An assignment model with local constraints: Competitive equilibrium and ascending auction

Economics Letters 188 (2020) 108905

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

An assignment model with local constraints: Competitive equilibrium and ascending auction✩ Lijun Pan a , Linyu Peng b , Yu Zhou b , a b

∗

School of Economics, Nanjing University, 22 Hankou Road, Nanjing, China Waseda Institute for Advanced Study, Waseda University, 1-6-1 Nishi Waseda, Shinjuku-ku, Tokyo 169-8050, Japan

article

info

Article history: Received 13 December 2019 Accepted 17 December 2019 Available online 30 December 2019 JEL classification: D44 D47 D82 Keywords: Assignment model Local constraints Competitive equilibrium Existence Lattice Ascending auction

a b s t r a c t We consider an assignment model where each agent has unit-demand quasi-linear preferences and may face some local constraint, i.e., her possible assignment is restricted to a subset of items. Our model takes the assignment models without and with outside options, e.g., Andersson (2007) and Andersson et al. (2013), as special cases. We show that local constraints may lead to the non-existence of competitive equilibrium (CE), and provide a sufficient and necessary condition that ensures its existence. We establish the lattice of CE prices. Besides, an ascending auction is proposed, either finding a CE or validating its non-existence in finitely many steps. It generalizes Andersson et al. (2013)’s auction by adjusting increments stepwise. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Auctions are used worldwide for the assignment of items to those who value them the most so that efficient outcomes are achieved. Examples include public procurement and online product selling. In many auctions, not all categories of items are available to bidders; Instead, some bidders are restricted to bidding on part of these categories. For example, in eBay, due to local commercial policies, some products are allowed to be sold in some countries but not in others. Thus not all eBay users have the chance to get those products. This note attempts to investigate the existence and dynamic implementation via auction of competitive equilibrium (CE) under such bidding \selling constraints. We consider an assignment model where each agent has unitdemand quasi-linear preferences and may face some local constraint, i.e., her possible assignment is restricted to a subset of items. Our model takes as special cases the assignment models without outside options such as task assignments, see, e.g., Andersson (2007) and Svensson (2009), and with outside options ✩ Zhou gratefully acknowledges financial support from the Grant-in-aid for Research Activity, Japan Society for the Promotion of Science (19K13653). We thank Dolf Talman for very careful reading and helpful comments. ∗ Corresponding author. E-mail addresses: [email protected] (L. Pan), [email protected] (L. Peng), [email protected] (Y. Zhou). https://doi.org/10.1016/j.econlet.2019.108905 0165-1765/© 2019 Elsevier B.V. All rights reserved.

such as item assignments, see, e.g., Mishra and Talman (2010), and Andersson et al. (2013). We show that with local constraints, a CE may fail to exist. We provide a condition, “the covering condition,” that is sufficient and necessary for the existence of CEs. The covering condition requires that any size of agents’ coalition should have enough items to be assigned. We also establish that the set of CE prices is a closed lattice bounded below. The lattice result indicates that the set of CE prices is nonempty if and only if there is a minimum CE price vector (MCEP). We propose an ascending auction that either finds an MCEP or validates the non-existence of CEs in finitely many steps. Our auction generalizes Andersson et al. (2013)’s auction by discretely adjusting increments stepwise, which allows agents’ valuations to be arbitrary real numbers. Our results are related to those results of CEs established in the assignment models without and with outside options for unitdemand quasi-linear preferences. Shapley and Shubik (1971) and Andersson and Svensson (2008) show the existence and lattice of CEs\ fair allocations w.r.t. some given price vector. However, in our settings, a CE may fail to exist and even under the covering condition, their results do not imply ours because our model covers cases excluded by their models, see, e.g., only part of agents have the outside options. In the assignment models without outside options, both Andersson (2007) and our auction find fair and optimal allocation

2

L. Pan, L. Peng and Y. Zhou / Economics Letters 188 (2020) 108905

prices. Andersson (2007)’s auction adjusts the increments continuously, different from our discrete adjustment. In the models with outside options, to find MCEPs, the auctions of Demange et al. (1986), Mishra and Talman (2010), and Andersson et al. (2013) neither allow agents’ valuations to be arbitrary real numbers nor set increments endogenously. Instead, the integer values of agents’ valuations and unit increment are assumed. Our auction has no restrictions to agents’ valuations and increments. Local constraints have been studied in different matching models where agents have strict preferences and no money is involved, see, e.g., Pycia (2016) and Afacan (2019). In the settings of exchange network, Pycia (2016) characterizes the top-trading cycle mechanisms. In the settings of object allocation with hierarchical endowment structures, Afacan (2019) introduces and characterizes the restricted trading cycles mechanisms. 2. The model A finite set M of items is assigned to a finite set N of agents.1 Each agent i ∈ N faces a non-empty local constraint Mi ⊆ M on items available to be assigned, i.e., agent i can only receive items from Mi . If Mi = M, agent i has no restriction on receiving items. For each i ∈ N, agent i receives one item and denote vi : Mi → R by her valuation function. For each m ∈ M, denote pm ∈ R and rm ∈ R by item m′ s price and reserve price. Let p = (pm )m∈M ∈ RM and r = (rm )m∈M ∈ RM be the corresponding price vectors. For each i ∈ N, denote agent i′ s utility at (m, pm ) ∈ Mi × R by ui (m, pm ) = vi (m) − pm . For each i ∈ N, let πi ∈ M be agent i′ s assigned item. An assignment π = (πi )i∈N ∈ (M)N is feasible if (i) for each i ∈ N, πi ∈ Mi and (ii) for each i, j ∈ N, if i ̸= j, then πi ̸= πj . Condition (i) says that each agent gets an item from her constrained item set. Condition (ii) says that no two distinct agents get the same item. Denote the set of feasible item assignments by Π . An allocation is a pair (π , p) ∈ Π × RM . Our model takes as special cases, the assignment models without outside options such as task assignments, and with outside options such as item assignments. Notably, our model covers cases excluded by those models, see, e.g., only part of agents have outside options. Example 2.1. The task assignment models, e.g., Andersson (2007) and Svensson (2009). There is a set M of tasks. Each agent i ∈ N can be assigned any task from M, i.e., Mi = M and the numbers of agents and tasks are the same, i.e., |N | = |M |. Each agent is assigned a task and there is no outside option. Example 2.2. The item assignment models, e.g., Demange et al. (1986), Mishra and Talman (2010), and Andersson et al. (2013). There is a finite set M ′ of items. Each agent i ∈ N either receives an item from M ′ or choose the outside option oi , i.e., Mi = M ′ ∪ {oi }, and M = M ′ ∪ {o1 , . . . , on }.2

Definition 3.1. An allocation (π, p) ∈ Π × RM is a competitive equilibrium (CE) if (i) for each i ∈ N, πi ∈ Di (p, Mi ); (ii) for each m ∈ M, pm ≥ rm , and pm > rm implies that there is i ∈ N such that πi = m. Definition 3.1 follows the standard manner in defining CE. In the models in Example 2.1, Definition 3.1 defines the fair allocation w.r.t. r, see, e.g., Andersson (2007). However, in the presence of local constraints, a CE may not exist, as illustrated below. Example 3.2. Let N = {1, 2, 3} and M = {a, b, c }. Let M1 = {a}, M2 = {a}, and M3 = {a, b, c }. At any assignment π ∈ Π , by πi ∈ Mi , we have π1 = π2 = a, contradicting the feasibility of π . We propose the following condition to dissolve the nonexistence problem. It says that any size of agents’ coalition should have enough items to choose, independent of prices. Definition 3.3. A tuple (N , (Mi )i∈N ) satisfies ⏐ ⏐ the covering condition if for each non-empty set N ′ ⊆ N, ⏐N ′ ⏐ ≤ |∪i∈N ′ Mi |. Theorem 3.4. The covering condition is sufficient and necessary for the existence of CEs. Proof. The necessary part is easy to see. The sufficient part follows from Theorem 4.6 below, which shows that if the covering condition holds, there is a CE. □ In the models in Examples 2.1 and 2.2 and Andersson and Svensson (2008), the covering condition holds trivially, and there is always a CE or a fair allocation w.r.t. r. As stated above, our model covers the cases excluded by those models so their existence results do not imply ours even under the covering condition. On the other hand, local constraints preserve the lattice property of CE prices. Formally, the set of CE prices is a lattice if for each pair p, p′ ∈ RM of CE prices, p ∧ p′ = (min{pm , p′m })m∈M and p ∨ p′ = (max{pm , p′m })m∈M are both CE prices. Theorem 3.5. The set of CE prices is a lattice that is closed and bounded below. Proof. Let (π, p) and (π ′ , p′ ) be two CEs. Let u = (ui )i∈N and u′ = (u′i )i∈N be the corresponding utility profiles at (π , p) and (π ′ , p′ ). Let N1 = {i ∈ N : ui > u′i }, N2 = {i ∈ N : ui < u′i }, and N3 = {i ∈ N : ui = u′i }. Let M1 = {m ∈ M : pm < p′m }, M2 = {m ∈ M : pm > p′m }, and M3 = {m ∈ M : pm = p′m }. For each i ∈ N1 , since (π ′ , p′ ) is a CE, ui > u′i implies vi (πi ) − pπi > vi (πi′ ) − p′π ′ ≥ vi (πi ) − p′πi . Thus p′πi > pπi and i

πi ∈ M1 . For each m ∈ M1 , by pm < p′m , there is i ∈ N such that πi′ = m. Since (π, p) is a CE, by pπi′ < p′π ′ , ui = vi (πi ) − pπi ≥ i vi (πi′ ) − pπi′ > vi (πi′ ) − p′π ′ = u′i . Thus i ∈ N1 . Thus |N1 | = |M1 | i

3. Competitive equilibrium For each i ∈ N, agent i′ s (locally constrained) demand set at p on Mi is Di (p, Mi ) = {m ∈ Mi : vi (m)−pm ≥ vi (m′ )−pm′ , for each m′ ∈ Mi }. Notice Di (p, Mi ) ̸ = ∅ and Di (p, Mi ) ⊆ Mi . We now define competitive equilibrium. 1

Items are allowed to be identical or heterogeneous or both.

2

The outside option oi is often normalized to be the dummy item 0.

holds. By similar reasoning, we have that each agent in N2 gets an item from M2 and |N2 | = |M2 |. Let p = p ∧ p′ and π be such that π i = πi if ui > u′i and π i = πi′ if ui ≤ u′i . We show that (π , p) is a CE. By π, π ′ ∈ Π and the above analysis, π ∈ Π holds. For each m ∈ M1 , pm = pm and for each m ∈ M2 ∪ M3 , pm = p′m . For each i ∈ N1 , π i = πi and p′π ≥ pπ i = p π i . Thus i vi (π i ) − p = ui ≥ vi (m) − pm = vi (m)−pm , for each m ∈ M1 πi

and vi (π i ) − p

πi

= ui ≥ u′i ≥ vi (m) − p′m = vi (m)−pm , for each

m ∈ M2 ∪ M3 . For each i ∈ N2 ∪ N3 , π i = πi′ and pπ i ≥ p′π =p i = u′i ≥ vi (m) − p′m = vi (m)− pm , for each π i . Thus vi (π i ) − p πi

L. Pan, L. Peng and Y. Zhou / Economics Letters 188 (2020) 108905

m ∈ M2 ∪ M3 and vi (π i ) − p

πi

= u′i ≥ ui ≥ vi (m) − pm = vi (m)−pm ,

for each m ∈ M1 . Thus Definition 3.1(i) holds. Since (π ′ , p′ ) is a CE, unassigned items and their prices at (π ′ , p′ ) remain the same at (π, p). Thus Definition 3.1(ii) holds. Let p = p ∨ p′ and π be such that π i = πi′ if ui > u′i and π i = πi if ui ≤ u′i . By similar reasoning as above, we have that (π, p) is a CE. Thus the set of CE prices is a lattice. The closedness comes from the continuity of the utility functions and it is straightforward that the CE prices are bounded below by r. □ A direct outcome of Theorem 3.5 is: Corollary 3.6. Consider the models in Examples 2.1 and 2.2 The set of CE prices or the set of prices induced by the fair allocations w.r.t. r is a closed lattice bounded below. In the models in Example 2.1, Svensson (2009) shows the same result as Corollary 3.6 In contrast, in the models in Example 2.2, Shapley and Shubik (1971) show a stronger result that the closed lattice is both bounded below and bounded above. Notice that in the models in Example 2.1 or in our settings, the lattice is not bounded above. 4. The ascending auction Theorem 3.5 indicates that the set of CE prices is non-empty if and only if there is a minimum competitive equilibrium price vector (MCEP). This section provides an auction either finding an MCEP or validating the non-existence of CEs in finitely many steps. The following concepts are central to define our auction. Definition 4.1. A non-empty set M ′ ⊆ M ⏐ is in excess demand at p if for each non-empty set M⏐′′ ⊆ M ′ , ⏐{i ∈ N : Di (p, Mi ) ⊆ M ′ ⏐ ⏐ and Di (p, Mi ) ∩ M ′′ ̸ = ∅}⏐ > ⏐M ′′ ⏐. Let E(p) be the set in excess demand with maximal cardinality at p and N(E(p)) = {i ∈ N : Di (p, Mi ) ⊆ E(p)} be the corresponding demanders.3 Definition 4.2. Given i ∈ N, (m, pm ) ∈ Mi × R, and m′ ∈ ′ Mi , Iim (m, pm ) is agent i’s indifference price of m′ at (m, pm ) if ′ vi (m′ ) − Iim (m, pm ) = vi (m) − pm . ′

For agent i, her utilities at (m′ , Iim (m, pm )) and (m, pm ) are the same. For each M ′ ⊊ Mi , recall Di (p, Mi \M ′ ) = {m ∈ Mi \M ′ : vi (m) − pm ≥ vi (m′ ) − pm′ , for each m′ ∈ Mi \M ′ }. Definition 4.3. The ascending auction is defined as: Step 1: The auctioneer sets t =: 0 and pt := r. Step 2: Each agent i reports Di (pt , Mi ). The auctioneer verifies E(pt ) and N(E(pt )). If E(pt ) = ∅, terminate the auction at pt . Otherwise, go to Step 3. Step 3: Each agent i ∈ N(E(pt )) reports her personalized increment Ii ∈ R such that in case Di (pt , Mi ) = Mi , she reports Ii = 0; In case Di (pt , Mi ) ⊊ Mi , she arbitrarily selects m ∈ Di (pt , Mi ) and m′ ∈ Di (pt , Mi \Di (pt , Mi )), and reports Ii = Iim (m′ , ptm′ ) − ptm . If for each i ∈ N(E(pt )), Iit = 0, the auctioneer sets the t increment δm = 0 for each item m. Otherwise, she sets the t increment δm for item m as follows:

{ δmt =

min

i∈N(E(pt )) s.t. Ii >0.

0

Ii if m ∈ E(pt ) otherwise.

3 Andersson et al. (2013) show that E(p) is unique and can be identified in polynomial time.

3

t Let δ t = (δm )m∈M . Then she sets t =: t + 1 and pt +1 =: pt + δ t . If pt +1 = pt , terminate the auction at pt +1 . Otherwise, return to Step 2.

Remark 4.4. In Step 3, in case Di (pt , Mi ) ⊊ Mi , by m ∈ Di (pt , Mi ) and m′ ∈ / Di (pt , Mi ), ui (m, ptm ) > ui (m′ , ptm′ ). Thus, Ii = Iim (m′ , ptm′ ) − ptm = ui (m, ptm ) − ui (m′ , ptm′ ) > 0. For each pair m, m′′ ∈ Di (pt , Mi ), ui (m, ptm ) = ui (m′′ , ptm′′ ). For each pair m′ , m′′′ ∈ Di (pt , Mi \Di (pt , Mi )), ui (m′ , ptm′ ) = ui (m′′′ , ptm′′′ ). Thus Ii is unique. The next example illustrates how the above auction works. Example 4.5. Let N = {1, 2, 3} and M = {a, b, c , d}. Let M1 = {a} and V 1 (a) = 5; M2 = {a, b}, V 2 (a) = 6, and V 2 (b) = 4; M3 = {b, c , d}, V 3 (b) = 4, V 3 (c) = 3, and V 3 (d) = 1. Let r = (ra , rb , rc , rd ) = (1, 1, 1, 2). The operation of the auction is summarized below. pt t=0 t=1 t=2

(1, 1, 1, 2) (3, 1, 1, 2) (4, 2, 1, 2)

D1 D2 D3 E(pt ) (·, M1 ) (·, M2 ) (·, M3 ) a a a

a a, b a, b

b b b, c

a a, b

∅

δt (2, 0, 0, 0) (1, 1, 0, 0) (0, 0, 0, 0)

We illustrate how to compute δ 0 and p1 . At Step 0, E(p0 ) = {a} and N(E(p0 )) = {1, 2}. For agent 1, by M1 \{a} = ∅, I1 = 0. For agent 2, M2 \{a} = {b} so I2 = I2a (b, 1) − pa = 2. Thus δa0 = 2 and δb0 = δc0 = δd0 = 0. Thus δ 0 = (2, 0, 0, 0) and p1 = (3, 1, 1, 2). Now we formally demonstrate the properties of the above auction. Theorem 4.6. (i) The proposed auction terminates at pT in finitely many steps, i.e., T < +∞. (ii) The proposed auction terminates in one of the following two cases: (ii-1) in case of pT = pT −1 , it validates the nonexistence of CEs; (ii-2) in case of E(pT ) = ∅, it identifies an MCEP pT . (iii) The covering condition holds if and only if E(pT ) = ∅. Proof. Part (i): By contradiction, suppose not. Then for each t ∈ N+ , E(pt ) ̸ = ∅ and pt ̸ = pt +1 . Thus, at each step t ∈ N+ , there t is i ∈ N(E(pt )) such that Ii = δm > 0 for each m ∈ E(pt ). Thus t +1 t Di (p , Mi ) ⊋ Di (p , Mi ). If i ∈ N(E(pt +1 )), E(pt +1 ) ̸ = E(pt ). If ′ ′ i∈ / N(E(pt +1 )), but i ∈ N(E(pt )) for some t ′ > t + 1, E(pt ) ̸= E(pt ). n Since M is finite, there are sequences {E(p )} and {N(E(pn ))} such that for each n ∈ N+ , E(pn ) ̸ = ∅ is unchanged and ′ N(E(pn )) ̸ = ∅. By the above analysis, at step n′ , for i ∈ N(E(pn )) n′ n′ such that Ii = δm > 0 for each m ∈ E(p ), we have i ∈ / N(E(pn )) for any n > n′ . Since N is finite, for a sufficiently large n, N(E(pn )) = ∅ holds, contradicting E(pn ) ̸ = ∅. Part (ii): By Definition 4.3, in case T = 0, the auction terminates if and only if E(p0 ) = ∅. In case T > 0, the auction terminates either pT = pT −1 or E(pT ) = ∅. Notice that E(pT ) = ∅ implies pT ̸ = pT −1 , which is mutually exclusive to pT = pT −1 . T T −1 (ii-1):⏐ By Definition 4.3, E(pT −1 ) ̸ = ∅. ⏐ ⏐ pT −1 ⏐= p T implies T −1 ⏐ T −1 ⏐ ⏐ ⏐ Thus N(E(p )) > E(p ) . By p = p ⏐ , Di (pT −1 , M⏐i ) = T −1 ⏐ ⏐ ⏐Mi holds ⏐ for ⏐each i ∈ ⏐ N(E(p )). Thus ∪i∈N(E(pT −1 )) Mi = ⏐E(pT −1 )⏐ < ⏐N(E(pT −1 ))⏐. There is no feasible assignment for agents in N(E(pT −1 )) at any prices. Thus there is no CE.

(ii-2): A non-empty set⏐ M ′ ⏐ ⊆ ⏐ M is overdemanded at p if ⏐ ⏐{i ∈ N : Di (p, Mi ) ⊆ M ′ }⏐ > ⏐M ′ ⏐. A non-empty set M ′ ⊆ M ′ ⏐is weakly underdemanded ⏐at p⏐ if ⏐[∀m ∈ M , pm > rm ] ⇒ ⏐{i ∈ N : Di (p, Mi ) ∩ M ′ ̸= ∅}⏐ ≤ ⏐M ′ ⏐. Step 1: p is an MCEP ⇐⇒ no set is overdemanded and weakly underdemanded at p.

4

L. Pan, L. Peng and Y. Zhou / Economics Letters 188 (2020) 108905

Step 1 follows the same reasoning in Theorem 2 of Mishra and Talman (2010). Step 2: E(pT ) = ∅ H⇒ no set is overdemanded at pT . By contradiction, suppose that there is a non-empty set M ′ ⊆ M that is overdemanded at pT . Then there is a non-empty set M ′′ ⊆ M ′ such that M ′′ is minimal overdemanded at pT .4 By Theorem 2 in Andersson et al. (2013), M ′′ ⊆ E(pT ), contradicting E(pT ) = ∅. Step 3: No set is weakly underdemanded at pT . We prove by induction. It is straightforward that no set is weakly underdemanded at p0 . Induction hypothesis: For some t < T , no set is weakly underdemanded at pt . Now we prove the case of t + 1. Consider an arbitrary non˜ ⊆ M such that for each m ∈ M, ˜ ptm+1 > 0. We show empty set M

˜ is not weakly underdemanded at pt +1 , i.e., that M ⏐ ⏐ ⏐ ⏐ ⏐{i ∈ N : Di (pt +1 , Mi ) ∩ M ˜ ̸= ∅}⏐ > ⏐˜ M⏐ .

(+ )

˜ = M ˜1 ∪ M ˜2 where M ˜1 = M ˜ ∩ E(pt ) and M ˜2 = M ˜∩ Let M t t ˜1 ̸= ∅, if (M \E(p )). For each i ∈ N(E(p )) such that Di (pt , Mi ) ∩ M Ii = 0, Di (pt , Mi ) = Di (pt +1 , Mi ) and if Ii ≥ minj∈N(E(pt )) s.t. Ij >0. Ij , Di (pt , Mi ) ⊆ Di (pt +1 , Mi ). Thus ⏐ ⏐ ⏐{i ∈ N : Di (pt +1 , Mi ) ∩ M ˜1 ̸= ∅}⏐ ⏐ ⏐ ˜1 ̸= ∅}⏐ ≥ ⏐{i ∈ N(E(pt )) : Di (pt +1 , Mi ) ∩ M ⏐ ⏐ ⏐ ⏐ ˜1 ̸= ∅}⏐ > ⏐˜ M1 ⏐ ≥ ⏐{i ∈ N(E(pt )) : Di (pt , Mi ) ∩ M

(∗)

The third strict inequality in (∗) comes from the definition of E(pt ). ˜2 , ptm = ptm+1 > 0 and for each m ∈ E(pt ), For each m ∈ M ptm+1 > ptm . Thus for each i ∈ N \N(E(pt )) such that Di (pt , Mi ) ∩ ˜2 ̸= ∅, Di (pt +1 , Mi ) = Di (pt , Mi ) and so M

⏐ ⏐ ⏐{i ∈ N : Di (pt +1 , Mi ) ∩ M ˜2 ̸= ∅}⏐ ⏐ ⏐ ˜2 ̸= ∅}⏐ ≥ ⏐{i ∈ N \N(E(pt )) : Di (pt +1 , Mi ) ∩ M ⏐ ⏐ ⏐ ⏐ ˜2 ̸= ∅}⏐ > ⏐˜ M2 ⏐ . = ⏐{i ∈ N \N(E(pt )) : Di (pt , Mi ) ∩ M

( ∗∗)

The third strict inequality in (∗∗) comes from the induction hypothesis.

4 A non-empty set M ′ ⊆ M is minimal overdemanded at p if (i) M ′ is overdemanded and (ii) no proper subset of M ′ is overdemanded.

˜2 = ∅, by (∗), (+) holds. In case M ˜1 = ∅, by (∗∗), (+) In case M ˜ ˜ holds. In case M1 ̸ = ∅ and M2 ̸ = ∅, by (∗) and (∗∗), (+) holds. Steps 2 and 3, together with Step 1, establish Part (ii). Part (iii): “If”: By Part (ii-2), there is an MCEP so a CE exists. The covering condition holds vacuously. “Only if”: By contradiction, suppose not. Thus Part (ii-1) holds, i.e., pT =⏐ pT −1 . Following ⏐ the reasoning in Part (ii-1), we can show ⏐∪i∈N(E(pT −1 )) Mi ⏐ < ⏐ same ⏐ ⏐N(E(pT −1 ))⏐, contradicting the covering condition. □ When obtaining an MCEP, the CE assignment can be identified by e.g., the augmenting path algorithm for maximum matching in the bipartite graph. Since models in Examples 2.1 and 2.2 satisfy the covering condition, Theorem 4.6 implies: Corollary 4.7. Consider the models in Examples 2.1 and 2.2 The proposed auction finds an MCEP/optimal and fair allocation prices in finitely many steps. In the models in Example 2.2, Andersson et al. (2013) propose an auction for the MCEP by additionally assuming that for each i ∈ N, vi (Mi ) ⊆ N and r ∈ NM . Their auction is a special case of ours: agents in N(E(pt )) do not report Ii and for each m ∈ E(pt ), δmt = 1. Applying our auction to the same models, for each i ∈ N(E(pt )), Ii > 0 always holds. However, in our general settings, it is not true since some i ∈ N(E(pt )) may report Ii = 0, e.g., agent 1 in Example 4.5. References Afacan, M.O., 2019. Matching with restricted trade. Internat. J. Game Theory 48, 957–977. Andersson, T., 2007. An algorithm for identifying fair and optimal allocations. Econom. Lett. 96 (3), 337–342. Andersson, T., Andersson, C., Talman, D., 2013. Sets in excess demand in simple ascending auctions with unit-demand agents. Ann. Oper. Res. 211, 27–36. Andersson, T., Svensson, L.G., 2008. Non-manipulable assignment of individuals to positions revisited. Math. Social Sci. 56 (3), 350–354. Demange, G., Gale, D., Sotomayor, M., 1986. Multi-item auctions. J. Political Econ. 94 (4), 863–872. Mishra, D., Talman, D., 2010. Characterization of the Walrasian equilibria of the assignment model. J. Math. Econom. 46 (1), 6–20. Pycia, M., 2016. Swaps on Networks. Working paper. Shapley, L.S., Shubik, M., 1971. The assignment game I: The core. Internat. J. Game Theory 1 (1), 111–130. Svensson, L.G., 2009. Coalitional strategy-proofness and fairness. Econom. Theory 40 (2), 227–245.