RETRACTED ARTICLE: Penalties and contagion in financial networks

Contagion and systemic risk, though in existence for quite a while, has seen increasing attention following the last Financial crises. In an attempt to contribute a little to the existing literature, this work focuses primarily on networks in which some paying-firms face systemic fragility through default cascade in specific settings where firms in cyclical connections defaults but defaults cascade without completing a cycle. To capture relevant dynamics to the equilibrium, this paper imposes a contractual system whereby firms who default incur an estimable amount of monetary penalties. Then it investigates under these conditions, the existence and dynamic properties of equilibrium. This paper investigates for strategic interactions between vulnerable Firms. Results show that in a setting where defaults do not feedback to source and where penalties are of considerable magnitude, there exists a unique, strictly dominant Nash equilibrium for Firms which depends on the Vulnerability index of Firms. A basic algorithm that computes the basic equilibriumdecisionof Firms is also proposed. Finally, the paper additionally analyses the role of the network of financial liabilities and systemic vulnerability on default decisions and on the free-riding behaviour as well as possible levels of payoffs for Firms involved. We find that when actions are limited, harsh penalties might harm some Firms that otherwise, would have achieved greater utility. Subjects: Economics; Economic Theory & Philosophy; Finance; Banking Dike Chukwudi Henry ABOUT THE AUTHOR Dike Chukwudi Henry is a Doctoral student of Finance and Economics at the School of Economics, University of Kent, Canterbury, UK. He specifically has his primary focus on behavioural economics, financial networks as well as games of strategy. With the growing interest in contagion and systemic risk, he focuses this specific study to introduce the concept of strategic network interaction where there exist incentives to avoid shortfalls in payment. This is as a means of creating a framework which based on its relatively simplistic assumptions, would open doors for more in-depth theoretical as well as empirical analysis into contagion and systemic risk facing financial systems. PUBLIC INTEREST STATEMENT The start of the twenty-first century has seen the rise of increasing attention given to the role to which networks and contagion play in decisions of economics agents of various forms. With the growing interest in financial linkages and spillovers in financial markets and as such, has increased studies that aim to understand these network effects and their role in promoting financial stability. This work introduces a system where contracts can be used as incentives to promote stability policies among Firms who are saddled with debt obligations. To do this, Firms who feel prone to default make costly decisions and to reduce such effect, observe their default network properties. Through spillover effects, they are able to verify their true position as a better guide in their policy-making decisions. The goal of this is to see how laws and regulations could be used as a tool to stimulate agents to act in ways that are less harmful to the overall financial system. Henry, Cogent Economics & Finance (2019), 7: 1575565 https://doi.org/10.1080/23322039.2019.1575565 © 2019 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Received: 10 August 2018 Accepted: 18 January 2019 First Published: 28 January 2019 *Corresponding address: Dike Chukwudi Henry, Department of Economics, Keynes College, University of Kent, Canterbury, United Kingdom E-mail: chd5@kent.ac.uk Reviewing editor: David McMillan, University of Stirling, UK Additional information is available at the end of the article Page 1 of 25 R E T A C T E D

Abstract: Contagion and systemic risk, though in existence for quite a while, has seen increasing attention following the last Financial crises. In an attempt to contribute a little to the existing literature, this work focuses primarily on networks in which some paying-firms face systemic fragility through default cascade in specific settings where firms in cyclical connections defaults but defaults cascade without completing a cycle. To capture relevant dynamics to the equilibrium, this paper imposes a contractual system whereby firms who default incur an estimable amount of monetary penalties. Then it investigates under these conditions, the existence and dynamic properties of equilibrium. This paper investigates for strategic interactions between vulnerable Firms. Results show that in a setting where defaults do not feedback to source and where penalties are of considerable magnitude, there exists a unique, strictly dominant Nash equilibrium for Firms which depends on the Vulnerability index of Firms. A basic algorithm that computes the basic equilibrium decision of Firms is also proposed. Finally, the paper additionally analyses the role of the network of financial liabilities and systemic vulnerability on default decisions and on the free-riding behaviour as well as possible levels of payoffs for Firms involved. We find that when actions are limited, harsh penalties might harm some Firms that otherwise, would have achieved greater utility.

Subjects: Economics; Economic Theory & Philosophy; Finance; Banking
Dike Chukwudi Henry ABOUT THE AUTHOR Dike Chukwudi Henry is a Doctoral student of Finance and Economics at the School of Economics, University of Kent, Canterbury, UK. He specifically has his primary focus on behavioural economics, financial networks as well as games of strategy. With the growing interest in contagion and systemic risk, he focuses this specific study to introduce the concept of strategic network interaction where there exist incentives to avoid shortfalls in payment. This is as a means of creating a framework which based on its relatively simplistic assumptions, would open doors for more in-depth theoretical as well as empirical analysis into contagion and systemic risk facing financial systems.

PUBLIC INTEREST STATEMENT
The start of the twenty-first century has seen the rise of increasing attention given to the role to which networks and contagion play in decisions of economics agents of various forms. With the growing interest in financial linkages and spillovers in financial markets and as such, has increased studies that aim to understand these network effects and their role in promoting financial stability. This work introduces a system where contracts can be used as incentives to promote stability policies among Firms who are saddled with debt obligations. To do this, Firms who feel prone to default make costly decisions and to reduce such effect, observe their default network properties. Through spillover effects, they are able to verify their true position as a better guide in their policy-making decisions. The goal of this is to see how laws and regulations could be used as a tool to stimulate agents to act in ways that are less harmful to the overall financial system.

Introduction
Financial contagion has increasingly gained attention by both theoretical and empirical economist as well as financial analyst in recent years following the 2008 financial crises. The G20 following these events recommended standards suggesting that more detailed attention be given to financial institutions which are of systemic importance and hence, potential risk they pose. In order to understand such implications of systemic risk in financial stability, some papers prior to this event such as Eisenberg and Noe (2001) have recently gained increased prominence. The systemic risk arising from a financial network was extensively introduced, and the concept of the fictitious default sequence was used to capture the domino effect/cascade emanating from a firm defaulting on other firms to which it has an obligation link to in an observable network. Equilibrium payments within this framework are deemed unique under mild regularity assumptions. A default clearing sequence was then proposed which, under regularity, clears under a total iteration equal to the amount of paying Firms in the network. Observing the nature and dynamics of such sequence gives so much details as to characteristics and behaviours of systemic risk. In observing such characteristics, recent works especially Demange (2016) has been able to capture systemically important financial institution at a baseline level where a third-party regulator is able to take advantage of contagion in intervening in a fragile financial system. The paper thus does not observe such dynamics under networks exhibiting systemic risk. This is so that by estimating a system involving defaults, regulatory authorities are able to use monetary intervention in the right areas such that it yields a system-wide stability.
Intuitions form the fictitious default sequence and clearing payment vector theorems have been used in various ways among which is an extension intended to capture its impact on assets prices Amini, Filipović, and Minca (2016) as well as a clearing system with liquidation strategies as they affect prices of illiquid assets (Feinstein, 2017). As another key of the motivation for this paper Dubey, Geanakoplos, and Shubik (2005) extensively deals with the problem of default punishment in general equilibrium analysis incorporating elements such as information asymmetry, adverse selection as well as moral hazard. Some similarities, as would be seen in the course of this work, are the ability for levels of default punishment to enhance coordination towards default. A notable drift, however, is that this model is analysed mainly on an Eisenberg and Noe (2001) foundation as it aims to unload the fact that even with homogeneous punishments to defaults, firms with default potential do not necessarily face the same problem. Information asymmetry is also assumed not to hold in this model as firms are aware of their network properties. Assets are fully liquid, and we do not assume recovery cost or change in price as a result of liquidation as proposed by Amini et al. (2016) and Feinstein (2017). Other key inspirations are Bramoullé and Kranton (2007) as well as Allouch (2015) whose aims were to provide behaviours to public goods in networks whereby Firms exert varying levels of effort given the available information they have and how other Firms might potentially benefit from those efforts. These concepts are then similarly adapted to the banking liquidation system by Denbee, Julliard, Li, Yuan, et al. (2014).
From the viewpoint of this paper, the intent is to understand the role of contagion in an environment whereby the regulator simply designs elements of debt contract in ways to spur Firms involved to act in the most stable form possible and as such, regulators need not make any cash injection policy. Not directly to any Firm involved at least. We rely on an environment characterized by evidence of non-feedback in the financial system which will be elaborated upon as this work proceeds as a primary focus. In other words, this work tends to see, as an alternative to monetary interventions, incidences whereby contracts are structured such that firms involved make attempt to salvage a fragile network by making investment decision with non-trivial opportunity costs. This is such that regulatory planners, through studying strategic interactions of agents R E T R A C T E D of a network, can simply aim to set specific elements of debt contract clause to achieve a certain level of stability as opposed to monetary intervention.
Models involving games of strategy in financial networks, however, have received less attention in comparison to other types of economic and social network. A recent work by Allouch and Jalloul (2017) captures the investment strategies of defaulting nodes in a financial network and found multiple Nash equilibrium arising from cyclical connections. Feinstein (2017) also investigates optimal liquidation strategy for illiquid assets in a network whose price is affected by total quantity liquidated in the economy. This paper improves on this existing literature by looking at a debtsystem of firms whereby defaulters face not only a zero equity value at the end of the clearing period, but also face additional implications of defaulting. These implications could range from restricted access to credit in the future, loss of goodwill, fall in credit rating, Legally enforced writs, could even be the cost of liquidation and closing down.
For simplicity, this work assumes that the financial contract terms are the same for all firms/ nodes in the network. The paper assumes a naive homogeneous cost measure for the implications of defaulting. The contract assumes that firms face a similar shaped payoff function. The paper then investigates for the presence of Nash equilibrium as well as conditions for uniqueness. The model also sets in a cyclical system whereby there are always non-defaulting nodes (Firms) in every loop. This is a very key attribute of this work as it seeks to explore the easiest form of Financial Network to understand the flow as well as optimality conditions and characteristics. It is hoped that intuitions from this work could yield transferable ideas relevant to other types of Financial network settings. Equilibrium strategy in this paper focuses on the amount of cash an agent might want to save exclusively for the network as opposed to consuming and the maximum utility thereof. Subsequently, it attempts to provide an algorithm that captures equilibrium decisions in this basic settings relaxing some earlier held assumptions.
Finally, studies specifically in integration, diversification as well as contagion have included Aldasoro and Angeloni (2015), Angeloni (2008) and most importantly, Elliott, Golub, and Jackson (2014) among many others have yielded varying conclusions. Elliott focused on the impact of connectivity on the valuation of a connecting firms asset. It also observed that sometime interconnections could lead to over-valuation of a firms asset and that the probability of cascade relied on Integration as well as diversification. This justifies designing our environment within one that has a relatively safe firm. What is considered a safe firm can manifest itself it different ways among which is lower amounts owed by other firms in the network. The simple simulation they ran showed how European countries 1 (in place of firms) who where debt bound in the last crises encountered default spillovers and thus all countries failed except Italy who simply had less network dependence and as such, was less integrated compared to other relatively larger countries. Integration covers the level of a firms exposure to another firm while diversification covers how spread-out the cross-holdings are. While an absence of integration means that contagion is unlikely to occur, the presence of integration actually decreases ability to default initially. One other useful point to highlight is that firms diversification could be bad at a stage (increase likelihood of cascade) and as it goes on further, it becomes more advantageous to the firms. Then in the later part of this work, attempts are made to capture the impact of integration as well as diversification as in Elliott et al. (2014) and capture the impact on strategic interactions of the Firms. At this point, we do not investigate within this model, a model with default friction as hinted by Deb (2016), Feinstein (2017) and specifically into the algorithm proposed by Rogers and Veraart (2013) as a form of dynamic implications of penalties on strategic decision but do feel that a careful approach to such could yield significant additional intuitions.

The model
In this section, we explore the unique environment to which our analysis is based upon. The aim is to build a basic foundation to understand the nature of strategic interactions in financial networks.

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We assume a Financial system made of N ¼ 1; :::; n; 1; . . . ; s f g which include n & N firms who have obligations to other firms in the networks which we going forward, refer to as paying firms, as well as s & N which represents firms who have no obligation to any other firm in the network, we will call this sink nodes. Let also n ¼ N j j representing the cardinality of N, then nominal liabilities are given by an n Â n matrix L. For the L, its row elements for a given firm i 2 N given j 2 N and jÞi contains ∑ j L ij indicating how much a firm i owes to other firm j. Then, for each, say Firm i, the nominal liability is given as L i ¼ ∑ j2N L ij thus implying the aggregate of Firm i 0 s obligation to other firms in the network while its claim to cash from the network is L i ¼ ∑ j2n L ji . A typical Firm i's size, then for the sake of our model is simply given as L i ∑ i2n L i which is the proportion total liability of a firm i compared the total amount owed in the system.
The parameter α ij is used to denote the ratio of a Firm i's debt to Firm j (i.e. α ij ¼ L ij L i ). This is so that the n Â n matrix α captures the relative liabilities of each of the firms of equal priority. 2 Let the sum of the row elements of L then be defined by a column vector L so that each element of L contains the full obligation of the firms in the network.
Networks are hence directional in our model. We importantly also assume that our network via liabilities creates at least one directional cycle of connection such that for at least one firm i 2 N, there exists a directed path to firm i through other firms. We call this subset of the network a ring. This is so that the network in Figure 1, for example, is said to have one ring which consists of firms i; j and k respectively For simplicity, these liabilities are redeemed (cleared) at about the same period which we denote as Period À 1 and we use Period À 0 to indicate the initial period. It is in this initial period that opportunities for various strategies of firms in the nodes arise. Choices once taken, cannot be revoked and would reflect on the clearing system in Period À 1 and thus, whatever decision is taken by the Firm i in Period À 0 amounts to the total exogenous cash in period À 1 and we denote it as Z i .
Thus, in line with Eisenberg and Noe (2001), if we use π i 2 ½0; L i to denote the payment of the Firm i given his total obligations and his total assets, then the clearing condition " Firm i would be denoted as: A simple interpretation of Equation (1) is that the Firm i pays the lesser between its total asset endowment 3 and its total obligation (liability). Let us denote the total endowment (particularly the left-hand side of the values within the bracket) as cf i for a given period, then Equation (1) can also be written as:

Categories of firms
We group firms under different categories depending on their given status in the network. To do this, we first establish some other useful definitions: Definition 1. An equity (denoted as eq) of a Firm is defined as the residual after payment in period 1. This is given for any Firm i 2 n as eq i ¼ cf i À L i Definition 2. A Defaulter is any Firm(s) in the network whose characteristics is that, for Firm i 2 n, Firms whose eq i < 0 are in the default set which we can call Dð0Þ and Firm i 2 Dð0Þ ( eq i < 0. The amount by which a Firm i 2 Dð0Þ defaults by is denoted as D i . This is such that Hence, it is possible to group Firms based on their status at the clearing period into two main broad types of classification, namely 1. Non-Defaulting Firms. these are firms who are unlikely to default at a given threshold of default by other connecting firms. Consider a Firm k 2 n whose eq k ! 0 and as such Firm 2. Defaulting Firms are Firms who are Defaulters.
This distinction is useful as we would focus primarily on decisions made by defaulting Firms as Firms who do not default need not take any further decision in this game. Figure 1 shows a prototype network for which its properties would be particularly useful in analysing the different types of networks based on the characteristics they emanate given the clearing system that follows the Fictitious default sequence. In this light, we define another important terminology as follows: Definition 3. A network is defined as non-feedback if no firm i 2 Dð0Þ defaults further as a result of its own default. This is because, with default cascade, networks with rings are prone to defaults which a firm incurs as a remnant cause of its very own default. Hence, feedback network. We, however, ignore such networks for the sake of this work.
Information: We also hold, for simplicity, that Firms in this network have perfect information. This implies that all profiles of each Firm is known to all other Firm in the network and thus, best replies can be estimated. This is key to understanding the dynamics of the system and can be relaxed later if need be.

Default penalties
As a special as well as pivotal feature this network has a special contractual obligation which includes an additional cost based on the level of default. We call this default penalty/punishment. For simplicity, we assume that the punishment parameter is estimated reliably at period À 0 and can be put purely monetary terms and this amount is applied with 100% certainty to defaults in period À 1. Also, punishment applies homogeneously to all set of connections 4 under a fixedpunishment rate denoted as; This model also assumes a payoff (utility) function " Firm i 2 Dð0Þ which are concave twice differentiable mappings of strategy profiles. In essence, an increasing amount of default warrants an even greater perceived punishment. While there exists possible alternative payoffs that may or may not yield robust results, the rationale behind this functional form is inspired by, among others, the system whereby a default leads to other business upheavals like litigation, claim for damages, restricted access to credit facilities going further, loss of goodwill thereby reducing ability to raise extra finance through equity, and so on. These implications are assumed to be of greater proportion as the size of the default become even more obvious.
A prime example would be the case of the collapse of the Lehman Brothers whose following its bankruptcy filing, led to a bigger collapse of the system and was one of the main drivers of the 2008 financial crises.

R E T R A C T E D
The previous expression of penalties above means that a Firm i treats all creditors as same with regards to punishment ability. Specifically, the cost function of a Firm i who is in default is denoted as: This is applied to any Firm in the default set regardless of the sources of defaults. 5 Firms could differ in size and L i in the expression above captures the fact that larger firms stand to loose greater in the case of default as compared to smaller ones (those with lower nominal obligations).

None dissolution
As further justification for the possible presence of Λ in a financial system, our framework assumes, particularly unlike the case of say Lehman Brothers collapse, that firms do not dissolve as a result of default in Period À 1. Also, Firms do not estimate a dissolution going further from period À 1. This is why Λ is the parameter used in capturing the estimated implications of default. This, in reality, could vary among Firms 6 but for simplicity, as stated earlier, we hold it constant (homogeneous) across all Firms in the network. Penalties/implications of default could range from future litigation, fall in credit-rating, refraining orders to access credit facilities over a period of time or even fall in the overall goodwill of the Firm.
Another reason for imposing Λ is because without it, firms pay the least they can afford with no implications past the period whatsoever. The presence of potential penalties on default creates a new (additional) optimization problem for the network where for a given Firm i who defaults this is explored in details later.

Endowments
So at period À 0, each firm i has a compulsory network reserve value given as Z e i . We assume that all firms in the network are rationale so that for a Firm i 2 Dð0Þ who seeks to minimize its default liability, it would have the choice of saving a level of cash which is denoted as Z s i . This saving choice comes from an outside network endowment which we define as Z A i . For the greater part of this work, we rely on the following assumption: However, decisions can only be made in period À 0 and we also assume that excess equity from savings in period-0 is of little to no value at period À 1, hence an irreversibility of choice. So for every Z s i injected into the network, Firms incur a cost of À Z s i .
For a more realistic approach, the latter part of the work naively examines a situation where Firms have a limited source of cash in period À 0 which we denote as Z A i . This now places a new constraint Z s i Z A i . In this unique scenario, we assume the value for whatever is consumed at Period À 0 to be the amount that is not saved for the network purpose. The amount to which it consumes is denoted as Z i c. This also means that if Z s i < Z A i , then it holds that: This means that Z i c represent the consumption which is exactly a one for one residual of the amount saved. This assumption is pretty weak but is done for simplicity in our analysis.

Strategy profiles
At Period À 0, the Firm i 2 Dð0Þ estimates its default based on the equation: which implies that the network endowment of the Firm i is given as cf For Firm i 2 Dð0Þ, the outcome of its decision at period À 0 would be the choice of saving the cash Z s i 2 R þ . This variable as a choice parameter depends on a number of factors. Among the ones we explore in this paper are the following; 1. Availability of cash outside Z e i : We already denoted as Z A i . This is the amount of cash with which the defaulting firm has to save from. We maintain assumption 1 for this part of the work.
2. Source(s) of Default: This has to do the two types of defaults; the Primary and Secondary. Both shall be explored in details.
3. The estimate default penalty parameter Λ: This captures the optimality conditions as well as uniqueness of equilibrium. But also more on that later.
The end product of the decision gives the cash endowment used in period À 1 which is hence given as:

Sources of defaults
Intuitively from Eisenberg and Noe (2001), we can categorize defaults of a Firm i 2 Dð0Þ in the network into two broad types, namely; 1. The Primary/First-wave Defaults: This is the amount of L i a Firm i 2 Dð0Þ is not able to repay purely as a result of factors outside the network. This is most likely to be as a result of owing much, expecting little and having little cash to meet up. In mathematical terms, it is when Notice that all connected Firms pays up their liability in full and the Firm i still defaults.
2. The Secondary/Higher-wave Defaults: This is the amount of L i the Firm i 2 Dð0Þ is not able to pay as a result of default cascades exclusively from the network and not as a result of factors outside the network. In essence, it is a case whereby Which indicates that had Firm i received its full payment from the network, it would not have defaulted.
Knowledge of these two sources is very useful in many ways. One of which includes the understanding of the nature of Financial network/system Firms are faced with. To capture some of the importance, we have the following proposition: Proposition 1. For a network N with default cascade to be termed a "non-feedback" network, then atleast one Firm i 2 Dð0Þ in every ring has a total default value equal to its first-wave default.
Given the nature of transition of default along different waves, we can extend from proposition 1 the following; Default cascade in a non-feedback network assumes, at maximum, a Hamiltonian cycle.
In essence, since out network takes a form of a directed cycle, then for the waves of default to be at a maximum of n À iterations, then it is assumed that a default value from a Firm i, touches all other Firms only once. This ensures that no residual from such default feeds back to Firm i because defaults decay completely in while cascading through other firms in the ring. A financial system that is non-feedback has the advantage in that it provides for easier intuition and understanding. However, in reality, systemic risk leads to breakdown and networks which fall into this category are financial networks with default feedback. But to provide a basic framework, we dwell for the most of this work, on a non-feedback financial system such that full systemic risk is avoided at this point.

Firms problem
Take a financial system as in Figure 1. For the sake of the paper, this would be our prototype nonfeedback network. This network has four Firms, namely, Firm i, Firm j, Firm k and Firm m. Firm m is a sink node and receives its proportion of whatever Firm k has. Based on the model set-up in the previous section, the Firms in "ash-colour" depict Firms who default (hence defaulters) and the orange indicates Firms who do not.
Assuming all the conditions held as previously described, we will analyse, in this section, the problem each paying Firms face individually as it represents different types of firms in the network as we will soon see.

A network with defaults
To get a foresight, assume a Firm i is a paying Firm, then without loss of generality, let δ i be an indicator variable that takes value  (1) can be written as Let J n,1 denote a length n column vector of ones.
I n denote the n Â n identity matrix.

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δ denote an n Â n matrix whose elements are the operator value δ ij such that if i as well as j are defaulters then δ ij ¼ δ ji ¼ 1. Otherwise δ ij ¼ δ ji ¼ 0.
d denote an n Â 1 column vector whose elements are made up of ½δ 1 ; ; δ n T : denote the Hadamard product.
We can solve explicitly for π i in terms of the actions of firms Z s : n;1 À dÞ L π 1 ¼ ðIn À δ α 0 Þ À1 ½d ðZ s þ Z e Þ þ ðIn À δ α 0 Þ À1 ½ðJ n;1 À dÞ L Let β :¼ ðIn À δ α 0 Þ À1 . Then we have: Observe from Equation (4) that β has an n Â n while the sum of the other parameters yields an n Â 1 dimension. It then means that the product of the two gives us an n Â 1 dimensional π 1 . Subsequently, for the firm i, it would be the i À th row of β that multiplies what is in the bracket so that for π 1 i , we now have:

Firm i
Since Firm i's only network inflow is from Firm k's payment and Firm k is not a defaulter, it implies that Firm i is a Primary defaulter. Given the model setup which includes Punishment as Λ and Cash breakdown into Z A as well as Z e , the problem for Firm i can hence be written so as to capture the decision of all connecting Firms and as follows: þ β i ½ðJ n;1 À dÞ L and 0 Z s i : Because from Figure 1, Firm k does not default, with reference to corollary-1, we rewrite its constraint as Since Firm i does not depend directly on Firm j but does so only through Firm k, then the need for the second as well as fifth constraint seizes to exist.
As pointed out earlier the first constraint shows that Firm i, even given the default penalty, only pays a maximum of L i or its total period À 1 regardless of cf 1 i . So it means we can rewrite the Firm i's objective function as: This equation above is now more straightforward and would be analysed in the next section, we move on to the next defaulting Firm j.

Firm j
The Firm j appears to be a defaulter as well. However, its source of cash through the network is solely from Firm i. The question that comes to mind is how much of Firm j's default comes from systemic cascade default of Firm i and how much comes entirely from Firm j (primary default)? To get the clear picture, Firm j sets its problem similar to Firm i as: and, 0 Z s j : As π k ¼ L k , Firm i does not benefit more than α ki L k from Firm k so its connection with Firm j is of no importance to its decision. Because assumption 1 still holds, then given λ and Firm i, Firm j's objective can be re-written as max Z s j U j ¼ À Λ 2 ðL j À π 1 j Þ 2 À Z s j subject to, and, 0 Z s j :

Best replies and equilibrium
The level of cash injection by a Firm into the network for sources outside captures firms risk aversion. Taking the First order conditions (FOC) of the programmes, the best reply function would be the value of Z s i that maps Z s j so as to maximize U i for a given Firm i. We can denote the optimal Z s i as Z sÃ i then under rationality, the clearing payment vector for a Firm i given Z sÃ is denoted as π 1 i . Still holding unto assumption 1 so that defaulting firms face no restriction as to how much they can save to avoid default, then we can use the generic definition of the Nash equilibrium as follows: Definition 4. A Firms strategy profile is a Nash-equilibrium if UðZ sÃ i Z sÃ j Þ ! UðZ s i Z sÃ j Þ .
So that the Nash of a firm in default is simply the level of savings that gives the maximum payoffs given its network status. We hence analyse each firm based on their different characteristics.

Firm i
From the objective function, the maximum Utility of the Firm i can be re-written as

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Observe from the constraint Z s i þ Z e i þ α ki L k ¼ π 1 i , That Z e i þ α ki L k is a constant. Best replies then under rationality implies the following lemma, Lemma 1. " Firm i 2 Dð0Þ, Firm i would aim for Z s i such that cf 1 , and Firm i's payoff means it looses Z s i so as to gain from potential punishment. Then if for cf i > π 1 i implies that π 1 i ¼ L i . This is so the more cf i is, then the greater is Z s i and as such U i ¼ ÀZ s i becomes more negative. Rationality implies firms strive for a corner utility. So defaulting firms aim for as much Z s i to get L i ¼ π 1 i which then implies cf 1 To get a better insight, take for example that for a given Firm i, L i ¼ 45 and π i ¼ 30 whilst Z A i ¼ 20, then the illustration above captures the fact that Firms facing an interior solution only increase their saving to the extent needed to reach their liability. It shows that once anything above Z s i ! 15 is saved, π 1 i ¼ L i ¼ 45 and never goes above that and this is captured in Figure 2 where we see that from savings above 15, payment never exceeds the liability 45 owing to the limited liability threshold. This means any further saving has no effect on payments. With the three constraints, we can estimate the best reply for Firm i either in terms of savings Z s i . Then given the utility function, the first order condition for maximization reflects that: This therefore implies that Firm i has the following best reply function.
So then given Firm i's best reply as in Equation (6) and then given that we make a further proposition: Proposition 2. Given assumption 1, the best reply function and Nash equilibrium decision are the same for Firm i Proof. This is particularly clear since all the parameters on the right hand side of Equation (6) are exogenous to the model.

□
In essence, given Firm i's is a primary defaulter, its decision depends solely on its asset availability as well as risk aversion given punishment and does not include the decision of any other firm in the network.

Firm j
Given Firm j's programme, the First order condition for maximization is given as; This therefore implies that Firm i has the following best reply function: This means that unlike Firm i, Firm j takes into account what the best reply of Firm i is given Firm i's strategies and forms its belief which reflects in its action set. The Firm j looks at the decision of Firm i reflected in Z s i and since Firm i has a dominant strategy equilibrium. It then estimates its own Nash equilibrium savings using Equation (7). Therefore, its Nash Equilibrium decision is: It is worth noting that given Z sÃ i if Z sÃ j > 0, Then given the network in Figure 1, Firm j is not entirely a "higher-wave defaulter". Firm jalso is, to a certain proportion, a primary defaulter. The amount of default D j purely attributable to Firm j(First wave default) which we, for convenience, denote asD j would now be calculated in terms of Figure 1 as: For a more general case as networks could contain more nodes whose connection flows to Firm j, the default liability attributable solely to Firm j (the primary default) in a network of N À Firms is given as: The basic implication is that with adequate knowledge on the cash availability of each Firm and default profile, Firms can estimate and possibly reduce the amount they save into the network to an amount equivalent to their first wave default. So that given the results so far, we have our first theorem as follows: Theorem 1. Given assumption 1, as Λ ! þ1,all defaulting firms store in for the network, an amount equivalent to their First Wave default/Primary default Value.
Proof. So " Firm i 2 Dð0Þ, Z A i ¼ þ1 and all other Firms are aware that each other share that similar position. Then for a pure first wave defaulter like Firm i in our example, 1 β i i Λ ! 0 as Λ ! þ1 so that its Nash decision would be to save equal to L i À α ji L j À Z e i which is exactly the value of its first wave default.
□ Figure 2. Movement of payment value with respect to increase in savings.

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Given that Firm j reliably estimates this, then similar rule would hold for Z s j .

Equilibrium characteristics
Given the original programme of a defaulting Firm i as Their complementary slackness condition is where μ i is the Kuhn-Tucker multiplier on the non-negativity constraint. 7 The left-hand side of Equation (10)  i . This value is dependent on the network of bilateral claims. ΛðL i À π 1 i Þ on the other hand is the marginal rate of substitution between the action Z s i and repayments π 1 i .
Existence of best replies for each Firm i 2 Dð0Þ depends on the behaviour of the payoff function. We assume that payoffs (utilities) are concave twice differentiable functions of Z s so that there exists a unique maximum point. From Equation (10), the FOC gives the condition for best replies which includes β i i . β i i also then includes the slope of the payoff with respect to Z s and for this reason, whether payoffs meet the existence criteria would depend on understanding the behaviour of diag β for every defaulting firm. To do this, let us assume that L a represents the loop (ring) in our Figure 1 example such that L a & N then we have the following lemma; Lemma 2. Let i 2 L a such and defaults cascades through such loop L a ,so far as networks are nonfeedback, then " i; β i i ¼ 1 Proof. Recall from Equation (5) that β ¼ ðI À δ α 0 Þ À1 Then assuming the network as in Figure 1 where by Firm k does not default, then the relative liability value under default is given as: Àα jk 1 0 @ 1 A then taking the determinant, we have: and as such

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This is intuitively because defaults do not cause more default to each firm within the clearing mechanism and as such, best reply slopes are principally function of the penalty Λ as opposed to any thing else. The second other condition is given as: In essence, in so far the punishment parameter Λ 2 R þ , then we are assured that a global maximum utility exists and it reflects the optimal saving decision. And thus, the reason the best reply functions are existent for defaulting firms and are given as previously stated. Strategic substitutability are captured in the best reply functions of both firms. Observable characteristics from the Nash equilibrium for both Firms are that actions are done in sequence such that Firm j, being a secondary defaulter, does not save until it is aware fully of the amount of savings Firm i is going to make and as such, substitutes for Firm i's actions. This is the core characteristics of savings decision under non-feedback network in eisenberg and as such, we have the following propose the following theorem which builds from lemma 2 □ Theorem 2. If a network reflects characteristics of a non-feedback under eisenberg, then given a game with default punishment, 9 a Firm i 2 D as well as a group of other firms À i 2 D where i 2 L a while À i & L a and i‚ À i such that its substitution to all other firms actions dZ sÃ i dZ s Ài ¼ 0.
Proof. Reasons are relatively straightforward. Non-feedback network implies on which for all firms in the network, defaults of any first-wave default firm does not feedback to the source firm. This implies one firm in the ring does not default and from our example in Figure 1, that is Firm k. Then Firm i is a pure first wave defaulter and thus, from its best replies in Equations 6 and 7, we have that the strategic substitute is given as: It is then more obvious from Equation (11)

□
The firm with these characteristics are the pure primary defaulters. With firms in the network, the theorem serves as the primary implication of a network where default does not feedback. Hence, so far as assumption 1 holds, the decision on choice savings becomes sequential whereby Firms substitute efforts of source defaulters and while one or more firms are then forced to save an amount equivalent to the full amount of their default due to inability to substitute.

Algorithm with limited endowment
In slightly more realistic settings, Firms are likely to have limited external endowment. This could include among many others: limited access to credit facilities, inadequate working capital, inability to raise cash by sales of debt and equity instruments, etc. For this section, we relax assumption 1 so that Z A is now in limited availability. Also, we assume at this point that the Firm now chooses how much to save into the network and consumes whatever is remaining. Z i c is thus used to capture the consumption decision of the Firm. This amount is, for ease, the direct residual of the amount saved such that " Firm i 2 Dð0Þ, With the presence of cash constraint, the defaulting Firm i now faces a possible interior utility function given as: Other constraints hold as well but then Firms decision are bounded by the additional constraint, The result from the First order conditions and best replies still hold, however since Firms are limited by cash, Nash equilibrium for the Firm i for example becomes, Which represents that Firms make savings not only because it is the best decision to make, but also given its capabilities. We assume this information is perfect for all Firms in the network. The intuition from the process of the Fictitious default algorithm is used to set up the following instructions that calculate Firms optimal savings decision given other Firms. It should be noted this algorithm works for non-feedback network settings alone.
The following steps are then proposed: 1. Assuming Z i ¼ Z e i , then use the fictitious default algorithm to compute for the clearing payment vector π i as well as the equity eq i: 2. Observe the FDA and ensure it clears in at most n À iterations. If it exceeds that, End. If it does not, proceed to the next step.
3. Any Firm whose eq ! 0 is not a defaulting Firm and thus, would be excluded from the game.
4. For the Firms whose eq 0, the Firm i 2 Dð0Þ. Search for two items which include • the first negative equity value for each firm. This is the same as the primary default which we denote asD 1 i ðkÞ. • the sum of all negative equities of each of the Firm throughout the iteration which we denote as eq min i ðkÞ and make a row vector of this maximum default, and 5. Based on values gotten, all Firm i's whose initial default value (the first negative value of equity) is at the first-wave would inject min Z A i ;D 1 i ðkÞ À 1 Λ È É .
6. For those Firms above, subtract the amount injected from the total default value and keep the balance which we denote as D res ðkÞ. This value would form the first wave default in the next iteration. Then, 7. Head to the next (second) wave defaulters. This is done by taking the new Z i into consideration and recomputing as in step 1 and find the new summation of all negative equity values of those firms. and then for new-defaulters, 8. For Firms whose eq min i ðk þ 1Þ ! 0, no further strategy is needed.
9. For Firms whose eq min i ðk þ 1Þ < 0, check for the amount of second wave default (D 2 i ðk þ 1Þ) and inject as much to erase it. i.e min Z A i ðk þ 1Þ;D 2 i ðk þ 1Þ À 1 Λ È É . Then still keep the residual balance as D res ðk þ 1Þ which forms the second wave default in the next iteration.
10. Repeat this till all waves of default are captured. Or until " Firm i with i : i 2 Dðk þ nÞ f g , Z A i ðk þ nÞ ¼ 0. Then end. 11. Z s i would then be the total amount injected by the Firm i across all iterations.
Lemma 3. In so far as Financial networks are non-feedback, the algorithm for Firms facing default punishment would clear at a total of at most n À iterations .
Proof. Non-feedback networks are guaranteed to clear in at most n À iterations. The algorithm proposed lets each firm make its decision using every wave of default given πð0Þ. Bringing back the following mapping, Z s ! FIXðΦ : ; α; π; Z e ; Z s Þ is concave, increasing and non-expansive which adapts

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from Lemma 5 of eisenberg where FIXðΦ:; α; π; Z e ; Z s Þ captures the new system given savings decision.
Then it means that since for every iteration, ΔZ s ðkÞ ! 0, then our algorithm sequence cannot create an increase in default and as such, cannot extend the number of waves of the original fictitious default algorithm. □ 7. Punishment levels and comparative statics So far, from the characteristics of equilibrium under clearing with default punishments it might seem like firms who are not first-wave defaulters might be able to avoid substantial default whilst making little effort to save any amount. However, the level of perceived punishment might, in a more realistic sense, be advantageous to the primary defaulter in comparison to others who default due to cascade. We aim to address the impact of lower to higher Λ on Firms equilibrium decision and most importantly, utility given their position in the default cycle.
In this regard, we address Firms based on their individual network characteristics.

The Firm i
Firm i always has a pure strategy to save enough to get π 1 i ¼ L i À 1 Λ . We refer to the part that includes the Λ component as the Optimal Default Value of any Firm so that for the Firm i, its optimal default value is 1 Λ . What then happens if the value of Λ is substantially low? Perceived punishment becomes lower. 1 Λ increases substantially. Though Z s i drops, payoff U i increases. This is mostly due to the fact that while the marginal benefit of savings reduces, the cost of savings remains equal. So relative to the previous benefit, savings become less desirable. Figure 3 shows us the ability for optimal savings to substantially drop following the fall in the amount of Λ. Hence, Firm i, being a primary defaulter is better-off increasingly defaulting.

The Firm j
The Firm j may not be just as fortunate. its pure strategy has the component which is also dependent on Firm i's action given as, 1Àα ij Λ . This eventually burdens Firm j with optimal default value of Firm i for each value successive default punishment parameter Λ takes. If however,

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A C T E D spillover access from Firm i to Firm j is low, then Firm j is increasingly better-off as well when Λ is low. But this also means that more of Firm j's default can be traced to its primary sources as lower spillover access means lower cascade. Recall that the magnitude of α ij captures the extent of default spillover. Hence, Problems with Integration (Acemoglu, Carvalho, Ozdaglar, & Tahbaz-Salehi, 2012) 7.1. Illustation 1 To effectively illustrate this, Assume the Financial system in Figure 4 above, other characteristics of the network are as follows: Other properties of the network are: Then given the First order condition for Firm i, 0 ¼ Λð20 À 1ðZ s i þ 5Þ À 0ðZ s j þ 5Þ À 0:5ð20ÞÞ For Firm j, it would be 0 ¼ Λð30 À 1ðZ s i þ 5 þ 0:5ð20ÞÞ À 1ðZ s j þ 5ÞÞ Therefore,

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and Z s j ¼ 10 À Z s i À 1 Λ : (1) Case 1: Λ ¼ 1:5 • Then the Best replies are Z s i ¼ 4:3 and Z s j ¼ 5. From the example, it is easy to see that Firm j suffers the more as the value of Λ falls. While its utility should have been higher, because Firm i is better off defaulting by increased amount, as is Firm j, the cascade from Firm i sees to it that Firm j looses more than it benefits by the fall in default. This problem is also heightened by the fact that Firm j depends more on Firm i as α ij ¼ 1 which is pretty high. Hence, a lower integration would have been better. But lower integration also means lower substitution for Firm j from Firm i's action. This leads to the following claim; Proof. As examined already from illustration 1, a secondary defaulter, say Firm j has as its optimal default value as 1Àα ij Λ . This is obviously an inverse function of α ij so that given a small value of Λ, then the higher α ij is, the lower 1Àα ij Λ in comparison to Firm i who benefits 1 Λ as it has to save less.
□ Claim 1. Firms with high integration 8 leading to potentially high secondary default suffer more relatively to those of primary defaulter as the value of punishment Λ falls.
Firms also substitute for greater amounts when integration is higher as they are exposed more to default cascades. This also poses the risk of the Firm loosing more in a case where the penalty is substantially low. While its utility might be better-off compared to that of the higher Λ, the rise in utility would be trivial compared to Firms who are primary defaulters. In such case, Secondary defaulters bear the brunt of the advantage lower Λ affords to a primary defaulter.

Observations
In Networks involving cyclical obligations, there might exist some potential for feedback-loops. In this case, firms would prefer to use, through their decisions, positions in the network to influence other firms decision to its own advantage. However, for the kind of network we have examined so far, this is hardly possible as whatever strategy a defaulting node takes that affects another defaulting firms decision, such decision would have no effect in return to the firm from whom such decision would have emanated from. Figure 5 captures this characteristic. We see that firm i Figure 5. An illustration of direction of influence. We see that firm i can influence firm j's decision. But because whatever firm j does thereof cannot get to firm i, firm i cannot take advantage over its power to influence.

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can affect firm j's decision but because there is no incoming link from firm j to firm i, firm i cannot benefit any further from any action it takes on firm j . Hence, the existence of one or more firm in every loop who is not defaulting implies that though some firms are at a position to affect other firms utility, such effect would have no firm attempting to affect others. From the figure above, we highlight the implication of savings decisions in Figure 1. The Firm i's decision can influence that of Firm j. But Firm i has no means of benefiting from any decision that harms Firm j. Let us take an example that Firm i decides to save less to clear its default with the hope that Firm j would, in turn, cover up for the deficit, Whatever decision Firm j takes would only benefit Firm k. And recall, prior to the decision, Firm k was not a defaulter. So it only means Firm k would have more cash available, and because Limited liability guarantees that Firm k still pays exactly what it owes, Firm i cannot influence j to its benefit even with the presence of cyclical connections.

The vulnerability index
Without this ability in place and with harsh penalties, Firms in default are hereby concerned about their source of vulnerability emanating from systemic fragility/vulnerability rather than the actual influence (threat) they pose to the system. So far, it is observable that such vulnerability is captured in the parameter β Ã i such that for i, β Ã i ¼ ∑ β i . For the defaulting firm i, we call this parameter the Vulnerability index. To elaborate, recall that β ¼ ðI À δ α 0 Þ À1 . Then let β Ã capture the n Â 1 column vector whose elements make up β Ã ð i ; . . . ; β Ã n and 1 n represent a column vector with element of ones. Then we have; Also for better comparison, Demange (2016)'s threat index matrix, μ is given as; μ ¼ ðI À δ αÞ À1 Á 1 n : So that while decisions made by third-party regulators captures the impact each defaulting Firm has on other defaulting Firms, our model means that defaulting firms are more concerned with how other defaulting firms decision affects them, hence the transpose which gives information on free-riding opportunities. Though the threat index is not a component of Nash decisions of Firms in default (as Firms make more self-centred decisions), the overall impact of every extra saved into clearing is the same for the entire network as the threat index suggests. This is so that while defaulting Firms cannot take any action that feedback to them, higher-wave defaulting Firms can free-ride on provisions from their default sources to a certain degree, thus granting them an otherwise, better utility, especially in a case whereby the Firm i is forced to and has the means to save in full. But should Firm i be at a position when it cannot help itself, the Firm j as a higher-wave defaulter suffers because it is vulnerable to Firm i. This is evident by the fact that from our illustration-1, β Ã i ¼ 1 while β Ã j ¼ 2. So typically, this indicates it is the Firm j who deducts the full effect of Firm i's decision in calculating his choice savings. This implies that though it is true that the threat index does measure the Bonacich 9 Centrality and as such power of a Firm, the vulnerability index does measure centrality as well, but this time, in reverse which captures the who is most affected by the decision of other defaulting Firms. Hence, the greater the defaulting Firm is vulnerably "Bonacich" central, the greater to which its decision is dependent on the action of other Firms. This explains why Firm j has an index of a 2.
Further implications indirectly observable from illustration-1 is that having a higher vulnerability index may not always guarantee better utility. From the viewpoint of varying intensity of penalties, we can see in the example that levels of penalties might lead to those who end of better-off relative to others. Penalties above opportunity cost of savings see that Firm i is made to save more. But even with such high punishments, restrictions in Z A i , it might see that not all defaults are covered. This would imply that Firm j's level of substitution is further restricted and as such, looses out as well due to its increased exposure to Firm i. But if Firm i is able to cover up, then Firm j is better-off when penalties are high. If penalties are lower than the opportunity cost of savings, Firm j is definitely at the loosing end as it would face a lower maximum utility from not saving. A situation where penalties are equal to R E T R A C T E D opportunity cost brings in uncertainty mostly for the Firm j. It is clear that Firm j, in essence, higherwave defaulter(s) is more likely to be at disadvantage relative to Firm i. Though not always, but most of the times first-wave defaulters are relatively more likely to be better off. But the main idea is that Firm j being at increasing disadvantage has no direct benefit to Firm i, this is owing to the fact Firm k is not a defaulter and thus pays a neutral amount regardless of Firm j's outcome.
It then becomes clear to a regulator who wishes to set up a system to incentives potential defaulting firms to strive towards paying in full might unintentionally affect other firms. Especially those to who are Vulnerably Bonacich central such that they are exposed to spillover consequence of incoming defaulting firms who might not pay off their debt in full. That is the typical trade-off to a high amount of penalty.

Conclusions
Payment systems with punishment on defaults could serve as a negative-incentive. Depending on the level of punishment, levels of defaults could reduce substantially and as such, Firms within a system would be forced to take necessary actions to guarantee efficiency. Given nonfeedback networks, firms might have limited substitution levels. Ability to substitute is explained, among other factors, by the Vulnerability Index of each defaulting Firm. Some actions can influence others while other actions might not. However, Firms in peril give more attention to how other Firms affect them rather than their influence on other Firms and as such, to the Firm, the systemic vulnerability it faces is more pivotal to its decision-making than its influence. Especially when such influence does not feedback to the Firm. Overall the utility got from saving in a default system depends on a lot of factors but is guaranteed to be higher the greater the avenue for substitution exist for the given Firm who is at risk of default. This is so that being vulnerable places a Firm at possibility of relatively higher or lower benefits corresponding a particular degree of penalty.
A core setback to this work would be a more realistic functional form to capture punishments. Concavity is adapted for this work due to not only computational convenience but also because intuitively, it is not impossible to have situations whereby a Firm faces an even greater proportion of punishment (of whichever sort) due to defaulting by a larger amount. That being said, functional forms of a defaulters payoff remains open for debate and has the potential to yield robust results. Robust results are also likely to be obtained should contract differ among edges such that the functional form of the objective function differs not only across Firms but even between contracts of the same Firm. Also, as with most decision-making model, limited information is bound to lead to different outcomes from the one gotten in this paper. In this case, adverse selection and moral hazard as hinted by Dubey et al. can see that a higher penalty leads to sub-optimal equilibrium. Non-feedback networks where cyclical properties lead to contagion as opposed to the default of all paying Firms in the network is used here to give basic reasoning as to Firms possible actions such that results under systemic risk are avoided. These kinds of network are just as prominent in the literature as some existing scholars as highlighted in Demange made attempts to rule out the frequency of systemic risk in real life, and this paper builds on similar intuitions. However, distinct information is likely to be obtained when investigating actions in existing default feedback financial systems. This, for initial understanding, has not been the primary focus of this paper. Another point worth considering is the fact that we assume all contracts are similar in the network with respect to payment as well as punishments such that default penalties are homogeneous across all edges. Heterogeneous penalties might be worth considering in future works.
But in all, we are still able to derive valuable intuitions. Penalties might be good or bad depending on the level perceived by Firms who are bounded by their respective network contracts and as such, network position is an important factor used in estimating the amount of effort Firms at risk make to the overall system. This is such that Firms, whose ability to meet its obligations depends more on the network as opposed to external sources, either benefit quite substantially from a harsh penalty or at the other extreme, be at great loss.