## 1 Introduction

This is to extend (Kazumura et al.

2020b) to a general case of an arbitrary number of agents and objects. They consider the multi-object allocation problem with monetary transfers where each agent obtains at most one object (unit-demand). A (consumption) bundle is a pair of object and payment. Each agent has a continuous preference relation over bundles satisfying the possibility of compensation and money monotonicity. Preferences are called classical if they satisfies object desirability. The classical domain is the class of all classical preferences The extended domain is the class of preferences which satisfies all properties but not object desirability.

An (allocation) mechanism, or simply mechanism chooses, for each preference profile, the object each agent receives and how much each agent pays. We focus on the following four properties of mechanisms. A mechanism is desirable if it satisfies those properties.

(i) Individual rationality requires that each agent’s bundle is at least as good as receiving nothing with no payment. Without this condition, agents does not participate the mechanism voluntarily. (ii) Non-wastefulness means that no agent prefers his own bundle to unassigned object with no payment. This condition is the property of weak efficiency. (iii) Equal treatment of equals requires that if there are two agents with same preferences, then their bundles are indifferent for their preferences. This is a weak condition of fairness. (iv) Strategy-proofness is the incentive compatible condition, which means that no agent has incentive to misreport his preference.

In multi-object allocation problem, for each preference profile, Walrasian equilibrium exists (Alkan and Gale

1990), and Demange and Gale (

1985) show that the set of Walrasian prices has a lattice structure; that is, there is the minimum price Walrasian equilibrium for each preference profile. The minimum price Walrasian mechanism is desirable (Demange and Gale

1985), and it satisfies efficiency and no subsidy (no-negative monetary transfer).

A subdomain of the classical domain is rich if for each object, there is a preference in the subdomain which demands only the object for some price and demands no object for some other price.

^{1} Kazumura et al. (

2020b) illustrate that various domains including the quasi-linear domain are rich, and show that in the case where the number of agents is greater than the number of objects, the minimum price Walrasian mechanism is the unique ex-post revenue maximizing mechanism among the desirable mechanisms satisfying no subsidy on a classical rich domain; moreover, the mechanism is also the unique ex-post revenue maximizing mechanism among the desirable mechanisms satisfying no bankruptcy on a domain including positive income effect preferences. We extend their results to a general case of an arbitrary number of agents and objects and on the subset of extended domain.

This article is organized as follows. Section

2 introduces the model and basic concepts, and checks the properties of minimum price Walrasian mechanisms. Our results are in Sect.

3. Section

4 provides proofs. Section

5 concludes.

## 2 The model

Let \(N=\left\{ 1,2,\ldots ,n\right\} \) be the set of agents and \(M=\left\{ 1,2,\ldots ,m\right\} \) be the set of different objects. Not consuming an object in M is called consuming the “null object”. Let \( L\equiv M\cup \left\{ 0\right\} \), where 0 denotes the null object. Each agent consumes at most one object. A typical (consumption) bundle for agent i is a pair \(z_{i}=\left( x_{i},t_{i}\right) \in L\times \mathbb {R}\): agent i receives object \(x_{i}\) and pays \(t_{i}\).

Each agent has a complete and transitive preference relation

\(R_{i}\) over

\(L \times \mathbb {R}\). Let

\(I_{i}\) and

\(P_{i}\) be the indifference relation and strict preference relation associated with

\(R_{i}\). A typical class of preferences is denoted by

\({\mathcal {R}}\). We call

\({\mathcal {R}}\) a

domain.

\(R_i\) is

classical if it satisfies the following assumptions:

1.
Continuity For each \(z_{i}\in L\times \mathbb {R}\), the sets \( \{ z^{\prime }_{i}\in L \times \mathbb {R} :z^{\prime }_{i}\,R_{i}\, z_{i}\} \) and \(\{ z^{\prime }_{i}\in L\times \mathbb {R}: z_{i}\,R_{i}\,z_{i}^{\prime }\} \) are closed.

2.
Possibility of compensation For each pair \(a,b\in L\) and each \( t\in \mathbb {R}\), there exist \(t^{\prime },t^{\prime \prime }\in \mathbb {R}\) such that \(\left( a,t\right) \,R_{i}\,\left( b,t^{\prime }\right) \) and \( \left( b,t^{\prime }\right) \,R_{i}\,\left( a,t^{\prime \prime }\right) \).

3.
Money monotonicity For each \(a\in L\) and each pair \( t,t^{\prime }\in \mathbb {R}\), if \(t<t^{\prime }\), then \(\left( a,t\right) \,P_{i}\,\left( a,t^{\prime }\right) \).

4.
Object desirability For each \(a\in M\) and each \(t \in \mathbb {R }\), \(\left( a,t\right) \,P_{i}\,\left( 0,t\right) \).

Let

\({\mathcal {R}}^{C}\) be the set of classical preferences—the

classical domain. Let

\({\mathcal {R}}^{E}\) be the set of preferences satisfying the above assumptions except for object desirability. We call

\( {\mathcal {R}}^E\) the

extended domain. Note that

\({\mathcal {R}}^{C} \subsetneq {\mathcal {R}}^{E}\). We assume that

\({\mathcal {R}} \subseteq \mathcal {R }^E\).

^{2}
A preference profile is a list of preferences \(R\equiv (R_{1},\ldots ,R_{n})\). Given \(i\in N\) and \(N^{\prime }\subseteq N\), let \(R_{-i}\equiv (R_{j})_{j\ne i}\) and \(R_{-N^{\prime }}\equiv (R_{j})_{j\in N{\setminus } N^{\prime }}\).

A (feasible) object allocation is an n-tuple \(x=\left( x_{1},x_{2},\dots ,x_{n}\right) \in L^{n}\) such that for each pair \(i,j\in N\) , if \(x_{i}=x_{j}\), then \(x_{i}=x_{j}=0\). Let A be the set of all object allocations. An allocation is a pair of an object allocation and a vector of payments, \(z=\left( \left( x_{1},x_{2},\dots ,x_{n}\right) ,\left( t_{1},t_{2},\dots ,t_{n}\right) \right) \in A\times \mathbb {R}^{n}\). Given \( z\in A\times \mathbb {R}^{n}\) and \(i\in N\), \(z_{i}=\left( x_{i},t_{i}\right) \in L\times \mathbb {R}\) denotes the bundle of agent i.

An (allocation) mechanism associates an allocation to each preference profile. Formally, a mechanism is a mapping \(f = (x,t) :{\mathcal {R}}^{n}\rightarrow A\times \mathbb {R}^{n}\). Given a mechanism f and a preference profile \(R\in {\mathcal {R}}^{n}\), agent i’s assignment under f at R is denoted by \(f_{i}\left( R\right) \). Moreover, we write \( f_{i}\left( R\right) \equiv (x_{i}(R), t_{i}(R)) \in L\times \mathbb {R}\), where \(x_{i}(R)\) denotes i’s object assignment and \(t_{i}(R)\) denotes his payment. We define \(f\left( R\right) \equiv \left( f_{1}\left( R\right) ,\ldots ,f_{n}\left( R \right) \right) \).

Non-wastefulness means that no agent prefers unassigned object with no payment to his own bundle. This condition is a weak condition of efficiency.

We say the allocation mechanism satisfies

no-wastage if for each

\(R \in {\mathcal {R}}^{n}\) and each

\(a \in M\), there is

\(i \in N\) such that

\( x_{i}(R) = a\). Note that when

\(n < m\), no mechanism satisfies no-wastage. Kazumura et al. (

2020b) define desirability by (i) individual rationality, (ii) no-wastage, (iii) equal treatment of equals, and (iv) strategy-proofness, which is different from our definition. However, since they assume that the number of agents is greater than the number of objects, no-wastage implies non-wastefulness. Thus, it is natural to relax no-wastage to non-wastefulness.

Let \(p=\left( p_{1},p_{2},\ldots ,p_{m}\right) \in \mathbb {R}_{+}^{m}\) be a price vector. We assume that the price of null object is equal to zero; that is, \(p_{0}=0\). Given \(i\in N\), \(R_{i}\in {\mathcal {R}}\), and \(p\in \mathbb {R} _{+}^{m}\), let \(D(R_{i},p)\equiv \{a\in L:\forall b\in L,\ (a,p_{a})\,R_{i}\,(b,p_{b})\}\) denote the demand set of agent i with \( R_{i} \) at p.

Next, we define the concept of Walrasian equilibrium. It is a pair of a price vector and an allocation such that for each agent it defines the object he demands and its price. Furthermore, the price of an unassigned object is zero.

Given

\(R\in {\mathcal {R}}^{n}\), let

\(W\left( R\right) \) be the set of Walrasian equilibria for

R, and define

$$\begin{aligned} Z(R) \equiv \{z \in A \times \mathbb {R}^{n}: \exists p \in \mathbb {R} ^{m}_{+} \ \text {s.t.}\ \left( z,p\right) \in W\left( R\right) \} \end{aligned}$$

and

$$\begin{aligned} P\left( R\right) \equiv \left\{ p\in \mathbb {R}_{+}^{m}:\exists z\in A\times \mathbb {R}^{n}\ \text {s.t.}\ \left( z,p\right) \in W\left( R\right) \right\} . \end{aligned}$$

Given

\(R \in {\mathcal {R}}^{n}\), we denote the minimum Walrasian price for

R by

\(p^{min}(R)\) and define

$$\begin{aligned} Z^{min}(R) \equiv \{ z \in A \times \mathbb {R}^{n}: (z,p^{min}(R)) \in W(R) \}. \end{aligned}$$

We say an allocation mechanism

f is a

minimum price Walrasian mechanism if for each

\(R \in {\mathcal {R}}^{n}\),

\(f(R) \in Z^{min}(R)\).

By the definition of Walrasian equilibrium, the minimum price Walrasian mechanism satisfies individual rationality, non-wastefulness, and equal treatment for equals, and so it is desirable; moreover, it satisfies no subsidy.

## 4 Proofs

Given \(i\in N\), \(R_{i}\in {\mathcal {R}}\), \(a\in L\) and \((b,t)\in L \times \mathbb {R}_{+}\), the compensated valuation \(V^{R_{i}}(a;(b,t))\) of a from (b, t) for \(R_{i}\) is the value such that \((a,V^{R_{i}}(a;(b,t))) \,I_{i}\,(b,t)\).

### 4.1 Proof of Theorem 1

Throughout this subsection, we assume that

\({\mathcal {R}}\) is rich. Our proof employes many results of Kazumura et al. (

2020b). We omit the proofs of such results, and write only the proofs of our new results.

We introduce the concept of favoring preferences. Given \(a\in M\), let \( {\mathcal {R}}^{a}\equiv \{R_{i} \in {\mathcal {R}}:\forall b\in L {\setminus } \{a\}\) , \((a,0)\,P_{i}\,(b,0)\}\). By the richness of domains, for each \(a\in M\), \( {\mathcal {R}}^{a} \ne \emptyset \). We say a preference \(R_{i}\in {\mathcal {R}} ^{a}\) is a-favoring.

We say a preference \(R_{i}\) is \((a,t)^{\varepsilon }\)-favoring if at price \(p\in \mathbb {R}_{+}^{m}\) such that \(p_{a}=t\) and \(p_{b}=0\) for each \(b\ne a\), agent i demands only object a, but when all objects’ prices slightly increase, then he demands nothing.

Given \(a\in M\) and a preference \(R_{i}\in \) \({\mathcal {R}}^{a}\), let \(t^{*}(R_{i},a)\equiv \min _{b\in L{\setminus } \{a\}}\{V^{R_{i}}(a; (b,0))\}\).

Note that even if all object price is zero except object a, agent i with preference \(R_{i}\in {\mathcal {R}}^{a}\) demands only object a if object a’s price is less than \(t^{*}(R_{i},a)\). Given \(a\in M\) and \(R_{i}\in {\mathcal {R}}^{a}\), \(t^{*}(R_{i},a)>0\) and we call \(t^{*}(R_{i},a)\) the supremum a-favoring payment for \(R_{i}\).

Facts

10 and

11 do not depend on the numbers of agents and objects. Moreover these facts hold whether

\({\mathcal {R}}\) satisfies object desirability or not.

We prove Claim by induction on k.

Base Case. Let

\(k=0\).

(i)
By assumption,

\(z_{1}\,P_{1}\,f_{1}(R)\). Thus, (i-a) holds. By (i-a),

\( z_i \in L \times \mathbb {R}_+\) and Fact

8,

\(x_{1}\ne 0\). Hence, (i-b) holds.

(ii)
By

\(z_{1}\,P_{1}\,f_{1}(R)\),

\(t_{1}<V^{R_{1}}(x_{1};f_{1}(R)))\). Thus, there is

\(\varepsilon _{1}>0\) such that

\(\varepsilon _{1}<\min (\{\underline{ p},V^{R_{1}}(x_{1};f_{1}(R))-t_{1}\}{\setminus } \{0\})\). By (i-b),

\(x_{1}\ne 0 \). Thus, by Fact

10, there is

\(z_{1}^{\varepsilon _{1}}\)-favoring preference

\(R_{1}^{\prime }\in {\mathcal {R}}\). Hence, (ii-a) holds. By

\(k=0\), (ii-b) holds vacantly.

(iii)
By (i-a), (ii-a) and Lemma

1 (ii),

\(x_{1}(R^{(1)})\ne x_{1}\). Thus since

\(R_{1}^{\prime }\) is

\(z_{1}^{\varepsilon _{1}}\)-favoring, by Remark

7,

\(z_{1}\,P_{1}^{\prime }\,f_{1}(R^{(1)})\).

(iv)
By \(k=0\), (iv) directly follows from (iii).

(v)
By

\(x_{1}(R^{(1)})\ne x_{1}\) and Lemma

2 (i), there is

\(j\in N{\setminus } \{1\}\) such that

\(x_{j}(R^{(1)})=x_{1}\). Without loss of generality, let

\( j\equiv 2\). We show that

\(z_{2}\,P_{2}\,f_{2}(R^{(1)})\). By Lemma

2 (ii) and

\((z,p)\in W(R)\),

\(t_{2}(R^{(1)})>t_{1}=p_{x_{1}}\). Thus,

$$\begin{aligned} z_{2}\,\underset{\text {by }(z,p)\in W(R)}{R_{2}}\,(x_{1},p_{x_{1}})\, \underset{\text {by }t_{2}(R^{(1)})>p_{x_{1}}}{P_{2}}\,(x_{1},t_{2}(R^{(1)})) \underset{\text {by }x_{2}(R^{(1)})=x_{1}}{=}f_{2}(R^{(1)})\text {.} \end{aligned}$$

Thus,

\(z_{2}\,P_{2}\,f_{2}(R^{(1)})\).

Inductive Hypothesis. Let

\(k\ge 1\). There are sets

\(N(k-1)\) and

N(

k) of distinct agents such that

\(N(k)\supseteq N(k-1)\),

\(\left| N(k-1)\right| =k-1\),

\( \left| N(k)\right| =k\), say

\(N(k-1)=\{1,2,\dots ,k-1\}\),

\( N(k)=\{1,2,\dots ,k\}\), and

\((\varepsilon _{j})_{j\in N(k)}\in \mathbb {R} _{++}^{k}\),

\(R^{(k-1)}\equiv (R_{N(k-1)}^{\prime },R_{-N(k-1)})\in \mathcal {R }^{n}\) and

\(R^{(k)}\equiv (R_{N(k)}^{\prime },R_{-N(k)})\in {\mathcal {R}}^{n}\) such that

(i-a-k)
\(z_{k}\,P_{k}\,f_{k}(R^{(k-1)})\) and

(ii-a-k)
\(\varepsilon _{1}<\min (\{\underline{p} ,V^{R_{1}}(x_{1};f_{1}(R))-t_{1}\}{\setminus } \{0\})\) and \(R_{1}^{\prime }\) is \(z_{1}^{\varepsilon _{1}}\)-favoring,

(ii-b-k)
for each \(j\in N(k)\), \(\varepsilon _{j}<\min \{ \varepsilon _{j-1}, t^{*}(R_{j-1}^{\prime },x_{j-1}), V^{R_{j}}(x_{j};f_{j}(R^{(j-1)}))-t_{j}\}\) and \(R_{j}^{\prime }\) is \( z_{j}^{\varepsilon _{j}}\)-favoring,

(iii-k)
\(x_{k}(R^{(k)}) \ne x_{k}\) and \(z_{k} \,P^{\prime }_{k} \, f_{k}(R^{(k)})\),

(iv-k)
\(x_{k}(R^{(k)})\notin \{x_{l}\}_{l\in N(k)}\), and

(v-k)
there is \(j \in N {\setminus } N(k)\) such that \(x_{j} \in \{x_{l}\}_{l \in N(k)}\) and \(z_{j} \,P_{j} \,f_{j}(R^{(k)})\).

Inductive Step.(i)
By (iv-k), there is

\(j\in N{\setminus } N(k)\) such that

\( z_{j}\,P_{j}\,f_{j}(R^{(k)})\). Without loss of generality, let

\(j=k+1\). Then, (i-a) holds. By (i-a),

\(z_i \in L \times \mathbb {R}_+\) and Fact

8,

\( x_{k+1}\ne 0\). Thus, (i-b) holds.

(ii)
The hypothesis (ii-a-k) is equivalent to (ii-a).

Next we show (ii-b). By Inductive Hypothesis,

\(\varepsilon _{k}>0\). By (i-a),

\(t_{k+1}<V^{R_{k+1}}(x_{k+1};f_{k+1}(R^{(k)})) \). By

\(t_{k} \ge 0\), Remark

8 (i) and (ii-b-k),

\(t^{*}(R_{k}^{\prime },x_{k})>0\). Thus, there is

\( \varepsilon _{k+1}>0\) such that

\(\varepsilon _{k+1}<\min \{ \varepsilon _{k}, t^{*}(R_{k}^{\prime },x_{k}),V^{R_{k+1}}(x_{k+1};f_{k+1}(R^{(k)}))-t_{k+1}\}\). By (i-b),

\( x_{k+1}\ne 0\). Thus, by Fact

10, there is

\(z_{k+1}^{\varepsilon _{k+1}}\) -favoring preference

\(R_{k+1}^{\prime }\in {\mathcal {R}}\). Hence by (ii-b-k), (ii-b) holds.

(iii)
By (i-a), (ii-a) and Lemma

1 (ii),

\(x_{k+1}(R^{(k+1)})\ne x_{k+1}\). Thus since

\(R_{k+1}^{\prime }\) is

\(z_{k+1}\)-favoring, by Remark

7,

\( z_{k+1}\,P_{k+1}^{\prime }\,f_{k+1}(R^{(k+1)})\).

(iv)
Suppose that

\(x_{k+1}(R^{(k+1)}) \in \{x_{l}\}_{l \in N(k+1)}\). By (iii), since

\(x_{k+1}(R^{(k+1)}) \ne x_{k+1}\), there is

\(j \in N(k+1) {\setminus }\{k+1\}\) such that

\(x_{k+1}(R^{(k+1)}) =x_{j}\). By

\( x_{k+1}(R^{(k+1)}) \ne x_{k+1}\) and (ii-b),

\(V^{R^{\prime }_{k+1}}(x_{k+1}(R^{(k+1)});(0,0)) < \varepsilon _{k+1}\). By individual rationality,

\(t_{k+1}(R^{(k+1)}) \le V^{R^{\prime }_{k+1}}(x_{k+1}(R^{(k+1)});(0,0)),\) and so

\(t_{k+1}(R^{(k+1)}) < \varepsilon _{k+1}\). By (ii-b),

\(\varepsilon _{k+1} < t^{*}(R^{\prime }_{j},x_{j}), \) and so

\(t_{k+1}(R^{(k+1)}) < t^{*}(R^{\prime }_{j},x_{j})\). However, by Lemma

2 (ii),

\(t_{k+1}(R^{(k+1)}) \ge t^{*}(R^{\prime }_{j},x_{j}).\) This is a contradiction. Thus,

\(x_{k+1}(R^{(k+1)}) \notin \{x_{l}\}_{l \in N(k+1)}\).

(v)
By (ii-a), (ii-b) and Lemma

2 (i), for each

\(i\in N(k+1)\), there is

\( j\in N\) such that

\(x_{j}(R^{(k+1)})=x_{i}\). By (iv), since

\( x_{k+1}(R^{(k+1)}) \notin \{x_{l}\}_{l \in N(k+1)}\), there is

\(j\in N{\setminus } N(k+1)\) such that

\(x_{j}(R^{(k+1)})\in \{x_{l}\}_{l\in N(k+1)}\).

By Lemma

2 (ii) and

\((z,p)\in W(R)\),

\( t_{j}(R^{(k+1)})>t_{j}=p_{x_{j}(R^{(k+1)})}\). Thus,

$$\begin{aligned}&z_{j}\quad R_{j}\quad (x_{j}(R^{(k+1)}),p_{x_{j}(R^{(k+1)})})\, \qquad \qquad \hbox {by }(z,p)\in W(R) \\&\quad P_{j}\quad (x_{j}(R^{(k+1)}),t_{j}(R^{(k+1)})) \quad \, \hbox { by }t_{j}(R^{(k+1)})>p_{x_{j}(R^{(k+1)})} \\&\quad =\,f_{j}(R^{(k+1)}). \end{aligned}$$

Hence

\(z_{j}\,P_{j}\,f_{j}(R^{(k+1)})\). The proof of Claim is completed.

By the above Claim, we derive a contradiction. For \(k=n-1\), by (i-b), for each \(i\in N\), \(x_{i}\ne 0\). If \(n>m\), it is impossible. Thus, we assume that \(n\le m\).

By (ii-a), (ii-b) and Lemma

2 (i), for each

\(i\in N\), there is

\(j\in N\) such that

\(x_{j}(R^{(n)})=x_{i}\). However, by Claim (iv), since

\( x_{n}(R^{(n)})\notin \{x_{i}\}_{i\in N}\), this is also impossible.

\(\square \)
### 4.2 Proof of Theorem 2

Throughout this subsection, we assume that \({\mathcal {R}}\supseteq {\mathcal {R}} ^C \cap {\mathcal {R}}^{++}\) and \(l\le 0\) is the associated lower bound of no bankruptcy.

Given \(a\in M\) and \(\delta >0\), define \({\mathcal {R}}^{a}(\delta )\equiv \{R_{i}\in {\mathcal {R}}:\forall b\in L{\setminus } \{a\},\ (a,0)\,P_{i}\,(b,-\delta )\}\). Note that for each \(a\in M\) and each \(\delta >0\), there is \(R_{i}\in {\mathcal {R}}^C \cap {\mathcal {R}}^{++}\) such that \( R_{i}\in {\mathcal {R}}^{a}(\delta )\). Thus, for each \(a\in M\) and each \(\delta >0\), \({\mathcal {R}}^{a}(\delta )\ne \emptyset \).

Given \(a\in M\), \(\delta > 0\) and \(R_{i} \in {\mathcal {R}}^{a}(\delta )\), let \( t^{*}(R_{i},a,\delta )\equiv \min _{b\in L{\setminus } \{a\}}\{V^{R_{i}}(a;(b,-\delta ))\}\).

For each \(a\in M\), each \(\delta >0\) and each \(R_{i}\in {\mathcal {R}} ^{a}(\delta )\), \(t^{*}(R_{i},a,\delta )>0\).

Given \((a,t)\in M\times \mathbb {R}_{+}\), \(\varepsilon >0\) and \(\delta >0\), let \({\mathcal {R}}((a,t),\varepsilon ,\delta )\) be the set of \( (a,t)^{\varepsilon }\)-favoring preferences in \({\mathcal {R}}^{a}(\delta )\).

Given \((a,t)\in M\times \mathbb {R}_{+}\), \(\varepsilon >0\) and \(\delta >0\),

let \({\mathcal {R}}^{++}_C((a,t),\varepsilon ,\delta ) \equiv {\mathcal {R}}^C \cap {\mathcal {R}}^{++} \cap {\mathcal {R}}((a,t),\varepsilon ,\delta )\).

Given \(R\in {\mathcal {R}}^{n}\) and \(l\le 0\), let \(\delta (R,l)\equiv n(\max _{k\in N}{\max _{b\in L}{V^{R_{k}}(b;(0,0))}})-l\).

To replace Lemma

1 with Lemma

4, Lemma

2 with Lemma

5, Remark

7 with Remark

10, Remark

8 with Remark

9, and Fact

10 with Fact

12, by the same logic in Proposition

1, we can prove the above claim.

\(\square \)
By extending the results of Kazumura et al. (

2020b), we showed that for an arbitrary numbers of agents and objects, the minimum price Walrasian mechanism is the unique ex-post revenue maximizing mechanism on rich domains among desirable mechanisms, and that no subsidy in this result can be replaced by no bankruptcy on the positive income effect domain.

There is the literature on auction with non-quasi-linear preferences. We conclude the paper by referring to some existing literature on auction with non-quasi-linear preferences and mentioning some lines for future research.

Most of the literature on auction with non-quasi-linear preferences focus on efficiency rather than on revenue maximization. Saitoh and Serizawa (

2008) and Sakai (

2008) show that in the cases of homogeneous objects and unit-demand preferences, the generalized Vickrey mechanism is the only mechanism satisfying strategy-proofness, efficiency, individual rationality, and no subsidy. Morimoto and Serizawa (

2015) extend these results to the case of heterogeneous objects by maintaining unit-demand preferences, and show that the minimum price Walrasian mechanism is the only mechanism satisfying the same four properties on classical domain. These works assume that the number of agents is greater than objects. Thus, it is an open question whether these results hold for an arbitrary numbers of agents and objects.

Zhou and Serizawa (

2018) also maintain unit-demand preferences, but study the special class of preferences, the common-tiered domains. It says that objects are partitioned into several tiers, and if objects are equally priced, agents prefer an object in the higher tier to one in the lower. They show that for each number of agents and objects, the minimum price Walrasian mechanism is the only mechanism satisfying same four properties on the common-tiered domains. Moreover, when the number of agents is less than the number of objects including null object, the minimum price Walrasian mechanism is also the only mechanism satisfying same four properties on the positive income effect domain. It is an open question whether their results also hold on the common-tiered domains exhibiting positive income effects for an arbitrary number of agents and objects.

There is also the literature on auction with non-quasi-linear preferences admitting multi-demand in various settings. Kazumura and Serizawa (

2016) study classes of preferences that include unit-demand preferences and additionally includes at least one multi-demand preference, and show that when the number of agents is greater than the number of objects, no mechanism satisfies the four properties on such a domain. Malik and Mishra (

2021) study the special classes of preferences, “dichotomous” domains. A preference is dichotomous if there is a set of objects such that the valuations of its supersets are constant and the valuations of other sets are zero. A dichotomous domain includes all such dichotomous preferences for a given set of objects. They show that when there are at least three agents and two objects, no mechanism satisfies the four properties on a dichotomous domain, but that for each number of agents and objects, the generalized Vickrey mechanism is the only mechanism satisfying the four properties on a class of dichotomous preferences exhibiting positive income effects.

Baisa (

2020) assumes that objects are homogeneous and shows that on the class of preferences exhibiting decreasing marginal valuations, positive income effect, and single-crossing property, if the preferences are parametrized by one dimensional types, there is a mechanism satisfying the above four properties, but that if types are multi-dimensional, no mechanism satisfies these properties. Shinozaki et al. (

2020) also assume the homogeneity of objects, and show that on the class of preferences including sufficiently various preferences exhibiting non-decreasing marginal valuations (minimal richness), the generalized Vickrey mechanism is the only mechanism satisfying the four properties, but that no mechanism satisfies these properties on the class of preferences that additionally includes at least one preference exhibiting decreasing marginal valuations.

These different results in various settings of multi-demand suggest that analyzing revenue maximization mechanisms in multi-demand settings would be technically challenging. However, such research is important in practical applications. Recently, Kazumura et al. (

2020a) develop methods to analyze strategy-proof mechanisms in general settings including multi-demand cases. Their method does not depend on the number of agents and objects. We believe that such methods would be useful to analyze revenue maximization mechanisms in multi-demand settings.

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