Bargaining set with endogenous leaders: A convergence result

Economics Letters 166 (2018) 10–13

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Bargaining set with endogenous leaders: A convergence result Javier Hervés-Estévez a , Emma Moreno-García b, * a b

Universidad de Vigo, RGEA-ECOBAS, Spain Universidad de Salamanca, IME, Spain

highlights • • • •

A new bargaining set for finite economies is introduced. This concept is relevant in replicated games or economies Differences with related concepts of bargaining sets are pointed out. This bargaining set converges to the set of Walrasian allocations when the economy is replicated.

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Article history: Received 24 October 2017 Received in revised form 17 January 2018 Accepted 26 January 2018 Available online 6 February 2018

a b s t r a c t We provide a notion of bargaining set for finite economies where the proponents of objections (leaders) are endogenous. We show its convergence to the set of Walrasian allocations when the economy is replicated. © 2018 Elsevier B.V. All rights reserved.

JEL classification: D51 D11 D00 Keywords: Bargaining sets Leader Coalitions Core Veto mechanism

1. Introduction The core of an economy is defined as the set of allocations which cannot be blocked or objected by any coalition. Thus, the veto mechanism that defines the core does not take into account that other agents in the economy may react to an objection and propose an alternative or counterobjection. Such two-step conception of the veto mechanism was considered by Aumann and Maschler (1964), who introduced the concept of bargaining set of a cooperative game.1 In the definitions by Aumann and Maschler (1964) and Davis and Maschler (1963), the original objection is proposed by a ‘‘leader’’ that must be excluded from any counterobjecting coalition. author. * Corresponding E-mail addresses: [email protected] (J. Hervés-Estévez), [email protected] (E. Moreno-García). 1 Maschler (1976) discussed the advantages that the bargaining set has over the core. https://doi.org/10.1016/j.econlet.2018.01.028 0165-1765/© 2018 Elsevier B.V. All rights reserved.

Geanakoplos (1978) considered sequences of transferable utility (TU) exchange economies with smooth preferences and modified the definition by Aumann–Davis–Maschler so that the ‘‘leader’’ was a group of agents containing a fixed (but small) fraction of the number of agents in the economy. Thus, as the number of agents grew along the sequence of economies, the number of individuals in the ‘‘leader’’ grew proportionately. Using nonstandard analysis, Geanakoplos showed that his bargaining set becomes asymptotically competitive as the number of agents grows. Shapley and Shubik (1984) showed that Aumann–Davis–Maschler’s bargaining set is approximately competitive in replica sequences of TU exchange economies with smooth preferences. Anderson (1998) extended both Geanakoplos’ result to nontransferable utility (NTU) exchange economies without smooth preferences and the Shapley and Shubik’s result to non-replica sequences of NTU exchange economies with smooth preferences. Mas-Colell (1989) considered (NTU) economies with a continuum of agents and proposed a modification of Aumann and

J. Hervés-Estévez, E. Moreno-García / Economics Letters 166 (2018) 10–13

Maschler’s bargaining set that does not involve the concept of a leader. Dutta et al. (1989) defined the consistent bargaining set arguing that the same requirement stated for objections should also be stated for counterobjections. Later on, Zhou (1994) defined a bargaining set by imposing restrictions on the coalition that counterobjects. Under conditions of generality similar to Aumann’s (1964) core equivalence theorem, Mas-Colell (1989) showed that his bargaining set characterizes the set of competitive allocations. Since both consistent and Zhou’s bargaining sets are subsets of Mas-Colell’s, they also equate the set of competitive allocations. In contrast to Debreu and Scarf’s (1963) core-convergence re sult,2 Anderson et al. (1997) showed that Zhou’s (1994) bargaining set, and consequently Mas-Colell’s (1989), not necessarily converge in replica sequences of economies, no matter how nice the preferences may be. However, it is worth noting that Anderson et al.’s counterexample does not show nonconvergence of the consistent Mas-Colell bargaining set (see Anderson, 1998; p. 4). In this paper, we provide a new definition of bargaining set. Our approach refers to scenarios where individuals are representatives of an institution, a trade union or an organization. Although our notion requires the presence of a ‘‘leader’’ in the objection process, it differs from the previous ones basically in two aspects. First, the leader proposing an objection has to be fully represented. This implies that no agent of the same type as the leader can participate in a counterobjection. Second, if an individual belongs to an objecting coalition, then any other agent of the same type that participates in a counterobjection is required to be better off than her homologue in the objection. These modifications of the leader models lead us to our bargaining set convergence result. We show that an allocation is Walrasian if and only if the corresponding equal treatment allocation defined in every replicated economy cannot be blocked by a justified objection. In other words, the set of Walrasian allocations is characterized by the intersection of bargaining sets of a sequence of replicated economies.3 Since our bargaining set is different from those considered in the related literature (Geanakoplos, 1978; Shapley and Shubik, 1984; Anderson, 1998) neither our convergence result can be deduced from the previous ones nor vice versa. The rest of the work is structured as follows. In Section 2, notations and preliminaries are stated. In Section 3 we present the notion of justified objections with leaders used to define our bargaining set. Section 4 contains our limit result and some concluding remarks. 2. Preliminaries, notations and some previous results Let E be an exchange economy with a finite set of agents N = {1, . . . , n}, who trade a finite number m of commodities. Each consumer i has a preference relation ≿i on the set of consumption 4 bundles Rm + , with the properties of continuity, strict convexity and strict monotonicity. Let ωi ∈ Rm denote the endowments of ++ consumer i. So the economy is E = (Rm + , (≿i , ωi )i∈N ). An allocation x is a consumption bundle xi ∈ Rm + for each ∑n agent i ∈ N . The allocation x is feasible in the economy E if i=1 xi ≤ 2 The core convergence is one the most commonly used tests of the price-taking assumption inherent in the definition of Walrasian equilibrium. See Anderson (1992, 2008) for surveys. 3 In a related paper, Hervés-Estévez and Moreno-Garcí (2017) obtained a convergence theorem for a bargaining set without any consideration of a leader, but under a necessary assumption of continuity of the Walrasian equilibrium correspondence instead. 4 This simplify the analysis and implies that any core allocation (and then any Walrasian allocation) is equal treatment. However, as in Debreu and Scarf (1963), our convergence result can be generalized by considering a weaker convexity that requires: if a consumption bundle z is strictly preferred to zˆ so is the convex combination λz + (1 − λ)zˆ for any λ ∈ (0, 1).

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∑n

i=1 ωi . A price system is an element of the (m − 1)-dimensional simplex of Rm + . A Walrasian equilibrium for E is a pair (p, x), where p is a price system and x is a feasible allocation such that, for every agent i, the bundle xi maximizes her preference relation ⪰i in the budget set Bi (p) = {y ∈ Rm + such that p · y ≤ p · ωi }. We denote by W (E ) the set of Walrasian allocations for the economy E . A coalition is a non-empty set of consumers. An allocation y ∑ is said to be attainable or feasible for the coalition S if y ≤ i i∈S ∑ mn i∈S ωi . Let x ∈ R+ be a feasible allocation in the economy E . The coalition S blocks x if there exists an allocation y which is attainable for S , such that yi ≿i xi for every i ∈ S and yj ≻j xj for some member j in S . When S blocks x via y we say that (S , y) is an objection to x. A feasible allocation is efficient if it is not blocked by the grand coalition, formed by all the agents. The core of the economy E , denoted by C (E ), is the set of feasible allocations which are not blocked or objected by any coalition of agents. For each positive integer r , the r-fold replica economy E r of E is a new economy with rn agents indexed by ij, with i = 1, . . . , n and j = 1, . . . , r, such that each consumer ij has a preference relation ≿ij = ≿i and endowments ωij = ωi . Note that a coalition

Sˆ in a replicated economy is formed by ri > 0 members identical ˆ to each agent i in a non-empty ⋃ subset S of {1, . . . , n}. Thus, S := {iji } is actually a coalition in {iji | i ∈ S , ji = 1, . . . , ri } = i∈S ji ∈{1,...,ri }

any replicated economy E r , for every r ≥ max{ri , i ∈ S }. It is known that, under the hypotheses above, the economy E has Walrasian equilibrium and that any Walrasian allocation belongs to the core (in particular, it is efficient). It is also known that if we repeat any Walrasian allocation when we enlarge the economy to r participants of each type, the resulting allocation is also Walrasian in the larger economy E r and consequently is in the core. Moreover, as Debreu and Scarf (1963) prove, any repeated non-Walrasian allocation is objected in some replicated economy (core convergence theorem). Addressing continuum economies, Mas-Colell (1989) provided a notion of bargaining set and showed its coincidence with the competitive allocations. This bargaining set can be straightforwardly translated to replicated economies as follows. An objection to the allocation (xij ) i∈N in the economy E r is de1≤j≤r

fined by a coalition Sˆ = {iji | i ∈ S , ji = 1, . . . , ri } , with max{ri , i ∈ S } ≤ r , and a collection of consumption bundles y = (yij )ij∈Sˆ ,

ωi and (ii) yij ≿i xij , for every ij ∈ Sˆ and yhk ≻h xhk for some hk ∈ Sˆ . A counterobjection to (Sˆ , y) is defined by a coalition Tˆ = {iji | i ∈ T , j∑ ai } in E r and i = 1, . . . , ∑ consumption plans (zij )ij∈Tˆ , such that (i) z ⩽ i∈T ai ωi , (ii) ij∈Tˆ ij ˆ ˆ ˆ ˆ zij ≻i yij if consumer ij ∈ T ∩ S and zij ≻i xij if ij ∈ T \ S .

such that (i)

∑

ij∈Sˆ yij

⩽

∑

i∈S ri

An objection is justified if it is not counterobjected by any coalition. BMC (E r ) is the set of feasible allocations in E r with no justified objection. To analyze convergence properties of bargaining sets for replicated economies, in the next section we will consider that each of the n agents of the economy E behaves as a representative of a large enough number of identical individuals. We will also use the fact that a finite economy E with n consumers can be associated to a continuum economy Ec with n types of agents as we specify next. The set of agents [ i−1 ini )the atomless economy [ n−1 ] Ec is I = [0, 1] = ⋃ n I , with I = , if i ̸ = n ; I = , 1 . All the agents in i i n i=1 n n n the subinterval Ii are of the same type i, that is, every agent t ∈ Ii has preferences ≿t = ≿i and endowments ω(t) = ωi . In this case, x = (x1 , . . . , xn ) is a Walrasian allocation in E if and only if the step function fx (defined by fx (t) = xi for every t ∈ Ii ) is a competitive allocation in Ec . Therefore, following Mas-Colell, if x = (x1 , . . . , xn ) is not Walrasian, then the step function fx does not belong to the bargaining set BMC (Ec ). Let B∗MC denote the set of equal treatment allocations in the Mas-Colell’s bargaining set. The equivalence between BMC (Ec ) and

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J. Hervés-Estévez, E. Moreno-García / Economics Letters 166 (2018) 10–13

competitive allocations in Ec implies that, under strict convexity of preferences, BMC (Ec ) and B∗MC (Ec ) coincide. We remind that Anderson et al. (1997) give an example with a unique Walrasian equilibrium but the measure of the set of individually rational Pareto optimal equal treatment allocations that have a justified objection tends to zero as the economy( is replicated. ) Therefore, the non-convergence for the sequence B∗MC (E r ) r ∈N is also shown. 3. Justified objections with endogenous leaders Next, we provide a notion of bargaining set where the leaders are endogenously determined in the objecting procedure. For this, let x be a feasible allocation in the economy E and consider an equal treatment objection to the allocation xr in E r given by xrij = xi for every j ∈ {1, . . . , r } and i ∈ N . That is, there are Sˆ = {iji |∑ i ∈ S , ji = 1∑ , . . . , ri } , with ri ≤ r , and y = (yi )i∈S , such that r y ≤ i i i∈S i∈S ri ωi and yi ≿i xi for every i ∈ S , with strict preference for some j ∈ S . We remark that without loss of generality we assume rh = r for some h ∈ S . Note that otherwise we can consider the objection (Sˆ , y) in the replicated economy r¯ E with r¯ = maxi∈S ri . Let LSˆ = {i ∈ S : ri = r }. In our framework, only a type in LSˆ can behave as a proponent of the objection (Sˆ , y).5 Indeed, given the objection (Sˆ , y), we say that it is justified if it is proposed by a leader ℓ ∈ LSˆ and (Sˆ , y) cannot be counterobjected by a coalition with no member of type ℓ. To be precise, we state the following definitions. Definition (Counterobjection). An objection (Sˆ , y) to xr in the economy E r , with Sˆ = {iji | i ∈ S , ji = 1, . . . , ri } and y = (yi )i∈S , is counterobjected if there exist Tˆ = {iji | i ∈ T , ji = 1, . . . , rˆi } and z = (z∑ i )i∈T , with rˆ∑ i ≤ r for every i ∈ T , such that ˆ (i) i∈T rˆi zi ≤ i∈T ri ωi and (ii) zi ≻i yi for every i ∈ T ∩ S and zi ≻i xi for every i ∈ T \ S . Definition (Justified Objection). The objection (Sˆ , y) to xr in the economy E r is justified if it is supported by a leader ℓ ∈ LSˆ and there is no coalition Tˆ = {iji |i ∈ T , ji = 1, . . . , ri } able to counterobject (Sˆ , y) in such a way that ℓ does not belong to T .

Theorem 4.1. The allocation x is Walrasian in the economy E if and only if xr belongs to the bargaining set of every replicated economy. That is,

⋂ r ∈N

Proof. Since ⋂ W (E ) ⊆ C r (E ) ⊆ Br (E ) for every r , it is immediate that W (E ) ⊆ r ∈N Br (E ). To show the converse, we define an auxiliary, larger bargaining set, B∗ (E r ), by considering counterobjections in any replicated economy. That is, the objection (Sˆ , y) to xr in the economy E r is justified∗ if there is a leader ℓ ∈ LSˆ such that, if a coalition T¯ = {iji | i ∈ T , ji = 1, . . . , r¯i } in any replicated economy r¯ E with r¯ ≥ r, counterobjects (Sˆ , y), then ℓ ∈ T . Let B∗r (E ) be the set of feasible r 6 allocations x in E such that xr ∈ B∗ (E⋂ ). Note that B(E r ) ⊆ B∗ (E r ) and then it is enough to show that r ∈N B∗r (E ) ⊆ W (E ). We will prove ⋂that if x is a non-Walrasian allocation in the economy E and x ∈ r ∈N B∗r (E ), then we obtain a contradiction. Note that since x is not Walrasian in E , the corresponding step function fx is not competitive in the associated continuum economy Ec and then it does not belong to BMC (Ec ), that is, there exists a (Mas-Colell) justified objection (S¯ , g) to fx . Let us denote αi = n µ(S¯ ∩Ii ) and S = {i ∈ N |αi > 0}. By convexity of preferences and Remark 5 in Mas-Colell (1989, p. 133) we can deduce that g is a step function given by g(t) = yi if t ∈ S¯ ∩ Ii , such that ∑ ∑ i∈S αi yi ≤ i∈S αi ωi , yi ≿i xi for every i ∈ S and yh ≻h xh for some h ∈ S . Moreover, αh = 1 and yi ∼i xi for every i such that αi < 1. We distinguish two cases depending on the number of types in S. If S = {h}, the coalition Sˆ r formed by r agents of type h blocks r xr in each E r , via the allocation ⋂ y∗r that assigns the bundle yh to r ˆ every member in S . If x ∈ r ∈N B (E ), one has that for every E r , there is a coalition Tˆ with no member of type h and an allocation z such that (Tˆ , z) counterobjects (Sˆ r , yr ). Therefore, we can find a counterobjection in Ec to the justified objection (S¯ , g), which is a contradiction. Now consider that S contains not only the type h. By continuity of preferences, we can take ε such that (1 − ε )yh ≻h xh . Let α = ∑ ˜ as follows: i∈S αi and define the allocation y i̸ =h

{ y˜ i =

Note that, for every r , an allocation is required to be equal treatment in order to belong to the bargaining set. Let Br (E ) denote the set of feasible allocations x in E such that xr ∈ B(E r ). Analogously, we denote by C r (E ) the set of feasible allocations x in E such that x r ∈ C (E r ) . 4. A convergence result In this section, we show that the bargaining set we have defined converges to the set of Walrasian allocations when the economy is replicated, in a similar way as Debreu–Scarf’s limit theorem for the core. 5 Note that in the original economy E , i.e. r = 1, every member of an objecting coalition S may become a potential leader since LS = S. Thus, in our approach, the relative size of the leader in any replicated economy is kept constant and equal to 1/n. See the interesting discussion on the relative size of the set of leaders in Anderson (1998, p. 17); .

(1 − ε )yi if i = h ε yh if i ̸ = h. yi +

α ∑

˜ By construction, i∈S αi yi ≤ i∈S αi ωi . Since preferences are monotone y˜ i ≻i xi for every i ∈ S . Actually, y˜ i ≻i yi ≿i xi , for every i ̸ = h. For every natural k ∈ N, let αik , i ∈ S be the smallest integer kα greater than or equal to kαi . Let us denote yki = ki (y˜ i − ωi ) + ωi .

∑

Definition (Bargaining Set). We say that the equal treatment allocation xr belongs to the bargaining set of E r if xr has no justified objection in E r . Let B(E r ) denote the bargaining set of E r .

Br (E ) = W (E ).

αi

Note that yki converges to y˜ i for every i ∈ S and then, by continuity of preferences, we have that yki ≻i xi for every i ∈ S and for all k ≥ k0 large enough. In addition, yki ≻i yi ≿i xi for every i ̸ = h and for all k ≥ k¯ 0 . We remark that ykj = (1 − ε )yj and αjk = k for every k. Let any k ≥ max{ko , k¯ 0 }. Then, the coalition Sˆ k , with αik agents of type i ̸ = h with i ∈ S , and k agents of type j, blocks x via yk , given by ykij =⋂ yki for each ij ∈ Sˆ , in the replicated economy E k . Therefore, ∗r if x ∈ r ∈N B (E ), one has that there exists a counterobjection ˆ (T , z), to the objection (Sˆ k , yk ) in some replicated economy E r with r ≥ k, such that Tˆ has no member of type h. That is, there are a coalition Tˆ = {iji | i ∈ T , ji = ∑ 1, . . . , βi } and ∑ consumptionkbundles z = (zi )i∈T , such that h ̸ ∈ T , i∈T βi zi ≤ i∈T βi ωi , zi ≻i yi ≻i yi for every i ∈ T ∩ S and zi ≻i xi for every i ∈ T \ S . This is a contradiction with the fact that the objection (S¯ , g) defines a justified objection to fx in the associated continuum economy. □ 6 We remark that B∗(r +1) (E ) ⊆ B∗r (E ) for every r ..

J. Hervés-Estévez, E. Moreno-García / Economics Letters 166 (2018) 10–13

Note that we show even more than Theorem 4.1 states: we prove a convergence result for a larger bargaining set where the counterobjection mechanism is considered in any replicated economy, which makes more difficult to obtain justified objections. Finally, we remark that our equivalence result holds for any bargaining set that lies between the one we consider in this paper and the set of Walrasian allocations. This is the case when the possibilities of objecting increase and/or when it is harder to counterobject, as it happens in the consistent bargaining set introduced by Dutta et al. (1989). This is also the case when the definition of bargaining set requires some additional properties as in Vohra (1991) and Serrano and Vohra (2002a, b), where an allocation is required to be efficient and individually rational in order to belong to the bargaining set. Acknowledgments We are grateful to Carlos Hervés-Beloso for his helpful comments. We also thank an anonymous referee for a careful reading and valuable suggestions that helped to improve this work. This work is partially supported by the Research Grants SA072U16 (Junta de Castilla y León), ECO2016-75712-P (AEI/ FEDER,UE) and RGEA-ECOBAS-Agrup2015/08 (Xunta de Galicia). References Anderson, R.M., 1992. The core in perfectly competitive economies. In: Aumann, R.J., Hart, S. (Eds.), Handbook of Game Theory with Economic Applications, Vol. I. North Holland, Amsterdam, Ch. 14. Anderson, R.M., 1998. Convergence of the Aumann–Davis–Maschler and Geanakoplos bargaining sets. Econom. Theory 11, 1–37.

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