1 Introduction

Consider allocating a set of discrete goods to a set of agents. In this study, we formalize an extended simultaneous eating algorithm (ESEA), where the fixed quota of each good is extended to a variable varying in a polytope. Base polyhedra defined in Sect. 2 are these polytopes that can be characterized by the greedy algorithm. We link these two algorithms, both the solutions and the order of the execution, based on the property of ordinal efficiency. This study is an extension of our former work (Fujishige et al. 2016, 2018, 2019; Sano and Zhan 2021; Zhan 2023).

1.1 Simultaneous eating algorithm

The simultaneous eating algorithm (SEA) was proposed by Bogomolnaia and Moulin (2001) in their seminal paper and is outlined informally as follows (a formal generalized definition will be given later).

Simultaneous Eating Algorithm (SEA): Let each good be an infinitely divisible object with a quota that agents eat during some time intervals. Each agent eats from his/her (remaining) favorite good at some speed, then proceeds to the next step when the good eaten by the agent(s) is completely exhausted.

Suppose that all agents have uniform eating speed, the above assignment mechanism is called probabilistic serial (PS). We assume that the value of definite integrals of the speed function is the same for all agents.

Now, consider a simple example of the assignment problem with the set of agents \(\{1,2,3\}\), set of goods \(\{a,b,c\}\). The preference profile is given as follows (Example 1 of Zhan 2023):

$$\begin{aligned} \begin{array}{cccc} 1 \quad \; &{} a \; \; &{} \;b &{} \ c \\ 2 \quad \; &{} b \; \; &{} a \; &{} \; \ c \\ 3 \quad \; &{} a\;\; &{} c &{} \; \; \ b \end{array} \end{aligned}$$

Here, agent 1 prefers a to b to c, agent 2 prefers b to a to c, and agent 3 prefers a to c to b.

Suppose that the quota of all goods is 1, by SEA, we get an output of the allocation as follows:

Note that the sum of each column, the quota of goods, is 1.

Next, we generalize the above problem. Let \(\rho : 2^E \rightarrow {\mathbb {Z}}_{\ge 0}\) be an integral set function on \(E=\{a,b,c\}\) defined as

$$\begin{aligned} \rho (X)={\left\{ \begin{array}{ll} \; 2 &{}\quad \text {if } |X |= 1 \\ |X |&{} \quad \text {if } |X |> 1 \end{array}\right. } \quad \forall X \subseteq \{e_1,e_2.e_3\}. \end{aligned}$$

And we let allocation P satisfy

$$\begin{aligned} \sum _{e_i \in X } x_P(e_i) \le \rho (X) \quad \forall X \subseteq \{e_1,e_2.e_3\}, \end{aligned}$$

where \(x_P \in {\mathbb {R}}^3 \) is a vector, and each entry of \(x_P(e_i)\) \((i \in \{1,2,3\}) \) is the column sum of allocation P. Then, we can obtain an allocation

One explanation of the above example is: Students are agents, and E is the type of specialists to advise these students. The value of \(\rho \) represents the number of specialists. The total number of three type specialists is 3. Each type of specialist is at most 2 (the sum of each column), the popular field can get more specialists.

Here is another one. Customers are agents. Each good in E is a shop renting some partitions of a tenant, where the value of \(\rho \) represents the number of the partitions rented by a shop. There are totally 3 partitions. To keep diversity, each shop can rent at most 2 partitions, while the popular shop can also get more partitions.

We have seen that the preferred good a shares larger portion 2 in the polymatroid constraints. Moreover, by swapping goods a and c in the above preference profile, good c can get a portion 2 from the total resource 3.

The desired allocation is yielded from the diminish return property of \(\rho : 2^E \rightarrow {\mathbb {R}}\)

$$\begin{aligned} \rho (X\cup \{e\})-\rho (X) \ge \rho (Y\cup \{e\})-\rho (Y) \quad \forall X \subseteq Y, \ e \in E \setminus Y \end{aligned}$$

called submodular function defined in Sect. 2. The above “specialist/partition" may also be other types. Preferred types obtain more copies of some goods among total resources.

1.2 Contributions

Besides the extension of variable quota, this study is directly motivated by ordinal efficiency of SEA.

Let \(N=\{1,\cdots ,n\}\) be a set of agents and E be a set of goods. An assignment is a matrix \(P\in {\mathbb {R}}^{N \times E}_{\ge 0}\), where the entry \(P(i,e)\ge 0\) denotes the portion or probability of good e agent i obtained. Assume each agent \(i \in N\) has a linear preference over goods given by

$$\begin{aligned} L^i:\ e_1^i\succ _i e_2^i\succ _i \cdots \succ _i e_m^i, \end{aligned}$$
(1)

where \(\{e_1^i,e_2^i,\cdots ,e_m^i\}=E\).

Consider two assignments P and Q. For each agent \(i\in N\) with preference \(L^i\), define a relation (sd-dominance relationFootnote 1) \(\succeq _i^{\textrm{sd}}\) between the ith rows \(P^i\) and \(Q^i\) of P and Q, respectively, as follows.

$$\begin{aligned} P^i \succeq _i^{\textrm{sd}} Q^i \quad \Longleftrightarrow \quad \forall \ell \in \{1,\cdots ,m\}:\ \ \sum _{k=1}^\ell P(i,e^i_k) \ge \sum _{k=1}^\ell Q(i,e^i_k). \end{aligned}$$
(2)

The assignment Q is sd-dominated by P if we have \(P_i \succeq _i^{\textrm{sd}} Q_i\) for all \(i\in N\) and \(P\ne Q\). We say that P is ordinally efficient if P is not sd-dominated by any other assignment (Bogomolnaia and Moulin 2001; Fujishige et al. 2018; Zhan 2023).

Let \({\mathscr {A}}\) denote all assignments. Ordinal efficiency is stronger than expected, i.e., it is equivalent to the following (welfare) summation

$$\begin{aligned} \max _{P \in {\mathscr {A}}} \sum _{i=1}^n \sum _{e \in E} P(i,e) u_i(e), \end{aligned}$$
(3)

where \(u_i:E \rightarrow {\mathbb {R}}_{\ge 0} \) is a utility function consistent with \(L_i\) (the formal definition will be given in Sect. 4.1) (McLennan 2002). During the execution of SEA, currently most preferred goods are selected to eat, the output is an assignment with the Pareto type of ordinal efficiency that is equivalent to a summation optimization (3), similar to the greedy algorithm. Additionally, there is room for an exact and unified characterization to cover agents’ different eating speed, to verify the role of polymatroid constraints playing in the assignment.

The main contribution of this study is clarifying these local and global relations in both the solutions and the order of the execution as follows:

  • Ordinal efficiency of extended SEA (ESEA)’s outcome is both Pareto efficient and the optimizer of welfare summation maximization (Theorem 5). The greedy algorithm coincides with ESEA in the execution, i.e., choosing currently best one to achieve the (global) summation optimization (Theorem 6).

  • Additionally, the above ESEA mechanism is formulated in Algorithm I, and its ordinal efficiency is characterized by the different eating speed (Theorem 2).

As a result, in the above example, good a followed by good b and the last null c eaten in ESEA is exactly the order selected by the greedy algorithm with an aggregated and consistent (with preferences) weight vector.

Note that polymatroids and submodularity are not strictly special constraints, they filter some redundant inequalities in the polyhedra defined by a system of linear inequalities (Fujishige 2005).

It should be pointed out that this study is closely related to our former work. The result directly used in this study is Theorem 1 from Fujishige et al. (2016). The above contributions are original.

1.3 Related work

Problems of allocating a set of goods according to agents’ ordinal preferences without monetary transfers have been investigated in several studies. There are many applications for such assignments, e.g., house allocations, kidney exchanges, assigning professors to lectures, specialists to tasks (Abdulkadiroğlu and Sönmez 2003; Aziz and Brandl 2022; Balbuzanov 2022; Budish et al. 2013; Harless 2019; Hashimoto et al. 2014).

In this subsection, after introducing our former studies to have a better understanding of the motivation and difference between this work and our former ones, we focus on ordinal efficiency. For other various extensions and characterizations of SEA and PS (Bogomolnaia and Moulin 2001), for example, refer to recent papers Aziz and Brandl (2022), Balbuzanov (2022), and references therein.

The main contribution of our previous work was to enlarge a fixed set of goods to a family of sets, in other words, extending the resource space from a (fixed) point to a polyhedron.

Our first related work is an unpublished working paper (Fujishige et al. 2016) which extended the PS to the matroidal resource space where the varying quota is bounded by one. We also proved the acyclic property of ordinal efficiency being kept on the extension which is the foundation of this study.

In Fujishige et al. (2018), the resource space is extended from matroids to polymadroids, i.e., quota one bound of goods is released. In this paper, the results about incentive-compatibility (i.e., agents can not obtain better allocation by not telling their true preferences) are presented which are the hardest part of these extensions. Various examples (in matching graphs) are provided to show the difference of our extensions and others, e.g., Budish et al. (2013) extended the probabilistic serial mechanism using a layer structure, which is a special case of our problem setting (in Sano and Zhan 2021, these are depicted in polytopes).

Sano and Zhan (2021) further extended PS to allow agents’ indifference preferences by combining the work of Fujishige et al. (2018) and Katta and Sethuraman (2006). Since submodularity is kept (in composition functions), non-trivial decisions about which good in each indifferent set can be taken by constructing parametric (independent flow) networks.

In Fujishige et al. (2019), the assignment problems are formulated and characterized as optimization ones. Zhan (2023) provides a non-algorithmic characterization of EPS based on a fairness property and a weak efficiency property.

Agents’ total demands are assumed to be larger than the resource in our problem settings. This reasonable assumption and applications in the assignment can be found in Thomson (2019). Insufficient resource guarantees that the outcome (the column sum of an assignment) of ESEA is a maximal vector in a resource polytope (called base polyhedron defined later) under a hereditary property (Fujishige 2005). This is a basic assumption of this study.

We turn to ordinal efficiency, the central concept of current study.

In Bogomolnaia and Moulin (2001), SEA is characterized by ordinal efficiency. Simultaneously, McLennan (2002) strengthened ordinal efficiency by showing that (Pareto type of) ordinal efficiency is equivalent to a welfare summation shown in (3). Mclennan proved the result using the separation theorem of convex theory (Rockafellar 1970) by applying the concepts in Abdulkadiroğlu and Sönmez (2003). Years later, Manea (2008) reproved McLennan’s result using a different approach, the acyclic property of ordinal efficiency. Manea introduced a weight vector on goods that matches agents’ preferences with the positive values of the assignment obtained from SEA.

Three enduring goals (efficiency, incentive-compatibility, and fairness) in mechanism design have been kept in our enlarged problem setting, where submodurality is essential (Fujishige et al. 2018). Our goal is to reveal that the diminishing return of the submodularity implies a higher efficiency when the preferred goods are chosen earlier (as shown in the example above). Manea’s approach was useful in achieving the goal. However, a notable difference exists since Manea’s proof assumes a unit quota of goods which is a challenging part of this study. The acyclic property in Fujishige et al. (2016) is fundamental to introduce an aggregated and consistent weight vector with agents’ preferences for our problem setting.

Additionally, in the last two decades, applications of submodular functions also appeared extensively in machine learning and artificial intelligence, refer to the summary in a recent paper Bilmes (2022).

The rest of the article is organized as follows: Sect. 2 presents the problem description. In Sect. 3, after introducing the acyclic property of ordinal efficiency, ordinal efficiency is characterized by ESEA. The main results are shown in Sect. 4. These include the definition of welfare maximization, introducing an associated weight function, and building the relation between ESEA and the greedy algorithm. Section 5 gives concluding remarks.

2 The model

Let \(N=\{1,2,\cdots ,n\}\) be a set of agents and E be a set of goods. Let \({{\mathscr {L}}}\) denote the profile of preferences \((L^i\) \( \mid i\in N)\), where \(L^i \) is defined in (1).

We suppose that a positive integral demand vector \(\textbf{d}\in {\mathbb {Z}}_{>0}^N\), agent i with demand \(d(i) > 1\) can also be viewed as d(i) sub-agents without loss of generality (Sano and Zhan 2021; Zhan 2023). Hence, we assume \( \mathbf{d=1} \), an all one vector in the sequel.

For the set E of goods, a (monotonic) submodular function \(\rho \) is defined as follows.

A pair \((E,\rho )\) of set E and rank function \(\rho : 2^E \rightarrow {\mathbb {R}}_{\ge 0}\) is called a polymatroidFootnote 2 if the following three conditions hold:

  1. (a)

    \(\rho (\emptyset ) = 0\),

  2. (b)

    for any \(X, Y\in 2^E\) with \(X \subseteq Y\) we have \(\rho (X) \le \rho (Y)\),

  3. (c)

    for any \(X,Y \in 2^E\) we have \(\rho (X) + \rho (Y) \ge \rho (X \cup Y) + \rho (X\cap Y) \).

Additionally, we assume \(\rho (E)>0\) to avoid the trivial case.

For a given polymatroid \((E, \rho )\), the submodular polyhedron \({\textrm{P}}(\rho )\) of \((E, \rho )\) is defined by

$$\begin{aligned} {\textrm{P}}(\rho )=\{x\in {\mathbb {R}}^E \mid \forall X\subseteq E: x(X)\le \rho (X)\} \end{aligned}$$

and its base polyhedron by

$$\begin{aligned} {\textrm{B}}(\rho )=\{x\in {\textrm{P}}(\rho )\mid x(E)=\rho (E)\}, \end{aligned}$$

where \(x(X)=\sum _{e\in X}x(e)\) for all \(X\subseteq E\). For a polymatroid \((E,\rho )\), we have \({\textrm{B}}(\rho ) \ne \emptyset \) (see Fujishige 2005; Oxley 2011 for more details).

Fix an \((N,E,{{{\mathscr {L}}}},\mathbf{d=1}, (E,\rho ))\). An assignment, or expected allocation, is a non-negative matrix \( P \in {\mathbb {R}}_{\ge 0}^{N\times E}\) satisfying

(i):

\( \sum _{e \in E} P(i,e) \le 1 \) for all \(i \in N,\)

(ii):

\( \sum _{i \in N} P^i \in {\textrm{B}}(\rho ), \)

where P(i, e) represents the probability that agent i receives good e, and the ith row, denoted by \(P^i\), is the total goods (or a bundle of goods) allocated to the agent i. We assume \(\rho (E) \le n,\) i.e., the resource is equal to or less than the agents’ total demands.

By our problem setting, the random assignment problem is to find an assignment matrix P with its column vector \(x_P \in {\mathbb {R}}^E\) satisfying

$$\begin{aligned} x_P \equiv \sum _{i \in N} P^i \in {\textrm{B}}(\rho ) \end{aligned}$$
(4)

and also meeting some desired economic properties.

We denote the random assignment problem by \(\textbf{RA}=(N,E,{{{\mathscr {L}}}}=(L^i \mid i \in N), \mathbf{d= 1},(E,\rho ))\). In the following, we write one element set \(\{e\}\) as e for simplicity.

For \(x \in {\textrm{P}}(\rho ) \), we define \({\mathscr {F}}(x) \subseteq 2^E \) by

$$\begin{aligned} {\mathscr {F}}(x) =\{ X \subseteq E \mid x(X)=\rho (X) \}, \end{aligned}$$
(5)

call \({\mathscr {F}}(x)\) x-tight sets. The following proposition is fundamental in the theory of polymatroids and submodular functions.

Proposition 1

For each vector \(x \in {\textrm{P}}(\rho )\), we have \({\mathscr {F}}(x)\) being closed under the set union and intersection, i.e., \(X\cup Y, \ X\cap Y \in {\mathscr {F}}(x)\) if \( X,Y \in {\mathscr {F}}(x)\).

We require some further notations in following definitions and characterizations.

Given a vector \(x\in {\textrm{P}}(\rho )\), by Proposition 1, there exists a unique maximal x-tight set, denoted by \({\textrm{sat}}(x)\) the saturation function, which is equal to the union of all x-tight sets,

$$\begin{aligned} {\textrm{sat}}(x) =\bigcup _{ X \subseteq E} \{ X \mid x(X) = \rho (X)\}, \end{aligned}$$
(6)

and \(x({\textrm{sat}}(x))=\rho ( {\textrm{sat}}(x))\). From the definition of \({\textrm{sat}}(x)\) we have (Fujishige 2005)

$$\begin{aligned} {\textrm{sat}}(x) =\{e\in E\mid \forall \alpha >0: x+\alpha \chi _e\notin {\textrm{P}}(\rho )\}, \end{aligned}$$
(7)

where, \(\chi _e\) means the characteristic function in \({\mathbb {R}}^E\) such that \(\chi _e(e')=1\) if \(e'=e\), \(\chi _e(e')=0\) if \(e' \in E \setminus e.\)

For \(x\in {\textrm{P}}(\rho )\) and \(e\in {\textrm{sat}}(x)\), we define the dependence function

$$\begin{aligned} {\textrm{dep}}(x,e)=\{e'\in E\mid \exists \alpha >0: x+\alpha (\chi _e-\chi _{e'})\in {\textrm{P}}(\rho )\}. \end{aligned}$$
(8)

It is known that (Fujishige 2005)

$$\begin{aligned} {\textrm{dep}}(x,e) = \bigcap _{e \in X \subseteq E} \{ X \mid x(X) = \rho (X) \}, \end{aligned}$$
(9)

and we have \(x({\textrm{dep}}(x,e))= \rho ({\textrm{dep}}(x,e))\) also by Proposition 1.

3 Two characterizations of ordinal efficiency

In this section, we present the generalization of two characterizations investigated in Bogomolnaia and Moulin (2001). Assignments treated in Bogomolnaia and Moulin (2001) are double stochastic matrices. Here, we consider matrices varying in column sum and subject to submodular function constraints.

3.1 Acyclicity

Fix \(\textbf{RA}=(N,E,{{{\mathscr {L}}}}, \textbf{1},(E,\rho ))\). For an expected allocation P, we define a (directed) graph \(H(P)=(V,A)\) with a vertex set V and an arc set A byFootnote 3

$$\begin{aligned} V= & {} \big (\bigcup _{i\in N}E^i\big )\cup E, \end{aligned}$$
(10)
$$\begin{aligned} A= & {} \big (\bigcup _{i\in N}A^i\big )\cup A_0\cup A^*, \end{aligned}$$
(11)

where each \(E^i\) is a disjoint copy of E, and \(A^i\), \(A_0\), and \(A^*\) are defined by

$$\begin{aligned} A^i= & {} \{(e^i_{k},e^i_{k+1})\mid k=1,\cdots ,m-1\}\qquad (\forall i\in N), \end{aligned}$$
(12)
$$\begin{aligned} A_0= & {} \{(e,e^i)\mid i\in N,\ e\in E, \ e^i\in E^i,\ e^i\ {\text {is a copy of }}e\}\nonumber \\{} & {} \cup \{(e^i,e) \mid i\in N,\ e\in E, \ e^i\in E^i,\ e^i\ {\text {is a copy of }}e,\ P(i,e)>0\}, \end{aligned}$$
(13)
$$\begin{aligned} A^*= & {} \{(e,e')\mid e, e'\in E,\ e\in {\textrm{dep}}(x^*_P,e')\setminus \{e'\}\}, \end{aligned}$$
(14)

where \(x^*_P \in {\textrm{B}}(\rho )\) and \(x^*_P(e) = \sum _{i \in N}P(i,e)\) for \(e \in E\). See Fig. 1 for an illustrative graph of H(P), where some of the broken arcs may not exist in \(A_0\). We considered that \(e^i_k\) \((k=1,\cdots ,m)\) appeared in (12) with the corresponding copies in \(E^i\) for each \(i\in N\).

Fig. 1
figure 1

The graph H(P) with \(e_1^1=f,\) \(e_m^n=f' \in {\textrm{dep}}(x^*_P,f)\)

Full size image

Theorem 1

(Theorem 3.1, Fujishige et al. 2016) An expected allocation P is ordinally efficient if and only if there exists no directed cycle containing at least one arc of \(\cup _{i\in N}A^i\) in \(H(P)=(V,A)\) defined above.

Here, we provide a basic fact from Theorem 1 which reveals an asymmetric structure of ordinal efficiency. It is crucial to obtain a reasonable weight vector on goods with the preference profile \({\mathscr {L}}\) being arbitrary.

Proposition 2

If an expected allocation P is ordinally efficient and there exist agents \(\ i,j \in N \, (i \ne j)\) with \(e \succ _i {{{\hat{e}}}}\) and \({{{\hat{e}}}} \succ _j e\), then we have \( P(i, {{{\hat{e}}}}) P(j,e)=0.\)Footnote 4

Proof

Conversely, suppose \(P(i, {{{\hat{e}}}}) P(j,e)>0\). Then we can construct a cycle \((e^i, {{{\hat{e}}}}^i, {{{\hat{e}}}}, {{{\hat{e}}}}^j, e^j, e^j, e, e^i) \) in H(P), as shown in Fig. 2. Here, the existence of arcs \(({{{\hat{e}}}}^i,{{{\hat{e}}}})\) and \((e^j,e) \in A_0\) of (13) follows from \(P(i, {{{\hat{e}}}}) P(j,e)>0\). This is a contradiction by Theorem 1. \(\square \)

Fig. 2
figure 2

A cycle in H(P) induced by \( P(i, {{{\hat{e}}}}) P(j,e)>0\) with \(e \succ _i {{{\hat{e}}}}\) and \({{{\hat{e}}}} \succ _j e\)

Full size image

Proposition 2 precludes a cyclic relation (\(e \succ _i {{{\hat{e}}}} \succ _j e\)) in a preference profile when restricted on the positive entries of an ordinally efficient assignment P. Hence, we can unify utilities in an order consistent with agents’ preferences on positive entries of allocations.

Given a preference profile \({\mathscr {L}}=(L^i \mid i \in N)\) and an expected allocation P, a strict domination via probability trade binary relation is defined as follows (Bogomolnaia and Moulin 2001; Manea 2008):Footnote 5

$$\begin{aligned} \forall e,e' \in E: e \rhd \hspace{-1mm} (P, {\mathscr {L}}) \ e' \Longleftrightarrow \{ \exists i \in N: e \succ _i e' \;{\textrm{and}} \; P(i,e') >0\}. \end{aligned}$$
(15)

The binary relation defined in (15) is characterized by ordinal efficiency given in the following Corollary 1.

Corollary 1

An expected allocation P is ordinally efficient if and only if there exists no directed cycle containing at least one arc \((e^i_k,e^i_{k+1})\) of \(\cup _{i\in N}A^i\) in \(H(P)=(V,A)\) with \(e^i_k \rhd \hspace{-1mm} (P,{\mathscr {L}})\ e^i_{k+1}\) for some \(1 \le k \le m-1\) and \(i \in N\).

Proof

Since each subgraph \((E^i,A^i)\) of H(P) is a directed path, a cycle contains an arc in \(\bigcup _{i\in N} A^i\) if and only if it contains an arc of a path in \(A^i\) adjacent to an arc of the circle in \(A_0\) with \(P(i,e^i_{k+1})>0\). Then the claim follows from Theorem 1. \(\square \)

3.2 Simultaneous eating algorithm of different eating speeds

In this subsection, we consider the relation between the extended SEA and ordinal efficiency for our problem setting. We adapt the approach proposed by Bogomolnaia and Moulin (2001).

Fix a random assignment problem \(\textbf{RA}=(N,E,{{{\mathscr {L}}}}, \textbf{1},(E,\rho ))\).

During the execution of the following algorithm, let \(\textbf{L}_q=( e^1(q), e^2(q), \dots , e^n(q) )\) be the list of goods eaten at step q, where \(e^i(q)\) \((i \in N)\) is agent i’s current most preferred available good. Note that \(e^i(1)=e^i_1\) and we may have \(e^i(q)=e^j(q)\) for \(i \ne j \).

We impose the following constraint, the same definite integral value of eating speed functions \(\omega _i: [0, \lambda ^*] \rightarrow {\mathbb {R}}_{\ge 0}\) (\(i \in N\)), i.e.:

$$\begin{aligned} \int _0 ^{\lambda ^*} \omega _i(t) dt =\lambda ^* \equiv \rho (E)/n \le 1\qquad {\mathrm{for \; all}} \; i \in N, \end{aligned}$$
(16)

where t denotes time. The assumption \(\lambda ^* \le 1\) in (16) follows from assumption \(\rho (E) \le n\) mentioned earlier. Let the expected allocation be denoted as \(P_{\omega } \in {\mathbb {R}}^{N \times E} _{\ge 0}\) when we emphasize different eating speeds \(\omega =(\omega _1,\dots ,\omega _n)\). Also, we simplify \(P_{\omega }\) to P when \(\omega = \textbf{1}\) if there is no ambiguity.

The following algorithm is a reformulation of the one given in Fujishige et al. (2018), Sano and Zhan (2021), Zhan (2023) where \(\omega = \textbf{1}\).

figure a

It should be pointed out that the (big) brackets in (17) and the formula below it mean vectors in \({\mathbb {R}}^E\). We have \(0<\lambda _{q-1} < \lambda _q\) (\(1\le q \le q^*\)) as the output of Algorithm I.

Example 1

Consider \(N=\{1,2,3,4\}\) and \(E=\{a,b,c,d\}\). Let \((E,\rho )\) be a polymatroid with the rank function defined by

$$\begin{aligned} \rho (X) = \left\{ \begin{array}{ll} 2|X| &{} \quad \text {if } |X| \le 2\\ 4 &{} \quad \text {if } |X| >2 \end{array} \right. \quad (\forall X\subseteq E). \end{aligned}$$
(18)

Suppose that the preference profile of agents is given as follows.

$$\begin{aligned} \begin{array}{cc} i\in N &{} {\mathrm{preference\ }}\ L^i \\ 1 &{} a\succ _1 b\succ _1 c\succ _1 d\\ 2 &{} a\succ _2 c\succ _2 b\succ _2 d\\ 3 &{} a\succ _3 c\succ _3 d\succ _3 b\\ 4 &{} b\succ _4 a\succ _4 d\succ _4 c \end{array} \end{aligned}$$

Agent 2’s eating speed is as shown in Fig. 3 while others eat at speed 1 throughout.

Fig. 3
figure 3

The eating speed of agent 2

Full size image

In Step \(q=1\) of Algorithm I, we have the current most preferred good list as \(\textbf{L}_1=(a,a,a,b) \).

From \(\textbf{L}_1\), \(\omega _1=\omega _3=\omega _4=1 \) and agent 2’s eating speed given in Fig. 3, the maximal \(\lambda \) computed by (17) is \( {\lambda _1 =4/5}\) since \(4/5+2/5+4/5=2 = \rho (\{a\}) \) by (18). In other words, at time 4/5, good a has been eaten away by agents as (current) most preferred one.

Then, \( x_1^1=x_1^3=(4/5,0,0,0), \, x_1^2=(2/5,0,0,0),\, x_1^4=(0,4/5,0,0), \, S_1=\{a\},\; x_1(S_1)=2 \) are obtained, good a is removed.

In Step \(q=2\), the most preferred good list is \(\textbf{L}_2=(b,c,c,b)\).

Now, the maximal \(\lambda \) of (17) is \( \, \lambda _2 =\lambda _1 + 1/5=1\) by \((1/5+3/5+1/5+1/5)+x_1({\{a\}})+x_1(\{b\}))=6/5+2+4/5=4= \rho (\{a,b,c\})=\rho (E).\) Since \(\rho (E)=4\) is the maximal value of function \(\rho \) on \(2^E\), it means that all resources are exhausted. Hence, Algorithm I stops at Step \(2^*\) for this example.

Thus, we obtained \( x^1_2 =(4/5, 1/5,0,0), \, x^2_2=(2/5,0,3/5,0), \, x^3_2=(4/5,0,1/5,0), x_2^4=(0,1,0,0), \,\, S_2=\{a,b,c,d\}=E, \; x_2(S_2)=4. \) The expected allocation is given by

Here, \(x_{q^*} \equiv x_{P_\omega }=(2,6/5,4/5,0) \in {\textrm{B}}(\rho )\). The \(x_{P_\omega } \)-tight sets are \(\emptyset , \{a\},\;\{a,b,c\}\),\( \{a,b,c,d\}\).

Example 2

Footnote 6 Suppose that \((N,E,{{{\mathscr {L}}}}, \textbf{1},(E,\rho ))\) is the same as that of Example 1, while all agents eat at uniform speed \(\omega _i=1, \forall i\in N\) during ESEA, i.e., we have EPS.

By EPS (Fujishige et al. 2018; Zhan 2023), first, agents 1, 2, and 3 eat good a, good b is eaten by agent 4. At time 2/3, good a is exhausted (\(\rho (\{a\})=3 \times 2/3=2\)) by the definition \(\rho \) of (18).

Next, agent 1 eats good b, agents 2 and 3 eat good c, and agent 4 continues eating good b. At time \(2/3 + 1/3\), total good resources 4 are exhausted.

Thus, we obtain the following assignment matrix:

which also has uniform 1 for each row sum, and the column sum vector is given by \(x_P=(2,4/3,2/3,0) \in {\textrm{B}}(\rho )\). The \(x_P\)-tight sets are the same as those of Example 2.

Two expected allocations obtained from Examples 1 and 2 are ordinally efficient, as will be verified in Theorem 2.

The outcome P of Example 2, EPS, is also envy-free, i.e., \(P^i \succeq _i^{\textrm{sd}} P^j\) for all \(i,j \in N\) Fujishige et al. (2018). Envy-freeness is a mostly used concept of fairness.

The following simple fact comes directly from Algorithm I. This property is also verified in a recent paper (Balbuzanov 2022) in a more general setting, polytopes defined by a system of linear inequalities with positive coefficients.

Lemma 1

During the exaction of Algorithm I, we have

$$\begin{aligned} \rho ( S_{q}) =x_p(S_q) = x_{P_\omega }(S_q) \equiv \sum _{e \in S_q} x_{P_{\omega }}(e), \quad \forall \ 0<q \le q^*. \end{aligned}$$
(19)

Proof

Let \(T_q = S_q \setminus S_{q-1}\). Since the goods eaten (or saturated) at each step do not affect later, we have

$$\begin{aligned} x_q^i(e) = x_s ^i(e) = x_{q^*}^i(e) \equiv P_\omega (i,e) \end{aligned}$$
(20)

for \(e \in T_q\), \(i \in N\) and \( q \le s \le q^* \). By summing all \(i\in N\) of (20) and \(S_q\) obtained in Algorithm I, we have (19). \(\square \)

We have the following theorem that characterizes the ESEA by ordinal efficiency, this is a generalization of Theorem 1 in Bogomolnaia and Moulin (2001).

Theorem 2

Let \(\textbf{RA}=(N,E, {{{\mathscr {L}}}},\textbf{1},(E,\rho ))\). The expected allocation \(P_{\omega }\) obtained from Algorithm I for any eating speeds \(\omega =(\omega _i \ | \ i \in N)\) is ordinally efficient. Conversely, if an expected allocation P is ordinally efficient, then there exist eating speeds \(\omega =(\omega _i \ | \ i \in N)\) such that \(P=P_{\omega }\), where \(P_\omega \) is an output of Algorithm I.

Proof

See the appendix. \(\square \)

As aforementioned in the introduction, ordinal efficiency is characterized by different eating speeds, and we try to employ a characterization that can cover the different speed.

4 Relation with the optimization of submodular functions

In this section, we present the main result, establishing the relation of ESEA and greedy algorithm (Edmonds 1970; Fujishige 2005) by ordinal efficiency.

We link the two algorithms by defining a summation optimization with a weight vector based on the acyclic characterization in Sect. 3. We will adapt the approach by Manea (2008) to extend McLennan (2002)’ result.

4.1 Welfare optimization

For a preference relation \(\succ _i\) (\(i \in N\)) over goods E, a utility function \(u_i: E \rightarrow {\mathbb {R}}\) is consistent with the preference if

$$\begin{aligned} u_i(e) \ge u_i(e') \;\; \Longleftrightarrow \;\; e \succ _i e', \quad \forall e,e' \in E. \end{aligned}$$
(21)

We call \(u_i\) satisfying (21) a von Neumann-Morgenstern (vNM) utility function.Footnote 7

Given an assignment \(P \in {\mathbb {R}}^{N \times E}_{\ge 0}\) and a vNM utility function \(u_i (i \in N) \), the expected utility of agent i is defined by

$$\begin{aligned} \textbf{E}_P(u_i) \equiv \sum _{e \in E } P(i,e)u_i(e). \end{aligned}$$
(22)

An assignment P is said to be ex ante Pareto optimal for vNM utilities \(u=(u_i \mid i\in N)\) if there does not exist an assignment Q such that \(\textbf{E}_Q(u_i) \ge \textbf{E}_P(u_i)\) for \(\forall i \in N\) and with strict inequality for at least one i.

Let \({{{\mathscr {A}}}}\) denote all assignments. We say that an assignment \(P\in {\mathbb {R}}^{N \times E}_{\ge 0}\) is ex ante welfare maximizer for utility \(u \in {\mathbb {R}}^N\) if

$$\begin{aligned} P = \arg \max _{Q \in {{{\mathscr {A}}}}} \sum _{i= 1}^n \textbf{E}_Q(u_i). \end{aligned}$$
(23)

We call a rank function \(\rho _m: 2^E \rightarrow {\mathbb {R}}_{\ge 0} \) modular if all submodular inequalities satisfying equalities \(\rho _m(X) + \rho _m(Y) =\rho _m(X \cup Y) + \rho _m(X\cap Y) \), \( \forall X,Y \in 2^E\). The base polyhedron \({\textrm{B}}(\rho _m)\) is a one point set in \({\mathbb {R}}^E_{\ge 0}\) (Sano and Zhan 2021).

The following theorem was shown in McLennan (2002). We present it here for completeness.

Theorem 3

(McLennan 2002) Fix \(\textbf{RA}=(N,E,{\mathscr {L}},\textbf{1},(E,\rho _m))\). The following conditions on the assignment \(P \in {\mathbb {R}}^{N \times E}_{\ge 0}\) are equivalent:

  1. (a)

    P is ordinally efficient;

  2. (b)

    There is a vector of utilities u consistent with \({\mathscr {L}}\) for which P is ex ante Pareto optimal;

  3. (c)

    There is a vector of utilities u consistent with \({\mathscr {L}}\) for which P is ex ante welfare maximizing.

Note that (c)\(\Rightarrow \)(b) and (b)\(\Rightarrow \)(a) follow from definitions. The essence of Theorem 3 is (a) \(\Rightarrow \) (c), i.e., ordinal efficiency is stronger than expected. From the results in Sect. 3, we will show the relation of (a) and (c) when the rank function \(\rho \) is generalized from \(\rho _m\).

4.2 Weight vector consistent with preferences

This subsection introduces a weight vector by adapting the approach given in Manea (2008). We begin by introducing a new binary operation to fit our problem setting.

Fix a \(\textbf{RA}=(N,E,{\mathscr {L}},\textbf{1},(\rho _m,E))\).

For an expected allocation P with preference profile \({\mathscr {L}}\), we defined a binary relation \(\rhd \hspace{-0.5mm} (P, {\mathscr {L}}) \) on E by (15) called a strict domination via probability trade. Here, a domination via submodular trade binary relation with \((E,\rho )\) is given by

$$\begin{aligned} {} e \bowtie _\rho \hspace{-1mm} (P, {\mathscr {L}})\ e' \Longleftrightarrow e \ntriangleright ( P, {\mathscr {L}}) \ e' \;\; {\textrm{and}} \;\; e \in {\textrm{dep}}(x_P,e') \setminus \{e'\}. \end{aligned}$$
(24)

Recall Fig. 1, \((e,e') \in A^*\) if \(e \in {\textrm{dep}}(x_P,e') {\setminus } \{e'\}\).

Now, we state Lemma 2 on the acyclicity with respect to \(\rhd \hspace{-0.5mm} (P, {\mathscr {L}}) \) and \(\bowtie _\rho \hspace{-1mm} (P, {\mathscr {L}}) \).

Lemma 2

Suppose that an expected allocation P is ordinally efficient. For each pair \((e,{{{\hat{e}}}})\) of goods with \(e \rhd \hspace{-0.5mm} (P, {\mathscr {L}}) {{{\hat{e}}}}\) or \(e \bowtie _\rho (P, {\mathscr {L}}) {{{\hat{e}}}}\), we have \({{{\hat{e}}}} \ntriangleright \hspace{-1mm} (P, {\mathscr {L}}) e \).

Proof

The case of \(e \rhd \hspace{-1mm} (P, {\mathscr {L}})\ {{{\hat{e}}}}\) follows from Proposition 2 and Corollary 1. We show \(e \bowtie _\rho (P,{\mathscr {L}}) {{{\hat{e}}}}\).

Conversely, suppose \({{{\hat{e}}}} \rhd (P, {\mathscr {L}}) e \). From the definition (15), there exists \(i \in N\) with \({{{\hat{e}}}} \succ _i e\) and \(P(i,e)>0\). Then, there exists a path from \({{{\hat{e}}}}^i\) to \(e^i\) in \(A^i\) and an arc \((e^i,e) \in A_0\) of H(P) in Fig. 1. The assumption \(e \bowtie _\rho (P,{\mathscr {L}}) {{{\hat{e}}}}\) means an arc \((e,{{{\hat{e}}}}) \in A^*\). Then, these arcs form a cycle in H(P), a contradiction to ordinal efficiency of P by Theorem 1. \(\square \)

Example 3

Let \(N=\{1,2,3\}\), \(E=\{a,b,c\}\), and \(\rho _m\) be a modular function with \(\rho _m(e)=1, \; \forall e \in E \) (i.e., \(\rho _m(E)=3\)). The preference profile is given as follows.

$$\begin{aligned} \begin{array}{cc} i\in N &{} {\textrm{preference}}\ \ L^i \\ 1 &{} a\succ _1 b\succ _1 c\\ 2 &{} a\succ _2 b \succ _2 c\\ 3 &{} c \succ _3 a\succ _3 b \end{array} \end{aligned}$$

Suppose that the expected allocation P is given by \(P(1,a)=1,P(2,b)=1,P(3,c)=1\) (all other entries are zero), we have \(\rhd (P, {\mathscr {L}})=\{ (a, b) \} \) since \(P(2,b) >0\) and \(a \succ _2 b\). (Note that \(\bowtie _\rho (P, {\mathscr {L}})=\emptyset \) for modular function since \(x_P(e) =\rho _m(e)\), \(\forall e \in E\).) It is clear that P is ordinally efficient since two of three agents got full allocation of the best preferred good and Lemma 2 is trivially satisfied (\(\bowtie _\rho \hspace{-1mm}(P, {\mathscr {L}}) \) is empty set).

Example 4

Consider \((N,E,{\mathscr {L}},\textbf{1},(E,\rho ) )\), the same as those given in Examples 1 and 2, and also the (non-zero entries of) expected allocation P by ESEA and EPS. Hence, P is ordinally efficient by Theorem 2, and we have \(\rhd (P, {\mathscr {L}})=\{(a, b), (a, c)\}\).

From the fact that \(S_2=\{a,b,c\} \subset E\) is the set with \(x_P(a) + x_P(b) + x_P(c) = \rho (S_2)=4\) and all proper subsets of \(S_2\) containing b, c are not tight, we have \((b,c),(c,b)\in \ \bowtie _\rho (P,{\mathscr {L}})\). Since good d is saturated finally, we have \(\{(a,d),(b,d),(c,d) \} \subset \; \bowtie _\rho (P,{\mathscr {L}})\). Thus, we obtained \(\bowtie _\rho (P,{\mathscr {L}})= \{(b,c),(c,b),(a,d),(b,d),(c,d) \}\).

Define the comprehensive domination via probability and submodular trades binary relation on E by a combination,

$$\begin{aligned} \unrhd _\rho (P,{\mathscr {L}}) \equiv \rhd (P,{\mathscr {L}}) \, \cup \, \bowtie _\rho \hspace{-1mm}(P,{\mathscr {L}}). \end{aligned}$$
(25)

We say that \(\unrhd _\rho (P,{\mathscr {L}}) \) is strictly acyclic if there exists no sequence of goods \(e_1,e_2, \dots ,e_k\) on E, such that \(e_1 \unrhd _\rho \hspace{-0.8mm}(P,{\mathscr {L}}) e_2\unrhd _\rho \hspace{-0.8mm}(P,{\mathscr {L}}) \cdots \unrhd _\rho \hspace{-0.8mm}(P,{\mathscr {L}}) e_k \rhd (P,{\mathscr {L}})e_1\).

The strict acyclicity of \(\unrhd _\rho (P,{\mathscr {L}}) \) in Example 3 is obvious. The \(\unrhd _\rho (P,{\mathscr {L}}) \) of Example 4 is strictly acyclic since P is ordinally efficient.

From the above discussions, Theorem 1 now can be rephrased as:

Proposition 3

An expected allocation P is ordinally efficient if and only if the relation \( \unrhd _\rho (P, {\mathscr {L}}) \) is strictly acyclic.

Let \(\rhd \) and \(\bowtie \) be two disjoint binary relations on E, and \(\unrhd \equiv \rhd \cup \bowtie \). A function \(w:\, E \rightarrow {\mathbb {R}}_{\ge 0}\) is a weak representation of \(\unrhd \) if

$$\begin{aligned}{} & {} e \rhd e' \Rightarrow w(e) \ge w(e') + 1 \qquad \;\forall e, e' \in E, \end{aligned}$$
(26)
$$\begin{aligned}{} & {} e \bowtie e' \Rightarrow w(e) \ge w(e') \qquad \qquad \forall e, e' \in E. \end{aligned}$$
(27)

The following proposition is a result from the choice theory.

Proposition 4

(Manea 2008) Any strict acyclic \( \unrhd \) on E admits a weak representation.

Associated with \(\unrhd \), an equivalence class can be obtained as follows. Defining a binary relation \(\triangleq \) on E,

$$\begin{aligned} e \triangleq e' \; \Longleftrightarrow \; \exists \ e_1 \unrhd e_2 \unrhd \cdots \unrhd e_k \unrhd e_1 \quad {\textrm{with}} \quad e, e' \in \{e_1, \cdots , e_k \}. \end{aligned}$$
(28)

The sequence \(e_1,e_2, \dots ,e_k\) may have repeated terms. Denote by \(E/\triangleq \), the set of equivalence classes of \(\triangleq \). Let [e] be the class containing e. Now define \(\triangleright > \) on \(E/\triangleq \) by

$$\begin{aligned}{}[e_1] \triangleright \hspace{-2mm} > [e_2] \; \Longleftrightarrow \; [e_1] \ne [e_2] \quad {\textrm{and}} \quad \exists e_1' \in [e_1], \;\; e_2' \in [e_2], \; \; e_1' \unrhd e_2'. \end{aligned}$$
(29)

Remark 1

The weak representation \(w \in {\mathbb {R}}^E\) can be interpreted as follows (Manea 2008). For \(e \in E\), we define w(e) as the length k of the longest chain of \(\triangleright \hspace{-2mm}>\) starting at [e], where \([e]=[e_1] \triangleright \hspace{-2mm}> [e_2] \triangleright \hspace{-2mm}> \cdots \triangleright \hspace{-2mm} > [e_k]\). Each w(e) is well defined if the relation \(\triangleright \hspace{-2mm}>\) is acyclic. The longest property of w implies \(w(e) \ge w(e')+1 \) for \([e] \triangleright \hspace{-2mm} > [e'] \).

Let \(\textbf{RA}=(N,E,{\mathscr {L}},\textbf{1},(E,\rho ))\). From the above discussions, a weak representation associated with an ordinally efficient assignment \(P \in {\mathbb {R}}^{N \times E}_{\ge 0}\) satisfies

$$\begin{aligned}{} & {} e \triangleright \hspace{-1mm}(P,{\mathscr {L}})e' \Longrightarrow w(e) \ge w(e') + 1 \quad \forall e,e' \in E, \end{aligned}$$
(30)
$$\begin{aligned}{} & {} e \ntriangleright \hspace{-1mm}(P,{\mathscr {L}})e', \ e \in {\textrm{dep}}(x_P,e') \Longrightarrow w(e) \ge w(e') \qquad \forall e,e' \in E. \end{aligned}$$
(31)

Example 5

Let the ordinally efficient allocation P be the same as that of Example 4 with \(E=\{a,b,c,d\}\). Observe that neither \(b \triangleright \hspace{-0.5mm}(P,{\mathscr {L}})c \) nor \( c \triangleright \hspace{-0.5mm}(P,{\mathscr {L}}) b\). Thus, the three equivalence classes of \( \triangleright \hspace{-2mm} > \) on E are \([a] \triangleright \hspace{-2mm}> [b]=[c]\triangleright \hspace{-2mm} >[d]\). Therefore, we have \(w(a)> w(b)=w(c) >w(d)\).

Remark 2

As presented in Example 5, let w be the weak representation associated with P, set

$$\begin{aligned} w(e) = w(e') \quad \forall e,e' \in T_q =S_q \setminus S_{q-1}, \; 1 \le q \le q^*, \end{aligned}$$
(32)

recall Algorithm 1, Examples 2 and 3 about the saturated set \(S_q\). For each pair \( e,e' \in T_q =S_q \setminus S_{q-1}\), we have \(e \ntriangleright \hspace{-1mm}(P,{\mathscr {L}})e' \). Otherwise, suppose that \(e \triangleright \hspace{-0.5mm}(P,{\mathscr {L}})e'\), i.e., there is \(i \in N\) such that \(e \succ _i e'\) (and \(P(i,e')>0\)), we obtain a contradiction by Algorithm I. If \(S_q\) is the minimal \(x_q\)-tight set containing both e and \(e'\), we can obtain (32) from (31).

The following Lemma 3 is a partial adoption of Theorem 1 given in Manea (2008).

Lemma 3

Suppose that \(P \in {\mathbb {R}}^{N\times R}_{\ge 0}\) is ordinally efficient and \(w \in {\mathbb {R}}^E\) is the associated weak representation. Then, there exist vNM utility functions \(u_i \in {\mathbb {R}}^E\) \((i \in N)\) satisfying

$$\begin{aligned}{} & {} u_i(e) = w(e) \qquad { if \,} P(i,e) >0, \end{aligned}$$
(33)
$$\begin{aligned}{} & {} u_i(e) \le w(e) \qquad \forall (i,e) \in N \times E. \end{aligned}$$
(34)

Proof

First, we show that the utility functions defined by (33) are consistent with the agents’ preferences. For each agent \(i \in N\), if \(e \succ _i e'\) with \(P(i,e)>0\) and \(P(i,e')>0\), then by the weak representation of w we have that \(u_i(e)=w(e)> w(e')=u_i(e')\).

Now, we extend \(u_i (i \in N)\) to all goods so that \(u_i\) is consistent with \( \succ _i\), and

$$ \begin{aligned}{} & {} u_i(e) < \; \min _{e' \in E} \;\; w(e') \quad {\textrm{if}} \, P(i,e)=0 \, {\textrm{and}} \, \{ e' | e \succ _i e' \& P(i,e') >0 \} = \emptyset , \end{aligned}$$
(35)
$$ \begin{aligned}{} & {} u_i(e) < \hspace{-1mm}\max _{\{e'| e \succ _i e' \& P(i,e')>0 \}} \hspace{-1mm} w(e')+1 \, {\textrm{if}} \, P(i,e)=0 \, {\textrm{and}} \, \{ e' | e \succ _i e' \& P(i,e') >0 \} \ne \emptyset . \nonumber \\ \end{aligned}$$
(36)

Such a representation satisfying (33)–(36) exists. From equations (33) and (35), we have (34) immediately. For (36), let \(e''\) be a maximizer of w on the right, then \(e \succ _i e''\), \(P(i,e'')>0\) (\(u_i(e'') =w(e'')\)). By the definition of the weak representation w, we have that \(u_i(e) < w(e'')+1 \le w(e).\) \(\square \)

For each expected allocation \(P' \in {\mathbb {R}}^{N\times R}_{\ge 0}\), Lemma 3 implies the following inequality

$$\begin{aligned} \sum _{i \in N} \sum _{e \in E} P'(i,e) u_i(e) \le \sum _{i \in N} \sum _{e \in E} P'(i,e) w(e)= \sum _{e \in E} w(e) x_{P'}(e). \end{aligned}$$
(37)

Equality in (37) is obtained if and only if \(P'(i,e)(w(e)-u_i(e))=0\) for all \(i \in N,e\in E\). When \(x_{P'}(e)=1\) (\(\forall e \in E\)) (i.e., the rank function is modular), then P satisfying (33) is the ex ante welfare maximizer for \(u=(u_i \mid i \in N)\) (Manea 2008).Footnote 8

In the next subsection, we show that expected allocation P in Lemma 3 is the maximizer of the following (38) for our general problem setting \(x_{P'}(e) \ne 1\) \((e \in E)\)

$$\begin{aligned} \max _{P' \in {{{\mathscr {A}}}}} \sum _{i \in N} \sum _{e \in E} P'(i,e) u_i(e) =\max _{P' \in {{{\mathscr {A}}}}}\sum _{e\in E} w(e) x_{P'}(e). \end{aligned}$$
(38)

4.3 Optimization and greedy algorithm

Fix a random assignment problem \(\textbf{RA}=(N,E,{{{\mathscr {L}}}},\textbf{1},(E,\rho ))\).

Suppose that an assignment \(P\in {\mathbb {R}}^{N\times R}_{\ge 0}\) is ordinally efficient with a weak representation \(w \in {\mathbb {R}}^E\). Then, by Lemma 3, the relations of (37) and (38) imply that P is the ex ante welfare maximizer for \(u \in {\mathbb {R}}^N\) if and only if P is the solution to the following problem

$$\begin{aligned} \max _{P' \in {{{\mathscr {A}}}}} \sum _{e \in E} w(e) x_{P'}(e). \end{aligned}$$
(39)

By the assumption \(n\ge \rho (E)\) (the resource is scarce), we have \(x_P \in {\textrm{B}}(\rho )\), and (39) can be rephrased as

$$\begin{aligned} \max _{x \in {\textrm{B}}(\rho )} \sum _{e \in E} w(e) x(e). \end{aligned}$$
(40)

The problem presented in (40) is well known in the theory of submodular functions that can be solved by the greedy algorithm (Edmonds 1970; Fujishige 2005). We also have a characterization as follows.

Theorem 4

(Theorem 3.16, Fujishige 2005) A base \(x \in {\textrm{B}}(\rho )\) is an optimal solution of (40) if and only if

$$\begin{aligned} w(e) \ge w(e') \qquad { for \; each} \; e \in {\textrm{dep}}(x,e'). \end{aligned}$$
(41)

First, we consider the solution of (39) associated with an ordinally efficient assignment.

Proposition 5

Suppose that an assignment \(P\in {\mathbb {R}}^{N \times E}_{\ge 0}\) is ordinally efficient, and \(w\in {\mathbb {R}}^E{}\) is a weak representation associated with P, then \(x_P\) is the optimal solution of problem (39).

Proof

Weak representation (30), or \( \bowtie _\rho (P, {\mathscr {L}})\), is exactly the relation (41) of Theorem 4.

We prove that weak representation (31), or \(\triangleright \hspace{-0mm}(P,{\mathscr {L}})\) also satisfies (41).

By the definition of \(e \triangleright (P,{\mathscr {L}}) e'\), there is \(i\in N\) with \(e \succ _i e'\) and \(P(i,e')>0\). Conversely, suppose \(e \notin {\textrm{dep}}(x_P,e')\). There are two cases.

Case-1: If \(e \succ _j e'\) for all \(j \in N\), from ordinal efficiency of P and Theorem 2, good e is eaten away before \(e'\) in Algorithm I. In other words, for each \( e' \in X \subseteq E\) with \(\rho (X)=x_P(X)\), we have that \(e \in X\), which contradicts \(e \notin {\textrm{dep}}(x_P,e')\) since \({\textrm{dep}}(x_P,e') \equiv \bigcap \{ Y \mid e' \in Y \subseteq E, \ \rho (Y)=x_P(Y)\} \).

Case-2: If there exists \(j \in N\) with \(e' \succ _j e\), then we have \(P(j,e)=0\) by Corollary 1. Hence, agent j can not eat good e after good \(e'\) has been eaten up. This implies that good e is contained in a saturated set containing \(e'\), a contradiction to \(e \notin {\textrm{dep}}(x_P,e')\) again. \(\square \)

From Lemma 3 and Proposition 5, we extended McLennan’s result as follows.

Theorem 5

Let \(\textbf{RA}=(N,E,{{{\mathscr {L}}}},\textbf{1},(E,\rho ))\). The following (a)\(\sim \)(c) on the expected allocation P are equivalent:

  1. (a)

    P is ordinally efficient,

  2. (b)

    there is a vector of utilities u consistent with \({{{\mathscr {L}}}}\) for which P is ex ante Pareto optimal,

  3. (c)

    there is a vector of utilities u consistent with \({{{\mathscr {L}}}}\) for which P is ex ante welfare maximizer.

Next, we link ESEA and the greedy algorithm. For simplicity, we consider the column with \(x_P(e)>0\).Footnote 9 We have the following lemma.

Lemma 4

Let \(w \in {\mathbb {R}}^E\) be a weak representation associated with an ordinally efficient allocation P from Algorithm I. We have for \( q=1, \cdots , q^*-1\),

$$\begin{aligned} w(e) > w(e') \qquad \qquad \forall e\in T_q, e' \in T_{q+1}, \end{aligned}$$
(42)

where \(T_q = S_q {\setminus } S_{q-1}\) for all \(q=1,\dots , q^*\).

Proof

Since \(x_P(e)>0\) for all \(e \in E\), there exists \(i \in N \) such that \(P(i,e)>0\). If for some \(i \in N\) with \(P(i,e')>0\) and \(e \succ _i e'\), where \(e \in T_q\) and \(e' \in T_{q+1}\) as (42), then we have \(w(e)>w(e')\) by the definition of the weak representation, and we are done. Suppose that for each \(i \in N\) with \(P(i,e')>0\) and \(e' \succ _i e\). From ordinal efficiency of P and Theorem 2, good e cannot be eaten before good \(e'\) during Algorithm I, contradicting \(e \in T_q\) and \(e' \in T_{q+1}\). \(\square \)

Note that Algorithm I, ESEA, works for a special case, uniform eating speed, i.e., the PS mechanism. Hence, goods eaten away earlier are more preferred by agents. Relation (42) of Lemma 4 implies that the weak representative w associated with the outcome P of ESEA represents aggregated preferences of agents. Since the greedy algorithm for (39) or (40) selects goods with larger weights (i.e., the aggregation of agents’ preference) of goods earlier, it coincides with ESEA.

Theorem 6

Consider an assignment problem \(\textbf{RA} =(N,E,{{{\mathscr {L}}}},\textbf{1},(E,\rho ))\), we have:

The outcome \(x_P\) of ESEA equals the optimal solution by the greedy algorithm for a weak representative vector \(w \in {\mathbb {R}}^E\) associated with the allocation P.

Moreover, the order of goods eaten away by ESEA coincides with the order selected by the greedy algorithm.

Proof

The first part comes from Propositions 5. The second part follows from Lemma 4, Remark 2, and the discussions above. \(\square \)

The following theorem is well-known.

Theorem 7

(Theorem 3.18, Fujishige 2005) The greedy algorithm works for (40) if and only if f is a submodular function on \(2^E\).Footnote 10

By Theorem 7, we can see that the submodularity, or equivalently the diminishing return property of \(\rho \) mentioned in the introduction, underlying the random assignment problem \(\textbf{RA}=(N,E,{{{\mathscr {L}}}},\textbf{1},(E,\rho )) \) of this study, is critical.

5 Concluding remarks

This study is an extension of our former work, extending the assignment mechanism proposed by Bogomolnaia and Moulin (2001) from a fixed quota of goods to variables varying in a base polyhedron in allocating goods to agents.

The aim is to build a relation between the ESEA assignment mechanism and greedy algorithm. The central concept underlying this study is a Pareto type optimality called ordinal efficiency possessed by the outcome of ESEA.

Our contributions are as follows.

  1. 1.

    The ESEA mechanism was formulated in Algorithm I. Ordinal efficiency was characterized using Algorithm I with different eating speeds (Theorem 2), which implies a possibility in employing a characterization to cover the different speed.

  2. 2.

    We presented the relation between the two algorithms, ESEA and the greedy algorithm, both in the solution (Theorem 5) and order of the execution (Theorem 6).

    These are possible since the Pareto optimality of ordinal efficiency is stronger than expected, which is a summation optimization with the weak representative weights consisting to agents’ preferences.

The challenging part in the proof is the column sum of the allocation matrices (the quota of each good) varying from one. The results also reveal that the submodularity, or diminishing return, plays a critical role (Theorem 7) in our problem setting.

Recently, the fact that a more general structure than polymatroid called greedoids works for greedy algorithms was revisited (Szeszlér 2021). Greedoids imply some orders, e.g, words, on a subset, similar to preferences. We hope that our study provides a view on these.