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Vertical syndication-proof competitive prices in multilateral assignment markets

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We consider a market comprising a number of perfectly complementary and homogeneous commodities. We concentrate on the incentives for firms producing these commodities to merge and form a vertical syndicate. The main result establishes that the nucleolus of the associated market game corresponds to the unique vector of prices with the following properties: (i) they are vertical syndication-proof, (ii) they are competitive, (iii) they yield the average of the buyers- and the sellers-optimal allocations in bilateral markets, and (iv) they depend on the traders’ bargaining power but not on their identity. The proof uses an isomorphism between our class of market games and the entire class of bankruptcy games.

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  1. We refer to Schmalensee (2007) for the vast industrial organization literature that deals with the incentives for the creation and/or destruction of syndicates.

  2. The assumption that technology markets have only three sectors is a defensible simplification, despite the fact that intermediary firms also participate in them.

  3. Economides (2001) analyzes the case of “US against Microsoft”.

  4. See, retrieved on 3-July-2015.

  5. See, retrieved on 29-June-2015.

  6. We do not claim that the nucleolus and the properties characterizing are the only reasonable possibilities.

  7. The impact of buyers’ bargaining power on their surplus was analyzed in Inderst and Wey (2007).

  8. See Thomson (2003, (2015) for an extensive survey of the subject.

  9. Given \(x,y\in \mathbb {R}^{n}\), we say \(x<_{Lex}y\) if there is some \(1\le i\le n\) such that \(x_{i}<y_{i}\) and \(x_{j}=y_{j}\) for \(1\le j<i\). Also, we say \(x\le _{Lex}y\) if \(x<_{Lex}y\) or \(x=y\).

  10. This assumption enables us to speak properly about the \(i^\mathrm{th}\) agent of sector \(k\in M\).

  11. For convenience, we define the bargaining power only in 2-regular markets. However, the definition can be generalized to arbitrary markets in \(\mathcal {BBM}\) by using Eq. (2) and the translation vector.

  12. Given a game (Nv) and a player \(i\in N\), the marginal contribution of i to the grand coalition is the amount \(v(N)-v(N{\setminus } \{i\})\).

  13. Theorem 2.1 is concerned with the core of the assignment game. Consequently, the translations that we present here are especially applicable when the rule that is being translated proposes core allocations. However, to study how properties in one setting are translated to the other setting, it is convenient to be able to translate any rule in \(\mathcal {BP}\) into a rule in \(\mathcal {BBM}\), and vice versa.

  14. This condition is usually imposed on the definition of a rule. However, to compare rules and properties of bankruptcy problems and markets, it is better to consider wd as a property of a rule in \(\mathcal {BP}\). In particular, note that the translation \(\Psi \) may give rise to a rule in \(\mathcal {BP}\) that is not well defined.

  15. We impose that \(\mathsf {f}\) satisfies cs to facilitate the explanation of the property.

  16. For the sake of understanding the property, we do not rename this unique sector of sellers, as it should be denoted by \(N^1\) instead of \(N^k\). An analogous comment applies to the sector of buyers. Also note that both the active seller and the inactive seller pay the same price. In particular, the inactive seller may be paying prices to other sellers that are below their costs.

  17. At this point, the question arises whether or not to require Eq. (10) to hold with a weak inequality and then use such a condition as a property instead of 2-vsp. The answer to this question falls outside the scope of the current paper. In Knudsen and Østerdal (2012) merging and splitting incentives are studied separately for arbitrary cooperative games.

  18. A point- or set-valued solution concept f is anonymous if for all \((N,v)\in \mathcal {G}\) and \(i\in N\), it holds that \(f_{\pi (i)}(N,\pi v)= f_i(N,v)\), where \(\pi :N \rightarrow N\) is a permutation and \(\pi v (S) = v(\pi (S))\) for all \(S\subseteq N\).

  19. We abuse notation and do not write the dependence of \(\mathbf {c}'\) and \(w'\) on the market \((\mathbf {c},w)\). This spares us the use of some cumbersome expressions.

  20. In line with part (i) of Lemma 5.2, it can be verified that the reduced market in the definition of svsp is a 2-regular market whenever \(\mathsf {f}\) satisfies cs and \((\mathbf {c},w)\in 2 \hbox {-}\mathcal {BBM}\).

  21. To be consistent with the assumptions made throughout this paper, dummy sellers might be added to sector k so that there is the same number of agents in all sectors.

  22. See, retrieved on 10-September-2016.


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This research received financial support from the Ministerio de Economía y Competitividad through Projects ECO2014-52340-P and MTM2014-53395-C3-2-P as well as from Generalitat de Catalunya through Project 2014-SGR-40. Discussions with Julio González-Díaz, Hans Gersbach, Marie Riekhof, and Clive Bell improved the paper and are gratefully acknowledged. We also acknowledge the comments received at the seminars and conferences in which this work was presented: Seminars at the Hebrew University of Jerusalem, Universitat de Barcelona, and University of Southern Denmark as well as at the SING9 and IWGTS2014 conferences in Vigo and São Paulo, respectively. Last but not least, we would like to thank the referees for their comments and suggestions which substantially helped improve a previous version of the paper. Finally, the usual disclaimer applies.

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Correspondence to O. Tejada.



Proposition 8.1

The set of assignment games associated with 2-regular markets, \( 2 \hbox {-}\mathcal {BBG}\), and the set of sectors games, \(\mathcal {SG}\), are isomorphic, i.e.,

$$\begin{aligned} 2 \hbox {-}\mathcal {BBG}\cong \mathcal {SG}. \end{aligned}$$

Proof of Proposition 8.1

Let \(N^1,\dots ,N^m\) be fixed throughout the proof, and recall that \(M=\{1,\dots ,m\}\) and \(N=N^1 \cup \cdots \cup N^m\). First, note that due to Remark 2.2,

$$\begin{aligned} \mathcal {SG}=\left\{ (M,v^{\mathbf {c},w}) : (\mathbf {c},w)\in 2 \hbox {-}\mathcal {BBM}\right\} , \end{aligned}$$

and, by Definition 2.3,

$$\begin{aligned} 2 \hbox {-}\mathcal {BBG}=\left\{ (N,\omega ^{\mathbf {c},w}) : (\mathbf {c},w)\in 2 \hbox {-}\mathcal {BBM}\right\} . \end{aligned}$$

Second, consider the following mapping:

$$\begin{aligned} \begin{array}{r r c l} \Gamma : &{} 2 \hbox {-}\mathcal {BBG}&{}\rightarrow &{} \mathcal {SG}\\ &{} (N,\omega ^{\mathbf {c},w}) &{}\rightarrow &{} (M,v^{\mathbf {c},w}) \end{array} \end{aligned}$$

Note that if \((N,\omega ^{\mathbf {c},w})=(N,\omega ^{\mathbf {c}',w'})\), then \((M,v^{\mathbf {c},w})=(M,v^{\mathbf {c}',w'})\), so \(\Gamma \) is well defined. Moreover, it is straightforward to check that \(\Psi \) is surjective. To check that it is also injective, let \((N,\omega ^{\mathbf {c},w}),(N,\omega ^{\mathbf {c}',w'})\in 2 \hbox {-}\mathcal {BBG}\) be such that

$$\begin{aligned} \omega ^{\mathbf {c},w}\ne \omega ^{\mathbf {c}',w'}. \end{aligned}$$

On the one hand, suppose that \(r(\mathbf {c},w)=0\). Then, by Eq. (28) it must be that \(r(\mathbf {c}',w')=1\). Thus, \(v^{\mathbf {c},w}=v_0\) and \(v^{\mathbf {c}',w'}\ne v_0\), which implies \(\Gamma (N,\omega ^{\mathbf {c},w})\ne \Gamma (N,\omega ^{\mathbf {c}',w'})\).

On the other hand, assume that \(r(\mathbf {c},w)=r(\mathbf {c}',w')=1\). From Eq. (28) it follows the existence of \(S\subseteq N\) minimal w.r.t. to inclusion such that \(\omega ^{\mathbf {c},w} (S)\ne \omega ^{\mathbf {c}',w'}(S)\). Furthermore, from the definition of the characteristic function of an assignment game, it must be that \(S=Z^R\) for some \(R\subseteq M\). Hence,

$$\begin{aligned} v^{\mathbf {c},w}(R)=\omega ^{\mathbf {c},w}(S)\ne \omega ^{\mathbf {c}',w'}(S)=v^{\mathbf {c}',w'}(R). \end{aligned}$$

Thus, \(\Gamma (N,\omega ^{\mathbf {c},w})\ne \Gamma (N,\omega ^{\mathbf {c}',w'})\), so \(\Gamma \) is injective.\(\square \)

Proof of Lemma 5.1

Let \(\mathsf {f}\) be a rule in \(\mathcal {BBM}\) satisfying cs, then it trivially satisfies part (ii). Part (i) then follows from Theorem 2.1. The reverse implication follows from Theorem 2.1 and the properties of the market \((\tilde{\mathbf {c}},\tilde{w})\) defined in Eq. (2).

Proof of Lemma 5.2

We only prove the first part, as the second part is trivial. Indeed, \(\vert N^{k} \vert = \vert N^{m} \vert = 2\) since \((\mathbf {c},w)\in 2 \hbox {-}\mathcal {BBM}\), and

$$\begin{aligned}&w_2- \alpha ^{k}_2(\mathbf {c},w,\mathsf {f}) \\ =&w_2 - c^{k}_2 - \sum _{l\in M \setminus \{k,m\}} \left( \mathsf {f}^{l}_1(\mathbf {c},w) + c^{l}_1\right) = w_2 - c^{k}_2- \sum _{l\in M \setminus \{k,m\}}c^{l}_1 - \sum _{l\in M \setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w)\\ \le&\left( w_2 - c^{k}_2- \sum _{l\in M \setminus \{k,m\}}c^{l}_1 \right) _+ - \left[ \mathsf {f}^{k}_2(\mathbf {c},w) + \mathsf {f}^{m}_2 (\mathbf {c},w)+\sum _{l\in M \setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w) \right] \le 0, \end{aligned}$$

where the penultimate inequality holds because \(\mathsf {f}^{k}_2(\mathbf {c},w) = \mathsf {f}^{m}_2(\mathbf {c},w) =0\)—due to Theorem 2.1—, the fact that \((\mathbf {c},w)\) is a 2-regular market, and because \(\mathsf {f}\) satisfies cs, while the last inequality holds since \((\mathsf {f}_k(\mathbf {c},w))_{k\in M}\) is a core allocation of \((N,v^{\mathbf {c},w})\).

\(\square \)

Proof of Lemma 5.3

By Lemma 5.2, the reduced market is a 2-regular market, so \(r( \alpha ^{k}(\mathbf {c},w,\mathsf {f}) ,w) \le 1\). Then it suffices to realize that cs implies \(\mathsf {f}^{k}_2(\mathbf {c},w)= \mathsf {f}^{k}_2 \left( \alpha ^{k}(\mathbf {c},w,\mathsf {f}) ,w \right) = 0 \) and

$$\begin{aligned} \mathsf {f}^{k}_1(\mathbf {c},w)&= \omega ^{\mathbf {c},w}(N)-\sum _{l\in M\setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w) - \mathsf {f}^{m}_1(\mathbf {c},w) \\&= \omega ^{\mathbf {c},w}(N)-\sum _{l\in M\setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w) - \mathsf {f}^{m}_1 \left( \alpha ^{k}(\mathbf {c},w,\mathsf {f}) ,w \right) \\&= \omega ^{\mathbf {c},w}(N)-\sum _{l\in M\setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w) - \omega ^{ \alpha ^{k}(\mathbf {c},w,{\mathsf{f}}),w}\left( N^{k}\cup N^{m}\right) + \mathsf {f}^{k}_1 \left( \alpha ^{k}(\mathbf {c},w,\mathsf {f}),w \right) \\&= \mathsf {f}^{k}_1 \left( \alpha ^{k}(\mathbf {c},w,\mathsf {f}),w \right) , \end{aligned}$$

where the first and the third equalities hold since \(\mathsf {f}\) yields efficient allocations for \((\mathbf {c},w)\) and \(( \alpha ^{k}(\mathbf {c},w,\mathsf {f}),w)\) respectively, the second equality holds since \(\mathsf {f}\) satisfies 2-vsp, and the last equality holds since by definition of the reduced market,

$$\begin{aligned} \omega ^{\alpha ^{k}(\mathbf {c},w,\mathsf{f}),w}\left( N^{k}\cup N^{m}\right)&= (w_1-\alpha ^{k}_1(\mathbf {c},w,\mathsf {f}))_+ \\&= \left( w_1-\sum _{l\in M \setminus \{m\}} c^l_1 -\sum _{l\in M \setminus \{k,m\}} \mathsf {f}^l_1 (\mathbf {c},w)\right) _+ \\&= \omega ^{\mathbf {c},w}(N)-\sum _{l\in M\setminus \{k,m\}} \mathsf {f}^{l}_1(\mathbf {c},w). \end{aligned}$$

\(\square \)

Proof of Claim A

Let \(\mathsf {f}\) be a rule in \( 2 \hbox {-}\mathcal {BBM}\) satisfying 2-cbs and let \((\mathbf {c},w),(\bar{\mathbf {c}},\bar{w})\in 2 \hbox {-}\mathcal {BBM}\) be two 2-sided markets satisfying \(w_1-c^1_1=\bar{w}_1-\bar{c}^1_1\) (same total surplus), \(c^1_2-c^1_1=\bar{c}^1_2-\bar{c}^1_1\), and \(w_1-w_2=\bar{w}_1-\bar{w}_2\) (same bargaining power). Then,

$$\begin{aligned} \bar{w}_2-\bar{c}^1_1&=(\bar{w}_1-w_1+w_2)-\bar{c}^1_1=(\bar{w}_1-\bar{c}^1_1)-(w_1-c^1_1)+(w_2-c^1_1)=w_2-c^1_1,\\ \bar{w}_1-\bar{c}^1_2&=\bar{w}_1-(c^1_2-c^1_1+\bar{c}^1_1)=(\bar{w}_1-\bar{c}^1_1)-(w_1-c^1_1)+(w_1-c^1_2)=w_1-c^1_2. \end{aligned}$$

Since \(\mathsf {f}\) satisfies 2-cbs, we obtain \(\mathsf {f}(\mathbf {c},w)=\mathsf {f}(\bar{\mathbf {c}},\bar{w})\) by inserting the above equations into the two expressions of Eq. (11).\(\square \)

Proof of Claim B

Let \(\mathsf {f}\) be a rule in \( 2 \hbox {-}\mathcal {BBM}\) satisfying 2-cbs and let \((\mathbf {c},w),(\bar{\mathbf {c}},\bar{w})\in 2 \hbox {-}\mathcal {BBM}\) be two 2-sided markets satisfying \(w_1-c^1_1=\bar{w}_1-\bar{c}^1_1\) (same total surplus), \(c^1_2-c^1_1=\bar{w}_1-\bar{w}_2\), and \(w_1-w_2=\bar{c}^1_2-\bar{c}^1_1\) (switched bargaining power). Then,

$$\begin{aligned} \bar{w}_2-\bar{c}^1_1&=(\bar{w}_1-c^1_2+c^1_1)-(\bar{w}_1-w_1+c^1_1)=w_1-c^1_2,\\ \bar{w}_1-\bar{c}^1_2&=(w_1-c^1_1+\bar{c}^1_1)-(w_1-w_2+\bar{c}^1_1)=w_2-c^1_1. \end{aligned}$$

Since \(\mathsf {f}\) satisfies 2-cbs, we obtain \(\mathsf {f}^1(\bar{\mathbf {c}},\bar{w})=\mathsf {f}^2(\mathbf {c},w)\) and \(\mathsf {f}^2(\bar{\mathbf {c}},\bar{w})=\mathsf {f}^1(\mathbf {c},w)\) by inserting the above equations into the two expressions of Eq. (11).\(\square \)

Proposition 8.2

The properties that characterize the Talmud Assignment Rule, \( \mathsf {T}\), are logically independent.


We consider the following four rules in \(\mathcal {BBM}\):

(i) Let the rule in \(\mathcal {BBM}\), \(\hat{\mathsf {T}}\), be defined for every \((\mathbf {c},w)\in \mathcal {BBM}\) as

$$\begin{aligned} {\hat{\mathsf {T}}}(\mathbf {c},w) = {\left\{ \begin{array}{ll} \mathsf {T}(\mathbf {c},w) &{} \text{ if } (\mathbf {c},w) \in 2 \hbox {-}\mathcal {BBM}, \\ \mathbf {0} &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

Then, \(\hat{\mathsf {T}}\) satisfies 2-vsp, 2-abp, and 2-cbs but not cs.

(ii) Let the rule in \(\mathcal {BBM}\), \(\hat{\mathsf {T}}\), assign for every \((\mathbf {c},w)\in \mathcal {BBM}\) the core-center (see González-Díaz and Sánchez-Rodríguez (2007)) of \((N,\omega ^{\mathbf {c},w})\). From Núñez and Rafels (2005) and Tejada and Núñez (2012), we know that this allocation differs from the nucleolus (of the assignment game) for the whole class of market games in \(\mathcal {BBG}\), but that they coincide for bilateral markets. Lastly, the core-center is an anonymous solution concept. Then, \(\hat{\mathsf {T}}\) satisfies cs, 2-abp, and 2-cbs, but not 2-vsp.

(iii) Let CEA be the constrained equal awards bankruptcy rule. This rule is defined for every \((M,E,d)\in \mathcal {BP}\) and \(i\in M\) by \(CEA_i (M,E,d) = \min \left\{ d_i,\lambda \right\} \), where \(\lambda \) is chosen so that \(\sum _{i\in M} CEA_i (E,d) = E\). The following properties are known:

  • (a) CEA chooses a core allocation of the bankruptcy game \((M,v^{E,d})\)—see Theorem 2 in Thomson (2003).

  • (b) CEA satisfies bc—see Theorem 1 in Herrero and Villar (2001).

  • (c) CEA does not satisfy cg—by Theorem 5.1 and because CEA satisfies wd by construction.

First, let \(\hat{\mathsf {T}} = \Phi (CEA)\). From Theorems 2.1 and I and statement (a), it holds that \(\hat{\mathsf {T}} \) satisfies cs. Second, following the same lines as in the proof of statement (ii) in Proposition 6.2, we can show that due to statements (a) and (b), \( \hat{\mathsf {T}}\) also satisfies 2-vsp. Third, the definition of \( \hat{\mathsf {T}} \) directly implies that it satisfies 2-abp. Fourth and last, statement (c), statement (ii) in Proposition 6.1, statement (i) in Proposition 6.2, and the fact that \(CEA= \Psi (\Phi (CEA))\) imply that \(\hat{\mathsf {T}} \) does not satisfy 2-cbs.

(iv) Consider the following subset of \( 2 \hbox {-}\mathcal {BBM}\):

$$\begin{aligned} \mathcal {V}=\left\{ ((c^1,c^2),w)\in \mathcal {BBM}: c^1_1=c^2_1=0,c^1_2=c^2_2=c,w_1=w_2<c\right\} . \end{aligned}$$

Since \(w_2-2c<0\) for all \((\mathbf {c},w)\in \mathcal {V}\), we have \(\mathcal {V}\subseteq 2 \hbox {-}\mathcal {BBM}\). Let the rule in \(\mathcal {BBM}\), \(\hat{\mathsf {T}}\), be such that for every \((\mathbf {c},w)\in \mathcal {V}\),

$$\begin{aligned} x_k = \hat{\mathsf {T}}^k_1 (N^1,N^2,N^3,\mathbf {c},w) = {\left\{ \begin{array}{ll} w_1 &{} \text{ if } k=1,\\ 0 &{} \text{ if } k=2,\\ 0 &{} \text{ if } k=3, \end{array}\right. } \end{aligned}$$

and \(\hat{\mathsf {T}}^k_2 (\mathbf {c},w)= 0\) for all \(k\in M=\{1,2,3\}\). Note that for every \((\mathbf {c},w)\in \mathcal {V}\)

$$\begin{aligned} v^{\mathbf {c},w}(\{1\})= & {} (w_2-c)_+=0, v^{\mathbf {c},w}(\{1,2\}) = (w_2)_+=w_2, \\ v^{\mathbf {c},w}(\{2\})= & {} (w_2-c)_+=0, v^{\mathbf {c},w}(\{1,3\}) = (w_1-c)_+=0, v^{\mathbf {c},w}(\{1,2,3\})=w_1.\\ v^{\mathbf {c},w}(\{3\})= & {} (w_1-2c)_+=0, v^{\mathbf {c},w}(\{2,3\}) = (w_1-c)_+=0, \end{aligned}$$

Hence, sectors 1 and 2 are symmetric players in the sectors game. Since the nucleolus is an anonymous solution, \(\mathsf {T}^1_1(\mathbf {c},w)=\mathsf {T}^2_1(\mathbf {c},w)\). Also, note that \(x_1+x_2 = w_1 \ge w_1\), so \(x \in C(M,v^{\mathbf {c},w})\), with \(M=\{1,2,3\}\). It then follows from part (i) of Theorem 2.1 that \(\hat{\mathsf {T}}(\mathbf {c},w)\in C(N,\omega ^{\mathbf {c},w})\) for all \((\mathbf {c},w)\in \mathcal {V}\). Next, let the rule in \(\mathcal {BBM}\), \( \tilde{\mathsf {T}}\), be defined for every \((\mathbf {c},w) \in \mathcal {BBM}\) by

$$\begin{aligned} \tilde{\mathsf {T}} (\mathbf {c},w) = {\left\{ \begin{array}{ll} \hat{\mathsf {T}} (\mathbf {c},w) &{} \text{ if } (\mathbf {c},w) \in \mathcal {V}, \\ \mathsf {T}(\mathbf {c},w) &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

By construction, \(\tilde{\mathsf {T}}\) satisfies 2-cbs and cs. Next we focus on 2-vsp. On the one hand, consider the reduced market \((N^1,N^3,\alpha ^{1}(\mathbf {c},w,\tilde{\mathsf {T}}),w)\), where \(\alpha ^{1}_1(\mathbf {c},w,\tilde{\mathsf {T}})=0\) and \( \alpha ^{1}_2(\mathbf {c},w,\tilde{\mathsf {T}})=c\). Then,

$$\begin{aligned} \tilde{\mathsf {T}}^3_1 (N^1,N^3,\alpha ^{1}(\mathbf {c},w,\tilde{\mathsf {T}}),w)&= \frac{w_1-0 + (w_1-c)_+-(w_2-0)_+}{2} = 0 \\&= \tilde{\mathsf {T}}^3_1 (N^1,N^2,N^3, \mathbf {c},w). \end{aligned}$$

On the other hand, consider the reduced game \((N^2,N^3, \alpha ^2 (\mathbf {c},w,\tilde{\mathsf {T}}),w)\), where \(\alpha ^{2}_1(\mathbf {c},w,\tilde{\mathsf {T}})=w_1\) and \(\alpha ^{2}_2(\mathbf {c},w,\tilde{\mathsf {T}})=w_1+c\). Then,

$$\begin{aligned} \tilde{\mathsf {T}}^3_1 (N^2,N^3,\alpha ^{1}(\mathbf {c},w,\tilde{\mathsf {T}}),w)&= \frac{w_1-w_1 + (w_1-w_1-c)_+-(w_2-w_1)_+}{2} = 0 \\&= \tilde{\mathsf {T}}^3_1 (N^1,N^2,N^3, \mathbf {c},w). \end{aligned}$$

That is, we have proved that \(\tilde{\mathsf {T}}\) satisfies 2-vsp. Lastly, since \(\tilde{\mathsf {T}}\) differs from \(\mathsf {T}\), it follows from Theorem II that \(\tilde{\mathsf {T}}\) cannot satisfy 2-abp.\(\square \)

Proof of Proposition 7.1

First, we show that \(\mathsf {T}\) satisfies svsp for 2-regular markets. Indeed, for such a market, the conditions in the definition of svsp easily follow by repeating the lines in the proof of Proposition 6.2 and the existence part of Theorem II using consistency instead of bilateral consistency. Finally, from Lemma 5.1 it follows that \(\mathsf {T}\) satisfies svsp for arbitrary markets.\(\square \)

Proof of Proposition 7.2

First, note that if a rule in \(\mathcal {BBM}\) satisfies 2-cbs, cs, 2-abp, and 2-vsp, by Theorem II it has to coincide with the Talmud Assignment Rule. Second, consider the bilateral 2-regular market \((\mathbf {c},w)\) defined by \(\mathbf {c}=c^{1}=(1,2)\) and \(w=(3,0)\), and let \(k=1\) and \(S=\{1\}\subseteq N^{1}\). Then, by Definition 6.1,

$$\begin{aligned} \mathsf {T}^{1}_1(c^{1},w)+\mathsf {T}^{1}_2(c^{1},w)= \frac{1}{2}\le 1 =\mathsf {T}^{1}_1(\gamma ^{1,S},w) + \mathsf {T}^{1}_2(\gamma ^{1,S},w), \end{aligned}$$


$$\begin{aligned} \mathsf {T}^{2}(c^{1},w)=\left( \frac{3}{2},0\right) \ne \left( 1,0\right) =\mathsf {T}^{2}(\gamma ^{1,S},w), \end{aligned}$$

which concludes the proof.\(\square \)

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Tejada, O., Álvarez-Mozos, M. Vertical syndication-proof competitive prices in multilateral assignment markets. Rev Econ Design 20, 289–327 (2016).

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