Abstract
We analyze the manipulability of competitive equilibrium allocation rules for the simplest manytomany extension of Shapley and Shubik’s (Int J Game Theory 1:111–130, 1972) assignment game. First, we show that if an agent has a quota of one, then she does not have an incentive to manipulate any competitive equilibrium rule that gives her her most preferred competitive equilibrium payoff when she reports truthfully. In particular, this result extends to the onetomany (respectively, manytoone) models the NonManipulability Theorem of the buyers (respectively, sellers), proven by Demange (Strategyproofness in the assignment market game. École Polytechnique, Laboratoire d’Économetrie, Paris, 1982), Leonard (J Polit Econ 91:461–479, 1983), and Demange and Gale (Econometrica 55:873–888, 1985) for the assignment game. Second, we prove a “General Manipulability Theorem” that implies and generalizes two “folk theorems” for the assignment game, the Manipulability Theorem and the General Impossibility Theorem, never proven before. For the onetoone case, this result provides a sort of converse of the NonManipulability Theorem.
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Notes
 1.
 2.
The competitive onetoone market was proposed in Gale (1960), who proved the existence of equilibrium prices in this market. Shapley and Shubik (1972) showed that the set of equilibrium prices form a complete lattice whose extreme points are the minimum and the maximum equilibrium prices. Sotomayor (2007) introduced the concept of a competitive equilibrium payoff for the multiplepartners assignment game and extended the previous results for this environment. She also proved that the set of competitive equilibrium payoffs is a subset of the set of stable payoffs and may be smaller than this set.
 3.
Our results are established for the competitive market game. For the assignment game, they can easily be transferred to the cooperative model because the core coincides with the set of competitive equilibrium payoffs (Shapley and Shubik 1972).
 4.
Demange and Gale (1985) extended the NonManipulability Theorem to a onetoone buyer–seller model where the utilities are continuous in money, but are not necessarily linear. They also extended it to any competitive equilibrium rule that maps the market defined by the true valuations to the buyeroptimal (or selleroptimal) competitive equilibrium for this market. Under the assumption that there are no monetary transfers between buyers (respectively, sellers), these authors proved that such a rule is collectively nonmanipulable by the buyers (respectively, sellers).
 5.
 6.
See also some more recent papers on dynamic auctions for multiple heterogeneous items (Ausubel 2006; Sun and Yang 2014). They consider bidders who can demand multiple heterogeneous items and have a general utility function over bundles (not necessarily separable or additive). They develop dynamic auctions that are efficient and strategyproof in the sense that sincere bidding by every bidder is an ex post perfect Nash equilibrium in the game.
 7.
The situation where there is no restriction on the number of objects a buyer can acquire is a particular case of our model, by making the quota of every buyer equal to the total number of objects. However, the quota makes sense in many situations. Consider, for example, a market with three sellers, \(s_{1}, s_{2} \) and \(s_{3}\). Seller \(s_{k} \) owns a number of cars of type k, for \(k = 1, 2, 3\). Suppose that buyer \(b_{j}\) has in hand an offer from a client who will purchase two cars, at most, of different types at the price of \(a_{jk}\) for \(k = 1, 2, 3\), should he obtain them in the market. Since buyer \(b_{j} \) knows that he can earn \(a_{jk}\) by reselling the car of type k, he will not buy at a higher price. And his quota is obviously 2.
 8.
For any set \(A \subseteq B \cup S\), we use the notation \(\sum _{A}\) for the sum over all elements in A.
 9.
\(\min _{k}\{u_{jk}\}\) and \(\min _{j}\{v_{jk}\}\) can depend on the maching x. However, as shown in Sotomayor (1992, 1999), they are independent of the matching for competitive equilibrium outcomes. For notational simplicity, we do not include a reference to the matching x in the expressions \(\min _{k}\{u_{jk}\}\) and \(\min _{j}\{v_{jk}\}\).
 10.
We will use “primes” to denote reported variables. For example, a is the true valuation matrix of the buyers, whereas \(a'\) is the reported valuation matrix of the buyers.
 11.
See also Demange and Gale (1985).
 12.
Barberà et al. (2016) show the equivalence between individual strategyproof and group strategyproof if the domain and the rule satisfy three properties. We cannot use their results because our framework does not satisfy the properties.
 13.
Consider a market with more than one optimal matching and only one competitive equilibrium price. Then, add any small \(\varepsilon >0\) to the valuation of any of the buyers who is matched under an optimal matching x. The maching x becomes the only optimal matching and, hence, the new market has more than one competitive equilibrium price.
 14.
Sotomayor (2008) also considers the case where the quotas of the agents are not integer numbers. In this case, the quotas can be interpreted as the amount of time available to each agent, which can be distributed among the partnerships the agent forms.
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We thank participants of seminars at the Universities of Salamanca, Valencia, Osaka, Waseda, East Anglia, and Seoul National, at the workshop on Game Theory in Rio de Janeiro, MOVEJerusalem, and ASSET meeting in Aix Marseille, as well as two reviewers and an Associate Editor for very helpful comments for very helpful comments. Marilda Sotomayor acknowledges financial support from CNPqBrazil. David PérezCastrillo is a fellow of MOVE. He acknowledges financial support from the Ministerio de Ciencia y Tecnología (ECO201231962 and ECO201563679P), Generalitat de Catalunya (2014SGR142), the Severo Ochoa Programme for Centres of Excellence in R&D (SEV20150563) and ICREA Academia. Previous versions of this paper, analyzing onetoone matching models only, circulated under the titles “Two Folk Manipulability Theorems in Onetoone Twosided Matching Markets with Money as a Continuous Variable” and “Two Folk Manipulability Theorems in the General Onetoone Twosided Matching Markets with Money”.
Appendix
Appendix
Proof of Lemma 4.2
(i) Suppose that \((u', v'; x')\) is feasible for \(M({{\underline{a}}})\). If \(x'\#_{j(q)k(h)} = 1\) then \(x'_{jk} = 1\) and \({{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}_{jk}\) by the Key Lemma. Then, \(u'\#_{j(q)}+v'\#_{k(h)}=u'_{jk}+v'_{jk} = {{\underline{a}}}_{jk} = {\underline{a}}\#_{j(q)k(h)}\). By definition, \(u'\#_{j(q)} \ge 0\) and \(v'\#_{k(h)} \ge 0\). Hence, \((u'\#, v'\#; x'\#)\) is a pairwisefeasible outcome for \(M({{\underline{a}}}\#)\), so it is feasible for \(M({{\underline{a}}}\#)\).
In the other direction, suppose that \((u'\#, v'\#; x'\#)\) is feasible for \(M({{\underline{a}}}\#)\). If \(x'_{jk} = 1\), then \(x'\#_{j(q)k(h)} = 1\) for some pair \((b_{j(q)}, s_{k(h)}) \in B\#\hbox {x} S\#\) by the Key Lemma and \({{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}_{jk}\) by the definition of \({{\underline{a}}}\#\). Then \(u'_{jk}+v'_{jk}=u'\#_{j(q)}+v'\#_{k(h)} = {{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}_{jk}\). By definition, \(u'_{jk} \ge 0\) and \(v'_{jk} \ge 0\). Hence, \((u', v'; x')\) is a pairwisefeasible outcome for \(M({{\underline{a}}})\), so it is feasible for \(M({{\underline{a}}})\).
(ii) Suppose that (\(u', v'; x'\)) is a competitive equilibrium for \(M({{\underline{a}}})\). Then, \(x'\#\) is an optimal matching for \(M({{\underline{a}}}\#)\). Also, \((u', v'; x)\) is a competitive equilibrium for \(M({{\underline{a}}})\) (see Theorem 1 of Sotomayor 1999). Moreover, if \(x \#_{j(q)k(h)} = 0\), we have either (1) \(x_{jk} = 0\), in which case \(\min _{d}\{u'_{jd}\} + \min _{l}\{v'_{lk}\}\ge {{\underline{a}}}_{jk}\) by Proposition 2.1, which implies \(u'\#_{j(q)}+v'\#_{k(h)} \ge \min _{d}\{u'_{jd}\} + \min _{l}\{v'_{lk}\} \ge {{\underline{a}}}_{jk }= {\underline{a}}\#_{j(q)k(h)}\), where the last equality follows from the definition of \({{\underline{a}}}\#\); or (2) \(x_{jk} = 1\), in which case \(u'\#_{j(q)}+v'\#_{k(h)} \ge 0 = {{\underline{a}}}\#_{j(q)k(h)}\). In either case, we obtain that \((u'\#, v'\#; x\#)\) is a competitive equilibrium for \(M({{\underline{a}}}\#)\). Then, by the optimality of \(x'\#, (u'\#, v'\#; x'\#)\) is also a competitive equilibrium for \(M({{\underline{a}}}\#)\).
(iii) Suppose that \((u'\#, v'\#; x'\#)\) is a competitive equilibrium for \(M({{\underline{a}}}\#)\). Then \((u'\#,v'\#; x\#)\) is also a competitive equilibrium for \(M({{\underline{a}}}\#)\). Moreover, \(x'\#\) is an optimal matching for \(M({{\underline{a}}}\#)\), which implies that \(x'\) is an optimal matching for \(M({{\underline{a}}})\). Then, if \(x_{jk}\) = 0 we have that \(x\#_{j(q)k(h)} = 0\) and \({{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}_{jk}\) for all \(b_{j(q)} \in B\#\) and all \(s_{k(h)} \in S\#\). The definition of (\(u'\#, v'\#)\) implies that \(\min _{d}\{u'_{jd}\} = u'\#_{j(q) }\) for some \(b_{j(q)} \in B\#\) and \(\min _{l}\{v'_{lk}\} =v' \# _{k(h)}\) for some \(s_{k(h)} \in S\#\). Hence, \(\min _{d}\{u'_{jd}\} + \min _{l}\{v'_{lk}\} = u'\#_{j(q)}+v'\#_{k(h)} \ge {{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}_{jk}\). By hypothesis, \(v'_{ik} = \min _{l}\{v'_{lk}\}\) holds for all \(b_{i} \in C(s_{k}, x)\). Then, Proposition 2.1 implies that \((u', v'; x)\) is a competitive equilibrium outcome for \(M({{\underline{a}}})\). Since \(x'\) is optimal for \(M({{\underline{a}}})\), Theorem 1 of Sotomayor (1999) implies that \((u', v'; x')\) is a competitive equilibrium outcome for \(M({{\underline{a}}})\). \(\square \)
Proof of Lemma 4.3
Define the outcome \((u', v'; x\)#) for \(M({{\underline{a}}}\#)\) as follows: \(v'_{k(z)} \equiv \min \{v_{k(1)},{\ldots }, v_{k(t(s_k))}\}\) for all \(s_{k(z)} \in S\#\) and the payoff vector \(u'\) is defined pairwisefeasibly. We claim that \((u', v'; x\#)\) is a competitive equilibrium outcome for \(M({{\underline{a}}}\#)\). First, it is clear that \((u', v'; x\#)\) is feasible for \(M({{\underline{a}}}\#)\) by definition. Second, \(u'_{j(q)} \ge u_{j(q)}\) for all \(b_{j(q)} \in B \#\) because \(v'_{k(z)} \le v_{k(z)}\) for all \(s_{k(z)} \in S \#\). Consider now a pair \((b_{j(q)}, s_{k(h)})\) with \(x \# _{j(q)k(h)} = 0\) and let \(l \le t(s_{k})\) such that \(v_{k(l)} =\min \{v_{k(1)},{\ldots }, v_{k(t(s_k))}\}\). There are two cases.
Case 1 \(x_{jk} = 0\), in which case \({{\underline{a}}}\#_{j(q)k(h)} = {{\underline{a}}}\#_{j(q)k(l)}\). It follows that \(u'_{j(q)}+v'_{k(h)} \ge u_{j(q)}+v_{k(l) } \ge {{\underline{a}}}\#_{j(q)k(l)} = {{\underline{a}}}\#_{j(q)k(h)}\), where we use Proposition 2.1 in the last inequality.
Case 2 \(x_{jk} = 1\), in which case \({{\underline{a}}}\#_{j(q)k(h)} = 0\). It follows that \(u'_{j(q)}+v'_{k(h)} \ge u_{j(q)}+v_{k(l) }\ge 0 = {{\underline{a}}}\#_{j(q)k(h)}\).
Given that \(u'_{j(q)}+v'_{k(h)} \ge {{\underline{a}}}\#_{j(q)k(h)}\) for all \((b_{j(q)}, s_{k(h)})\) with \(x\#_{j(q)k(h)} = 0\), the result that \((u', v'; x\#)\) is a competitive equilibrium outcome for \(M({{\underline{a}}}\#)\) follows from Proposition 2.1.
We now use Lemma 4.2 to obtain that the outcome \((u'', v''; x)\) related to \((u', v'; x\#)\) is competitive for \(M({{\underline{a}}})\). Then, by the optimality for S of \(({\underline{u}}, {{\overline{v}}} )\) in \(M({{\underline{a}}})\),
On the other hand, let (\({\underline{u}}', {{\overline{v}}}'; x\#\)) be the competitive equilibrium outcome related to \(({\underline{u}}, {{\overline{v}}} ; x)\) according to Lemma 4.2. By the optimality for \(S\#\) of (u, v; x#) in \(M({{\underline{a}}}\#)\) we have that \({{\overline{v}}}'_{k(h)} \le v_{k(h)}\) for all \(s_{k(h)} \in S\)#. We also have that \(v_{k}=v'_{k}\) for all \(s_{k}\) with \(t(s_{k}) = 1\). Therefore, if \(t(s_{k}) = 1\) we must have that
From (A1) and (A2) we obtain that \({{\overline{v}}} _{k}=v''_{k}\) so the inequality in (A2) is in fact an equality and so \({{\overline{v}}} _{k}=v_{k}\) for all \(s_{k}\) with a quota of one, which concludes the proof. \(\square \)
Proof of Lemma 4.4
If \(B+\ne \varnothing \), Lemma 9.20 of Roth and Sotomayor (1990), due to Demange and Gale (1985), implies that there exist \(b_{j(q) }\in B \#B+\) and \(s_{k(h)} \in x(B+)\) such that \((b_{j(q)}, s_{k(h)})\) blocks (u, v; x). Since \(b_{j(q)} \in B\#B+, u_{j(q)} \le u^{+}_{j(q)}\). Also, since \(s_{k(h)} \in x(B+)\) there exists some \(b_{l(m)} \in B+\) such that \(x_{l(m)k(h)} = 1\). Then, \(u_{l(m)} > u^{+}_{l(m)}\) so \(v_{k(h)} \le v^{}_{k(h)} \le \hbox {v}^{+}_{k(h)}\) by the competitiveness of \((u^{+}, v^{})\). Therefore, \(s_{k(h)} \notin S+\). \(\square \)
Proof of Lemma 4.6
(i) Suppose by contradiction that \(x(SS') = \{b_{0}\}\). Given that \(s_{0} \in SS', x(SS') = \{b_{0}\}\) implies that if \(s_{k} \in S'\) and \( x_{jk} = 1\), then \(b_{j} \ne b_{0}\). Moreover, every \(b_{j} \ne b_{0}\) fills her quota and does that with sellers in \(S'\). That is, \(x(S')=B\{b_{0}\}\). Then take \(\lambda >0\) such that \({\underline{v}}{}_{{k}} \lambda >0\) for all \(s_{k} \in S'\) and define (u, v; x) as follows:

\(v_{k}={\underline{v}}{}_{{k}}  \lambda \;\hbox { if }\; s_{k} \in S'\;\hbox { and }\; v_{k}={\underline{v}}{}_{{k}},\;\hbox { otherwise};\)

\(u_{jk}={{\overline{u}}} _{jk}+ \lambda \;\hbox { if }\; b_{j} \in x(S')=B\{b_{0}\}\;\hbox { and }\; x_{jk} = 1;\)

\(u_{0k}={{\overline{u}}} _{0k} = 0\;\hbox { if }\; x_{0k} = 1.\)
We claim that (u, v; x) is a competitive equilibrium outcome for M(a, r). In fact, the feasibility of (u, v; x) and condition (iii) of Proposition 2.1 are clearly satisfied. To show that \(\min _{d}\{u_{jd}\} + v_{k} \ge a_{jk} r_{k}\) for all \((b_{j}, s_{k}) \in B\)xS, use the construction of (u, v; x) and the fact that (\({{\overline{u}}} , {\underline{v}}; x\)) is a competitive equilibrium outcome for the case when \(b_{j} \ne b_{0}\). When \(b_{j}=b_{0}\), it is enough to verify the inequality when \(s_{k} \in S'\). In this case, we have that \(\min _{d}\{u_{jd}\} + v_{k} = {\underline{v}}{}_{{k}} \lambda > 0 = a_{0k} r_{k} = a_{jk}r_{k}\). However, \(v_{k} < {\underline{v}}{}_{{k}}\) for all \(s_{k} \in S'\), and \(S' \ne \varnothing \), which contradicts the fact that \(({{\overline{u}}} , {\underline{v}}; x)\) is a selleroptimal competitive equilibrium outcome for M(a, r). Hence, \(x(SS') \ne b_{0}\).
(ii) Arguing by contradiction, suppose \({{\overline{u}}} _{jt}+{\underline{v}}{}_{{k}} > a_{jk}r_{k}\) for all (\(b_{j}, s_{k}) \in x(SS')\)x\(S'\) with \(b_{j} \ne b_{0}\) and \(x_{jk} = 0\), and for all \( s_{t} \in SS'\) with \(x_{jt}= 1\). Then there exists \(\lambda >0\) such that \({\underline{v}}{}_{{k}} \lambda > 0\) for all \(s_{k} \in S'\) and also such that for all \((b_{j}, s_{k}) \in x(SS')\)x\(S'\), with \(b_{j} \ne b_{0}\) and \(x_{jk} = 0\), and for all \(s_{t} \in SS'\) with \(x_{jt} = 1\), the parameter \(\lambda \) satisfies
Now, define (\(u', v'; x\)) as follows:
We claim that \((u', v'; x)\) is a competitive equilibrium outcome. The argument is similar to the one used in part (i). We only need to check that \(\min _{d}\{u'_{jd}\} + v'_{k} \ge a_{jk} r_{k}\) for all \(s_{k} \in S'\) and \(x_{jk} = 0\). Then, let \((b_{j}, s_{k}) \in B\)x\(S'\), with \( x_{jk} = 0\) and let \(\min _{k}\{u'_{jk}\} = u'_{jt}\), for some \(s_{t} \in C(b_{j}, x)\). If \(s_{t} \in S'\) then the result follows from the competitiveness of \(({{\overline{u}}} , {\underline{v}}; x)\); if \(s_{t} \in SS'\) the result follows from (A3).
However, \(v'_{k}< {\underline{v}}{}_{{k}}\) for all \(s_{k} \in S'\), and \(S'\ne \varnothing \), which contradicts that (\({{\overline{u}}} , {\underline{v}}; x\)) is buyeroptimal for M(a, r). \(\square \)
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PérezCastrillo, D., Sotomayor, M. On the manipulability of competitive equilibrium rules in manytomany buyer–seller markets. Int J Game Theory 46, 1137–1161 (2017). https://doi.org/10.1007/s001820170573y
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Keywords
 Matching
 Competitive equilibrium
 Optimal competitive equilibrium
 Manipulability
 Competitive equilibrium rule
JEL
 C78
 D78