On the unification of centralized and decentralized clearing mechanisms in financial networks

Abstract

We analyze clearing mechanisms in financial networks in which agents may have both monetary individual assets and mutual liabilities. A clearing mechanism prescribes mutual payments between agents in a financial network to settle their mutual liabilities. The corresponding payments, summarized in a payment matrix, are made in accordance with agent-specific claims rules that stem from the vast literature on claims situations. The bottom payment matrix is the payment matrix that contains the minimal amount of payments required to clear the network. We explicitly characterize the bottom payment matrix as the result of a recursive centralized mechanism. We subsequently show that several types of decentralized clearing mechanisms lead to the bottom payment matrix as well. In fact, we show that this unification of centralized and decentralized clearing mechanisms relies on the composition property of the underlying agent-specific claims rules that dictate the payment mechanism.

1 Introduction

Unlike classical bankruptcy situations in which claimants claim from a single bankrupt entity, bankruptcy situations in a network setting are more intricate because of the interdependence between agents. Exogenous or endogenous shocks can disrupt a financial system and can cause agents to default which may trigger default of other agents or systems. This phenomenon is known as financial contagion, see, for example, Glasserman and Young (2016), Jackson and Pernoud (2021), and Elliott and Golub (2022). Although the network perspective became increasingly important as a result of the global financial and banking crises between mid 2007 and early 2009, the recent COVID-19 pandemic stresses the importance of studying bankruptcy situations in a network setting once more.

Our model of a financial network follows the seminal paper of Eisenberg and Noe (2001). A financial network comprises a finite set of agents and is characterized by a non-negative estates vector containing each agent’s monetary estate and a non-negative claims matrix containing mutual rightful claims between pairs of agents. The unilateral setting is characterized by a single agent’s estate and a vector of rightful claims on the estate. A bankruptcy situation arises if this agent is insolvent and we need to allocate the available estate among the claimants. After the early work of O’Neill (1982) on bankruptcy situations, many so-called claims rules have been introduced, see Thomson (2019) for an overview. A claims rule provides, for each bankruptcy situation, an allocation of the estate among the claimants. In essence, claims rules are clearing mechanisms in the unilateral setting, and thus naturally form the basis for clearing mechanisms in the network setting. We follow this convention, and, as in Eisenberg and Noe (2001), let claims rules dictate the payments between agents in a financial network. However, we emphasize that clearing mechanisms without underlying claims rules can be analyzed as well, see, for example, Ketelaars (2020).

In this paper, we study clearing mechanisms in financial networks. A clearing mechanism prescribes mutual payments between agents in a financial network to settle their mutual liabilities. The corresponding payments, summarized in a payment matrix, are made in accordance with claims rules. Most of the literature on financial contagion follows Eisenberg and Noe (2001) and uses the proportional rule as the underlying payment mechanism. This is in line with current insolvency proceedings in which payments are in accordance with the pari passu principle—Latin for “equally and without preference.” However, the proportional rule does not fully encompass insolvency proceedings in, for instance, the United States and the European Union. In practice, as debts may be of different seniority, claimants in insolvency proceedings are partitioned into classes of different priority and within each class there is pari passu treatment.¹ Moulin (2000) provides an axiomatic characterization of such priority claims rules in bankruptcy situations, and Elsinger (2009) considers priority classes in clearing in a network setting. As we want to take such priority claims rules and the individuality of agents into account, we allow for general agent-specific claims rules like Csóka and Herings (2018). Nevertheless, contrary to Csóka and Herings (2018) but in line with all research on financial contagion, we consider financial networks in the perfectly divisible setup.

In centralized clearing mechanisms, an independent authority is in charge of the clearing process. In accordance with most literature on financial contagion, we characterize a payment matrix of a centralized clearing mechanism by a fixed point of an appropriate mapping. We call payment matrices corresponding to centralized clearing mechanisms $\phi$-transfer schemes, in which $\phi$ is a vector of claims rules. A $\phi$-transfer scheme satisfies three criteria. First, payments are made in accordance with claims rules, for instance those dictated by insolvency law. Second, the total payment by an agent cannot exceed the amount it has at its disposal, a limited liability requirement. Third, each agent either pays off all its debts, or everything it has at its disposal is paid to its claimants, an absolute priority of debt over equity requirement. The set of $\phi$-transfer schemes is a complete lattice so that there exists a bottom $\phi$-transfer scheme and a top $\phi$-transfer scheme, see, for example, Csóka and Herings (2018) and Csóka and Herings (2024). Despite the existence of (infinitely) many $\phi$-transfer schemes, Groote Schaarsberg et al. (2018) shows that any two $\phi$-transfer schemes lead to the same transfer allocation. A transfer allocation is a redistribution of the estates vector according to a payment matrix. The literature on contagion is concerned with computing the top $\phi$-transfer scheme. For example, Eisenberg and Noe (2001) and Rogers and Veraart (2013) propose efficient algorithms to compute the top $\phi$-transfer scheme, in which $\phi$ comprises proportional rules. Chen et al. (2016) proposes a different efficient algorithm that also clears a network by using maximal payments which are consistent with the same proportionality requirement.² More recently, Csóka and Herings (2022) presents a centralized clearing mechanism by formulating a programming problem that determines the top $\phi$-transfer scheme.

On the other hand, the bottom $\phi$-transfer scheme has received little attention in the existing literature. Interestingly, we show that the bottom $\phi$-transfer scheme is the limit of a recursive procedure for a centralized clearing mechanism that generates a monotonically increasing sequence of payment matrices. Moreover, we show that the bottom $\phi$-transfer scheme can also be obtained as the result of several types of decentralized clearing mechanisms under the condition that the underlying claims rules satisfy composition.³ Composition (Young 1988) is a property that pertains to bankruptcy situations in which an agent allocates a provisional estate value but later learns that the true value is larger than expected. If a claims rule satisfies composition, then, for any bankruptcy situation, allocating the surplus value and adding this to the initial allocation is equivalent to simply reallocating the actual (larger) value. Thus, our result is one of unification: the centralized clearing mechanism based on $\phi$-transfer schemes and several types of decentralized clearing mechanisms generate the bottom $\phi$-transfer scheme as an outcome.

In decentralized clearing mechanisms, agents as individuals are in charge of the clearing process. One advantage of such mechanisms is that it relies on local information only. As far as we know, there exist two types of decentralized clearing mechanisms in the literature that adhere to the aforementioned criteria.⁴

First, the $\phi$-based individual settlement allocation procedure (ISAP) put forward in Ketelaars et al. (2020). In each step of this recursive procedure, each agent pays the other agents by allocating its current estate on the basis of its own claims rule. As a result of these payments, there is a redistribution of the individual estates and a reduction in the mutual liabilities. Such a procedure need not be finite, but has a corresponding limiting payment matrix nonetheless. We show that the payment matrix that is the result of $\phi$-based ISAP is the bottom $\phi$-transfer scheme if the underlying claims rules satisfy composition.

Second, the $\phi$-based decentralized clearing processes put forward in Csóka and Herings (2018), but in a discrete setup. We first generalize this process for a perfectly divisible setup. In each step of this recursive procedure, exactly one agent is selected that essentially communicates to the other agents what it would pay according to its own claims rule based on what it currently has at its disposal. The amount each agent has at its disposal in a step is its initial estate plus provisional payments from the other agents. The selection process need not be deterministic. In general, the selection process can be history dependent, stochastic or both. Csóka and Herings (2018) shows that in the discrete setup any $\phi$-based decentralized clearing process terminates in a finite number of iterations and converges to the bottom $\phi$-transfer scheme. In the perfectly divisible setup, a $\phi$-based decentralized clearing process need not be a finite process but we show that it still converges to the bottom $\phi$-transfer scheme, irrespective of the selection process. The actual payments that take place in a $\phi$-based decentralized clearing process are thus those with respect to the bottom $\phi$-transfer scheme.

In addition to the two above types of decentralized clearing mechanisms, we introduce one that is a variation on $\phi$-based ISAP, a procedure in which agents pay simultaneously in each step. Now, exactly one agent is selected in each step that pays the other agents by allocating its current estate on the basis of its own claims rule, which reduces the claims of the other agents on this agent. Notice that such a mechanism differs from a $\phi$-based decentralized clearing process because actual transfers between agents, that is, those that reduce mutual liabilities, take place in each step. We show that also this variation on $\phi$-based ISAP prescribes the bottom $\phi$-transfer scheme if the underlying claims rules satisfy composition, irrespective of the selection process.

This paper is organized as follows. In Sect. 2, we recap existing claims rules and mutual claims rules. Section 3 contains our main result on the unification of the centralized clearing mechanism based on $\phi$-transfer schemes, the decentralized clearing mechanism based on $\phi$-based ISAP, and their corresponding $\phi$-based decentralized sequential clearing mechanisms. Section 4 concludes and provides a discussion of our results in regard to their application and economic relevance.

2 Preliminaries

2.1 Claims rules

A claims situation is a pair $(e,c) \in \mathbb {R}_{+} \times \mathbb {R}^N_{+}$ in which N is a finite set of claimants, e is an estate, and $c = (c_i)_{i \in N}$ is a vector of rightful claims on the estate. The class of all claims situations on N is denoted by $\mathcal {C}^N$. A subclass of claims situations on N in which the sum of claims exceeds the value of the estate, that is, $\sum _{i \in N} c_i > e$, is the class of bankruptcy situations and is denoted by $\mathcal {B}^N$.

A claims rule $\varphi :\mathcal {C}^N \rightarrow \mathbb {R}^N$ prescribes how the estate in each claims situation will be allocated among the claimants. For all $(e,c) \in \mathcal {C}^N$, the allocation vector $\varphi (e,c)$ satisfies

• (i) $0 \le \varphi _i(e,c) \le c_i$ for all $i \in N$,

• (ii) $\displaystyle \sum _{i \in N} \varphi _i(e,c) = \min \{e,\sum _{i \in N} c_i \}$.

The second condition (ii) boils down to $\sum _{i \in N} \varphi _i(e,c) = e$ in bankruptcy situations. On the other hand, if the estate can cover all claims, then conditions (i) and (ii) imply $\varphi (e,c) = c$.

In this paper, we assume that a claims rule satisfies estate monotonicity, which requires that no claimant should receive less than what it did receive initially when it turns out there is more to be allocated. All of the three claims rules which will be defined later on satisfy estate monotonicity.

Definition 2.1

A claims rule $\varphi$ on $\mathcal {C}^N$ satisfies estate monotonicity if, for all $(e,c) \in \mathcal {C}^N$ and $(\tilde{e},c) \in \mathcal {C}^N$ with $e \le \tilde{e}$, it holds that $\varphi (e,c) \le \varphi (\tilde{e},c)$.⁵

Estate monotonicity implies estate continuity (see Proposition A.1 in the appendix).

Definition 2.2

A claims rule $\varphi$ on $\mathcal {C}^N$ satisfies estate continuity if, for all $(e,c) \in \mathcal {C}^N$ and for any sequence of non-negative estates $(e^k)_{k=1}^{\infty }$ that converges to e, the sequence $(\varphi (e^k,c))_{k=1}^{\infty }$ converges to $\varphi (e,c)$.

In particular, by assuming that the claims rules in this paper satisfy estate monotonicity, we implicitly assume that they also satisfy estate continuity.

Composition (Young 1988) is a property of a claims rule that pertains to situations in which an agent allocates a provisional estate value but later learns that the true value is larger than expected. If a claims rule satisfies composition, then, for any claims situation, allocating the surplus value according to the claims rule and modified claims and adding this to the initial allocation is equivalent to simply reallocating the actual (larger) value.

Definition 2.3

A claims rule $\varphi$ on $\mathcal {C}^N$ satisfies composition if, for all $(e,c) \in \mathcal {C}^N$ and $(\tilde{e},c) \in \mathcal {C}^N$ with $e \le \tilde{e}$, it holds that

$$\begin{aligned} \varphi (\tilde{e},c) = \varphi (e,c) + \varphi (\tilde{e}-e,c-\varphi (e,c)). \end{aligned}$$

Composition implies estate monotonicity,⁶

The following three claims rules, which will be used in examples later on, are well known in the literature on claims situations. For an extensive survey on claims rules, see Thomson (2019). The constrained equal awards rule CEA interprets equality in terms of gains. It allocates the estate among the claimants as equal as possible provided that no claimant receives more than it claims. For all $(e,c) \in \mathcal {B}^N$, it is defined by

$$\begin{aligned} \textrm{CEA}_i(e,c) = \min \{c_i, \alpha \} \end{aligned}$$

for all $i \in N$, with $\alpha \ge 0$ such that $\sum _{j \in N} \min \{c_j, \alpha \} = e$. The constrained equal awards rule satisfies composition.

The natural dual of the constrained equal awards rule is the constrained equal losses rule CEL in the sense that it interprets equality in terms of losses with respect to the claims. The losses are incurred as equal as possible by the claimants provided that no claimant’s loss exceeds its claim. For all $(e,c) \in \mathcal {B}^N$, it is defined by

$$\begin{aligned} \textrm{CEL}_i(e,c) = \max \{0, c_i - \beta \} \end{aligned}$$

for all $i \in N$, with $\beta \ge 0$ such that $\sum _{j \in N} \max \{0,c_j -\beta \} = e$. The constrained equal losses rule satisfies composition.

We complement these two rules with a third rule, namely the Talmud rule TAL (Aumann and Maschler 1985). If the claims are relatively low, the Talmud rule follows a dual procedure; if the claims are relatively high, the Talmud rule destroys half the claims and prescribes an allocation using the constrained equal awards rule. For all $(e,c) \in \mathcal {B}^N$, it is defined in terms of CEA as follows:

$$\begin{aligned} \textrm{TAL}(e,c) = {\left\{ \begin{array}{ll} \begin{aligned} &{} c - \textrm{CEA}\left( \sum _{j \in N} c_j - e, \frac{1}{2}c\right) &{} \quad &{} \text { if } \sum _{j \in N} c_j \le 2e, \\ &{} \textrm{CEA}\left( e,\frac{1}{2}c\right) &{} \quad &{} \text { if } \sum _{j \in N} c_j > 2e. \end{aligned} \end{array}\right. } \end{aligned}$$

The Talmud rule does not satisfy composition.

2.2 Mutual claims rules

Financial networks generalize claims situations by allowing for multiple estates and mutual claims. A financial network is a pair $(E,C) \in \mathbb {R}_{+}^N \times \mathbb {R}_{+}^{N \times N}$ in which N is a finite set of agents, $E = (e_i)_{i \in N}$ is an estates vector, and $C = (c_{ij})_{i,j \in N}$ is a claims matrix. Each coordinate $e_i$ of E represents the non-negative estate belonging to agent $i \in N$. The claims matrix C represents mutual liabilities between agents. Each cell $c_{ij}$ of C represents the rightful claim of agent $j \in N$ on agent $i \in N$. Row i in C thus captures creditors of agent i, whereas column i of C captures debtors of agent i. By assumption, agents have no claim on themselves, so, for all $i \in N$, $c_{ii} = 0$. No additional conditions are imposed on the claims matrix, in particular, there is no condition on the relation between claims $c_{ij}$ and $c_{ji}$ for $i \ne j$. We denote the i-th row of C by $\overline{c}_i = (c_{ij})_{j \in N}$.

We will first show how a claims situation can be interpreted as a financial network. To this end, we introduce an artificial agent, agent zero, with non-negative estate. If $(e,c) \in \mathcal {C}^N$ is a claims situation with $N=\{1,2,\dots ,n\}$, then (e, c) can be associated with a financial network $(E,C) \in \mathcal {F}^{\{0\} \cup N}$, which is given by

$$\begin{aligned} E = \begin{pmatrix}e \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \text { and } C = \begin{bmatrix} 0 &{}\quad c_1 &{}\quad \dots &{}\quad c_n \\ 0 &{}\quad 0 &{}\quad \dots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \dots &{}\quad 0 \end{bmatrix}. \end{aligned}$$

Financial networks can be used to study bankruptcy situations in a network setting. If the total of the estates is insufficient to completely settle all mutual claims, then the problem arises of how the total of the estates should be allocated among the agents. The analysis, however, is more complex than bankruptcy situations in a unilateral setting because the bilateral payments on which the allocation is based depend not only on the individual estates. The extent to which an agent can pay its creditors depends on its individual estate as well as the payments it receives from the other agents. The focus of this paper is on the minimal amount of payments required to prescribe an allocation.

The class of all financial networks on N is denoted by $\mathcal {F}^N$. A mutual claims rule $\mu :\mathcal {F}^N \rightarrow \mathbb {R}^N$ prescribes how the total of the estates in a financial network will be allocated among its agents. For all $(E,C) \in \mathcal {F}^N$, the allocation vector $\mu (E,C)$ satisfies

• (a) $\mu _i(E,C) \ge 0$ for all $i \in N$,

• (b) $\displaystyle \sum _{i \in N} \mu _i(E,C) = \displaystyle \sum _{i \in N} e_i$.

Condition (a) is a non-negativity condition and condition (b) implies that a reallocation of the estates neither generates nor destroys value. The case for which the clearing procedure is not frictionless, that is, $\sum _{i \in N} \mu _i(E,C) < \sum _{i \in N} e_i$, is beyond the scope of this work, because in that case we lose the direct connection with claims rules. Clearing procedures that take bankruptcy costs into account are analyzed in, for example, Rogers and Veraart (2013).

All mutual claims rules analyzed in this paper rely on an underlying payment matrix. A payment matrix is a non-negative matrix $P = (p_{ij})_{i,j \in N}$ in which cell $p_{ij}$ indicates the payment of agent $i \in N$ to agent $j \in N$. A (bilateral) transfer scheme is a specific type of payment matrix that contains feasible and reasonable (bilateral) payments.

Definition 2.4

Let $(E,C) \in \mathcal {F}^N$. The payment matrix $P = (p_{ij})_{i,j \in N}$ is called a transfer scheme for (E, C) if,

• (I) for all $i,j \in N$, $0 \le p_{ij} \le c_{ij}$,

• (II) for all $i \in N$, $\displaystyle \sum _{m \in N} p_{im} \le e_i + \sum _{m \in N}p_{mi}$.

The set of all possible transfer schemes for (E, C) is denoted by $\mathcal {P}(E,C)$.

Condition (I) bounds the payment of agent $i \in N$ to agent $j \in N$ by zero and the claim of agent j on agent i; condition (II) is a limited liability assumption in the sense that the total payment by agent $i \in N$ cannot exceed the amount agent i has at its disposal.

A transfer scheme can directly be translated into a transfer allocation, in which the allocation to each agent equals its initial estate plus its net payments.

Definition 2.5

Let $(E,C) \in \mathcal {F}^N$ and let $P \in \mathcal {P}(E,C)$. The vector $\alpha ^P = (\alpha _i^P)_{i \in N}$ is called a transfer allocation if, for all $i \in N$,

$$\begin{aligned} \alpha ^P_i = e_i + \sum _{j \in N}p_{ji} - \sum _{j \in N} p_{ij}. \end{aligned}$$

(1)

Note that condition (II) of a transfer scheme implies that the transfer allocation is non-negative.

In the definition of a mutual claims rule, we did not specify how the reallocation of the total of the estates, in the form of transfers (e.g., payments) between agents, should take place. One way to pin down the clearing mechanism is to impose that transfers between agents constitute a $\phi$-transfer scheme which then gives rise to a transfer allocation.

Let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules, in which $\varphi ^i$ is the claims rule associated with agent $i \in N$. Correspondingly, a payment matrix is called a $\phi$-transfer scheme if $\phi$ forms the basis for a payment matrix with the additional requirement that payments satisfy a consistency condition.

Definition 2.6

Let $(E,C) \in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules. The payment matrix $P = (p_{ij})_{i,j \in N}$ is called a $\phi$-$transfer scheme$ for (E, C) if, for all $i,j \in N$,

$$\begin{aligned} p_{ij} = \varphi ^i_j\left( e_i + \sum _{m \in N} p_{mi}, \overline{c}_i\right) . \end{aligned}$$

(2)

The set of all possible $\phi$-transfer schemes for (E, C) is denoted by $\mathcal {P}^{\phi }(E,C)$.

The payments in a $\phi$-based transfer scheme are consistent in the sense that, for each agent, the payments to the other agents follow from allocating its estate plus its incoming payments in accordance with its claims rule.

The set of $\phi$-transfer schemes for a financial network is a subset of the set of transfer schemes for that same network.

Proposition 2.1

Let $(E,C) \in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules. Then, $\mathcal {P}^{\phi }(E,C) \subseteq \mathcal {P}(E,C)$.

Proof

Let $P \in \mathcal {P}^{\phi }(E,C)$, and let $i \in N$. Then, for all $j \in N$, condition (i) of the claims rule $\varphi ^i$ implies that

$$\begin{aligned} 0 \le p_{ij} = \varphi ^i_j\left( e_i + \sum _{m \in N} p_{mi},\overline{c}_i\right) \le c_{ij}; \end{aligned}$$

furthermore, condition (ii) of the claims rule $\varphi ^i$ implies that

$$\begin{aligned} \sum _{j \in N} p_{ij}&= \sum _{j \in N} \varphi ^i_j\left( e_i + \sum _{m \in N} p_{mi},\overline{c}_i\right) = \min \left\{ e_i + \sum _{m \in N} p_{mi}, \sum _{j \in N} c_{ij}\right\} \\&\le e_i + \sum _{m \in N} p_{mi}. \end{aligned}$$

$\square$

One of the main results of Groote Schaarsberg et al. (2018) is that any two $\phi$-transfer schemes lead to the same transfer allocation. However, in Groote Schaarsberg et al. (2018) agents share a common claims rule, that is, $\phi = (\varphi ^i)_{i \in N}$ with $\varphi ^i = \varphi$ for all $i \in N$, though it is readily verified that their result carries over to the situation with agent-specific claims rules.

Theorem 2.1

(cf. Groote Schaarsberg et al. 2018) Let $(E,C) \in \mathcal {F}^N$, let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules, and let $P, P' \in \mathcal {P}^{\phi }(E,C)$. Then, $\alpha ^{P} = \alpha ^{P'}$.

The above theorem implies that the resulting allocation vector depends only on $\phi$ and not on the underlying $\phi$-transfer scheme.

Definition 2.7

A mutual claims rule $\rho ^{\phi }$ on $\mathcal {F}^N$ is called a $\phi$-$based allocation rule$ if, for all $(E,C) \in \mathcal {F}^N$,

$$\begin{aligned} \rho ^{\phi }(E,C) = \alpha ^P, \end{aligned}$$

in which P is a $\phi$-transfer scheme for (E, C).⁷

The previous clearing mechanism based on $\phi$-transfer schemes is centralized. A clearing mechanism that is decentralized is the $\phi$-based individual settlement allocation procedure (ISAP). In such a procedure, agents settle their claims individually on the basis of $\phi$ which eventually leads to a unique redistribution of the estates. Here, the definition of ISAP differs slightly from Ketelaars et al. (2020) as we allow for agent-specific claims rules.

Definition 2.8

Let $(E,C)\in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules. The individual settlement allocation procedure (ISAP) generates a sequence of estates vectors $(E^k)_{k \in \mathbb {N}}$, a sequence of claims matrices $(C^k)_{k \in \mathbb {N}}$ and a sequence of payment matrices $(\Phi ^k)_{k \in \mathbb {N}}$ with $\Phi ^k = (\Phi ^k_{ij})_{i,j \in N}$ in the following way.

• 1. Initially, set $E^1 = (e^1_i)_{i \in N} = E$ and $C^1 = (c^1_{ij})_{i,j \in N} = C$.

Then, recursively for $k=2,3,\dots$

• 2. For each agent $i\in N$ the payment to agent $j \in N$ in step $k-1$ is equal to

  $$\begin{aligned} \Phi ^{k-1}_{ij} = \varphi ^i_j\left( e^{k-1}_i,\overline{c}^{k-1}_i\right) , \end{aligned}$$

  in which $\overline{c}^{k-1}_i \in \mathbb {R}^N_{+}$ is the i-th row of claims matrix $C^{k-1}$.

• 3. Subsequently, update the estates vector to $E^k = (e^k_i)_{i\in N}$ with

  $$\begin{aligned} e_i^k = e_i^{k-1} + \sum _{m \in N} \Phi ^{k-1}_{mi} - \sum _{m \in N} \Phi ^{k-1}_{im}. \end{aligned}$$

• 4. Correspondingly, the claims matrix is updated to $C^k = (c^k_{ij})_{i,j\in N}$ with

  $$\begin{aligned} c^k_{ij} = c^{k-1}_{ij} - \Phi ^{k-1}_{ij}. \end{aligned}$$

ISAP is a finite procedure if and only if there exists a step $k \in \mathbb {N}$ such that either $e_i^k = 0$ or $\overline{c}^k_i =0$ for all $i \in N$. Even if ISAP takes an infinite number of steps, the limit of the sequence of estates vectors it generates exists.⁸

Theorem 2.2

(cf. Ketelaars et al. 2020) Let $(E,C) \in \mathcal {F}^N$ be a financial network and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules. Then, the limit of the sequence $(E^k)_{k \in \mathbb {N}}$ generated by the individual settlement allocation procedure exists.

A $\phi$-based ISAP allocation rule is then defined to be equal to the limit of the sequence of estates vectors.⁹

Definition 2.9

A mutual claims rule $r^{\phi }$ on $\mathcal {F}^N$ is called a $\phi$-based ISAP allocation rule if, for all $(E,C) \in \mathcal {F}^N$,

$$\begin{aligned} r^{\phi }(E,C) = \lim _{k \rightarrow \infty } E^k, \end{aligned}$$

in which $(E^k)_{k \in \mathbb {N}}$ is the sequence generated by ISAP for $(E,C) \in \mathcal {F}^N$ with respect to a vector of claims rules $\phi = (\varphi ^i)_{i \in N}$.

Alternatively, the allocation vector of a $\phi$-based ISAP allocation rule $r^{\phi }$ can be characterized as a specific transfer allocation. Note that the payment matrix that is the result of ISAP is a transfer scheme, however, as is also shown in Ketelaars et al. (2020), it need not necessarily be a $\phi$-transfer scheme.

Proposition 2.2

(cf. Ketelaars et al. 2020) Let $(E,C) \in \mathcal {F}^N$, let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules, and let the payment matrix $P=(p_{ij})_{i,j \in N}$ be given by

$$\begin{aligned} p_{ij} = \sum _{k = 1}^{\infty } \Phi ^k_{ij} \end{aligned}$$

for all $i,j \in N$, in which $(\Phi ^k)_{k \in \mathbb {N}}$ with $\Phi ^k = (\Phi ^k_{ij})_{i,j \in N}$ is the sequence of payment matrices generated by ISAP for (E, C) with respect to $\phi$. Then, $r^{\phi }(E,C) = \alpha ^P$.

3 On the unification of clearing mechanisms

In this section, we show that several centralized and decentralized clearing mechanisms can be unified by the payment matrix that contains the minimal amount of payments required to clear the network, the so-called bottom $\phi$-transfer scheme.

In the previous section, we did not discuss the existence of $\phi$-transfer schemes. For our specific purposes, we show existence of a specific $\phi$-transfer scheme by means of the following recursive procedure.¹⁰

Let $(E,C) \in \mathcal {F}^N$. A $\phi$-transfer scheme $P=(p_{ij})_{i,j \in N}$ with respect to (E, C) can be constructed recursively as follows. Define for all $i\in N$ and $k \in \mathbb {N}$,

$$\begin{aligned} \gamma _i(k+1) = e_i + \sum _{j \in N} \varphi ^j_i(\gamma _j(k),\overline{c}_j), \end{aligned}$$

(3)

with $\gamma _i(1) = e_i$. Then, for all $i,j \in N$, set

$$\begin{aligned} p_{ij} = \lim _{k \rightarrow \infty } \varphi ^i_j(\gamma _i(k),\overline{c}_i). \end{aligned}$$

(4)

First, let us verify that the limit in (4) exists. Let $i \in N$ and note that

$$\begin{aligned} \gamma _i(1) = e_i \le e_i + \sum _{j \in N} \varphi ^j_i(\gamma _j(1),\overline{c}_j) = \gamma _i(2). \end{aligned}$$

Let $k \in \mathbb {N}$ and assume that $\gamma (k) \le \gamma (k+1)$. Estate monotonicity of the claims rules in $\phi$ then implies that

$$\begin{aligned} \gamma _i(k+1) = e_i + \sum _{j \in N} \varphi ^j_i(\gamma _j(k),\overline{c}_j) \le e_i + \sum _{j \in N} \varphi ^j_i(\gamma _j(k+1),\overline{c}_j) = \gamma _i(k+2). \end{aligned}$$

By induction it follows that (3) constitutes a monotonically increasing sequence

$$\begin{aligned} \gamma _i(1) \le \gamma _i(2) \le \gamma _i(3) \le \dots , \end{aligned}$$

that is bounded from above by $e_i + \sum _{j \in N} c_{ji}$. By the monotone convergence theorem, the sequence $(\gamma _i(k))_{k \in \mathbb {N}}$ has a limit.

Now, to verify that P is in fact a $\phi$-transfer scheme, one needs to check condition (2). Let $i, j \in N$, then

$$\begin{aligned} p_{ij}&= \lim _{k \rightarrow \infty } \varphi ^i_j(\gamma _i(k),\overline{c}_i) \\&= \varphi ^i_j\left( \lim _{k \rightarrow \infty }\gamma _i(k),\overline{c}_i \right) \\&= \varphi ^i_j\left( e_i + \lim _{k \rightarrow \infty } \sum _{m \in N} \varphi ^m_i(\gamma _m(k),\overline{c}_m), \overline{c}_i\right) \\&= \varphi ^i_j\left( e_i + \sum _{m \in N} p_{mi}, \overline{c}_i\right) . \end{aligned}$$

The first equality follows from (4), the second equality follows from estate continuity of $\varphi ^i$, which follows from estate monotonicity of $\varphi ^i$, the third equality follows from (3) and the last equality follows from (4).

The $\phi$-transfer scheme that is constructed as before is henceforth called the bottom $\phi$-transfer scheme. The correctness of this terminology will be shown in Theorem 3.1.

Definition 3.1

Let $(E,C) \in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules. The payment matrix $\underline{P}^{\phi }(E,C) = (p_{ij})_{i,j \in N}$ is called the bottom $\phi$-transfer scheme for (E, C) with respect to $\phi$ if, for all $i,j \in N$,

$$\begin{aligned} p_{ij} = \lim _{k \rightarrow \infty } \varphi ^i_j(\gamma _i(k),\overline{c}_i), \end{aligned}$$

in which, for all $i \in N$ and $k \in \mathbb {N}$,

$$\begin{aligned} \gamma _i(k+1) = e_i + \sum _{j \in N} \varphi ^j_i(\gamma _j(k),\overline{c}_j), \end{aligned}$$

with $\gamma _i(1) = e_i$.

The recursive procedure is illustrated in the following example.

Example 3.1

Consider the financial network $(E,C) \in \mathcal {F}^N$ given by $N = \{1,2,3\}$,

$$\begin{aligned} E = \begin{pmatrix}2 \\ 1 \\ 1 \end{pmatrix} \text { and } C = \begin{bmatrix} 0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 5 &{} 2 &{} 0 \end{bmatrix}. \end{aligned}$$

Let $\phi = (\textrm{CEA},\textrm{CEL},\textrm{TAL})$. Initially, we have $\gamma (1) = E = (2,1,1)$. To determine $\gamma (2)$, we compute the agents’ payments on the basis of $\gamma (1)$, which are given by

Hence, $\gamma (2) = (2,1,1) + (1,1\frac{1}{2},1\frac{1}{2}) = (3,2\frac{1}{2},2\frac{1}{2})$. Next, the payments under $\gamma (2)$ are given by

Hence, $\gamma (3) = (2,1,1) + (2\frac{1}{2},2,3) = (4\frac{1}{2},3,4)$. We see that under $\gamma (2)$ agents 1 and 2 have paid off all their debts, which means that their payments do not change in subsequent steps. Because agent 3 has not yet paid off all its debts, its payment will change under $\gamma (3)$:

Hence, $\gamma (4) = (2,1,1) + (4,2,3) = (6,3,4)$. Note that agent 3 can only allocate a maximum of 4 because agents 1 and 2 have paid off all their debts. Therefore, no more updates will take place in subsequent steps. We have $\gamma (4) = \gamma (5) = \gamma (6) = \dots$, so that the limit of $(\gamma (k))_{k\in \mathbb {N}}$ is obtained in a finite number of steps. The corresponding $\phi$-transfer scheme is given by

$$\begin{aligned} P = \begin{bmatrix} 0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 3 &{} 1 &{} 0 \end{bmatrix}; \end{aligned}$$

the transfer allocation is given by

$$\begin{aligned} \rho ^{\phi }(E,C) = (2,1,1) + (4,2,3) - (3,2,4) = (3,1,0). \end{aligned}$$

The following theorem justifies the term bottom $\phi$-transfer scheme.

Theorem 3.1

Let $(E,C) \in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules such that, for all $i \in N$, $\varphi ^i$ satisfies estate monotonicity. Then, $\underline{P}^{\phi }(E,C) \le P$ for all $P \in \mathcal {P}^{\phi }(E,C)$.

Proof

Let $(P^k)_{k \in \mathbb {N}}$ be the sequence of matrices in which, for each $k \in \mathbb {N}$, $P^k = (p_{ij}^k)_{i,j \in N}$ is given by

$$\begin{aligned} p_{ij}^k = \varphi ^i_j(\gamma _i(k),\overline{c}_i), \end{aligned}$$

for $i,j \in N$. Let $i \in N$. Then, as we have seen before,

$$\begin{aligned} \gamma _i(1) \le \gamma _i(2) \le \gamma _i(3) \le \dots , \end{aligned}$$

and, by estate monotonicity of $\varphi ^i$,

$$\begin{aligned} P_i^1 \le P_i^2 \le P_i^3 \le \dots , \end{aligned}$$

(5)

in which, for all $k \in \mathbb {N}$, $P_i^k$ is the i-th row of matrix $P^k$. Then, for the bottom $\phi$-transfer scheme it follows that $\underline{P}^{\phi }(E,C) = (\lim _{k\rightarrow \infty } p_{ij}^k)_{i,j\in N} = \lim _{k \rightarrow \infty } P^k$.

Now, let $P \in \mathcal {P}^{\phi }(E,C)$ and let, for all $i \in N$, $P_i$ and $P_i^{\phi }$ denote the i-th rows of $P = (p_{ij})_{i,j\in N}$ and $\underline{P}^{\phi }(E,C) = (p_{ij}^{\phi })_{i,j \in N}$, respectively. We will prove that $\underline{P}^{\phi }(E,C) \le P$.

Assume that $\underline{P}^{\phi }(E,C) \le P$ does not hold. Then, there exists at least one agent $i \in N$ such that $p_{ij}^{\phi } > p_{ij}$ for at least one $j \in N$. Without loss of generality, let $i = 1$. If $p_{1j}^{\phi } > p_{1j}$ for at least one $j \in N$, then by condition (2) of a $\phi$-transfer scheme

$$\begin{aligned} p_{1j}^{\phi } = \varphi _j^1\left( e_1 + \sum _{m \in N} p_{m1}^{\phi },\overline{c}_1\right) > \varphi _j^1\left( e_1 + \sum _{m \in N} p_{m1},\overline{c}_1\right) = p_{1j}. \end{aligned}$$

Consequently, estate monotonicity of claims rule $\varphi ^1$ implies that we must have

$$\begin{aligned} e_1 + \sum _{m \in N} p_{m1}^{\phi } > e_1 + \sum _{m \in N} p_{m1}, \end{aligned}$$

and thus $p_{1j}^{\phi } \ge p_{1j}$ for all $j \in N$. Equivalently, in vector notation, we have $P_1^{\phi } > P_1$. We know from (5) that the sequence $(P_1^k)_{k \in \mathbb {N}}$ is monotonically increasing and also converges to $P_1^{\phi }$, so there must exist a $K_1 \in \mathbb {N}$ such that $P_1^{K_1} \le P_1 < P_1^{K_1+1}$. Condition (2) of a $\phi$-transfer scheme states that

$$\begin{aligned} P_1 = \varphi ^1\left( e_1 + \sum _{m \in N} p_{m1}, \overline{c}_1\right) < \varphi ^1\left( e_1 + \sum _{m \in N} p_{m1}^{K_1}, \overline{c}_1\right) = P_1^{K_1+1}. \end{aligned}$$

Therefore, estate monotonicity of $\varphi ^1$ implies that agent 1 has received less under P than under $P^{K_1+1}$, that is, we have

$$\begin{aligned} \sum _{m \in N} p_{m1} < \sum _{m \in N} p_{m1}^{K_1}. \end{aligned}$$

Hence, $p_{m1} < p_{m1}^{K_1}$ for some $m \in N \setminus {\{1\}}$. Without loss of generality, let $m = 2$ so that $p_{21} < p_{21}^{K_1}$. This means that $p_{21}^{\phi } > p_{21}$ so that we can apply the previous arguments to the case of agent 2. Likewise agent 1, we find that $P_2^{\phi } > P_2$. Therefore, by (5), there must exist a $K_2 \in \mathbb {N}$ such that $P_2^{K_2} \le P_2 < P_2^{K_2+1}$. In particular, because $p_{21} < p_{21}^{K_1}$, $K_2 + 1$ can be at most $K_1$, that is, $K_2 < K_1$. Condition (2) of a $\phi$-transfer scheme states that

$$\begin{aligned} P_2 = \varphi ^2\left( e_2 + \sum _{m \in N} p_{m2}, \overline{c}_2\right) < \varphi ^2\left( e_2 + \sum _{m \in N} p_{m2}^{K_2}, \overline{c}_2\right) = P_2^{K_2+1}. \end{aligned}$$

Therefore, estate monotonicity of $\varphi ^2$ implies that agent 2 has received less under P than under $P^{K_2+1}$, that is, we have

$$\begin{aligned} \sum _{m \in N} p_{m2} < \sum _{m \in N} p_{m2}^{K_2}. \end{aligned}$$

Hence, $p_{m2} < p_{m2}^{K_2}$ for some $m \in N {\setminus }{\{1,2\}}$. Agent 1 is excluded because $p_{12} \ge p_{12}^{K_1} \ge p_{12}^{K_2}$, which follows from $P_1^{K_1} \le P_1$, (5) and $K_1 > K_2$. The premise is that we must be able to find an agent that is different from agents 1 and 2. Without loss of generality, let $m = 3$. By repeatedly applying the same arguments, we arrive at a contradiction because N is finite, that is, we eventually run out of agents to select. $\square$

Ketelaars et al. (2020) stresses the importance of the composition principle (see Definition 2.3) in establishing the equivalence between a $\phi$-based allocation rule $\rho ^{\phi }$ and a $\phi$-based ISAP allocation rule $r^{\phi }$. In fact, we will show that this equivalence relationship holds for their underlying clearing mechanisms as well.

Using the composition principle we are able to provide an explicit connection between $\phi$-transfer schemes and the payments made in ISAP. Composition of the underlying claims rules guarantees that, for any financial network, the sequence of cumulative payment matrices generated by ISAP with respect to $\phi$ converges to the bottom $\phi$-transfer scheme.

Theorem 3.2

Let $(E,C) \in \mathcal {F}^N$, let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules such that, for all $i \in N$, $\varphi ^i$ satisfies composition, and let the payment matrix P be given by $P = \sum _{k = 1}^{\infty } \Phi ^{k}$, in which $(\Phi ^k)_{k \in \mathbb {N}}$ is the sequence of payment matrices generated by ISAP for (E, C) with respect to $\phi$. Then, $P = \underline{P}^{\phi }(E,C)$.

Proof

We first show that $P \le \underline{P}^{\phi }(E,C)$. Next, we show that $P \in \mathcal {P}^{\phi }(E,C)$ so that also $\underline{P}^{\phi }(E,C) \le P$ (Theorem 3.1) which then implies that $P = \underline{P}^{\phi }(E,C)$.

Let $(E^k)_{k \in \mathbb {N}}$, $(C^k)_{k \in \mathbb {N}}$ and $(\Phi ^k)_{k \in \mathbb {N}}$ be the sequences generated by ISAP for (E, C) with respect to $\phi$. Set $E^k = (e^k_i)_{i \in N}$, $C^k = (c_{ij}^k)_{i,j \in N}$ and $\Phi ^k = (\Phi _{ij}^k)_{i,j \in N}$. For all $k \in \mathbb {N}$, let $P^k = (p_{ij}^k)_{i,j \in N}$ with $p_{ij}^k = \sum _{\ell =1}^k \Phi _{ij}^{\ell }$ for all $i,j \in N$. Here, $p_{ij}^k$ is the accumulated payment of agent $i \in N$ to agent $j \in N$ up to step $k \in \mathbb {N}$ in ISAP.

Before we start, we show three general characteristics of ISAP which are stated in (6), (7) and (8). In combination with the composition principle, this leads to (9) which is used frequently later on. Fix some step $k \in \mathbb {N}$ and some $i \in N$. First, the estate of agent i at step k is equal to its initial estate plus its net payments in steps $1,2,\dots ,k-1$, that is,

$$\begin{aligned} e_i^k&= e_i^{k-1} + \sum _{m \in N} \Phi _{mi}^{k-1} - \sum _{m \in N} \Phi _{im}^{k-1}, \end{aligned}$$

which implies that

$$\begin{aligned} e_i^k&= e_i^1 + \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{mi}^{\ell } - \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{im}^{\ell }. \end{aligned}$$

(6)

Second, in each step of ISAP the payments adhere to the limited liability requirement. In particular, the accumulated outgoing payments of agent i up to step k are at most its initial estate plus accumulated incoming payments up to step $k-1$, that is,

$$\begin{aligned} \sum _{m \in N} \sum _{\ell = 1}^{k} \Phi _{im}^{\ell } \le e_i^1 + \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{mi}^{\ell }. \end{aligned}$$

(7)

Inequality (7) follows from combining equation (6) and the fact that the total outgoing payment of agent i at step k is at most its estate at step k, that is,

$$\begin{aligned} \sum _{m \in N} \Phi _{im}^k&= \sum _{m \in N} \varphi _m^i(e_i^k,\overline{c}_i^k) = \min \left\{ e_i^k,\sum _{j \in N} c_{ij}^k\right\} \le e_i^k, \end{aligned}$$

in which the first equality follows from the definition of ISAP and the second equality follows from the definition of a claims rule. Additionally, estate monotonicity of $\varphi ^i$ and (7) imply that

$$\begin{aligned} \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 1}^{k} \Phi _{im}^{\ell } ,\overline{c}^1_i \right) \le \varphi _j^i\left( e_i^1 + \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{mi}^{\ell } ,\overline{c}^1_i \right) \end{aligned}$$

(8)

for all $j \in N$.

Now, we show that composition of $\varphi ^i$ implies that the left-hand side of (8) is in fact equal to the accumulated payment of agent i to agent $j \in N$ up to step k, that is, for all $j \in N$, we have

$$\begin{aligned} p_{ij}^k = \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 1}^{k} \Phi _{im}^{\ell } ,\overline{c}^1_i \right) . \end{aligned}$$

(9)

To this end, we first show that, for all $\ell \in \mathbb {N}$, we have

$$\begin{aligned} \varphi ^i\left( \sum _{m \in N} \Phi _{im}^{\ell },\overline{c}_i^{\ell }\right) = \varphi ^i\left( e_i^{\ell },\overline{c}_i^{\ell }\right) . \end{aligned}$$

(10)

Let $\ell \in \mathbb {N}$ and recall that $\sum _{m \in N} \Phi ^{\ell }_{im} = \min \{e_i^{\ell },\sum _{j \in N}c_{ij}^{\ell } \}$. If $e_i^{\ell } \le \sum _{j \in N} c_{ij}^{\ell }$, then (10) is immediate. Otherwise $e_i^{\ell } > \sum _{j \in N} c_{ij}^{\ell }$, which means that the estate at step $\ell$ can cover all claims of agent i at step $\ell$. Therefore,

$$\begin{aligned} \varphi ^i\left( \sum _{m \in N} \Phi _{im}^{\ell },\overline{c}^{\ell }_i\right)&= \varphi ^i\left( \sum _{j \in N} c_{ij}^{\ell },\overline{c}_i^{\ell }\right) = \overline{c}_i^{\ell } = \varphi ^i\left( e_i^{\ell },\overline{c}_i^{\ell }\right) . \end{aligned}$$

Using composition of $\varphi ^i$ and (10), we find that, for all $j \in N$, the left-hand side of (8) can be rewritten as

$$\begin{aligned}&\varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 1}^{k} \Phi _{im}^{\ell },\overline{c}^1_i \right) \\&\quad = \varphi _j^i\left( \sum _{m \in N} \Phi ^1_{im} ,\overline{c}^1_i \right) + \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 2}^{k} \Phi _{im}^{\ell } ,\overline{c}^1_i - \varphi ^i\left( \sum _{m \in N} \Phi ^1_{im} ,\overline{c}^1_i \right) \right) \\&\quad = \varphi _j^i\left( e_i^1 ,\overline{c}^1_i \right) + \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 2}^{k} \Phi _{im}^{\ell } ,\overline{c}^1_i - \varphi ^i\left( e_i^1 ,\overline{c}^1_i \right) \right) \\&\quad = \varphi _j^i\left( e_i^1 ,\overline{c}^1_i \right) + \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 2}^{k} \Phi _{im}^{\ell } ,\overline{c}_i^2 \right) \\&\quad = \sum _{\ell = 1}^{k} \varphi _j^i\left( e_i^{\ell },\overline{c}_i^{\ell }\right) \\&\quad = \sum _{\ell = 1}^k \Phi _{ij}^{\ell } \\&\quad = p_{ij}^k. \end{aligned}$$

The first equality follows from composition of $\varphi ^i$; the second equality follows from (10); the third equality follows from the definition of ISAP; the fourth equality from follows from repeatedly applying the previous arguments; the fifth equality follows from the definition of ISAP. This finishes our preparations.

Let $\underline{P}^{\phi }(E,C) = (p^{\phi }_{ij})_{i,j \in N}$. We will show by induction on k that $P^k \le \underline{P}^{\phi }(E,C)$ for all $k \in \mathbb {N}$. Clearly, this implies $P = \lim _{k\rightarrow \infty } P^k \le \underline{P}^{\phi }(E,C)$. Let $k = 1$. Then, for all $i,j \in N$, it holds that

$$\begin{aligned} p_{ij}^1 = \varphi _j^i\left( e_i^1,\overline{c}^1_i\right) \le \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi}^{\phi },\overline{c}^1_i\right) = p_{ij}^{\phi }. \end{aligned}$$

The inequality follows from estate monotonicity of $\varphi ^i$ and the second equality follows from $\underline{P}^{\phi }(E,C) \in \mathcal {P}^{\phi }(E,C)$ (see (2)). Next, let $k \in \mathbb {N}$ and assume that $P^{k} \le \underline{P}^{\phi }(E,C)$. Now consider $k+1$. Then, for all $i,j \in N$, it holds that

$$\begin{aligned} p_{ij}^{k+1}&= \varphi _j^i\left( \sum _{m \in N}\sum _{\ell = 1}^{k+1}\Phi _{im}^{\ell },\overline{c}_i^1\right) \\&\le \varphi _j^i\left( e_i^1 + \sum _{m \in N}\sum _{\ell =1}^{k} \Phi _{mi}^{\ell },\overline{c}_i^1\right) \\&= \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi}^{k} ,\overline{c}^1_i\right) \\&\le \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi}^{\phi } ,\overline{c}^1_i \right) \\&= p_{ij}^{\phi }. \end{aligned}$$

The first equality follows from (9); the first inequality follows from (8); the second inequality follows from the induction hypothesis and estate monotonicity of $\varphi ^i$; the last equality follows from $\underline{P}^{\phi }(E,C) \in \mathcal {P}^{\phi }(E,C)$ (see (2)).

Finally, to show that $P \in \mathcal {P}^{\phi }(E,C)$, we need to show that (see (2)), for all $i,j \in N$,

$$\begin{aligned} p_{ij} = \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi},\overline{c}^1_i\right) . \end{aligned}$$

So let $i,j \in N$. We distinguish between two mutually exclusive cases. Define the set

$$\begin{aligned} B = \left\{ p \in N :e_p^k < \sum _{m \in N} c_{pm}^k \text { for all } k \in \mathbb {N} \right\} \end{aligned}$$

as the set containing agents that never have sufficient funds to pay off their remaining debts in ISAP.

First, if $i \in B$, then in each step $k \in \mathbb {N}$ of ISAP agent i allocates its full estate. That is, for all $k \in \mathbb {N}$, we have $\sum _{m\in N} \Phi _{mi}^k = e_i^k$, or equivalently, using (6),

$$\begin{aligned} \sum _{m \in N}\sum _{\ell =1}^k \Phi _{im}^{\ell } = e^1_i + \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{mi}^{\ell }. \end{aligned}$$

(11)

Consequently, we have

$$\begin{aligned} p_{ij}&= \lim _{k \rightarrow \infty } p_{ij}^k \\&= \lim _{k \rightarrow \infty } \varphi _j^i\left( \sum _{m \in N} \sum _{\ell = 1}^{k} \Phi _{im}^{\ell } ,\overline{c}^1_i \right) \\&= \lim _{k \rightarrow \infty } \varphi _j^i\left( e_i^1 + \sum _{m \in N} \sum _{\ell = 1}^{k-1} \Phi _{mi}^{\ell } ,\overline{c}^1_i \right) \\&= \varphi _j^i\left( e_i^1 + \sum _{m \in N} \left( \lim _{k \rightarrow \infty } p_{mi}^{k-1} \right) ,\overline{c}_i^1\right) \\&= \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi} ,\overline{c}^1_i \right) . \end{aligned}$$

The second equality follows from (9); the third equality follows from (11); the fourth equality follows from estate continuity of $\varphi ^i$.

Second, if $i \notin B$, then there exists a $K \in \mathbb {N}$ such that $e_i^K \ge \sum _{m \in N} c_{im}^K$ and, as a consequence, $\sum _{m \in N} \Phi _{im}^K = \sum _{m \in N} c_{im}^K$. In other words, agent i pays off all its remaining debts at time moment K, which means that $\Phi _{ij}^k = 0$ for all $k \in \{K+1,K+2,\dots \}$.

It suffices to prove that $p_{ij}^K = c_{ij}^1$. If $p_{ij}^K = c_{ij}^1$, then clearly $p_{ij} = c_{ij}^1$ because $\Phi _{ij}^k = 0$ for all $k \in \{K+1,K+2,\dots \}$. Moreover, if $p_{ij}^K = c_{ij}^1$, we also have

$$\begin{aligned} c_{ij}^1 = p_{ij}&= p_{ij}^K \\&= \varphi _j^i\left( \sum _{m \in N} \sum _{\ell =1}^{K} \Phi _{im}^{\ell },\overline{c}_i^1\right) \\&\le \varphi _j^i\left( e_i^1 + \sum _{m \in N} \sum _{\ell =1}^{K-1} \Phi _{im}^{\ell },\overline{c}_i^1\right) \\&= \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{im}^{K-1},\overline{c}_i^1\right) \\&\le \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{im},\overline{c}_i^1\right) \le c_{ij}^1, \end{aligned}$$

which implies that all inequalities are equalities and therefore, in particular,

$$\begin{aligned} p_{ij} = \varphi _j^i\left( e_i^1 + \sum _{m \in N} p_{mi},\overline{c}_i^1\right) . \end{aligned}$$

Here, the third equality follows from (9); the first inequality follows from (8); the second inequality follows from estate monotonicity of $\varphi ^i$; the last inequality follows from the definition of a claims rule.

To show that $p_{ij}^K = c_{ij}^1$, we first show that the accumulated payments of agent i up to K are in fact equal to its total claims. That is, by definition of ISAP,

$$\begin{aligned} \begin{aligned} \sum _{m \in N} \sum _{\ell = 1}^K \Phi _{im}^{\ell }&= \sum _{m \in N} \sum _{\ell = 1}^{K-1} \Phi _{im}^{\ell } + \sum _{m \in N} \Phi ^K_{im}\\&= \sum _{m \in N} \sum _{\ell = 1}^{K-1} \Phi _{im}^{\ell } + \sum _{m \in N} c_{im}^K \\&= \sum _{m \in N} \sum _{\ell = 1}^{K-1} \Phi _{im}^{\ell } + \sum _{m \in N} \left( c_{im}^{K-1} - \Phi _{im}^{K-1}\right) \\&=\sum _{m \in N} \sum _{\ell = 1}^{K-2} \Phi _{im}^{\ell } + \sum _{m \in N} c_{im}^{K-1} \\&= \sum _{m \in N} c^1_{im}. \end{aligned} \end{aligned}$$

In the second equality, we use the fact that $i \notin B$ which implies that $\sum _{m \in N} \Phi _{im}^K = \sum _{m \in N} c_{im}^K$ as argued before. From this we can conclude that

$$\begin{aligned} p_{ij}^K&= \varphi _j^i\left( \sum _{m \in N} \sum _{\ell =1}^{K} \Phi _{im}^{\ell },\overline{c}_i^1\right) \\&= \varphi _j^i\left( \sum _{m \in N} c_{im}^1,\overline{c}_i^1\right) \\&= c_{ij}^1. \end{aligned}$$

$\square$

The following corollary is a direct consequence of Proposition 2.2 and Theorem 3.2. It says that the equivalence relationship between a $\phi$-based allocation rule $\rho ^{\phi }$ and a $\phi$-based ISAP allocation rule $r^{\phi }$ also holds with agent-specific claims rules, thereby generalizing the main result of Ketelaars et al. (2020).

Corollary 3.1

Let $(E,C) \in \mathcal {F}^N$ and let $\phi = (\varphi ^i)_{i \in N}$ be a vector of claims rules such that, for all $i \in N$, $\varphi ^i$ satisfies composition. Then,

$$\begin{aligned} \rho ^{\phi }(E,C) = \alpha ^{\underline{P}^{\phi }(E,C)} = r^{\phi }(E,C). \end{aligned}$$

In addition to the previous two $\phi$-based clearing mechanisms, there exist two types of $\phi$-based decentralized sequential clearing mechanisms for which payments converge to the bottom $\phi$-transfer scheme (cf. Theorem 3.1 and Theorem 3.2).

In each step of a sequential clearing mechanism, exactly one agent is selected that makes a payment to the other agents on the basis of a claims rule $\varphi$. Like the individual settlement allocation procedure, sequential clearing mechanisms need not terminate in a finite number of iterations. Therefore, we require that the selection process is such that each agent is in principle selected an infinite number of times.¹¹ The selection process need not be deterministic. In general, the selection process can be history dependent, stochastic or both. We represent a realization of a selection process by an ordering of the agents $\sigma :\mathbb {N} \rightarrow N$, in which $\sigma (k) \in N$ is the agent that is selected to pay at step $k \in \mathbb {N}$. The set of all realized orderings $\sigma$ of N that are the direct consequence of all possible selection processes with the property that $|\{k \in \mathbb {N} :\sigma (k) = i\}| = \infty$ for all $i \in N$ is denoted by $\Pi (N)$.

We will now outline two types of sequential clearing mechanisms, both of which are defined for any $\sigma \in \Pi (N)$. The first $\phi$-based sequential clearing mechanism is a variation on the $\phi$-based individual settlement allocation procedure, a procedure in which agents pay simultaneously in each step. To accommodate for sequential payments based on a realized ordering $\sigma \in \Pi (N)$, we need only change the second component of ISAP of Definition 2.8 in the following way:

• 2. The payment of agent $\sigma (k-1)=i$ to agent $j \in N$ in step $k-1$ is equal to

  $$\begin{aligned} \Phi ^{k-1}_{ij} = \varphi ^i_j\left( e^{k-1}_i,\overline{c}^{k-1}_i\right) , \end{aligned}$$

  in which $\overline{c}^{k-1}_i \in \mathbb {R}^N_{+}$ is the i-th row of claims matrix $C^{k-1}$; for all $i \ne \sigma (k-1)$, it holds that $\Phi ^{k-1}_{ij} = 0$ for all $j \in N$.

The other steps in Definition 2.8 remain the same. Correspondingly, given a selection procedure and a corresponding realized ordering $\sigma \in \Pi (N)$, we call such a $\phi$-based sequential clearing mechanism a $\phi$-based asynchronous ISAP.

Theorem 3.3

If each coordinate of $\phi$ is a claims rule that satisfies composition, then the resulting payment matrix of a $\phi$-based asynchronous ISAP is the bottom $\phi$-transfer scheme, irrespective of the realized ordering $\sigma \in \Pi (N)$.

This statement follows from a direct modification of the proof of Theorem 3.2; the crux of the proof is establishing (9).

The second $\phi$-based sequential clearing mechanism is a variation on the recursive procedure provided in Definition 3.1 of the bottom $\phi$-transfer scheme. Given a realized ordering $\sigma \in \Pi (N)$, the clearing mechanism generates a sequence of payment matrices $(P^k)_{k \in \mathbb {N}}$ with $P^k = (p^k_{ij})_{i,j\in N}$ as follows. Initially, set $P^0 = 0$. Define, for all $i \in N$ and $k \in \mathbb {N}$,

$$\begin{aligned} \delta _i(k+1) = e_i + \sum _{j \in N} p_{ji}^k, \end{aligned}$$

(12)

with $\delta _i(1) = e_i$, in which, for all $j \in N$,

$$\begin{aligned} p_{ji}^{k} = {\left\{ \begin{array}{ll}\varphi _i^j(\delta _j(k),\overline{c}_j) \quad &{} \text { if } j = \sigma (k), \\ p_{ji}^{k-1} \quad &{} \text { if } j \ne \sigma (k). \end{array}\right. } \end{aligned}$$

(13)

Then, define $P = (p_{ij})_{i,j \in N}$ by setting, for all $i,j \in N$,

$$\begin{aligned} p_{ij} = \lim _{k\rightarrow \infty } p_{ij}^k. \end{aligned}$$

(14)

In fact, the above $\phi$-based sequential clearing mechanism is the analogue of a decentralized clearing process introduced in Csóka and Herings (2018) for the discrete setup.¹²

Correspondingly, given a selection procedure and a corresponding realized ordering $\sigma \in \Pi (N)$, we call such a $\phi$-based sequential clearing mechanism a $\phi$-based decentralized clearing process. The process can be interpreted as follows. Each agent keeps track of the amount at its disposal, that is, its initial estate plus the payments it has received so far, as represented by (12). Each time an agent is selected, it makes a (possible) incremental payment according to its claims rule by allocating what it currently has at its disposal.

Theorem 3.4

If each coordinate of $\phi$ is a claims rule that satisfies estate monotonicity, then the resulting payment matrix P given in (14) of a $\phi$-based decentralized clearing process is the bottom $\phi$-transfer scheme, irrespective of the realized ordering $\sigma \in \Pi (N)$.

To see this, we first argue that P is a $\phi$-transfer scheme. Because each claims rule in $\phi$ is estate monotone, (12) constitutes a monotonically increasing sequence $(\delta (k))_{k \in \mathbb {N}}$ that is bounded from above. Hence, its limit exists as a result of the monotone convergence theorem. Consequently, estate continuity of claims rules in $\phi$ implies that (13) constitutes a monotonically increasing sequence

$$\begin{aligned} P^1 \le P^2 \le P^3 \le \dots , \end{aligned}$$

which has a limit as well. By construction of a $\phi$-based decentralized clearing process, it follows that the resulting payment matrix P given in (14) is a $\phi$-transfer scheme, irrespective of the realized ordering $\sigma \in \Pi (N)$. The assertion that P is the bottom $\phi$-transfer scheme then follows directly from the proof of Theorem 3.1.

The following example illustrates how $\phi$-based (asynchronous) individual settlement allocation procedures and $\phi$-based decentralized clearing processes differ in interpretation. In each step of a $\phi$-based decentralized clearing process, agents essentially communicate what they will pay to each other, thereby not updating their mutual liabilities. The only transfers between agents that take place are those with respect to the resulting limiting payment matrix, that is, the bottom $\phi$-transfer scheme. This contrasts with $\phi$-based (asynchronous) individual settlement allocation procedures in which actual transfers between agents take place in each step.

Example 3.2

Reconsider the financial network $(E,C) \in \mathcal {F}^N$ of Example 3.1 given by $N = \{1,2,3\}$,

$$\begin{aligned} E = \begin{pmatrix}2 \\ 1 \\ 1 \end{pmatrix} \text { and } C = \begin{bmatrix} 0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 5 &{} 2 &{} 0 \end{bmatrix}. \end{aligned}$$

Again, let $\phi = (\textrm{CEA},\textrm{CEL},\textrm{TAL})$. Consider the following realized ordering of the agents $\sigma = (1,3,2,1,3,2,\dots )$. We will illustrate the corresponding $\phi$-based asynchronous ISAP and $\phi$-based decentralized clearing process in conjunction.

Let, for all $k \in \mathbb {N}$, $\hat{P}^k = \sum _{\ell =1}^k \Phi ^{\ell }$ denote the accumulated payments at step k under $\phi$-based asynchronous ISAP. Furthermore, in regard to the $\phi$-based decentralized clearing process, let $P^k$ be as defined in (13) for all $k \in \mathbb {N}$.

Initially, set $E^1 = E = \delta (1)$ and $P^0 = 0$. Agent 1 is first selected to pay. We have $\textrm{CEA}(2,(0,1,2)) = (0,1,1)$ in both clearing mechanisms, so

$$\begin{aligned} \hat{P}^1 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{bmatrix} \text { and } P^1 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{bmatrix} . \end{aligned}$$

Hence,

$$\begin{aligned} E^2&= (2,1,1) + (0,1,1) - (2,0,0) = (0,2,2), \\ \text { and } \delta (2)&= (2,1,1) + (0,1,1) = (2,2,2). \end{aligned}$$

Agent 3 is selected next and pays $\textrm{TAL}(2,(5,2,0)) = (1,1,0)$ in both mechanisms because $E_3^2 = \delta _3(2) = 2$, so

$$\begin{aligned} \hat{P}^2 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \end{bmatrix} \text { and } P^2 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \end{bmatrix} . \end{aligned}$$

Hence,

$$\begin{aligned} E^3&= (2,1,1) + (1,2,1) - (2,0,2) = (1,3,0), \\ \text { and } \delta (3)&= (2,1,1) + (1,2,1) = (3,3,2). \end{aligned}$$

In the third step, agent 2 is selected and pays $\textrm{CEL}(3,(1,0,1)) = (1,0,1)$ in both mechanisms because $E_2^3 = \delta _2(3) = 3$, so

$$\begin{aligned} \hat{P}^3 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{bmatrix} \text { and } P^3 = \begin{bmatrix}0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{bmatrix} . \end{aligned}$$

Hence, $E^4 = (2,1,1)$ and $\delta (4) = (4,3,3)$. Moreover, the claims matrix in step 4 for $\phi$-based asynchronous ISAP is equal to

$$\begin{aligned} C^4 = C - \hat{P}^3 = \begin{bmatrix}0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 4 &{} 1 &{} 0 \end{bmatrix}. \end{aligned}$$

In subsequent steps, agent 2 will not make any incremental payments as it is debt free. In the fourth step, agent 1 is selected and becomes debt free by paying $\textrm{CEA }(E_1^4,\overline{c}_1^4) = \textrm{CEA}(2,(0,0,1)) = (0,0,1)$ under $\phi$-based asynchronous ISAP and by paying $\textrm{CEA}(\delta _1(4),\overline{c}^1_1) = \textrm{CEA}(4,(0,1,2)) = (0,1,2)$ under the $\phi$-based decentralized clearing process. Consequently,

$$\begin{aligned} \hat{P}^4 = \begin{bmatrix}0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{bmatrix} \text { and } P^4 = \begin{bmatrix}0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{bmatrix}, \end{aligned}$$

so that $E^5 = (1,1,2)$ and $\delta (5) = (4,3,4)$.

Note that the matrices $\hat{P}^4$ and $P^4$ are still equal, however they will become different in the next step. Because agents 1 and 2 have become debt free, there will be one more payment by agent 3 after which both procedures essentially terminate. Agent 3 is selected in the next step and pays $\textrm{TAL}(E^5_3,\overline{c}^5_3) = \textrm{TAL}(2,(4,1,0)) = (1\frac{1}{2},\frac{1}{2},0)$ under $\phi$-based asynchronous ISAP and pays $\textrm{TAL}(\delta _3(5),\overline{c}_3^1) = \textrm{TAL}(4,(5,2,0)) = (3,1,0)$ under the $\phi$-based decentralized clearing process. Therefore, the limiting payment matrices $\hat{P}^5 = \hat{P}^6 = \dots = \hat{P}$ and $P^5 = P^6 = \dots = P$ are given by

$$\begin{aligned} \hat{P} = \begin{bmatrix} 0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 2\frac{1}{2} &{} 1 \frac{1}{2} &{} 0 \end{bmatrix} \text { and } P = \begin{bmatrix} 0 &{} 1 &{} 2 \\ 1 &{} 0 &{} 1 \\ 3 &{} 1 &{} 0 \end{bmatrix}, \end{aligned}$$

respectively. The payment matrix P equals the bottom $\phi$-transfer scheme obtained in Example 3.1. Note that the Talmud rule does not satisfy composition and thus the payment matrix $\hat{P}$ may differ from the bottom $\phi$-transfer scheme as is the case here. Correspondingly, the transfer allocations are unequal:

$$\begin{aligned} \alpha ^{\hat{P}} = \left( 2\frac{1}{2},1\frac{1}{2},0\right) \ne (3,1,0) = \alpha ^P. \end{aligned}$$

4 Conclusion and discussion

In the context of financial networks, this paper introduces a recursive procedure for a centralized clearing mechanism that leads to the bottom $\phi$-transfer scheme as a bilateral payment matrix. Moreover, we show that several types of decentralized clearing mechanisms prescribe the bottom $\phi$-transfer scheme as well. Composition of the underlying claims rules turns out to be essential for this unification of the centralized clearing mechanism based on $\phi$-transfer schemes, the $\phi$-based individual settlement allocation procedure, and $\phi$-based decentralized sequential clearing mechanisms.

The rise of bitcoin, among other peer-to-peer electronic currencies, manifest the decentralization in finance by offering a way for one party to transfer money to another without relying on a trusted third party, see, for example, Satoshi Nakamoto’s whitepaper on bitcoin (Nakamoto 2008). Our result of unification extends this notion of decentralization to clearing procedures. For example, a central authority is superfluous when settling debts in a network in which agents pay their creditors in accordance with insolvency law, that is, pari passu treatment within each priority class, because a centralized prescription of the payments is equivalent to a decentralized one. In this way, similar to cryptocurrencies, a decentralized clearing mechanism also limits the exchange of, possibly sensitive, information required to specify how payments between agents in a network should take place.

The choice of a clearing mechanism is dependent on the situation. The centralized clearing mechanism based on $\phi$-transfer schemes as well as the decentralized $\phi$-based ISAP lend themselves for static situations, such as an evaluation of the solvency of the agents in a network. That is, by assessing the resulting transfer allocation one can pinpoint agents that will default on their debt obligations, after which appropriate preemptive measures can be taken. The advantage of $\phi$-based ISAP in this sense is that it offers a way to do this decentralized, as may be required by technologies built on the blockchain. Sequential clearing mechanisms are arguably more suitable for dynamic situations, for example, actual insolvency proceedings, because at each point in time one agent makes a payment to the other agents. A business’ assets are typically sold in stages, meaning that payments to creditors occur in stages too. If all agents use claims rules that satisfy composition, then the time at which assets are sold as well as the order in which agents pay are irrelevant for the eventual payments. Furthermore, even the method of clearing is irrelevant. Agents may either transfer money between each other at each step, which reduces the mutual liabilities, or merely communicate what they will pay to each other at each step and only make payments at the very end, which are prescribed by the bottom $\phi$-transfer scheme.

Estate monotonicity implies estate continuity

Proposition A.1

If a claims rule $\varphi$ on $\mathcal {C}^N$ satisfies estate monotonicity, then $\varphi$ satisfies estate continuity.

Proof

Assume that $\varphi$ satisfies estate monotonicity. Let $(e,c) \in \mathcal {C}^N$, and suppose that $\varphi (e,c)$ is not continuous with respect to e. Then, there exists an $i \in N$ and sequences $(\underline{e}^k)_{k \in \mathbb {N}}$ and $(\overline{e}^k)_{k \in \mathbb {N}}$, in which, for all $k \in \mathbb {N}$, $0 \le \underline{e}^k \le \underline{e}^{k+1} \le e \le \overline{e}^{k+1} \le \overline{e}^k$ with $\lim _{k \rightarrow \infty } \underline{e}^k = e$ and $\lim _{k \rightarrow \infty } \overline{e}^k = e$, such that $\lim _{k \rightarrow \infty } \varphi _i(\underline{e}^k,c) \ne \lim _{k \rightarrow \infty } \varphi _i(\overline{e}^k,c)$. Let $\varphi (e^{-},c) = \lim _{k \rightarrow \infty } \varphi (\underline{e}^k,c)$ and $\varphi (e^{+},c) = \lim _{k \rightarrow \infty } \varphi (\overline{e}^k,c)$. In fact, because, for all $k \in \mathbb {N}$, $\underline{e}^k \le \overline{e}^{k+1} \le e \le \overline{e}^{k+1} \le \overline{e}^k$, estate monotonicity of $\varphi$ implies that $\varphi (\underline{e}^k,c) \le \varphi (\underline{e}^{k+1},c) \le \varphi (e,c) \le \varphi (\overline{e}^{k+1},c) \le \varphi (\overline{e}^k,c)$. Hence, also $\varphi (e^{-},c) < \varphi (e^{+},c)$ and $\varphi (e^{-},c) \le \varphi (e,c) \le \varphi (e^{+},c)$.

First, assume that $\varphi (e^{-},c) < \varphi (e,c) \le \varphi (e^{+},c)$. Then, for all $k \in \mathbb {N}$, $\varphi (\underline{e}^k,c) < \varphi (e,c)$ and $\sum _{j \in N}\varphi _j(\underline{e}^k,c) < \sum _{j \in N} \varphi _j(e,c)$. From $\sum _{j \in N} \varphi _j(e,c) \le \sum _{j \in N} c_j$ and condition (ii) of a claims rule it follows that, for all $k \in \mathbb {N}$, $\sum _{j \in N}\varphi _j(\underline{e}^k,c) = \underline{e}^k$. At the same time, however,

$$\begin{aligned} e&= \lim _{k \rightarrow \infty } \underline{e}^{k} = \lim _{k \rightarrow \infty } \sum _{j \in N}\varphi _j(\underline{e}^k,c) < \lim _{k \rightarrow \infty }\sum _{j \in N} \varphi _j(e,c) = \min \left\{ e,\sum _{j \in N} c_j\right\} \le e, \end{aligned}$$

which is a contradiction.

Second, assume that $\varphi (e^{-},c) \le \varphi (e,c) < \varphi (e^{+},c)$. Then, in a similar fashion as for the previous case, for all $k \in \mathbb {N}$, $\varphi (e,c) < \varphi (\overline{e}^k,c)$ and $\sum _{j \in N} \varphi _j(e,c) < \sum _{j \in N} \varphi _j(\overline{e}^k,c) \le \sum _{j \in N} c_j$. Therefore, by condition (ii) of a claims rule, $\sum _{j \in N} \varphi _j(e,c) = e$. Consequently, we obtain

$$\begin{aligned} e&= \lim _{k \rightarrow \infty } \overline{e}^k \ge \lim _{k \rightarrow \infty } \min \left\{ \overline{e}^k,\sum _{j \in N} c_j\right\} = \lim _{k \rightarrow \infty }\sum _{j \in N}\varphi _j(\overline{e}^k,c)> \lim _{k \rightarrow \infty }\sum _{j \in N} \varphi _j(e,c) =e, \end{aligned}$$

which is again a contradiction. $\square$