Simultaneous Nash bargaining with consistent beliefs

Simultaneous Nash Bargaining with Consistent Beliefs Roberto Burguetyand Ramon Caminalz This draft: November 2010 Abstract We propose and analyze a new solution concept, the Rsolution, for three-person, transferable utility, cooperative games. In the spirit of the Nash Bargaining Solution, our concept is founded on the predicted outcomes of simultaneous, two-party negotiations that would be the alternative to the grand coalition. These possibly probabilistic predictions are based on consistent beliefs. We analyze the properties of the Rsolution and compare it with the Shapley value and other concepts. The Rsolution exists and is unique. It belongs to the bargaining set and to the core whenever the latter is not empty. In fact, when the grand coalition can simply execute one of the three possible bilateral trades, the Rsolution is the most egalitarian selection of the bargaining set. Finally, we discuss how the Rsolution changes important conclusions of several well known Industrial Organization models. Keywords: cooperative games, bargaining, endogenous fallback options, consistent beliefs, Rsolution. JEL classi…cation numbers: C71, C78, L14. We thank Albert Banal, Olivier Compte, Matthew Ellman, Jordi Massó, Clara Ponsatí, Debraj Ray and József Sákovics for useful comments. Also, we acknowledge the support of the Barcelona GSE, Generalitat de Catalunya, and Spanish Ministry of Science and Innovation (project ECO2008-01850). yInstitut d’Anàlisi Econòmica CSIC, and Barcelona GSE, e-mail: [email protected] zInstitut d’Anàlisi Econòmica CSIC, and Barcelona GSE, e-mail: [email protected] 1 1 Introduction When bilateral bargaining is one of the components of an economic model, most authors use the Nash Bargaining Solution (NBS) as a reduced form that maps the fundamentals of the model into negotiated outcomes. Since we often know very little about how agents actually bargain in the real world, a black-box approach seems justi…ed. After all, the principles and intuitions implicit in the NBS are very convincing. However, such broad consensus does not exist when bargaining involves three players and di¤erent pairs of players can achieve by themselves di¤erent agreements.1This is the case when one (or more) player(s) may trade or reach an agreement with two alternative, potential partners. When analyzing such problems, some authors take a non-cooperative approach and assume a particular bargaining protocol. An alternative is to invoke solution concepts borrowed from cooperative game theory. The Shapley value is the most popular choice, as a simple value characterized by seemingly natural axioms. Yet, the Shapley value predicts outcomes that in some cases are controversial, to say the least.2 This paper presents a new solution concept for three-player cooperative games that can be readily applied to predicting the outcome of three-party negotiations. Instead of attempting to identify sensible axioms that single out one outcome or considering a particular protocol that would do the job, our approach is based on a few mainstream ideas in economics. The …rst is that the NBS is a satisfactory prediction for two-player bargaining or in general for what are called pure bargaining games, where the only coalition that adds some surplus is the grand coalition.3The second is that when players bargain they also form beliefs about what would happen if agreement is not reached in that particular negotiation. The third one is 1Examples of economic models that include three-player bargaining abound. In Section 4 we discuss in detail some particular examples. 2See, for instance, De Meza and Selvaggi (2007), page 89. 3See Krishna and Serrano (1996) for a non-cooperative motivation for this solution. 2 that these beliefs should satisfy some notion of consistency with payo¤s. 4 Consider one of the simplest of these three-person bargaining situations, that of a buyer that has to choose among two potential sellers. A prediction for any such model should include a (possibly probabilistic) prediction of which of the two trades will take place and how players would split the surplus in each of the two potential trades.5Also, if the latter prediction is to be made according to the NBS, then disagreement points for each of the two negotiations should be speci…ed. For the buyer, the disagreement payo¤s should be endogenous. Indeed, the fallback option in each negotiation is the possibility to trade with the alternative seller. As we allow for more complex interactions, we will need to consider the case where all two-player negotiations result in some positive surplus. This is known as the three-player/three-cake problem (see Binmore, 1985). In this case, disagreement points and payo¤s will need to be simultaneously and endogenously determined for all three players in all three alternative twoplayer negotiations. Moreover, now the (possibly probabilistic) prediction of what negotiation will end in an agreement will be necessary in order to consistently calculate (expected) fallback options. Finally, what is predicted for the three-player/three-cake problem may leave gains that the three players may realize by coordinating. In other words, the total surplus that the grand coalition can realize may exceed the surplus expected from bilateral negotiations. That may be so because of synergies that can be realized only with the participation of all three players or just because, absent coordination, players anticipate that ine¢ cient bi4In our previous research on labor contracting (Burguet et al., 2002) we also had to decide how to predict the outcome of negotiations among three players. In fact, in the Appendix of that paper we timidly started to outline some of the ideas that we fully develop here. 5The Shapley value predicts that the buyer will buy from the most e¢ cient seller, yet the non trading seller will still receive a positive payment at the expense of the trading partners. Such a positive payo¤ is sometimes interpreted as the bribe that the non-trading seller receives in order to allow the implementation of the e¢ cient trade. We will show that such a justi…cation makes sense only in some games but not in this particular example. 3 lateral agreements may occur with positive probability. In this case, players may be able to avoid ine¢ cient outcomes through three-party negotiations and then we expect them to share the extra surplus according to the (generalized) NBS.6In particular, the disagreement point for this three-player negotiation should be the players’expected payo¤s in the alternative to the grand coalition agreement: the predicted outcome for bilateral negotiations. As we have mentioned, our solution concept requires that agents form (and share) beliefs on the probabilities of success of each alternative negotiation. This is an important feature of our concept. In addition, we will impose a consistency requirement on this system of beliefs: parties should not expect a two-player negotiation to succeed when both parties to that negotiation prefer their alternative one. In Section 2 we present our solution concept, the Rsolution, as a formalization of these ideas. We show that the Rsolution exists and is unique. That is, it turns out that these simple ideas are su¢ cient to predict the division of surplus in these games. Moreover, computing the Rsolution is a straightforward exercise. We provide these computations for all parameter values. The idea that disagreement points in three-party negotiations should emanate from the alternative to these negotiations, that is, the predicted outcomes of simultaneous, bilateral negotiations, is probably non controversial. The same applies to assuming that disagreement points in simultaneous, bilateral negotiations should be endogenous. Moreover, the ideas are not novel. Bennett’s (1997) approach to the analysis of such negotiations is the closest to ours in spirit (also, see Binmore, 1985, and references in Bennett, 1997). Indeed, Bennett also argues that disagreement payo¤s should be obtained endogenously, but in her solution players do not form and share beliefs about the probability of success of each bilateral negotia6Three-party negotiations may not be feasible due to outside constraints. In Section 4, we consider one case when this is so. 4 tion.7Indeed, in Bennett’s approach, when two parties negotiate both use as a fallback option their own agreement with the third player. That is equivalent to assuming that di¤erent players assign probability one to two di¤erent, mutually exclusive outcomes. On the contrary, a central piece of our concept is the endogenously determined, coherent system of beliefs that players use to compute their endogenous fallback options.8 We analyze the properties of the Rsolution in Section 3. We show that the Rsolution satis…es symmetry, e¢ ciency, and the dummy player axioms. Thus, it has to violate the additivity axiom since the Shapley value is the only solution concept that satis…es all four. Indeed, the Rsolution is not additive. We argue that, rather than a weakness, this non additivity is a desirable property of the concept for problems like the one discussed above. The seemingly innocuous additivity axiom implicitly imposes too much structure on what "protocols" are feasible for the players. For instance, in our one-buyer, two-sellers example, it implicitly imposes that the buyer cannot attempt bundling or make joint o¤ers for two goods when dealing with the same two potential sellers of these two goods. The Rsolution lets the primitives of the problem speak about such possibilities. Contrary to the Shapley value, the Rsolution is a selection of the core when the latter is not empty. When the core is empty, the Aumann-Maschler bargaining set (BS) is the most popular generalization. The BS contains the core and is never empty. We show that, again contrary to the Shapley value, the Rsolution is a selection of the BS. In fact, for superadditive, 7In Bennett (1997), a solution should specify the division of surplus in each alternative bilateral negotiation. The disagreement point in each negotiation is the payo¤ that each player would obtain in her alternative negotiation. Thus, the disagreement point in some negotiations may be outside the feasible set of that negotiation, which Bennett interprets as failure of the negotiation. A predicted outcome speci…es what negotiation will succeed and then sharing of the surplus according to the NBS (or any other concept) given the corresponding disagreement point. 8In Section 3 we also discuss alternative approaches to endogenizing fallback options, which are implicit in the notion of consistency proposed by Hart and Mas-Colell (1989) and Serrano and Shimomura (1998). 5 three-player TU-games, the BS (for the grand coalition) coincides with the core when the latter is not empty, and is a singleton when the core is empty. Thus, the Rsolution coincides with the BS in the latter case. Moreover, if bilateral bargaining is all there is in the game, that is, if the grand coalition does not add any additional surplus, the Rsolution is the most egalitarian selection in the BS. Thus, it is more egalitarian than other, di¤erent selections of the core or the BS, like the nucleolus.9 We postulate the Rsolution as a satisfactory, unifying concept that can be used to analyze models that include three-party negotiations. In Section 4 we illustrate the use of our concept in some leading models in the Industrial Organization literature. Exclusive contracts (Segal and Whinston, 2000), endogenous mergers (Horn and Persson, 2001), and the property-rights theory of the …rm (Hart and Moore, 1990) have been analyzed in models with a renegotiation stage, but using some other, diverse solution concepts. In Section 4 we also discuss the use and implications of the Rsolution in these cases. Section 5 o¤ers some closing discussions. Finally, most of the proofs are relegated to an Appendix. 2 The Rsolution of a three-person game Let N=f1;2;3gbe the set of players, and let 2Nrepresent the set of subsets of N. An element Z22Nrepresents a coalition. A TU game in characteristic form is the pair (N; v), where v: 2N!Rsatis…es v(?) = 0. We assume vto be superadditive. Assumption 1 (superadditivity): If Z; Z02Nand Z\Z0=?, then v(Z) + v(Z0)v(Z[Z0). To save some space, we will use an abbreviated notation for the vfunction. Thus, we will let vij =v(fi; jg),vi=v(fig)and V=v(f1;2;3g). 9The nucleolus is also a selection of the BS. Thus, when the core is empty, the nucleolus and the RSolution coincide. However, when the core is not empty and set-valued, the two concepts di¤er. More on this in Section 3. 6 Also, every time we write "for all i; j" or "for all i; j; k" we mean for all i; j = 1;2;3; i 6=j, and for all i; j; k = 1;2;3,i6=j6=k; i 6=k, respectively. That is, di¤erent sub/superindices in the same expression will always denote di¤erent players. Without loss of generality, we will assume that v12 v1v2v13 v1v3v23 v2v3. In other words, coalition f1;2gis the (weakly) most "e¢ cient" among the two-player coalitions and coalition f2;3gis the (weakly) least e¢ cient. The heart of our solution concept is a prediction of the outcomes of the three possible bilateral negotiations, including a prediction of which of these negotiations would succeed (with what probability), should threeplayer negotiations fail.10 In many cases this is in fact all that will be needed for predicting the outcome of the whole game. We begin by de…ning this prediction for the outcome of simultaneous, bilateral negotiations. For each player iin each bilateral negotiation ij, we denote i’s predicted payo¤ by uij i. Also, we represent by pij the predicted probability that players iand jare the ones whose negotiation succeeds and then "trade". Finally, since our concept is based on the two-player NBS, for each player iin each bilateral negotiation ij, we will de…ne i’s disagreement payo¤ or fallback option, which we will represent by tij i. Before de…ning our solution, we explain the consistency requirements on these values that will de…ne our solution concept for simultaneous, bilateral negotiations. i) Given the fallback options, tij i, players iand jshare any extra surplus equally, provided this surplus is positive. That is, uij i=tij i+1 2vij tij itij j= 1 2vij +tij itij j, if vij tij i+tij j. However, if their disagreement payo¤s sum up to an amount in excess of the worth of the coalition, vij < tij i+tij j, then players will not be willing to reach an agreement. In this case, uij i=tik i. In the next paragraph we discuss the reasons and interpretation of this spec10 As in the one-buyer/two-sellers example or in the three-player/three-cake game, we assume that only one of the two-player coalitions could form, if the grand coalition cannot form. See Sections 4 and 5 for more on this. 7 i…cation. ii) The disagreement payo¤s are computed according to the payo¤s predicted in, and the probability distribution over alternative, two-party negotiations. In particular, assume that the negotiation between iand j‡ounders, and players contemplate their options in the larger picture of all two-player negotiations. As players calculate what they expect to get in this scenario, tij i, they predict that, (a) with probability pij what they face is precisely this default, tij i; (b) with probability pik coalition (i; k)will reach an agreement, and player i’s payo¤ will be uij i; and (c) with probability pjk it will be coalition (j; k)who will agree, and hence i’s payo¤ will be vi. Thus, tij i=pijtij i+pikuik i+pjkvi. If pij <1we can rewrite this expression as: tij i=pikuik i+pjkvi 1pij : Thus, player i0s fallback option in her negotiation with jis the expected payo¤ in alternative negotiations, where the expectation is "conditional" on her negotiation with jhaving come to a halt.11 If the sum of the disagreement points in the negotiation between players iand kexceeds the worth of that coalition, vik < tik i+tik k, then uij i=tik i, and then the de…nition above implies that tij i=vi. In other words, if an agreement between iand kis not viable, then when players iand jnegotiate they anticipate that if they do not reach an agreement then players jand kwill do so and share vjk with probability one, so that player i’s payo¤ will be vi.12 Thus, player i’s payo¤ when (hipothetically) dealing with kif her negotiation with jare suspended coincides with her payo¤ when dealing with junder the same assumption, i.e., tij i. iii) pij is (virtually) zero if uik iuij iand ujk juij j, with one strict in11 In contrast to our approach, Benett (1997) assume that players iand jbelieve that each one of them will be able to reach an agreement with player kwith probability one, in case negotiations between iand jfail. 12 Note that if coalition (i; k)is not viable then uik iwill not enter into the computations of expected payo¤s, and will only matter in the determination of tij i: 8 equality. That is, an agreement between players iand jcannot be reached (with non-negligible probability) if both players prefer their alternative agreement, one of them strictly. Thus, we will build on the NBS by de…ning endogenous fallback options for each negotiation. Often, our solution will predict that some coalition would form with probability one, should the three-player coalition fail to form. However, probability one events leave too many degrees of freedom with respect to what are consistent outcomes in the rest of events. In order to avoid this indeterminacy, we will proceed in the standard way of …rst considering only probability distributions that assign to each two-player negotiation a probability of success bounded away from 1. De…nition 1 For  > 0, an Prediction for simultaneous, bilateral negotiations for the three-player game (N; v),PSBN for short, is a triple nuij i(); tij i(); pij ()oi;j=1;2;3that satis…es: 1) uij i() = (1 2vij +tij i()tij j()if vij tij i() + tij j(); tik i()otherwise; 2) tij i() = pij ()tij i() + pik ()uik i() + pjk ()vi,for all i; j; k; 3) p12 () + p13 () + p23 ()=1;pij ()1for all i; j; and for all i; j; k,pij ()<  if uij i()uik i()and uij j()ujk j(), with one strict inequality. Our prediction for simultaneous, bilateral negotiations is the limiting value of predictions as the upper bound on pij tends to 1. De…nition 2 A Prediction for simultaneous, bilateral negotiations for the three-player game (N; v), PSBN for short, is a triple nuij i; tij i; pijoi;j=1;2;3 that satis…es lim!0nuij i(); tij i(); pij()oi;j=1;2;3=nuij i; tij i; pijoi;j=1;2;3. 9 the grand coalition coincides with the core if the latter is not empty. If the core is empty, then the bargaining set of the grand coalition is a singleton. The proof of this popular lemma is given in the Appendix. This lemma allows us to consider only the relationship between the Rsolution and the BS. Proposition 2 The Rsolution belongs to the bargaining set (for the grand coalition) and so to the core if the latter is not empty. Proof. First, we study the core. An element of the core is a positive vector (x1; x2; x3)such that: (i) x1+x2+x3=Vand (ii) xi+xjvij for all i; j. Adding up these last three conditions, we obtain x1+x2+x3v12+v13+v23 2, which combined with condition (i) gives: Vv12 +v13 +v23 2:(2) When v12 v13 +v23, i.e., in Regions 1 and 2, this is satis…ed trivially. It is then immediate to check that the Rsolution satis…es (i) and (ii) in Regions 1 and 2. Thus, in Regions 1 and 2 the Rsolution belongs to the core and then to the BS. In Region 3 the core may be empty, that is, (2) may not hold. Thus, we will show that the Rsolution belongs to the BS. Remember that in Region 3 Ui=V+vij +vik2vjk 3. Since Uivifor all i, and since the grand coalition cannot be part of an objection, we need only consider objections that use two-player coalitions. Thus, consider an objection of iagainst j, for i= 1;2;3, and j6=i. That is, consider a division of vik,x= (xi; xk) where k6=i; j:xi+xk=vik, such that xi> Ui, and xk> Uk. We show that there is a counter-objection of j, that is, a division y= (yj; yk)of vjk where yj+yk=vjk, such that yjUjand ykxk. Consider in particular yj=Uj, so that yk=vjk Uj. If xi> Ui, then xk=vik xi< vik Ui. But then ykxk> vjk Uj(vik Ui) = 0: 16 Thus, if xis an objection then yis a counter-objection. QED. In Region 3, when V < v12+v13+v23 2, since the BS is a singleton and the Rsolution belongs to the BS, we conclude that the Rsolution coincides with the BS, and so with any selection or subset of the BS, in particular the nucleolus and the kernel (Nash set). We next discuss the Rsolution with regard to these concepts and the "consistency" motivations behind them. For the rest of this section, let us restrict attention to the case V=v12. i.e., suppose that the grand coalition does not add surplus. In this domain, the Rsolution can be characterized from a perhaps surprising perspective. Indeed, let us label an allocation as the most egalitarian in a set if it Lorentzdominates the rest of allocations in the set. Proposition 3 If V=v12, the Rsolution coincides with the selection of the most egalitarian allocation in the bargaining set. Thus, it also coincides with the selection of the most egalitarian allocation in the core, when the core is not empty. Proof. Note that U1U2U3. Thus, a more egalitarian allocation would require to increase the payo¤ of player 3or, at least, to increase the payo¤ of player 2by reducing the payo¤ of player 1. We show …rst that in Region 1and Region 2any allocation xin the BS or, equivalently in these regions, in the core assigns a payo¤ x3= 0. Assume otherwise x3>0. Then x1+x2=v12x3< v12, so that the allocation would not be in the core. This immediately proves that the Rsolution is the most egalitarian allocation in the BS for Region 1. Now suppose that we are in Region 2and that there is an allocation xthat is more egalitarian than the Rsolution. Since x3= 0, this implies that x2> v12v13, so that x1+x3=v12 x2< v13 violating the conditions for xto be in the core. Thus, the Rsolution is the most egalitarian allocation in the BS in Region 2. Finally, in Region 3 the core is empty, so that the BS is a singleton. Thus, the Rsolution is the only allocation in the BS. QED 17 Thus, the Rsolution is the most egalitarian among the stable (in the sense of Aumman-Maschler) allocations. That is, the most egalitarian among the allocations that cannot be blocked in the sense of the (grand coalition) BS. An alternative selection in the BS is the nucleolus. For these games, the nucleolus is also a selection of the kernel, itself a subset of the BS. Thus, as we mentioned above, in Region 3 the four concepts, BS, nucleolus, kernel, and Rsolution, coincide. In regions 1 and 2 and when v12 =V(the core is not empty), the nucleolus is (x1; x2; x3) = 1 2(v12 +v13 v23);1 2(v12 +v23 v13);0.18 Both the kernel and the Shapley value coincide with the NBS for twoplayer, TU games. Moreover, each of the two concepts has been shown to be the unique generalization of the NBS, in the sense that each satis…es a different concept of internal consistency (Serrano and Shimomura, 1998; Hart and Mas-Colell, 1989).19 For the present discussion, the concept of internal consistency means that if x= (x1; x2; x3)is the corresponding solution it satis…es the following property: for any pair of players i; j,(xi; xj)is the NBS of a reduced game (N0; v0), where N0=fi; jgand v0(N0) = xi+xj. The difference between the two consistency criteria lies in what v0(fig)and v0(fjg) are. That is, the disagreement point in the reduced negotiation between i and j. Keeping the normalization vi= 0, for the kernel (and nucleolus, since for these games both concepts coincide), v0(fig) = maxfvik xk;0g(Serrano and Shimomura, 1998), whereas for the Shapley value v0(fig) = 1 2vik (Hart and Mas-Colell, 1989). That is, in both cases if two players i; j bargain over how to share the total that the solution allocates to them, xi+xj, they still agree on the division (xi; xj), provided the disagreement point is as speci18 See Leng and Parlar, 2010. 19 Compte and Jehiel (2010) de…ne another extension of the NBS, the Coalitional Nash Bargaining Solution, as the allocation that maximizes the product of payo¤s in the core. Note that, with three players and v12 =V, all core allocations give a product of payo¤s equal to 0. When the core contains an interior (which requires V > v12 ), the Coalitional Nash Bargaining Solution and the most egalitatian selection of the core coincide. Yet the Rsolution is not the most egalitarian selection in this case. 18 …ed. This latter point is the crucial di¤erence between the kernel and the Shapley value on one hand, and the Rsolution on the other. Just like in the concepts proposed by Bennett (1997), neither of the disagreement payo¤s for the negotiation between iand jde…ned above come from a feasible, alternative agreement. For instance, for i; j = 1;2, the disagreement point that sustains the kernel is (v13; v23)and the one that sustains the Shapley value is (v13 2;v23 2). Although they imply a di¤erent division of the surplus with player 3, in both cases the disagreement payo¤s result from player 1 and also player 2"trading" with player 3. But those two trades are mutually exclusive, and in that sense players’expectations are not consistent. Instead, in the Rsolution, if two players i; j bargain over how to share their total payo¤, Ui+Uj, they still agree on the division (Ui; Uj)provided that the disagreement point is a lottery over the payo¤s (vik Uk;0) and (0; vjk Uk), where the lottery is part of the solution. That is, the NBS and the Rsolution are also consistent, but in a way that is itself based on consistently computed disagreement points.20 4 Applications In this section we study in some detail how our solution concept changes the predictions of well-known Industrial Organization models in which bargaining among three players plays a crucial role. We start with a model (Segal and Whinston, 2000) that …ts perfectly within the set of games considered in previous sections. Next, we discuss an example (Horn and Persson, 2001) where bilateral agreements generate externalities (the worth of an individual coalition depends on whether or not the other two players reach an agreement). We argue that the Rsolution can also be applied to this type of games (partition function form) by simply taking into account the value of 20 In Region 3, all concepts coincide. The reason is that in that case the "feasible" disagreement point and the "infeasible" one lie on the same 45 degree line in the payo¤ space for any pair i; j. 19 individual coalitions conditional on the agreement between the other two players. Moreover, in this example the grand coalition cannot form and hence in this case the natural solution concept is not the Rsolution but the PSBN. Finally, we argue that the ideas contained in the Rsolution can easily be extended to match the three-player example discussed by Hart and Moore (1990). The main issue in this example is that bilateral trades are not mutually exclusive. 4.1 Exclusive contracts Segal and Whinston (2000), SW, study the impact of exclusive contracts. Their main insight is that an exclusive contract enhances the ability of the incumbent seller to capture rents in the ex-post bargaining game, but it is irrelevant in protecting his relation-speci…c investment, unless such investment generates an externality on the entrant. This is a somewhat counter intuitive result that contradicts the conventional wisdom (see, for instance, Klein, 1988, Marvel, 1982, or Masten and Sneyder, 1993). Here we discuss a version of the model presented in their Section 2. There are three players B; S; and E: Player B(buyer) derives a potential utility of 1from one unit of the good that can be provided by either S(the incumbent seller) or by E(the entrant). There are three periods: 0;1;and 2. In period 0,Sand Bmay or may not sign an exclusive contract. In period 1, player Stakes a costly investment decision, x2[0;1], which a¤ects the incumbent seller’s costs. Also in period 1, once xis …xed, players learn the realization of a random variable y2[0;1], which in‡uences the entrant’s cost and is distributed according to the cumulative function H(y)and has expectation by. In period 2production and trade take place, and players receive their payo¤s. Players Sand Ecan produce one unit of the good at a cost cs(x)and ce(y), respectively. For simplicity, we assume cs(x)=1x and ce(y)=1y. If in period 0players Sand Bhad signed an exclusive 20 contract, then in period 2player Bcannot purchase from Ewithout S’s permission. In both cases, with and without an exclusive contract, B; S; and Ebargain in period 2about who produces the good and how the surplus is distributed. In their general model SW use a generalization of the Shapley value as the solution concept for the renegotiation in period 2:As an illustration of their ideas let us apply the Shapley value to the above simple version of their model.21 In the absence of any contract, the worth of various coalitions is as follows: V= max fx; yg; vSB =x; vBE =y: (3) The rest of coalitions have a worth of 0. Under the exclusive contract, the only di¤erence is that vBE = 0: According to the Shapley value, without exclusivity S0s payo¤ equals Une S=1 3max fxy; 0g+x 6:Thus, S’s marginal return on investment is 1 3H(x)+ 1 6:Note that the marginal return on investment for the pair (B; S) is 2 3H(x) + 1 3. Hence, from the point of view of the pair (B; S)there is underinvestment (the classic hold up problem). Surprisingly, under the Shapley value an exclusive contract does not help reducing the underinvestment problem. More speci…cally, under exclusivity player S’s payo¤ is equal to Ue S=1 3max fx; yg+x 6and the marginal return on investment is also 1 3H(x) + 1 6:22 Conclusion 1 Under the Shapley value, an exclusive contract does not affect investment incentives. 21 In their Section 2, SW consider the case of a competitive entrant who is willing to supply the good at a price pe= 1 y; and given such an outside option players Band Sengage in bargaining and the outcome is determined by the NBS. It turns out that exclusivity is also neutral with respect to investment incentives. 22 The marginal social return on investment is H(x):Hence, the equilibrium level of investment may be below or above the …rst best level. 21 It is important to emphasize that the neutrality result hinges on the speci…c way the Shapley value is computed. An exclusive contract only changes player S’s payo¤ by changing his marginal contribution to the grand coalition. In the absence of exclusivity S0s marginal contribution to the grand coalition is max fxy; 0gand under exclusivity it is max fx; yg:The di¤erence between these two values is y: Hence, under exclusivity S0spayo¤ increases by 1 3y(where 1 3is the weight of the grand coalition in payo¤s), but S’s marginal return on investment remains unchanged. Let us now analyze the same problem when we use the Rsolution to predict payo¤s in period 2. In this case, player S’s payo¤ is Une s=8 < : x 2;if yx 2 xy; if xyx 2 0;if x < y Thus, S’s marginal return on investment is H(x)1 2Hx 2. Once again, there is underinvestment from the point of view of the pair (B; S): the marginal return on investment for the pair (B; S)is equal to min fH(2x);1g: Under exclusivity, player S’s payo¤ is the same that we found when we used the Shapley value: Ue S=x 2;if yx; x 6+y 3, if yx: Thus, S’s marginal return on investment is 1 3H(x) + 1 6. Therefore, under exclusivity investment incentives may be enhanced or depressed with respect to the no contract case. Note that under exclusivity the Rsolution and the Shapley value coincide. Hence, we need to understand why these two solution concepts deliver di¤erent payo¤s in the absence of a contract. In the latter case, if y > x the Shapley value grants player Sa payo¤ of x 6. If we think in terms of the sequential arrival interpretation of the Shapley value, such payo¤ results from the fact that Smakes a positive contribution in case he arrives second after player B. However, according to the Rsolution player Sis redundant 22 and should get a zero payo¤ (he is player 3and the only thing that he might do is to in‡uence the way yis split between Band E). Thus, under the Rsolution incentives to invest will be enhanced if y > x is a likely scenario; i.e., if investment costs are relatively high so that xis low. However, if x 2< y < x then player S’s marginal contribution to the coalition with B is x(weight 1 6) and the marginal contribution to the grand coalition is xy (weight 1 3). Hence, according to the Shapley value Sis able to appropriate one half of his investment e¤orts. In contrast, the Rsolution grants player Sa payo¤ of xy(in this case player Sis player 2), and hence he is able to appropriate the entire return on investment. Thus, under the Rsolution incentives to invest are depressed if x 2< y < x is a likely scenario; i.e., if investment costs are relatively low and xis high. In this case, the paradoxical result obtained by SW is magni…ed.23 Conclusion 2 Under the Rsolution, an exclusive contract enhances investment incentives if the cost of investment is relatively high, but the opposite holds if the cost is relatively low. In other words, under the Rsolution exclusivity helps protecting relationspeci…c investments only when the seller’s competitive position is su¢ ciently weak. Exclusivity is useful only when there is a lot to protect.24,25 4.2 Endogenous mergers Horn and Persson (2001), HP, present a model of endogenous merger formation. Here we focus on the example discussed in their Section 2.1, which 23 If y < x 2S’s marginal return on investment is equal to 1 2;under both the Shapley value and the Rsolution. 24 If investment costs are su¢ ciently high (xlow);then the level of investment under exclusivity is ine¢ ciently high. In other words, from a social point of view an exclusive contract may actually overprotect relation-speci…c investments. 25 De Meza and Selvaggi (2007) also show that SW’s conclusions are not robust to changes in the solution concept for the bargaining game. They set up a non-cooperative bargaining game that delivers di¤erent predictions than the Rsolution and show that exclusivity always enhances investment incentives. 23 considers a market initially populated by three oligopolistic …rms, 1, 2, and 3. They are allowed to merge, but not to form a monopoly. In other words, there are four possible market structures: no merger, 1 and 2 merge, 1 and 3 merge, and 2 and 3 merge. Although …rms are symmetric before any merger, the synergies generated by alternative mergers are asymmetric. Firms’pro…ts in the no merger case are normalized to 0. Pro…ts of the …rm resulting from the merger between …rms iand j, and the non-merged …rm kare denoted by ij and krespectively and are: 12 = 70; 3= 50; 13 = 100; 2= 0; 23 = 90; 1= 5: In previous sections we de…ned the Rsolution for games in characteristic form, where the value of a coalition is independent of the agreements reached by players not included in the coalition. However, in HP the value of stand-alone coalitions do depend on whether or not the other two players have reached an agreement. Thus, this model can be described as a game in partition function form (Lucas and Thrall, 1963). In a three-player game, we need to specify what player ican obtain if no coalition is formed, wi ffig;fjg;fkgg;and what player iobtains if the other two players do form a coalition, wi ffig;fj;kgg. Myerson (1977) extended the Shapley value for partition function form games. In his extension, player i’s payo¤ depends on both, wi ffig;fjg;fkgg and wi ffig;fj;kgg. On the contrary, the de…nition of the Rsolution already takes into account possible externalities. The standalone worth plays a role only in the de…nition of the values tij iand tik i. These values are obtained as a probability distribution over the events that can be expected as an alternative to iforming coalition with jor k, respectively. The only such event that has istanding alone is the formation 24 of coalition fj; kg. Thus, only wi ffig;fj;kgg matters, and then vishould be interpreted as this value. Summarizing: Remark 2 The Rsolution de…ned for games in characteristic form can also be applied to games in partition function form, simply by replacing the worth of individual coalitions, vi, with the worth of individual coalitions conditional on the other two players forming a coalition, wi ffig;fj;kgg: The net surplus created by each merger is given by: 12 12= 65; 13 13= 45; 23 23= 35: Thus, the most e¢ cient merger (from the point of view of …rms’pro…ts) is the one between …rms 1and 2. HP use as a solution concept the set of market structures that are not dominated from the point of view of decisive players. In other words, in an Equilibrium Ownership Structure (EOS) the sum of pro…ts achieved by all decisive players must be at least as high as in any other market structure. Since in this example all players are decisive when we compare alternative duopolies (all …rms have a di¤erent position in each possible market structure resulting from a merger) then the only market structure which is undominated is the resulting from the merger between …rms 1and 2:In other words, HP predict that the most e¢ cient market structure will occur with certainty. Conclusion 3 Under the notion of Equilibrium Ownership Structure the e¢ cient merger occurs with probability one. HP do not allow any transfer between the merged and non-merged …rms. Hence, since the grand coalition cannot be formed, we cannot directly apply the Rsolution to this particular model. However, we can still predict 25 Moore (1990) paper. In the spirit of our solution concept, the set of possible events when the grand coalition fails to form should in‡uence the way parties share the surplus if the grand coalition does form. In a companion paper, we extend the Rsolution for games with any such set of possible events. In general, the information contained in the characteristic function of a game is not su¢ cient to determine that set. Therefore concepts that are de…ned on only the information contained in the characteristic function, like the Shapley value, will be insensitive to variations in the set of possible events. The study of games involving more than three players poses new questions that are not present in the current analysis. One set of such questions has to do with the hierarchy of coalitions and is related to the discussion in the previous paragraph. As we have just mentioned, in this paper we have assumed that if the grand coalition breaks down then only one trade between two players can be realized. In fact, there are three alternative two-player coalitions and each one of them is expected to strike a deal with certain probability. Therefore, computing the fallback option of each player in each coalition is relatively straightforward. However, in a four-player game, if the grand coalition fails then the relevant alternatives are not so easy to obtain even if we impose that only disjoint coalitions can form. The alternative to the grand coalition may be a one three-player coalition, excluding the fourth player but it may also be two disjoint two-player coalitions. Specifying the fallback option of a particular player in an arbitrary coalition can still be done along the lines discussed in Subsection 4.3, but it involves a higher degree of complexity. We leave the analysis of games with more than three players for future research. 6 References Bennett, E. (1997), "Multilateral Bargaining Problems", Games and Economic Behavior 19, 151-179. 32 Binmore, K. (1985) "Bargaining and Coalitions." Chapter 13 in Gametheoretic Models of Bargaining, ed. Alvin Roth. Cambridge: Cambridge University Press, 269-304. Binmore, K., A. Shaked, and J. Sutton (1989) "An Outside Option Experiment", Quarterly Journal of Economics 104 (4), 753-770. Binmore, K., M. Osborne, and A. Rubinstein (1992) "Noncooperative Models of Bargaining", Chapter 7 in Handbook of Game Theory, Vol. 1, ed. R. Aumann and S. Hart, Elsevier Science Publishers, 179-225. Burguet, R., R. Caminal, and C. Matutes (2002), "Golden cages for showy birds: optimal switching costs in labor contracts", European Economic Review 67 (7), 1153-1185. Compte, O., and P. Jehiel (2010), "The Coalitional Nash Bargaining Solution", Econometrica, 78 (5), 1593-1623. Chiu, S. (1998), "Noncooperative Bargaining, Hostages, and Optimal Asset Ownership", American Economic Review 88 (4), 882-901. De Meza, D. and B. Lookwood (1998), "Does Ownership Always Motivate Managers? Outside Options and the Property Rights Theory of the Firm", Quarterly Journal of Economics 113 (2), 361-386. De Meza, D. and M. Selvaggi (2007), "Exclusive Contracts Foster RelationshipSpeci…c Investment", The RAND Journal of Economics 38 (1) (Spring), 85-97. Grossman, S., and O. Hart (1986), “The Costs and Bene…ts of Ownership: A Theory of Vertical and Lateral Integration”, Journal of Political Economy 94, 691-719. Hart, O. and J. Moore (1990), "Property Rights and the Nature of the Firm", Journal of Political Economy 98 (6), 1119-1158. Hart, S. and A. Mas-Colell (1989), "Potential, Value, and Consistency", Econometrica 57 (3), 589-614. Horn, H. and L. Persson (2001), "Endogenous mergers in concentrated 33 markets", International Journal of Industrial Organization 19, 1213-1244. Klein, B. (1988), "Vertical Integration as Organizational Ownership: The Fisher Body-General Motors Relationship Revisited." Journal of Law, Economics and Organization 4, 199-213. Krishna, V. and R. Serrano (1996), "Multilateral Bargaining", Review of Economic Studies 63, 61-80. Leng, M. and M. Parlar (2010), "Analytic Solution for the Nucleolus of a Three-Player Cooperative Game", Naval Research Logistics 57, 667672. Lucas, W., and R. Thrall (1963), "n-Person Games in Partition Function Form", Naval Research Logistics Quarterly 10, 281-298. Marvel, H.P (1982), "Exclusive Dealing." Journal of Law and Economics 25, 1-25. Masten, S.E. and E.A. Snyder (1993), "United States versus United Shoe Machinery Corporation: On the Merits." Journal of Law and Economics 36, 33-70. Myerson, R. (1977), "Values of Games in Partition Function Form", International Journal of Game Theory 6 (1), 23-31. Segal, I. and M. Whinston (2000), "Exclusive Contracts and Protection of Investments", The RAND Journal of Economics 31, (4) (Winter), 603633. Serrano, R. and K. Shimomura (1998), Beyond Nash Bargaining Theory: The Nash Set, Journal of Economic Theory 83, 286-307. Shaked, A. and J. Sutton (1984), "Involuntary unemployment as a perfect equilibrium in a bargaining model", Econometrica 52, 1351-1364. Spier, K. and M. Whinston (1995), On the E¢ ciency of Privately Stipulated Damages for Breach of Contract: Entry Barriers, Reliance, and Renegotiation, The RAND Journal of Economics 26, (2) (Summer), 180-202. Winter, E (2002) "The Shapley value", chapter 53 in R.J. Aumann & S. Hart (ed.) Handbook of Game Theory with Economic Applications, 202534 2054. Weber, R. (1988) "Probabilistic Values for Games", chapter 7 in A. E. Roth (ed.) The Shapley value: essays in honor of Lloyd S. Shapley, 101-120. 7 Appendix 7.1 Proof of Proposition 1: First we propose an PSBN for the game (N; v)for small enough. This will show existence. To save in notation, we will dispose of the ()index of the solution, and specify if we refer to the limit instead. 1) Let 1 2v12 v13. 1.a) If 1 2v12 > v13 (so that v12 v13 +v23 is also satis…ed), consider u12 1=u12 2=1 2v12, and uij i= 0 for all other values of i; j. Also, let p12 = 1, p13 =p23 = 2. Finally, let t12 1=t12 2=ti3 3= 0 and ti3 i=1 2v12 for i= 1;2. Note that lim!01 2v12 =1 2v12 > v13 v13. Thus, for su¢ ciently small, this satis…es the de…nition of an PSBN. 1.b) If 1 2v12 =v13 > v23, consider u12 1=u12 1=1 2v12 (=v13), and u23 2=u23 3= 0. Also, let p12 = 1 and p13 = 0,p23 =. Then, t12 1= t23 3=t12 2=t13 3= 0 and t23 2=1 2v12 > v23. To complete the de…nition of an PSBN we need only t13 1= (1 )1 2v12,u13 1=1 2(v13 +t13 1t13 3) = (1  2)v13 and u13 3= 2v13. 1.c) If v13 =v23 =1 2v12, consider u12 1=u12 2=1 2v12 (=vi3,i= 1;2), p12 = 1 and p13 =p23 = 2. Then ti3 i=(1)v12 2< vi3,i= 1;2. Also, consider u13 3=u23 3=A > 0. Thus, ti3 3,i= 1;2, will have to satisfy: ti3 3=A 2;and A=1 2vi3(1 )v12 2+A 2; and solving for Ataking into account that 1 2v12 =vi3, we obtain A=vi3 43; which is smaller than vi3for small . Note that for small ti3 3+ti3 i< vi3, i= 1;2. Also, note that given these values for ui3 i, we should de…ne t12 1= t12 2=  2(vi3A) =(v122A) 4, and t12 1+t12 2< v12. This satis…es the de…nition of an PSBN. 2) If v12 v13 +v23 but v13 >1 2v12, then consider u12 1=u13 1=u1, to be obtained later, with 0< u1< v13, and u23 2=u23 3= 0. Thus, u12 2= v12 u1> u23 2and u13 3=v13 u1> u23 3. Consequently, let p23 =. Then 35 p12 = 1 p13. Finally, u23 2=u23 3= 0 implies that t12 2=t13 3= 0, and we can then check that t12 2+t12 2< v12, whereas t13 1+t13 3=p12u12 2 1+p13u13 3 1 = (v12 u1)p13(v12 v13) 1: We will propose u1su¢ ciently close to v13 so that t13 1+t13 3v13. In that case, ushould satisfy u1=1 2(v13 +(1 p13)u1 1p13 ) = 1 2(v12 +p13u1 +p13 ): This is a system of two equations with two unknowns. Note that if we have a (valid) solution to this system, then as approaches 0the …rst equation approaches u1=1 2(v13 +u1)whose only solution is v13 =u1. (For positive , indeed u1< v13.) Thus, for small enough, t13 1+t13 3< v12u1=v122v13 + v13 + (v13 u1)and the right hand side converges to v12 2v13 +v13 < v13. Also, solving for u1, we can write the system as v13 1 +  p13 +=v12 1 +  1p13 : This is a quadratic equation in p13 with one positive root that converges to 0as converges to zero. Thus, we have an PSBN for small enough. And for small, p12 is close to 1. 3) If v12 < v13 +v23, then propose uij i=uik i=ui>0, for all i; j; k. Then the de…nition of uij irequires that ui+uj=vij for all i; j. This is a system of three linear (independent) equations with solution ui=vij +vikvjk 2. Also, tij i=pikui 1pij . Finally, pshould satisfy ui=1 2(vij +pikui pik +pjk pjkuj pik +pjk ) for all i; j; k. Taking into account ui+uj=vij, these equations can be written as p13u2+p23u1= 0; p12u3+p13u2= 0; p12u3+p23u1= 0: Note that the third equation is simply the sum of the previous two. That is, there are only two linearly independent equations. Thus, two of these equations plus p13 +p23 +p23 = 1 form a linear system with a unique solution. The solution is a probability distribution, since all three variables take positive values. Indeed, the …rst two equations can be written as p13 u1= 36 p23 u2and p12 u2=p13 u3, so that all solution vectors to these two equations have either all positive components or all negative. And no solution with all negative components satis…es the equation p13 +p23 +p23 = 1. Finally, note that tij j+tij i=pjkuj pjk+pik pikui pjk+pik , so that since both uj; ui< vij, indeed tij j+ tij i< vij. This concludes the proof of existence. Next, we can simply check that if we select the PSBN that we have just characterized for each possible values of vij for all ij, then the lim!0fu(); t(); p()gis as stated in the Proposition. Thus, we only need showing that there is no other triple fu; t; pgthat is the limit of a sequence of PSBN as approaches 0. First we prove a handy result. Lemma 2 In a PSBN, cycles cannot occur. That is, it cannot be that uij iuik i;ujk juij j;uik kujk kfor some values of i; j; k. Moreover, uij i= uik i;ujk j=uij j;uik k=ujk kcan only occur if v12 v13 +v23. Proof of Lemma: First, assume that we have such cycle with at least one strict inequality, and such that tij i+tij jvij for all ij. In any such cycle, uij i=1 2vij +tij itij jfor all i; j; k. Substituting for vij =uij i+uij j, and also substituting for tij i=pik 1pij uik i(6) we can write this expression as (uij iuij j)(1 pij) = pikuik ipjkujk j(7) Adding these three equations, for all three pairs, this implies that (uij iuij j)+(uik kuik i)+(ujk jujk k) = 0; that is, uij i+uik k+ujk j=uij j+uik i+ujk k, which violates the inequalities de…ning the cycle if there is one that is strict. Second, assume that tij i+tij j> vij for some ij, but tik i+tik kvik, and tjk j+tjk kvjk. Given the cycle, this implies that uij i=uij j=uik i= 0, so that also uik k=vik. Thus, equations (7) for the pair jk become (ujk jujk k)(1 pjk) = pikvik. Since pjk <1, that implies ujk kujk j. Note, however, that tjk j= 0, since uij j= 0, so that ujk jujk k. These two inequalities then imply both ujk j= ujk k=vjk 2, and pik = 0. Since the cycle inequalities include uik kujk k, then we must have vik vjk 2. But substituting for ujk k=vjk 2and pik = 0 in 37 (6) corresponding to tik k, we also have that tik k=pjk vjk 2<vjk 2vik. This contradicts that uik k=vik. Third, assume that tij i+tij j> vij and tik i+tik k> vik for some ij and ik but tjk j+tjk kvjk. That implies that uij i=uij j=uik i=uik k= 0, which implies that tjk j=tjk k= 0, so that ujk k=vjk 2> uik k, which contradicts the inequalities in the cycle. Thus, the only cycle that may exist is uij i=uik i;ujk j=uij j;uik k=ujk k, with tij i+tij jvij for all ij. But the system ui+uj=vij, for all ij has a valid solution only in Region 3, and coincides with the one found above. QED Thus, an PSBN must satisfy: uij iuik i;ujk juij j;uik kujk k;(8) and except for the one we used in 3) above, at least two inequalities must be strict. Also, given part three of the de…nition of PSBN, pik <  unless uij i=uik iand uik k=ujk k. Thus, in any but the PSBN constructed in 3) above, pik < . Thus, in a sequence that converges as !0, we must have lim!0pik = 0. Consider such a sequence of PSBN so that lim!0pij >0and lim!0pjk > 0. From (8) and part three of the de…nition of PSBN, that implies that for small uij j=ujk j. Thus, since at least two inequalities need to be strict, uij i> uik iand uik k< ujk k. These last inequalities imply that ujk k+ujk j=vjk and uij i+uij j=vij. Also, as approaches 0,pjk pjk+pik approaches 1, as does pij pij +pik , so that applying part one of the de…nition of a PSBN, uij j=ujk j!uj=1 2(vij +uj) = 1 2(vjk +uj): This cannot occur unless vij =vjk. In the latter case, uj=vij =vjk, which implies that both uij iand ujk kconverge to 0, and so tik i+tik kconverges to 0, in which case uik iconverges to vik 2> uij ifor small and when vik >0. This is a contradiction unless vik = 0. But if vij =vjk,vik = 0, the limit of such a sequence coincides with the PSBN constructed in 3) above. Thus, we must have that both lim!0pik = 0, and either lim!0pjk = 0 or lim!0pij = 0. But if lim!0pij = 0 then lim!0pjk >0, and this contradicts part 3 of the de…nition of an PSBN since uij iuik iand uij jujk j with at least one inequality. Thus, assume that lim!0pik = lim!0pjk = 0. We consider two possible cases: 1) Assume that tjk j+tjk k> vjk in all the terms of the sequence29 as  converges to 0, so that ujk j=ujk k= 0 = tij j, for each small enough in the 29 Note, in general, that except in trivial cases, either this is satis…ed for close to 0or else the sequence cannot converge. 38 sequence considered. Thus, from (8), we must also have that uik k= 0. Since only one inequality in (8) may be non strict, and ujk k=uik kwe must have uij i> uik i, and since ujk j= 0, we must also have that uij j> ujk j. These two inequalities imply that uij i+uij j=vij. Since tij j= 0, we must then have that uij ivij 2. Thus: 1.a) If vij 2> vik, since tik iconverges to uij ivij 2, then for small we must also have that tik i+tik k> vik, so that uik i= 0, and then tij j=tij i= 0, and then uij j=uij i=vij 2. Note that tjk juij jand tjk k= 0. Thus, for tjk j+tjk k> vjk, it must be that vij 2> vjk. This requires that ij = 1;2and also that we are in Region 1. Thus, the limit of such sequence is the one stated in the Proposition. 1.b) If vij 2vik, as before, if tik i+tik k> vik, then uij j=vij 2, and since tik k= 0, this would imply that vij 2tik i> vik which is a contradiction. Thus, we must have tik i+tik kvik. Thus, since uik i= 0, we must have uik k=vik. Since ujk k= 0, this contradicts the inequality uik kujk kin (8) unless vik = 0. In the latter case, since vij 2vik,vij = 0 and we have a contradiction with tjk j+tjk k= 0 > vjk. 2) Assume that tjk j+tjk kvjk in all the terms of the sequence as  converges to 0. 2.a) If tij i+tij jvij, then ujk jtjk j=pij pij +pik uij j; where the right hand side converges to uij j. From, (8),uij jujk j. Thus, the limit of any such sequence should satisfy lim!0uij j= lim!0ujk j= lim!0tjk j. That implies that lim!0ujk k= 0, and requires that vij vjk, and lim!0uij i=vij vjk. Since ujk kuik k, then we also have lim!0uik k= 0. But if lim!0ujk k= 0, then lim!0tik k= 0, whereas tik iuik i. Thus, if vik > vij vjk, then lim!0tik i+tik k< vik and then lim!0uik k>vik(vij vjk) 2>0, which is a contradiction. Therefore, vik vij vjk. Since lim!0tik i= lim!0uij i=vij vjk, then lim!0uik k= 0 = lim!0ujk k. Thus, lim!0uik i> 0, only if lim!0uik i=vij vjk. In this case, we would have ui= lim!0uij i= lim!0uik i,uk= lim!0uik k= lim!0ujk k, and uj= lim!0uij j= lim!0ujk j. This equation, together with ui+uk=vik,ui+uj=vij uj+uk=vjk has a solution only in Region 3(vik =vij vjk). Thus, if vik < vij vjk,uik i= 0 for small, so that tik i= 0, so that uij jvij 2, and then vjk vij 2. This is Region 2, and the limit coincides with the one stated in the Proposition. 2.b) If tij i+tij j> vij, then uij i(=uij j)= 0, so that tik i= 0, and since from (8) uij iuik i, then uik i= 0. On the other hand, tik kapproaches 0as !0, 39 and then tik i+tik kapproaches 0, which contradicts uik i= 0 unless vik = 0. Moreover in this latter case tjk j=tjk k= 0, so that ujk j=ujk k=vjk 2, so that ujk j> uij jwhich contradicts (8). 7.2 Proof of Lemma 1 Without loss of generality, assume that vi= 0, for all i= 1;2;3. Assume the core is not empty, that is, condition (2) holds, and that xdoes not belong to the core. We will show that xdoes not belong to the BS of the grand coalition. We do not need to consider allocations where xi<0for some i, or where x1+x2+x3< V , since they cannot be in the BS. Thus, assume that xi+xj< vij for some i; j, so that xk> V vij , for k6=i; j: Consider an objection yof iagainst kwhere yi+yj=vij, with yi> xiand yj> xj. A counter-objection zof kagainst iwould have to satisfy that zjyj, and zj+zk=vjk, so that zkvjk yj=vjk (vij yi). Also, zkxk> V vij. Therefore, if vjk (vij yi)< V vij; or yi< V vjk; then the objection ywould have no counter-objection and xwould not belong to the BS. If xi< V vjk we can always construct such y, and then a necessary condition for xto belong to the BS is that xiVvjk. Switching the subscripts iand j, we could consider an objection y0of jagainst k, and repeat the argument to show that a necessary condition for xto belong to the BS is that xjVvik. Thus, a necessary condition is that xi+xj2Vvjk vik vij; where the last inequality follows from condition (2). This contradicts that xi+xj< vij and proves that the BS coincides with the core when the latter is not empty. Now assume that condition (2) is not satis…ed. In particular, this implies that we are in Region 3. We have shown above that the Rsolution belongs to the BS. So we only need to show that any other allocation does not belong to the BS. Note that (2) implies that for any feasible allocation (including the e¢ cient ones), if xi=Ui+(in Region 3), then xj+xkvjk , for any  > 0. So, consider an e¢ cient allocation such that this is the case for some , and an objection yof jagainst i, with yj=xj+ 2and yk=vjk yj=vjk xj 2. A counter-objection zof i against jshould satisfy that zkykbut also zixi, so that zkvik xi. Thus, for ito indeed have a counter-objection against jit is required that vik xi=vik Uiyk=vjk xj 2; that is, xjvjk vik +Ui+ 2=Uj+ 2, where the last equality follows from the de…nition of Ui. Thus, this is a necessary condition for xto be in the BS. Switching the subscripts jand k, we would also conclude that another necessary condition is that xkUk+ 2. Thus, a necessary condition is that xi=VxjxkVUjUk=Ui. And this contradiction proves the result. QED 40 7.3 Proof of Remark 4 Existence: For small, let u1i 1=u1i i=v1i 2for i= 2;3,p= 12, and p1i= for i= 1;2. Thus, t1i i= 0, and u1i i=u1i 1=v1i 2. Also, t23 i= (1 )v1i 2, so that t23 2+t23 3= (1 )v12+v13 2. If v12+v13 2> v23, then for small u23 i= 0. Hence, t12 2=t13 3= 0. If v12+v13 2=v23, then it must be that v12 =v13 =v23. Then u23 i=v23 2. Still, t12 2=t13 3= 0. In both cases, we have an PSBN with uindependent of . In the limit, p= 1. Thus, applying the de…nition of the Rsolution, U1=v12+v13 2, and Ui=v1i 2. Uniqueness: Consider a limit of PSBN where u12 1=u12 2= 0. This means that for small t12 1+t12 2=t12 2=p23u23 2 1(p+p12)> v12. This is a contradiction, unless p23 =p13 = 0. since u23 2v23 v12 and p23 1(p+p12). Also p23 =p13 = 0 implies p+p13 = 1, which violates the third condition in the de…nition of an PSBN, so indeed u12 1=u12 2= 0 cannot be the limit of a sequence of PSBN. For the same argument, we cannot have u13 1=u13 3= 0. Thus, since t1i 1= 0, we have that u12 2v12 2, and u13 3v13 2. Also, u1i 1>0, since p23 1, and so t1i i< v23 for all  > 0. Thus, p1i , for i= 2;3. Thus, t23 2+t23 3=(p+p12)u12 2 1p23 +(p+p13)u13 3 1p23 converges to u12 2+u13 3 as converges to 0, so that for small, u12 2+u13 3>v12 2+v13 2v23. Thus, for small, u23 i= 0, and then t1i i= 0, for i= 2;3, and p23 . Thus u1i 1; u1i ishould converge to v1i 2, and pshould converge to 1. This completes the proof. QED 41