity between equilibrium concepts with slight amounts of incomplete information and the limit of full information, which seems symptomatic of the tools of game theory rather than a real phenomenon. This does not contradict the fact that when both the sequence of priors and its limit are in the interior of the distribution space, the set of sequential equilibria satisfies upper hemi-continuity (Kreps and Wilson (1982), Proposition 2, p.876). Here upper-hemi continuity fails since we are converging to complete information. Given the importance of the complete information case in the theory and (its approximation) in practice, the failure of upper hemi-continuity is important and disturbing. 19