Entropy in Financial Contagion Research
Michael Stutzer
Prof. of Finance
University of Colorado, Boulder
michael.stutzer@colorado.edu
KEYWORDS: contagion, stability, entropy, bankruptcy resolution
ABSTRACT:
A large literature has evolved to study the reasons and ramifications of default in liabilities
networks, e.g. the network of interbank loans, or their customers’ funds’ transfers among each
other. Much attention has focused on the prospect for default cascades in which one party’s
inability to meet its obligations can prevent its creditors from doing the same, which in turn
cause other defaults further downstream. This is a type of financial contagion.
The matrix of liabilities between the agents must be estimated. Entropic estimators have been
used to estimate the unobserved cells; we argue that this is a reasonable procedure under the
circumstances envisioned. In addition, entropy is used to construct new indices of default and
contagion severity, and a new index of the inequity inherent in a default resolution. Using the
latter in conjunction with the game-theoretic theory of conflict resolution, we demonstrate
advantages associated with deviation from the common default resolution procedure that requires
a defaulting party to pay its creditors the same proportion of what each is owed.
Key Points:
• Entropy is used to devise new indices of default severity, contagion severity, and inequity
in default resolution procedures
• A simple example is used to illustrate these indices, and is bootstrapped to illustrate
reasons why entropic liability matrix estimators do not generally bias the default,
contagion, and inequity measures in one direction or the other.
• Relaxing the usual restriction that the default initiator must pay all its claimants the same
proportion (say, 80% of what each is owed) is justifiable within the game theory of
bargaining (e.g. the Nash Bargaining solution), and can both increase efficiency while
lowering inequity in the default resolution procedure.
1
I. Introduction
The standard representation of a payments network starts with a snapshot of gross liabilities
owed by each agent (bank, firm, trader, etc.) to each other agent, in the form of a matrix L:
 L11
L
 21
 

 LN 1
L12
L22

LN 2
… L1N 
… L2 N 
  

… LNN 
in which Lij is an amount that agent i owes to agent j. These are gross rather than net liabilities,
so that L ji need not be − Lij ; in fact, all elements are nonnegative. The entries could represent
outstanding loan balances, or loan payments that are due, or checks drawn on one bank that must
be deposited in accounts at another bank, or payments owed as a result of mutual trading
activities, etc. Define the row sum li , column sum a j , and their corresponding shares of the
grand totals Li = li / ∑ lk and Aj = a j / ∑ ak .
k
k
We will use the following numerical example for illustration throughout:
l
#2
#3 # 4
 Agent #1
 #1
0
0
10
0
10

 #2
30
0
20
20 70

Lij ≡  #3
10
30
0
10
50
 #4
10
0
20
0
30

50
30
50
30 160
 a
 A
.313 .187 .313 .187

L 
.063
.437 

.313
.187 




(1)
Now suppose for the moment that are no estimation errors. We see that Agent #2 owes l2 = 70
but is owed only a2 = 30 . Without some additional funds to use, it cannot pay all its liabilities,
and hence will have to default on some of them. Agent #2 owes 20 to Agent #3, who has no
surplus available from its a3 = 50 in assets to pay its l3 = 50 in total liabilities, and hence will
also have to default on some payments if it doesn’t receive payment from the defaulting Agent
#2. In this way, default by one agent may trigger defaults by others. A cascade of defaults that
is triggered by a single default is a type of financial contagion. Here the contagion was triggered
by some situation that resulted in Agent #2 owing more in the aggregate than it was due to
receive, without outside resources to draw upon. With other matrices, there may be more than
one agent initially in default, and those may trigger subsequent defaults.
2
This aspect of credit/payment systems is not only relevant, it may have motivated the advent of
bankruptcy law centuries ago. As noted in Kadens (2010, pp.1237-8):
“The merchant or trader who relied on credit lived constantly on the edge. The
still relatively primitive state of communication, travel, and production meant that
he could be sure when he would receive the next shipment or the next payment on
which his ability to pay his own creditors depended. His goal was to `synchronize
the payments being made to him as a creditor with those he had to make as a
debtor’, and this he could never do with complete assurance. As all merchants and
traders who depended on credit existed in this state of financial instability, the
insolvency of one person who owed significant debts could lead to the failure of
many others.”
One might conjecture that financial contagion is less likely when the liabilities are more evenly
distributed. The intuition is that a default by one agent will cause many agents to lose small
amounts, insufficient to cause any of them to default on their own obligations. In order to study
this conjecture, one needs to define what is meant by “liabilities are more evenly distributed”.
The entropy defined in section II can be used for this purpose.
Then one needs to define a procedure for resolving defaults. Is an agent who is stiffed by another
permitted to renege on its own obligations to it? Can a bankruptcy judge determine who it will
renege on? Following Elimam, et. al. (1996) and Eisenberg and Noe (2001), the literature has
focused on a default resolution process in which defaulting agents are required to proportionally
renege on all their respective creditors. In their words (op.cit., p.239) “all claimant nodes are
paid by the defaulting node in proportion to the size of their nominal claim on firm assets.” In
other words, after any default cascade has ended, an agent that can pay only θ % of its total
liabilities must pay exactly θ % of the funds it owes to each of its creditors. 1 The equilibrium
resolution procedure is described in section III, and used to formulate equilibrium, entropic
indices of both default and subsequent contagion severity. In section IV, we show why there
needn’t be any particular relationship between the entropy of a liability matrix and these default
and contagion severity indices is explored in section IV. Perhaps fairness considerations might
favor different default resolution procedures in some circumstances, and laws could be changed
to permit or even encourage bankruptcy courts to implement different procedures on grounds of
fairness. In section V, an entropy-based index of inequity is proposed. Section VI demonstrates
beneficial effects on both default severity and inequity that stem from relaxing the proportional
payment rule for an agent who initiates a default cascade. Section VII both reviews and adds
new results to the significant game-theoretic results that support alternative bankruptcy
resolution procedures, because the theorems previously developed are for single debtor vs.
multiple claimant situations, and are incorrect in financial networks. Section VIII concludes.
1
The Proportional Rule was retained when Rogers and Veraart (2013, p. 884) extended the Eisenberg and Noe
(op.cit.) framework “by introducing costs of default if loans have to be called in by a failing bank” (p.882).
3
II. The Entropy of the Liabilities Matrix
The survey paper by Upper (2011, sec. 4.2) describes a widely-used procedure to define an
entropy of the liabilities matrix L is to normalize it by dividing each of its cells by the grand total
of all cells, i.e. define Pij = Lij / ∑∑ Lij , and compute the Shannon entropy of the normalized
i
j
matrix H = −∑∑ Pij log Pij . By adopting the convention 0 log 0 = 0 , so cells with zeros, e.g.
i
j
those along the diagonal, contribute nothing to entropy, and hence are omitted from the double
sum, i.e. we calculate H = − ∑ Pij log Pij In our example (1), the Shannon entropy H = 2.10 .
ij ;i ≠ j
Suppose that all cells off the diagonal are unknown, but that the column and row sums of L are
observed. This situation was faced by early researchers with access to financial reports that
listed total liabilities and assets of agents without breaking out the bilateral specifics. The
unknown cells have been estimated by maximizing this entropy subject to the constraints that
row and column sums have fixed values (typically known by observation). That is:
maxPi≠ j −
∑ P log P ij ;i ≠ j
ij
ij
subj. to=
Aj=
: ∑ Pij L=
; ∑ Pij 1
i ; ∑ Pij
j ≠i
i≠ j
(2)
ij ;i ≠ j
The problem re-defined with sums over all cells has the solution Pij = Li Aj , i.e. the joint
distribution with maximum entropy would be the product of the marginals,, as-if we had
assumed the distribution of liabilities was independent of the distribution of assets. 2 But that
Pii Li Ai ≠ 0 . In accord with this problem, Upper and Worms
would imply the counterfactual=
(2004) reformulate the problem as one of finding the joint distribution as close to the product of
the marginals as possible (measured by the entropy of the former relative to the latter) when
Pii = 0 . Formally, one minimizes the mutual information subject to the known row and column
totals and the constraints Pii = 0 :
min Pi≠ j
Pij
∑ P log L A
ij ;i ≠ j
ij
i
j
.t. ∑ Pij L=
; ∑ Pij 1
s=
Aj=
i ; ∑ Pij
j ≠i
i≠ j
(3)
ij ;i ≠ j
The objective function in (3) (i.e. mutual information) is seen to be the relative entropy of the
offdiagonal probabilities Pij with respect to the product measure Lij . Using illustrative example
(1), suppose that we did not observe values of the cells but did observe the row totals li and
2
This is a consequence of Theorem 2.6.6 in Cover and Thomas (1991, p.28).
4
column totals a j needed to calculate the marginal probabilities Li and Aj listed in (1). The
Excel Solver® quickly found a solution to (3), from which the liabilities Lij = Pij / ∑ li are
i
recovered. Rounded to two decimal places (causing some minor adding-up errors), they are 3:
#1
#2
#3
#4
l
 Agent
 #1
0
3.13 4.69 2.17 10

 #2
22.36
0
32.56 15.08 70

18.89 18.36
0
12.74 50
 #3
 #4
8.75 8.50 12.74
0
30

50
30
50
30 160
 a
 A
.313 .187 .313 .187

L 
.063
.437 

.313
.187 




(4)
Comparing (4) to (1) illustrates how the entropic estimator (3) spreads the liabilities more
evenly. Three off-diagonal cells in (1) were zeroes. None of them are zero in (4). In (1), Agent
#1 owed all liabilities to a single agent (#3). In (4), that was spread across all three other agents.
The mutual information in (4), i.e. the value of the objective function in (3), is only 0.288. The
(higher) mutual information in (1) is 0.466.
This estimator is rationalized by appealing to the principle that one should incorporate all
information that is known, and nothing else. Here, what is known are the respective row and
column sums, and that diagonal elements are zero (no agent owes payments to itself). Had the
diagonal elements also been unrestricted, the solution to (3) would indeed be Pij = Li Aj achieving
the global minimum mutual information equal to zero, and hence Lij = Pij / ∑ li . If more
i
information is known, e.g. some of the individual cells’ values are observed, we need only
subtract them from their respective row and column totals, and then drop the corresponding
probabilities from being estimated by (3).
Upper (op.cit.) notes that such entropic estimators will not place zeros in any off-diagonal cell,
counterfactual with evidence garnered from more complete datasets. Interbank lending has a
tiered structure, in which numerous, smaller banks are linked to fewer, much larger “core” banks,
which in turn are linked to each other. In this “two-tiered” structure, the large number of small
banks are not linked closely with each other, introducing a degree of sparseness in L which
would not be reproduced by the above procedure. But he also notes that the analyst could
introduce additional constraints on the problem to better represent these aspects. One could
make use of statistics that characterized more complete datasets in analogous circumstances.
Again, the foundation of the resulting estimators is to incorporate known information – but
nothing else – into the constrained minimization.
3
(4) would still result had we maximized the constrained Shannon entropy rather than minimized the constrained
mutual information.
5
III. Entropic Indices of Default and Contagion Severity
In calculations concerning default, we first must consider the simultaneity problem in the
liabilities network : an agent owes funds to others, but in turn is owed funds by them. Suppose
we assume the proportional payment rule requiring each bankrupt agent i to pay a maximal
proportion 0 ≤ θi < 1 of its separate liabilities to each of its creditors, i.e. a pro rata distribution,
when it can’t fully pay all of them (if it can fully pay of them, then it does not default, i.e.
θi = 1 ). How do we know that an equilibrium clearing vector
=
θ (θ1 , θ 2 , …, θ N ) of recovery
shares exists? In other words, how do we know that the proportionality rule can be applied to all
agents simultaneously? Eisenberg and Noe (op.cit.) proved that a clearing vector does exist,
provided conditions under which it is unique, and showed that a linear programming problem
can be solved to find a maximal clearing vector.
We follow the lucid exposition of Demange (2015) to define the clearing vector problem
and its solution. The solution specifies that agent i pays agent j X ij = θi Lij , so that the aggregate
of payments from agent i will be
∑X
j ≠i
∑X
j ≠i
ji
ij
= θi li , while the aggregate of payments to agent i will be
= ∑ θ j L ji . To focus sole attention on the role of the liabilities matrix, in what follows I
j ≠i
assume that the agent has no exogenous resource ei available to pay shortfalls in the event that
the aggregate of payments made to agent i are insufficient to cover its aggregate liabilities. 4 So
the equilibrium requirement is that a vector θ between 0 and 1 satisfies the linear inequalities:
θi li − ∑ θ j L ji ≤ 0; i =
1, …, N
j ≠i
Eisenberg and Noe (op.cit.) prove that a maximal equilibrium clearing vector θ* must maximize
any increasing function of θ , in particular it could maximize the aggregate liabilities paid
throughout the network:
θ* = arg maxθ1 ,…,θ N
∑θ l
i i
i
s.t. θi li − ∑ θ j L ji ≤ 0; i =
1, …, N
(5)
j ≠i
4
Letting agents have exogenous funds to cover defaults only complicates the issues addressed here. In actuality,
rules or laws must be mutually or externally enacted and enforced to ensure that agents maintain fixed levels of
collateral that can be assigned to cover defaults, so such analyses will be situation-dependent. Also, if such
collateral requirements are high enough, there will be no initial bankruptcies, much less contagion. When
collateral requirements are less than that, simulations of default and contagion would be dependent on both the
magnitudes and the distribution of the collateral, complicating our goal of understanding the relationships
between the structure of the liability matrix L, and the default, contagion, and inequity measures developed
herein, as well as the alternative default resolution procedures analyzed herein. That understanding is enhanced
by assuming situations in which default is not a rare event, as it will be when assumed collateral is high enough.
Readers interested in estimates for a particular financial network can easily modify the analysis herein to
incorporate that network’s distribution of assignable collateral.
6
In our illustrative example (1), the Excel Solver® quickly found the solution
θ* = (1.0, 0.148, 0.344, 0.213) . We see that Agent #1 does not default, and hence pays the full 10
that it owed. As described earlier, Agent #2 had to default, because it owed 70 but was
scheduled to receive only 30 from others. In equilibrium Agent #2 only pays θ 2 = 14.8% of its
l2 = 70 total liability. This triggered a cascade in which both Agent #3 and Agent #4 defaulted,
respectively paying only 34.4% and 21.3% of their total liabilities. They both had originally
owed amounts equal to what was due from others, and hence were vulnerable to the impossible
situation of Agent #2. Only Agent #1 avoided default, due to its healthy originally situation in
which it owed 40 less than was due from others. After clearing, the realized payments X ij are:
 Agent

 #1
 #2

X ij ≡  #3
 #4
 *
 a
 A*

#4
l*
L* 

0
10
.228
2.951 10.328 .235

3.443 17.213 .392 
2.131
0
4.262
0
6.393 .146 

10
10.328 17.213 6.393 43.934


.228
.235
.392
.146

#1
#2
0
0
4.426
0
3.443 10.328
#3
10
2.951
0
(6)
Examination of the Proportional Rule’s resolution payments matrix (6) reveals a number of the
rule’s properties. We see that the defaulting Agents #2, #3, and #4 all pay out the amounts they
each receive, i.e. l *i = a*i for all of them, so the corresponding constraints in (5) are binding.
Thus the solution to (3) incorporates the common bankruptcy provision that receivables of
defaulting agents are fully paid out to creditors; nothing is left for others. While (1) shows that
40 offset by the bankruptcy triggering Agent #2’s
Agent #1 had positive “net worth” a1 − l1 =
−40 , the default clearing process wipes those out, i.e.
negative net worth a2 − l2 =
l *1 − a*=
10 − 10 and l *2=
− a*2 10.328 − 10.328 so the first and second constraints in (5) are
1
binding. While ex-ante aggregate liabilities (and hence aggregate assets) owed both totaled 160,
ex-post clearing the (objective function) total liabilities paid is only 43.934. Moreover, the
distribution of liabilities and assets (i.e. the vectors L and A) have changed. Note that the
fraction of liabilities paid ex-post by defaulting Agents #2 and #4 fell from the fractions they
owed ex-ante, while the opposite occurred for Agents #1 and #4 – despite the default of Agent
#4.
How do we quantitatively measure default and contagion? First, we define measures of default
and contagion frequency. The total fraction of defaulting agents, both those initially defaulting
and those whose subsequent defaults are triggered by them, is denoted
D = card {θi* ;θi* < 1} / N
7
(7)
In our example, that is D = 3/4. While this captures the frequency of default, it does not measure
the frequency of contagion. Two of the four agents were in the default cascade, and we define
Dc =2/4 as the contagion analog of (7).
The severity of default and contagion is an additional measure that augments the mere number of
agents defaulting, because the agents who default may be relatively large and consequential. I
propose a new, entropic measure of default severity. Recall that the shares of total liability
originally owed by the agents is the distribution L = (l1 / ∑ l j , …, lN / ∑ l j ) . After clearing, the
j
j
vector of fractions actually paid is denoted Q ≡ (θ L , …, θ LN ) . Note that the
*
1 1
*
N
elements of Q are also nonnegative, and are bounded above by the corresponding elements of L,
but do not sum to one unless there is no default (i.e. θ* ≡ 1 ). We can still define the nonnegative,
relative entropy-like Default Severity Index:
L
S ≡ ∑ Li log  i
i
 Qi

 1 
 = ∑ Li log  *  ≥ 0
 i
 θi 
(8)
which is zero if and only if all θi* = 1 , i.e. no agents default. We see that an agent contributes
more to the default severity index (8) when the agent’s share of total liabilities owed is higher
and the fraction of its liabilities paid is lower, in accord with the connotation of severity. In
addition, we have:
Proposition 1: Minimizing the Default Severity Index (8) can serve as a substitute objective
function in (5) to produce the maximal clearing vector θ * .
Proof: Eisenberg and Noe (op.cit.) proved that constrained maximization of any increasing
function of the vector θ produces the maximal clearing vector θ * . Hence it can also be
produced by constrained minimization of any decreasing function of θ , like (8). QED
The aforementioned properties make the Default Severity Index (8) is a useful measure of default
consequences. In our example (1), S = 1.46.
We can also separately measure the severity of any contagion that is triggered by a default. We
just calculate the fraction of (8) attributable to subsequent defaults triggered by initial default(s).
In the case of (1), the defaults of Agents #3 and #4 were triggered by the default of Agent #2. So
we calculate the Contagion Severity Index:
Sc ≡
L3 log(1/ θ3* ) + L4 log(1/ θ 4* )
∑ Li log(1/ θi* )
i
which is 42.7% in Example 1.
IV. Is the Mutual Information Related to Default and Contagion?
8
(9)
What are the relationships, if any, between the Shannon entropy or mutual information in the
liabilities matrix and these measures of default severity? The issue originally arose when
estimating the full liabilities matrix knowing only its row and column totals. Note that matrices
(1) and (4) have the same row and column totals. The fear was that entropic estimation of the
unknown liabilities matrix might introduce a predictably signed estimation bias when estimating
default and contagion severity from the estimated liabilities matrix.
If lower mutual information is achieved by spreading the liabilities out across agents, default by
an agent i may adversely affect more agents, but perhaps each of those agents can absorb
relatively small losses better than in matrices with higher mutual information, in which the
defaulting agent’s liabilities could (but don’t have to) be more concentrated. This would suggest
a possible positive bias indicated by a positive relationship between mutual information and
default or contagion severity, at least when comparing liabilities matrices with the same row and
column totals. But suppose the more evenly-spread liabilities in the matrix with lower mutual
information are larger than what the other agents can absorb. Then contagion may be worse than
had the large liabilities been concentrated on just one or a few other agents. This suggests a
possible negative relationship between mutual information and default or contagion severity,
among matrices with the same row and column totals.
In his survey, Upper (op.cit., sec. 4.2) summarizes the findings of earlier studies. Such studies
used different simulation methodologies and different measures of default and contagion severity
than will be used herein. He reported mixed findings among those studies. A complex set of
simulation experiments (conducted post-Upper) was conducted by Sachs (2014). She randomly
generated liability matrices in which all banks had equal total assets. She found that the
relationship between entropy and her measure of contagion (not based on the equilibrium
clearing vector) depended on the sparsity of the liability matrix.
What is the connection in our example, using our equilibrium clearing vector-based definitions?
The Shannon entropy of (1) is 2.10, while that of (4) is 2.8, while the mutual information in (1) is
0.466 vs. 0.288 for the minimal mutual information matrix (4). We saw that the lower mutual
information in (4) was indeed achieved by spreading liabilities more evenly across cells. While
in both cases one agent initially defaults and triggers subsequent defaults of two other agents, the
Default Severity Index (8) rose to S = 1.56 from 1.46 when the liabilities matrix in (4) was
substituted for the matrix in (1). We see that in this example, the mutual information (Shannon
entropy) was negatively (positively) related to the Default Severity Index (8). The Contagion
Severity Index (9) behaves similarly. It increased a bit, from 43% using matrix (1) to 45% using
the minimum mutual information matrix (4). Hence in this example, the mutual information in
the liabilities matrix and measures of default and contagion severity are negatively related,
despite the somewhat more evenly distributed liabilities occurring in the lower mutual
information matrix (4) with the same row and column totals. Hence, we see that intuition about
the effects of spreading-out liabilities can be misleading.
9
To generate more evidence, a simple, easily replicable way to simulate liabilities matrices is now
adopted. 5 First, we permute the off-diagonal elements in (1), to produce other possible liabilities
matrices with identical numbers in them. Note that permuting the off-diagonal elements will
result in matrices with the same Shannon entropy, because permutation of matrix elements will
permute the labels of the various Pij , but won’t change the sum of products defining the entropy.
Hence there cannot be any relationship between the Shannon entropy and default or contagion
measures produced from matrices resulting from these permutations. But because the row and
column totals will not be preserved by these permutations, the mutual information of these
matrices will differ, and hence in principle can be related to default and contagion severity. In
order to provide evidence based on comparisons to matrices with identical row and column
totals, each matrix produced by permutation is paired with the minimum mutual information
matrix produced from its row and column totals, i.e. its version of (4). Another advantage of this
procedure is that it fixes the network’s total liabilities (and hence network total assets) in each
pair to be the same as in the base example.
Example 1 has total liabilities of 160. 500 paired matrices were produced by permuting the offdiagonal cells to produce an analog of (1), and then using each to produce the corresponding
analog of (4). In none of the 500 pairs does the minimal mutual information matrix result in more
contagion frequency than its higher mutual information counterpart. 77% of the pairs have the
same contagion frequency, 19% have one fewer contagion default, while the rest have two fewer
contagion defaults. But the total number (i.e. initial + contagion) defaults are the same in each
pair, i.e. less contagion defaults are countered by more initial defaults. Hence, among matrices
with the same row and column totals, it would be misleading to infer that total default frequency
is lower when the mutual information is lower. Moreover, there is no relationship between the
change in mutual information and the change in default severity (8) or contagion severity (9).
To illustrate, Figure 1 plots the (positive, by construction) decrease in mutual information
between each pair and the corresponding change in Default Severity Index (8). Across the 500
pairs, the mean change in default severity was near zero (0.03) with a standard deviation over 5
times that. The scatterplot shows little connection between the two, with a correlation coefficient
of -11%. Because there is no theoretical reason to believe the connection is linear, perhaps
Kendall’s rank correlation τ is a better measure of correlation, but it is only -3.6%. Similar
findings occur when substituting contagion severity (9) for default severity (8). On average over
the pairs, it dropped a small amount (0.06), but with a standard deviation more than twice that.
The Kendall τ rank correlation between the decrease in mutual information and change in
contagion severity is only 9.6%.
Now suppose we do not wish to hold anything constant except the actual numbers appearing in
Example 1, i.e. instead of permuting the elements, we produce a different 500 pairs by
bootstrapping the off-diagonal elements in (1). That is, we sample the off-diagonal elements in
(1) with replacement rather than without, again pairing each resulting matrix with its minimum
5
The more complex simulation methodologies used in the cited papers are more extensive, but not as easily
replicable by outside parties, and still rely on specific examples or heroic assumptions, e.g. Sachs’ (op.cit.)
assumption holding fixed the distribution of assets across agents.
10
mutual information estimate having the same row and column totals. In contrast to the
permutations, this will produce liabilities matrices with different total network liabilities. We
still found no connection between the change in mutual information and the change in either the
severity or contagion indices. The Kendall τ correlations were all below 5%. Corroborating
visual evidence in seen in Figure 2, a scatter plot of the positive decrease in mutual information
and the corresponding change in default severity for each of the 500 pairs.
Why Isn’t Mutual Information More Closely Related to Default and Contagion?
The mutual information is a measure of the dependence between the row proportions vector
L1 , …, LN and the column proportions vector A1 , …, AN considered as two random variates
determined by a random liability matrix L . While the mutual information is zero when the row
and column proportions are independent, it is always positive regardless of whether the
dependence (e.g. correlation) is positive or negative. But there is a signed dependency measure
that is closely connected to the severity and contagion indices. That characteristic is the rank
correlation between agent liabilities and agent assets. The (sound) intuition is that default and
contagion will be more severe when agents with relatively high total liabilities have relative low
total assets from which to pay them. Because there is no reason to expect a linear connection, we
surmise that the Kendall rank correlation τ L , A between the agents’ respective shares of liabilities
and assets will be negatively related to the Default Severity and Contagion Severity Indices.
Evidence for that is now provided.
Figure 3 uses the same 500 bootstrap replications of Example 1 to illustrate the negative
relationship between the Default Severity Index S and τ L , A . 6 Figure 4 shows that the Contagion
Severity Index Sc in (9) is positive much more often when that rank correlation is negative than
when it is positive, in accord with the above intuition.
The mutual information of the liabilities matrix is a measure of dependence between the row
(liability) and column (asset) totals. But isn’t a signed measure of dependence, like the Kendall
τ rank correlation. We have
Proposition 2: Figures 3 and 4 show that negative dependence (i.e. τ L , A < 0) between agents’
liabilities and assets is directly related to default and contagion. While mutual information is a
measure of that dependence, it is nonnegative so can be high when dependence is positive as
well as negative. This is why there is no definitive link between mutual information and default
and contagion.
Because the two vectors have only 4 elements apiece, Kendall’s τ can only assume a small number of values.
This accounts for the discreteness of the horizontal axis values plotted in Figures 3 and 4.
6
11
V. Inequity of Bankruptcy Resolutions: An Entropy Measure
Given the strict proportional payment clearing rule, the clearing vector θ * minimized the default
severity, which could be taken as a measure of efficiency in bankruptcy resolution. But what
about fairness? There can be dimensions to fairness of default resolution other than preventing
defaulting agents from paying different percentages of what it owes to its creditors. In example
(1), the agents’ respective shares (subject to a bit of rounding error) of the 160 in total liabilities
originally owed are L = (6.3%, 43.7%,31.3%,18.7%) . Ex-post clearing, (6) shows that their
corresponding shares of the 43.93 in total liabilities paid are L* = (22.8%, 23.5%,39.2%,14.5%) .
How fair is it that these share vectors are different? Perhaps the fairest outcome would be if the
distribution of ex-post shares paid L* = L , which is the distribution of ex-ante shares owed.
Barring the feasibility of that, it is proposed that the entropy of L relative to L* be used to
measure the inequity of the bankruptcy resolution. That is, our measure of inequity I is:
I = ∑ Li log
i
Li
L*i
(10)
Plug the last columns of (6) and (1) into (10) to calculate I = 0.168. A little algebra shows that
the Inequity Index (10) and Default Severity Index (8) are related by
I= S + log(∑ θi* Li ) ≡ S + log(∑ θi*li / ∑ li ) .
i
i
(11)
i
That is, the Inequity Index is the Default Severity Index, plus the log of total liabilities paid as a
fraction of what was originally owed. We see that it embodies a tradeoff between the two
desiderata of low default severity and high total liabilities paid (the objective function in (5)),
when the latter is weighted by its log. 7
As an alternative to maximizing
∑θ l in (5) (or equivalently, minimizing the Default Severity
*
i i
i
Index (8)), we can consider finding a clearing vector that minimizes the Inequity Index (10). In
Example (1), it turns out that the respective solutions are the same. To see this, we first re-write
the strict proportional payment clearing constraints in (5) as the system of linear
inequalities M θ ≤ 0 . Eisenberg and Noe (op.cit., Theorem 1) prove that even if there is more
than clearing vector θ , the negative slack (if any) in each inequality i (which is li* − ai* ) is the
same, and there is no slack for any defaulting agent (i.e. θi < 1 ). For the clearing
vector θ * solving (5) and used to produce (6), only Agent #1 does not default, and that constraint
also has no slack (like the three defaulting agents’ constraints). Hence even if there were
multiple clearing vectors, they must be solutions of the linear equations M θ = 0 , i.e. they must
be in the null space of M. In example (1), this null space turns out to be one dimensional,
7
If the latter weren’t weighted by its log, there would be no tradeoff, because Proposition 1 shows that
constrained minimization of the default severity index produces the same clearing vector as constrained
maximization of total liabilities paid.
12
spanned by the vector (61/ 13,9 / 13, 21/ 13,1) . Thus a clearing vector must be proportional to
this vector. Because the constant of proportionality cancels when one computes the vector L*
used in (10), any clearing vector yields the same value of (10), i.e. I = 0.168 .
VI. Would Deviations from Strict Proportionality Be Desirable?
Perhaps deviation from the strict proportional payment rule would permit clearing with less
default severity (8) and/or less inequity (10). In Example (1), suppose the cascade-triggering
Agent #2 is permitted to pay different fractions of what it owes to the other agents. Each other
agent is still required to pay a constant proportion. Denote a resulting (not necessarily
proportional) clearing vector by θ = (θ1 , θ 21 , θ 23 , θ 24 , θ3 , θ 4 ) , in which case the clearing vector must
satisfy the modified inequalities from (5):
0 −10 −10 
 10 −30 0
 0
30 20 20 −30 0 

 −10 0 −20 0
50 −20 


0
0 −20 −10 30 
 0
 θ1 
θ 
 21 
θ 23 
 ≤
θ 24 
 θ3 
 
 θ 4 
0 
0 
 
0 
 
0 
0 
 
0 
(12)
This also necessitates modifying the objective function in (5) to be the inner product of the 6component θ with the vector (10,30,20,20,50,30), whose 2nd – 4th components reflect the
possibility of separate fractions payable by Agent #2 to Agents #1, #3, and #4. The constrained
maximum of that subject to (12) is
θ max = (1, 0, 1, 7.1%, 71.4%, 28.6%) . From θ max we see that Agent #1 still pays 100% of the 10
that it originally owed. Agent #2 pays none of the 30 it owed to Agent #1, while paying 100% of
the 20 it owed to Agent #3, and 7.1% of the 20 it owed to Agent #4, summing to 21.429 of the 70
it originally owed the others. Agent #3 pays 71.4% of the 50 that it originally owed, and Agent
#4 pays 28.6% of the 30 that it originally owed. Aggregate to find that of the 160 originally
owed by them, the agents respectively paid Q = (6.3%, 13.4%, 22.3%, 5.4%) of it. They
originally owed shares of that 160 equal to L seen in the last column of (1). Using these vectors
Q and L, the Default Severity Index (8) is 0.858, much lower (i.e. better) than its value with
proportional clearing (1.46). The Contagion Severity Index (9) is 39.6%, which is also lower
(i.e. better) than its value with strict proportional clearing (42.7%). Hence, we see that both
default and contagion severity improve when Agent #2 is allowed to pay different percentages of
the liabilities it owes to the other agents, i.e. when Agent #2 is permitted to violate the
Proportional Rule. The liabilities X ij paid after this modified clearing procedure are reproduced
below:
13
 Agent

 #1
 #2

X ij =  #3
 #4

 a max
 max
 A
L* 

0
0
10
0
10
.132 
0
0
20
1.429 21.429 .283

7.143 21.429
0
7.143 35.714 .472 
2.857
0
5.714
0
8.571 .113


10
21.429 35.714 8.571 75.714


.132
.283
.472
.113
#1
#2
#3
#4
l
max
(13)
Total liabilities of 75.714 are paid in the modified clearing equilibrium, vs. 43.93 paid under the t
Proportional Rule. So that performance measure is better, too. But what happens to inequity, i.e.
how close is (10) to zero? Plug the last columns of (13) and (1) into (10) to calculate
that I = 0.110 , compared to I = 0.168 under the proportional clearing rule. Hence inequity is
lowered (i.e. improved) under the modified clearing procedure. Treatment of the initially
defaulting Agent #2 seems reasonable, too. Comparing (13) to (6) shows that the initially
defaulting Agent #2 pays far more to Agent #3 and only somewhat less to Agents #1 and #4,
resulting in total payments close to 11 higher than occurred under strict proportionality.
In summary, permitting the cascade-triggering Agent #2 to deviate from the proportional
payment rule resulted in less default and contagion severity and less inequity in recovery.
How common is this phenomenon? Among our 500 bootstrapped liability matrices, there are 39
in which the only initial default is by Agent #2. In each case, the default severity index is lower
when the proportional payment requirement is relaxed for Agent #2 in the way described above.
The decline in default severity averaged 31%. In no case did the default by Agent #2 trigger
more defaults. In 21 of the 39 cases the number of triggered defaults remained the same, one
less agent defaulted in 14 of the cases, while two less agents defaulted in the other 4 cases.
Hence contagion frequency was never greater, and frequently was lower. Not surprisingly,
contagion severity also declined in 37 of the 39 cases (the other two cases already had very low
contagion severity), by an average of 68.7%. The inequity index (10) also declined in 32 of the
39 cases. The average decline was a modest 19.1%. We have
Proposition 3: The decrease in default and contagion severity enabled by relaxing the
proportional payment requirement for the initial defaulting Agent #2 is not occurring at the
expense of higher inequity. The upshot of this simulation is that the previous calculations with
Example 1 are not misleading.
Moreover, the null space of the matrix in (12) has dimension 3. As such, it is possible that
constrained minimization of the inequity index (10) or (11) will yield a clearing vector different
than the clearing vector θ max = (1, 0, 1, 7.1%, 71.4%, 28.6%) by maximizing aggregate liabilities
paid when the proportional payment rule was relaxed for Agent #2. The constrained
minimization of (10) yields θ min = (1, 0, 68.5%, 25.1%, 62.4%, 37.6%) that achieves the
minimum I = .098 versus the slightly more inequitable I = 0.110 achieved by maximizing
14
aggregate liabilities paid. The liabilities paid, after clearing that minimizes the Inequity Index,
are reproduced below:
 Agent
#1
#2
#3
#4
l*

0
0
10
0
10
 #1
 #2
0
0
13.704 5.026 18.730

X ij =  #3
6.243 18.730
0
6.243 31.217
 #4
3.757
0
7.513
0
11.270
 *
10 18.730 31.217 11.270 71.217
 a
 A*
.140
.263
.438
.158

L* 

.140 
.263

.438
.158




(14)
Comparing (13) to (14), we see that the tradeoff for achieving the modest decrease in inequity
(from .110 to .098) is a modest loss of 75.714 - 71.217= 4.5 in aggregate liabilities paid
corresponding to a modest increase in default severity from 0.858 to 0.909.
Comparing (13) to (14), we see that permitting the cascade-triggering Agent #2 to deviate from
the proportional payment rule resulted in different bankruptcy resolutions, depending on
whether one wants the highest feasible aggregate payments (equivalently, the lowest feasible
default severity index) or the lowest feasible inequity.
VII. Is the Proportional Rule Really Equitable? Game-Theoretic Insights
Results in the previous section raise questions about the Proportional Rule’s merit. Fortunately,
there is a large literature that examines the normative basis for the Proportional Rule and
plausible alternative rules. Thomson (2003 and 2015) has authored two surveys of this large
literature. The literature considers a single agent i who owes amounts Lij > 0 to N separate
claimants, but only has resources ai < ∑ Lij ≡ li available to pay them. Generalization of this
j ≠i
set-up to our networks -- in which debtors and claimants are not mutually exclusive -- has not
heretofore been done, but will be later in this section.
Theorem 1 in Thomson (2003) summarizes results in Dagan and Volij (1993) that use gametheoretic bargaining models to characterize both the Proportional Rule and the very plausible
alternative of equalizing dollar (rather than percentage) payments across the sole defaulter’s
claimants. Thomson refers to the latter as the Constrained Equality (CE) Rule. Using X ij ≤ Lij to
denote the payments made by the sole defaulting agent i after resolution, the CE Rule identifies a
constant Ci such that :
X ij
min
=
( Lij , Ci ) s.t. ∑ X ij ai and X ij ≤ Lij (CE)
(15)
j ≠i
as opposed to our previously modeled Proportional Rule, which identified a constant θi such
that X ij θ=
ai and X ij ≤ Lij . In (15), the min operator ensures that no claimant is
=
i Lij s.t . ∑ X ij
j ≠i
15
paid more than that claimant is owed (i.e. X ij ≤ Lij ), and that all claimants j are paid the same
amount Ci when their respective inequalities are strict, i.e. claimants not made whole receive the
same dollar (rather than percentage) payment. 8
In our example (1), the literature applies when considering the plight of Agent i = 2 in isolation,
=
L21 30,
=
L23 20, and
=
L24 20 but only has a2 = 30 available to pay them. The CE
who owes
= X 23
= X 24= C=
10 , i.e. the claimants split the
Rule (15) resolves the situation by setting X 21
1
defaulter’s assets equally. Had a2 = 65 instead, the CE resolution would have been
X 21 = 25, X 23 = X 24 ≡ C1 = 20 , because the third and fourth agents can’t be paid more than they
are owed (i.e. 20 apiece).
Consider the bargaining game between the sole defaulter, denoted i, and the multiple
claimants j ≠ i who cannot all be made whole. A constrained feasible vector of payments X i
satisfies X ij ≤ Lij and ∑ X ij = ai 9. In the absence of a bargaining solution, it is assumed that
j ≠i
each claimant receives nothing. A Nash Bargaining solution is a feasible vector of payments
that maximizes the product of claimants’ utility gains ∏ [U j ( X ij ) − U j (0)] , where U j is a (not
j ≠i
necessarily strictly) concave utility function. A weighted Nash Bargaining solution for
w
weights wij maximizes ∏ [U j ( X ij ) − U j (0)] ij . Theorem 1 in Thomson (2003) cites the Dagan
j ≠i
and Volij (1993) results that when utility functions U j are linear with U j (0) = 0 , (i) the CE
Rule (15) is a Nash Bargaining solution, while (ii) the Proportional Rule is a weighted Nash
Bargaining solution when the weights wij = Lij . The former result reflects the presumption that
the creditors have equal bargaining weight over the default resolution process, while the latter
reflects the presumption that those who are owed more have logarithmically more weight in
proportion to what they are owed. On this basis and others summarized in Thomson (2003), the
CE Rule deserves consideration as an alternative to the Proportional Rule heretofore analyzed.
As noted, the formulations in Dagan and Volij (op.cit.) use linear utility functions U ij = X ij ,
while our formulations above assumed concave utilities; moreover, their proofs are informal. It
is insightful to see the following formal proof of this more general result.
Levinthal’s (1918) history of early bankruptcy law notes that ancient Jewish law required equal dollar (rather than
percentage) payments to creditors, subject to the constraints that this would not compensate any creditors more
than they were owed (op.cit, p.234). Indeed, Thomson (2003) writes that this was advocated by Moses
Maimonides, the influential 12 th Century Jewish Sage. This is not what is sometimes called the “Talmud Rule”,
see Thomson (2003, p.256).
8
9
Replacing the equality with the less than or equal to sign is possible, but the desideratum of Pareto Optimal
payments implies that rules will result in payments that sum to ai .
16
Proposition 4: When there is a single defaulting agent i , and claimants j ≠ i have concave
utilities U j ( X ij ) with identical threat points U j (0) = 0 , the Nash Bargaining solution is a CE
rule.
Proof:
The Lagrangian for the problem is
∏

=
j ≠i


U j ( X ij ) − ∑ λ j ( X ij − Lij ) − γ  ∑ X ij − ai 
j ≠i
 j ≠i

We assume U j ( X ij > 0) > 0 and that U j ' > 0 and U "j ≤ 0 . Then one can replace the objective
function
U ′j ( X ij* )
*
ij
U j (X )
∏
j ≠i
U j ( X ij ) := log ∑ U j ( X ij ) in the Lagrangian. The first order conditions are:
j ≠i
=
λ *j + γ * , j ≠ i
(16)
*
2
d U ′j ( X ij )  U ′′jU j − U ′j
Because we assume concave utility, U ≤ 0 so
=
< 0 . Hence the left


dX ij U j ( X ij* ) 
U2
''
j
hand side of each equation j in the first order condition is monotone (decreasing) and must have
an inverse, dubbed I j . So=
X ij* I j (λ *j + γ * ) . If X ij* < Lij , the complementary slackness condition
implies that λ *j = 0 in (16), in which case the equations yield X ij* =
I j (γ * ) ≡ Ci , ∀j:X*ij < Lij . In
other words, all creditors of agent i who are not made whole receive equal dollar payments Ci.
So X ij* = min( Lij , Ci ) in accord with the CE Rule (15). QED
While Proposition 4 shows that the Dagan and Volij (1993) proof can be modified to
accommodate heterogeneous concave utilities rather than just their linear utilities, their result
equating (liability-) weighted Nash Bargaining with the Proportional Rule does not similarly
generalize. To see this, take the log of the weighted objective function to find the Lagrangian for
that problem:
=
l
∑
j ≠i
Lij log(U j ( X ij )) − ∑ j ≠i λ j ( X ij − Lij ) − γ (∑ X ij − E )
j ≠i
With first order conditions:
U ′j ( X ij* ) λ *j + γ *
=
, j≠i
U j ( X ij* )
Lij
17
As before, if X ij* < Lij , λ *j = 0 , and upon inverting the first order conditions one
obtains X ij* = I j (
γ*
Lij
) . Because I j is monotone decreasing, X ij* is a monotone increasing function
of Lij , denoted X ij* = f j ( Lij ) . For some other agent j’, if X ij* ' < Lij ' , f j ' is a possibly different
increasing function of Lij ' . This is because it is derived from its own agent’s marginal log utility,
and hence may not be the same function. But the Proportional Rule requires that these be the
same increasing linear function X ij* = θi* Lij , for j , j ′ .
Shummer and Thomson (1997) provided alternative constrained optimization characterizations
of the CE Rule. Their Proposition 3 characterizes the CE required payments as the feasible
payments minimizing ``the largest amount received by any agent and the smallest such amount”
while their Proposition 4 characterizes them as the feasible payments minimizing “the variance
of the results received by all the agents”. 10 Due to the close connection between variance and
entropy, and the fact that the unconstrained maximum entropy distribution is uniform, the latter
characterization suggests the possibility that there is also an entropic characterization of the CE
rule. This is indeed the case, as shown below.
Proposition 5: Define the (unnormalized) entropy
∑ −X
j ≠i
ij
log X ij of the payments X ij . The CE
Rule (15) produces payments that solve the following constrained maximization of entropy:
max − ∑ X ij log X ij s.t.
j ≠i
X ij ≤ Lij , ∀j ≠ i
∑X
j ≠i
ij
(17)
= ai
Proof: The Lagrangian for the problem (17) is


 −∑ X ij log X ij + ∑ λ j ( X ij − Lij ) + γ  ∑ X ij − ai 
=
j ≠i
j ≠i
 j ≠i

The first order conditions result in the following equations for the solutions X ij* :
λ*
X ij* = e j eγ
*
−1
. If X ij* < Lij , the complementary slackness condition implies that λ * j = 0 , in
=
which case the equations
yield X ij* eγ
*
−1
≡ Ci , ∀j:X*ij < Lij . In other words, all creditors not
10
Readers of these papers should note that my use of the term “feasible” incorporates the constraint that no
claimant receive more than it is owed. The propositions in those papers also impose that constraint, but not in
their definition of feasible allocations. As such, there is no loss in generality when imposing it in the feasibility
condition.
18
made whole receive equal dollar payments. So X ij* = min( Lij , Ci ) in accord with the CE Rule
(15). QED
Generalizations to Financial Networks: Some Important Differences
In a network like our example (1), the systemic consequences of agent #2’s default must also be
considered. When implementing the Proportional Rule in Section III, we depended on the
existence proof of Eisenberg and Noe (op.cit., p.240) . They created a mapping Φ on the feasible
set of payments vectors, whose maximal fixed point determines the Proportional Rule solution.
Existence of a maximal (and a minimal) fixed point is implied by the Knaster-Tarski Fixed Point
Theorem.
While their mapping does not straightforwardly extend to the CE Rule, a different mapping can
be used for this purpose. Under the CE rule, the feasible set is defined by a set of nonnegative
numbers C1 , …, CN ≡ C such that ∑ min( Lij , Ci ) ≤ ∑ min( L ji , C j ) i =
1, …, N . The left hand
j ≠i
side is
∑X
j ≠i
∑X
j ≠i
ji
ij
j ≠i
≡ li (C) , i.e. aggregate payments made by Agent i, while the right hand side is
= ai (C) , i.e. the aggregate of payments received by it from the other agents. We prove:
Proposition 6: In a network under the CE Rule, the feasible set is nonempty. Moreover, the
C * (C1* , …, CN* ) defining its payments X ij* = min( Lij , Ci* ) can be found by solving
maximal =
max X ij
∑∑ X
i
j ≠i
ij
≡ max C1 ,…,CN
∑∑ min( L , C ) s.t.
i
j ≠i
ij
i
1, …, N
li (C) ≡ ∑ min( Lij , Ci ) =
ai (C) ≡ ∑ min( L ji , C j ), i =
j ≠i
(18)
j ≠i
Proof: Using the above notation for aggregate liabilities paid and assets received by the agents,
define the vector-valued maps l (C) ≡ (l1 (C), …, lN (C)) and a (C) ≡ (a1 (C), …, aN (C)) on the
1, …, N . This subset is a complete lattice
subset S of vectors C in which Ci ∈ [0, max j ≠i Lij ] , i =
with the usual ordering ≤ of N-vectors. l (C) is monotone increasing on S , and hence has an
inverse. Because a (C) is monotone nondecreasing on S , the map f : S → S ; f (C) = l −1 (a (C)) is
monotone on the complete lattice S . A fixed point of f satisfies the constraints in (18). By
the Knaster-Tarski Fixed Point Theorem 11 , the map has a set of fixed points which is also a
complete lattice (and hence nonempty), and hence has a maximal element. So it can be found by
solving (18). QED
2
1

Solving (18) with the liabilities matrix in our example (1), find C = 10, 1 , 5, 3  . The
3
3

payments matrix X ij = min( Lij , Ci ) is shown below:
11
See https://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem
19
 Agent #1
#2
#3
#4
l*

0
0
10
0
10
 #1
 #2
12
0 12
12
5
3
3
3

 #3
5
5
0
5
15

 #4
31
0
31
0
62
3
3
3

*
 a
10
5
15 6 2
36 2 3
3

 A*
.273 .136 .409 .182
L* 

.273
.136 

.409 

.182 




(19)
Comparing the CE Rule payments (19) to the Proportional Rule payments (6), we see that total
liabilities paid were less under the CE Rule (36 2/3 vs. 43.934). The major contributor to this
difference is the initially defaulting Agent #2, who owed 70 and paid 10.3 under the Proportional
rule, but only 5 under the CE rule. The Inequity Index (10) is the entropy of the distribution of
agent liabilities originally owed L = (6.3%, 43.7%,31.3%,18.7%) to the distribution of what is
paid: from (19) that is L* = (27.3%,13.6%, 40.9%,18.2%) . Using (10), the Inequity Index
I = .340 , substantially higher than its value I = .168 under the Proportional Rule. This is not
surprising, because the Inequity Index (10) penalizes deviation of the distribution of percentage
liabilities paid from the distribution of original percentages owed. The Proportional Rule
resulted in the initially defaulting Agent #2 paying around 24% of total liabilities paid, much
closer to the 43.8% of liabilities originally owed than what is required by the CE Rule, which
required Agent #2 to pay only about 14% of liabilities paid. This is the major contributor to the
increase in the Inequity Index arising from the CE Rule.
While Proposition 6 characterizes the extension of the CE Rule to networks, it does not imply
that the CE Rule is the Nash Bargaining solution within the network, as it is when there is only a
single exogenous defaulter with multiple creditors (see Proposition 4). With a network, the
Nash Bargaining solution solves the following problem:


max X ij ∏i U i  ai ≡ ∑ X ji  s.t.
j ≠i


X ij ≤ Lij
li ≡ ∑ X ij =
j ≠i
∑X
j ≠i
ji
(20)
≡ ai , i= 1, …, N
The Lagrangian is:
=

∏ U (∑
i
i
j ≠i
)


X ji − ∑ ∑ λij ( X ij − Lij ) − ∑ γ i  ∑ X ij − ∑ X ij 
i
i j ≠i
j ≠i
 j ≠i

As in the proof of Proposition 4, we can substitute the log of the product of utilities and write the
first order conditions with respect to each X ij as:
20


U ′j  ∑ X ji* 
 j ≠i
 λ * + γ * − γ * , ∀i, j ≠ i
=
ij
i
j

* 
U j  ∑ X ji 
 j ≠i

(21)
As in the proof of Proposition 4, for any j we can invert the left hand side to derive
∑
j ≠i
X *ji =
I j (λij* + γ i* − γ *j ), j ≠ i
Now if Agent i pays two agents j and j’ less than they are respectively owed, complementary
slackness implies that λij* = λij*′ = 0 . But due to the presence of γ j * and γ j '* in system (21), this
does not force X
X ij′ ≡ Ci as it does in the first order conditions (16) that lead to Proposition
=
ij
4. That is, an agent i does not have to pay the same amount to two other agents who each
receive less than owed. In networks, the Nash Bargaining solution does not imply the CE rule.
To illustrate this using our example (1), we again assume linear utilities and numerically solve
(20) to find:
 Agent #1 # 2 #3 # 4
l*

0
0
10
0
10
 #1
 #2
0
0
10
20 30

0
30
0
10
40
 #3
 #4
10
0
20
0
30
 *
10
30
40
30 110
 a
 A*
.091 .273 .363 .273

L* 

.091
.273

.363
.273




(22)
The Nash Bargaining solution (22) is neither the Proportional Rule solution (6) nor the CE Rule
solution (19). Total liabilities paid is higher in the Nash Bargaining solution than in either of
them (110 vs. 43.93 and 36 2/3, respectively), and accordingly the Default Severity Index (8) is
much lower (.440 vs. 1.68 and 1.81). Moreover, recall that the Inequity Index (10) is the relative
entropy of the distribution of agent liabilities owed L = (6.3%, 43.7%,31.3%,18.7%) relative to
the distribution of what is paid: from (17) that is L* = (9.1%, 27.3%,36.3%, 27.3%) . The Nash
Bargaining solution’s distribution of liabilities paid appears closer to the distribution of what was
originally owed than do the distributions resulting from the Proportional Rule
L* = (22.8%, 23.5%,39.2%,14.5%) and the CE Rule L* = (27.3%,13.6%, 40.9%,18.2%) . Not
surprisingly the relative entropy, i.e. our Inequity Index (10), is only I = .066 with the Nash
Bargaining solution, which indeed is lower than the Inequity Index values resulting from the
Proportional Rule ( I = .168 ) and the CE Rule ( I = .340 ). We summarize our findings as
Proposition 7:
21
Proposition 7: In a network, the Nash Bargaining solution to (20) does not implement either the
CE Rule nor the Proportional Rule, even when utilities are linear. At least in our example, it
produced both higher efficiency (i.e. higher total liabilities paid or lower Default Severity Index
(8)) and lower inequity (10) than those rules did.
It is interesting to compare the Nash Bargaining solution with linear utilities (22) to the liabilityweighted Nash Bargaining solution computed from the same liability matrix (1). It is:
 Agent #1
#2
#3
#4
l*
L* 


0
0
10
0
10
.091
 #1
 #2
0
0 13.75 16.25
30
.273


3.75 30
0
10
43.75 .398
 #3
 #4
6.25 0
20
0
26.25 .278
 *

10
30 43.75 26.25 110
 a

 A*

.091 .273 .398 .278


(23)
In general, the liability-weighting should shift payments toward those who were originally owed
more. Comparing (23) to (22), we see that liability-weighting resulted in slightly more going to
Agent #3 and slightly less going to Agent #4. This is not surprising, because (1) shows that
Agent #3 was owed 50 while Agent #4 was owed only 30. The total liabilities paid are 110, the
same as in the (un-weighted) Nash Equilibrium, and the Default Severity Index (8) is .437.
However, there are multiple solutions, because (22) achieves the same value of the liabilityweighted Nash objective function as does (23). Not surprisingly, the liability-weighted solution
(23) is inconsistent with the Proportional Rule, e.g. Agent #2 pays 0% of the 30 it owed to Agent
#1, while paying around 69% of the 20 it owed Agent #3, and around 81% of the 20 it owed
Agent #4. In summary, the single-debtor characterization of the CE (Proportional) Rule as a
Nash (liability-weighted Nash Bargaining) solution does not generalize to financial networks.
An Extension: Permitting Netting Before Resolution
Mokal (2015) highlights bankruptcy provisions in many countries that provide priority
treatment to counterparties of bankrupt agents who have entered into swaps, repos, and/or some
other derivative securities with them. The counterparties are permitted to net their claims against
each other before general bankruptcy resolution procedures take place.
In our example, suppose the initially bankrupt Agent #2 and is permitted to net its
promised cash flows before a bankruptcy resolution procedure. For convenience, the original
liabilities matrix (1) is reprinted below:
22
#2
#3 # 4
l
 Agent #1
 #1
0
0
10
0
10

 #2
30
0
20
20 70

Lij ≡  #3
10
30
0
10
50
 #4
10
0
20
0
30

50
30
50
30 160
 a
 A
.313 .187 .313 .187

L 
.063
.437 

.313
.187 




Note that Agent #2 owes 20 to Agent #3, who in turn owes 30 to Agent #2. If Agent #2
is permitted to net that liability against what it is owed, Agent#2 will owe nothing to Agent #3,
while Agent #3 will owe only 10 to Agent #2:
Lnet ij
l
#2
#3 # 4
 Agent #1
 #1
0
0
10
0
10

 #2
30
0
0
20 50

≡  #3
10
10
0
10
30
 #4
10
0
20
0
30

50
10
30
30 120
 a
 A
.313 .187 .313 .187

L 
.083
.417 

.250 
.250 




(24)
Mokal (op.cit.) argues that some policymakers have encouraged such netting prior to
bankruptcy proceedings, based in part on the claim that it serves to mitigate contagion. Mokal
cites guidance from the U.N. Commission on International Trade Law (2005), which wrote:
“Without the ability to close out, net, and set-off obligations…a debtor’s failure to
perform its contract…could lead the counterparty to be unable to perform its financial contracts
with other market participants. The insolvency of a significant market participant could result in
a series of back-to-back transactions, potentially causing financial distress to other market
participants, and in the worst case, resulting in the financial collapse of other counterparties,
including regulated financial institutions. This domino effect is often referred to as systemic risk,
and is cited as a reason as a significant policy reason for permitting participants to close out,
net, and set off obligations in a way that normally would not be permitted by insolvency law.”
While Mokal (op.cit.) persuasively argues that this source of contagion was not a significant risk
factor in the Financial Crisis of 2008 12, it is interesting to study its effects within the context of
our example, meant to model a relatively severe contagion situation that could arise in payments
and trading networks.
12
See Mokal (op.cit., p. 45). In fact, Mokal (op.cit.) further argues that an alternative contagion channel
(precipitated by adverse common shocks, resulting in asset sale-induced decreases in collateral values and
subsequent bankruptcies induced by inadequate collateralization) did play an important role in the Financial Crisis,
and that netting exacerbates that contagion channel. Hence Mokal opposes netting prior to bankruptcy
resolution.
23
Substituting (24) for (5) will affect the bankruptcy resolution payments matrix X ij . If the
Proportional Rule is used, substitute (24) into (5) and solve to find the clearing fractions paid
θ*net = (100%,9.7%, 48.4%, 22.6%) vs. the previously calculated fractions
θ* = (100%, 14.8%, 34.4%, 21.3%) . Instead of the resolution matrix (6), netting prior to
bankruptcy will produce the resolution matrix:
 Agent

 #1
 #2

X ij ≡  #3
 #4
 *
 a
 A*

L* 

.277 
.134 

.402 
2.258
0
4.516
0
6.774 .1875

10
4.839 14.516 6.774 36.129


.277 .134
.402
.188

#1
#2
0
0
2.903
0
4.839 4.839
#3
10
0
0
#4
l*
0
10
1.935 4.839
4.839 14.516
(25)
So in this example, permitting the bankruptcy triggering Agent #2 to net its exposures prior to
resolution does not deter contagion – Agents #2, #3, and #4 still wind up defaulting. In fact,
netting before applying the Proportional Rule results in a slight increase in the Default Severity
Index S from 1.46 (calculated using the original liabilities matrix (1) and the Proportional Rule’s
resolution matrix (6)) to 1.53 (calculated using the netted liabilities matrix (24) and the
Proportional Rule’s resolution matrix (25)), and an increase in the Inequity Index I from 0.168 to
0.326.
Of course, the Nash Bargaining rule analyzed earlier will also result in a different resolution
payments matrix when netting is permitted. In our example, the Nash Bargaining solution
yields:
 Agent #1 # 2
#3 # 4

0
0
10
0
 #1
 #2
0
0
0
10

X ij ≡  #3
10
10
0
10
 #4
0
0
20
0
 *
10
10
30
20
 a
 A*
.143 .143 .428 .286

l*
10
10
30
20
70
L* 

.143
.143

.428
.286 




(26)
Netting before applying the Nash Bargaining Rule results in an increase in the Default Severity
Index S from 0.440 (calculated using the original liabilities matrix (1) and the Nash Bargaining
resolution matrix (22)) to 0.772 (calculated using the netted liabilities matrix (24) and the Nash
Bargaining resolution matrix (26)), and an increase in the Inequity Index I from 0.066 to 0.233.
Hence in this example, netting does not improve default severity or inequity of its resolution.
24
VIII. Conclusions
The mutual information statistic is defined using the Kullback-Leibler relative entropy. It has
heretofore been used as an objective function in constrained minimization problems for
estimating unknown cells in inter-agent payments matrices, and analogous matrices arising in the
social sciences. Inter-agent payments matrices are inputs to studies estimating the frequency and
severity of default, which includes both initial defaults and secondary defaults induced by the
initial defaults (a.k.a. contagion). The mutual information is a measure of dependence between
the distribution of agents’ respective liabilities (money owed) and assets (money that is owed to
them), but it is positive regardless of whether that dependence is positive or negative. It is
argued that only negative dependence (e.g. negative Kendall rank correlation) clearly results in
more default and contagion severity. So it isn’t surprising that the simulations conducted in this
paper did not demonstrate a fixed relationship between the mutual information in the liabilities
matrix and measures of default and contagion severity, including new, relative entropy-based
measures devised for and used throughout this paper.
The standard default resolution procedure forces a defaulting party to pay the same percentage of
what it owes each of its claimants. For example, if a defaulting party has enough funds to pay
80% of its total claims, the standard procedure requires it to pay 80% of what it owes to each of
its claimants. This is dubbed the Proportional Rule. When a single agent’s default triggers other
defaults (i.e. there is contagion), it is shown that relaxing this restriction it will often result in
both lower default and contagion severity as well as less inequity in recovery. Minimizing
inequity rather than default severity is a feasible alternative objective, but may not be nearly as
important as relaxing the strict proportional payments constraint on agents that trigger defaults.
In light of these findings, investigation of the game-theoretic bargaining literature uncovers
reasons favoring alternative default resolution rules. But that literature has heretofore only
considered the case where only one agent is a debtor to multiple claimants. In that limited
setting, the Nash Bargaining solution requires equal dollar payments (when not greater than what
is owed), rather than percentage payments. This is termed the Constrained Equality (CE) Rule.
Herein, the game-theoretic analysis was extended to financial networks, where potentially all
agents are both debtors and claimants. A fixed point argument was used to prove that there will
be a CE Rule resolution payments matrix, but it was also shown the Nash Bargaining solution no
longer implies that it be used. In fact, the Nash Bargaining solution will not feature either fixed
percentage or fixed dollar payments. The same negative result is obtained when comparing the
Proportional Rule to the liability-weighted Nash Bargaining solution. A computed example
shows that the Nash solutions can dominate both of these rules, in terms of less default severity
and less inequity in bankruptcy resolution payments.
25
REFERENCES
Demange, G. (2015). Contagion in financial networks: A Threat Index (CESifo Working Paper
No. 5307). Retrieved from www.CESifo-group.org/wp
Dagan, N. and Volij, O. The bankruptcy problem: a cooperative bargaining approach.
Mathematical Social Sciences, 26, 287–297.
Eisenberg, L. and Noe, T. (2001). Systemic risk in financial systems. Management Science,
47(2), 236-249.
Elimam, A., Girgis, M., and Kotab, S. (1996). The use of linear programming in disentangling
the bankruptcies of Al-Manakh stock market crash. Operations Research, 44(5), 665-676.
Kadens, E. (2010). The last bankrupt hanged: balancing incentives in the development of
bankruptcy law. Duke Law Journal, 59(7), 1229-1319.
Levinthal, L. (1918). The early history of bankruptcy law. University of Pennsylvania Law
Review, 66, 223-250.
Mokal, R. J. (2015). Liquidity, systemic risk, and the bankruptcy treatment of financial
contracts. Brooklyn Journal of Corporate, Financial, and Commercial Law (forthcoming).
Rogers, L.C.G. and Veraart, L.A.M. (2013). Failure and rescue in an interbank network.
Management Science, 59(4), 882-898.
Sachs, A. (2014). Completeness, interconnectedness and distribution of interbank exposures—a
parameterized analysis of the stability of financial networks. Quantitative Finance, 14(9), 16771692.
Thomson, W. (2003). Axiomatic and game-theoretic analysis of bankruptcy and taxation
problems: a survey. Mathematical Social Science, 45, 249-297.
Thomson, W. (2015). Axiomatic and game-theoretic analysis of bankruptcy and taxation
problems: an update. Mathematical Social Science, 74, 41-59.
Upper, C. (2011). Simulation methods to assess the danger of contagion in interbank markets.
Journal of Financial Stability, 7, 111-125.
Upper, C. and Worms, A. (2004). Estimating bilateral exposures in the German
interbank market: Is there a danger of contagion? European Economic Review, 48, 827-849.
26
FIGURES
27
28