Network-Aware Strategies in Financial Systems
aa r X i v : . [ q -f i n . R M ] F e b Network-Aware Strategies in Financial Systems
Pál András Papp
ETH Zürich, [email protected]
Roger Wattenhofer
ETH Zürich, [email protected]
Abstract
We study the incentives of banks in a financial network, where the network consists of debt contractsand credit default swaps (CDSs) between banks. One of the most important questions in such asystem is the problem of deciding which of the banks are in default, and how much of their liabilitiesthese banks can pay. We study the payoff and preferences of the banks in the different solutions tothis problem. We also introduce a more refined model which allows assigning priorities to paymentobligations; this provides a more expressive and realistic model of real-life financial systems, whileit always ensures the existence of a solution.The main focus of the paper is an analysis of the actions that a single bank can execute ina financial system in order to influence the outcome to its advantage. We show that removingan incoming debt, or donating funds to another bank can result in a single new solution that isstrictly more favorable to the acting bank. We also show that increasing the bank’s external fundsor modifying the priorities of outgoing payments cannot introduce a more favorable new solutioninto the system, but may allow the bank to remove some unfavorable solutions, or to increaseits recovery rate. Finally, we show how the actions of two banks in a simple financial systemcan result in classical game theoretic situations like the prisoner’s dilemma or the dollar auction,demonstrating the wide expressive capability of the financial system model.
Theory of computation → Network games; Applied computing → Economics; Theory of computation → Algorithmic mechanism design
Keywords and phrases
Financial network, credit default swap, creditor priority, clearing problem,prisoner’s dilemma, dollar auction .A. Papp and R. Wattenhofer 1
The world’s financial system is a complex network where financial institutions such as banksare connected via various kinds of financial contracts. If some financial institutions gobankrupt, then others might suffer as well; the financial network might experience a rippleeffect. Two of the most common financial contracts are (i) debt contracts (some bank owesa specific amount of money to another bank) and (ii) Credit Default Swaps (CDSs). A CDSis a simple financial derivative where the payment obligation depends on the defaultingof another bank in the system. The combination of debt contracts and CDSs turns outto provide a simple and yet expressive model, which is able to capture a wide range ofinteresting phenomena in real-life financial markets.Given a set of banks and a set of payment obligations between these banks, one of themost natural questions is to decide which of the banks can fulfill these obligations, andwhich of them cannot, and hence are in default. The problem of deciding what portionof obligations banks can fulfill is known as the clearing problem . One can easily encountera situation when this problem has multiple different solutions in a financial system. It isnatural to study how much the individual banks prefer these solutions, i.e. what is theirpayoff in specific solutions of the system.In this paper we study the problem from the point of view of a single bank v . We analyzewhether some simple actions of v can improve its situation in the network. In a financialsystem, the complex interconnection between the banks can easily result in situations wherebanks can achieve a better outcome in surprising and somewhat counterintuitive ways. Forexample, being on the receiving end of a debt contract is generally considered beneficial,because the bank obtains payment from this contract. However, in a system with debts andCDSs, it is also possible that if a bank v nullifies a debt contract as a creditor, then (througha number of intermediate steps in the network) this results in an even higher total payoff for v . Such phenomena are crucial to understand, since if banks indeed execute these actions toobtain a better outcome, then these opportunities will determine how the financial systemchanges and evolves in the future.We begin with a description of the financial system model recently developed by Schulden-zucker et. al. [21], which serves as the base model for our findings. We then introduce amore refined version of this model which also assigns priorities to each contract, and assumesthat banks have to fulfill their payment obligations in the order defined by these priorities.We show that besides being more expressive and realistic, this augmented model still ensuresthe existence of a solution.Our main contribution is an analysis of various different actions that banks in the systemcan execute in order to increase their final payoff when the system is cleared. We first showthat by removing an incoming debt (partially or entirely) or by donating extra funds toanother bank, a bank might be able to increase its payoff. We then show that investingmore external assets or reprioritizing its outgoing payments can also allow a bank to influencethe system. However, these actions do not allow a bank to introduce more favorable newsolutions, but they can allow the bank to remove unfavorable solutions from the system, orincrease its own recovery rate.Finally, we present some simple examples where two banks try to influence the financialsystem simultaneously, resulting in situations that are identical to the classical prisoner’sdilemma or dollar auction game. This suggests that financial systems in this model canexhibit very rich behavior, and if two or more banks execute these actions simultaneously,this can easily lead to complex game-theoretic settings. Network-Aware Strategies in Financial Systems
Numerous studies on the properties of financial systems are directly or indirectly basedon the financial system model introduced by Eisenberg and Noe in [11]. This model onlyassumes simple debt contracts between banks. Different studies have later also extended thismodel with default costs [20], cross-ownership relations [24, 12] or so-called covered CDSs[17]. The related literature has studied the propagation of shocks in many different variantsof these models [2, 8, 5, 4, 1, 13].One disadvantage of these models is that they can only describe long positions of bankson each other, meaning that a worse situation for one bank is always worse (or the same) forany other bank. For example, if a bank is unable to pay its debt, then its creditor receivesless money, and it might not be able to pay its debts either. This already enables the modelto capture many interesting phenomena, e.g. how a small shock causes a ripple effect in thenetwork. However, long connections imply that there is a solution in these systems which issimultaneously the best for all banks. As such, the models cannot represent the opposinginterests of banks in many real-world situations, and thus these models are not so interestingfrom a game-theoretic point of view.On the other hand, a more realistic model was recently introduced by Schuldenzucker,Seuken and Battiston [21]; we assume this model of financial systems in our paper. Besidesdebt contracts, this new model also allows credit default swaps between banks, which areessentially financial derivatives where banks are betting on the default of another bank.CDSs are a prominent kind of derivative that played a significant role in the 2008 financialcrisis [14]; as such, they have been studied in various works in the financial literature [10,18, 9]. While the model still remains relatively simple with these two kind of contracts, itnow also allows us to model short positions , when it is more favorable for a bank if anotherbank is worse off. This increases the expressive power of the model dramatically, allowingus to capture a wide range of properties of practical financial systems.The work of Schuldenzucker et. al. analyzes their model from a complexity-theoreticperspective. The authors show that in the base variant of this model, each system has at leastone solution; however, if we also assume so-called default costs, then some systems mightnot have a solution at all. In case of default costs, they also describe sufficient conditionsfor the existence of a solution. Their follow-up work shows that it is computationally hardto decide if a solution exists, and also to find or approximate a solution of the system [22].However, to our knowledge, the model has not been analyzed from a game-theoreticperspective before. Our paper aims to lay the foundations of such an analysis, by evaluatinga variety of simple (and yet realistic) actions that allow nodes to influence the network due tothe presence of short positions. Since banks often have conflicting interests in these systems,these actions can easily lead to interesting game-theoretical dilemmas.The only similar game-theoretic analysis we are aware of is the recent work of Bertschinger et. al. [6], set in the original model of Eisenberg and Noe. Instead of having institutionalrules for payment obligations in case of default, [6] assumes that banks can freely selectthe order of paying their outgoing debts, or even decide to make partial payments in somecontracts. The paper discusses the properties of Nash-Equilibria and Social Optima in thissetting. While this has a connection to our observations in Section 5.3, we analyze theresults of such actions in a significantly more complex model with CDSs.In general, measuring the sensitivity or complexity of a financial network has also beenexhaustively studied [15, 3, 5, 4]. The topic also has a major importance for financialauthorities in practice, who regularly conduct stress tests to analyze real-world financial .A. Papp and R. Wattenhofer 3 systems. The clearing problem, in particular, also plays an important role in the EuropeanCentral Bank’s stress test framework [7], for example.
The model introduced by [21] describes a financial network as a set of banks (i.e. nodes),denoted by V , with different kinds of financial contracts (i.e. directed edges) between specificpairs of banks. Banks in our examples are usually denoted by u , v or w . Every bank in thesystem has a predefined amount of external assets , denoted by e v for bank v . We assume that each contract in the system is between two specific banks u and v . Acontract obliges u (the debtor) to pay a specific amount of money to bank v (the creditor),either unconditionally or based on a specific event. The amount of payment obligation inthe contract is the weight (in financial terms: the notional) of the contract.While these contracts might be connected to earlier transactions between the banks (e.g.a loan offered by v to u in the past which results in a debt contract from u to v in thepresent), we assume that these initial payments are implicitly represented in the externalassets of banks, and thus the external assets and the contracts together provide all thenecessary information to describe the current state of the system.The outgoing contracts of bank v altogether specify a given amount of total paymentobligations for v . If v is unable to fulfill all these obligations, then we say that v is indefault . In this case, we are interested in the portion of liabilities that v is still able to pay,known as the recovery rate of v and denoted by r v . The definition shows that we alwayshave r v ∈ [0 , v is in default exactly if r v <
1. The recovery rates of all banks isrepresented together in a recovery rate vector r ∈ [0 , V .The model allows two kinds of contracts between banks in the system. In case of a simple debt contract, u has to pay a specific amount to v unconditionally, i.e. in any case. On theother hand, credit default swaps ( CDSs ) are ternary financial contracts, made in referenceto a third bank w known as the reference entity . A CDS describes a conditional debt whichonly requires u to pay a specific amount to v if w is in default. In particular, if the weightof the CDS is δ and the recovery rate of w is r w , then the CDS incurs a payment obligationof δ · (1 − r w ) from u to v .In practice, CDSs often describe an insurance policy on debt contracts for the creditorbank. If v is the creditor of a debt coming from w , and v suspects that w might go intodefault and thus will be unable to pay some of its debt, then v can enter into a CDS as acreditor with some other bank u in the system, in reference to w . If w indeed defaults andcannot pay its liabilities to v , then v instead receives some payment from u . Nonetheless,there could be other reasons for banks to enter CDS contracts, e.g. speculative bets aboutfuture developments in the market. Since payment obligations in CDSs depend on the recovery rate of other banks, the assetsand liabilities of a bank are defined as a function of the vector r . The liability of u towards v is the sum of payment obligations from all simple debt contracts and CDSs, i.e. l u,v ( r ) = c u,v + X w ∈ V c wu,v · (1 − r w ) , Network-Aware Strategies in Financial Systems where c u,v denotes the weight of the simple debt from u to v , and c wu,v denotes the weightof the CDS from u to v with reference to w (understood as 0 if the contracts do not exist).The total liabilities of u is then the sum of liabilities to all other banks, i.e. l u ( r ) = X v ∈ V l u,v ( r ) . In contrast to this, the actual payment from u to v can be lower than l u,v ( r ) if u is indefault. In this case, the model assumes that u makes payments based on the principle ofproportionality , i.e. it uses all of its assets to make payments to creditors, in proportionto the respective liabilities. In practice, this means that u can pay an r u portion of eachliability, and thus its payment to v is defined as p u,v ( r ) = r u · l u,v ( r ) . On the other hand, the assets of v is the sum of its external assets and its incomingpayments, i.e. a v ( r ) = e v + X u ∈ V p u,v ( r ) . Recall that a recovery rate describes the portion of liabilities that a bank can pay. Hencegiven the assets and liabilities of each bank v , the recovery rate r v must satisfy r v = 1 if a v ( r ) ≥ l v ( r ), and r v = a v ( r ) l v ( r ) otherwise. A vector r is called a solution (in financial terms: aclearing vector) if it describes an equilibrium point for these equalities, i.e. if for each bank v , r v satisfies this constraint for the assets and liabilities defined by r . Previous work hasexpressed this by defining the update function f : [0 , V → [0 , V as f v ( r ) = ( , if a v ( r ) ≥ l v ( r ) a v ( r ) l v ( r ) , if a v ( r ) < l v ( r ) , and defining a solution as a fixed point of the update function.In order to model the utility function of nodes in the system, we define the payoff (infinancial terms: equity) of a bank v as the amount of remaining assets after payments if anode is not in default, and 0 otherwise, i.e. q v ( r ) = max( a v ( r ) − l v ( r ) , . We assume thatthe aim of each bank is to maximize its own payoff.Note that assets, liabilities and payoffs are always defined with regard to a certain recov-ery rate vector r . However, in order to simplify notation, we do not show r explicitly whenit is clear from the context, and instead we simply write e.g. a v or q v .Figure 1 shows an example financial system with three banks u , v and w , with a consistentnotation to that of [21, 22]. The system has e u = 2, e v = 1 and e w = 0. There are twodebts of weight 2 in the system: one from u to v , the other from u to w . Finally, the systemcontains a CDS from w to v (also of weight 2), which is in reference to bank u .Regardless of recovery rates, bank u has liabilities l u = 4 and assets a u = 2, so r u = inany case. This implies that u can only make payments of r u · v and w . Given r u = , the CDS induces a liability of 2 · (1 − r u ) = 1 from w to v . Since w receives anincoming payment of p u,w = 1 from u , we have a w = l w = 1, so w can still pay its liabilityand has a recovery rate of r w = 1. Finally, v has incoming payments p u,v = 1 and p w,v = 1,external assets e w = 1, and no liabilities. This implies a v = 3 and l v = 0, and thus r v = 1.Hence ( r u , r v , r w ) = ( , ,
1) is the only solution of the system, providing a payoff of q u = 0, q w = 0 and q v = 3 to the banks.We also use two sanity assumptions introduced by previous work to exclude degeneratecases [21]. First, we assume that no bank enters into a contract with itself or in reference toitself. Furthermore, since CDSs are regarded as an insurance on debt, we require that if abank w is a reference entity of some CDS, then w is the debtor of at least one debt contractof positive weight. .A. Papp and R. Wattenhofer 5
22 2 u vw
Figure 1
Example financial system with three banks. External assets are shown in rectanglesbesides the nodes, simple debt contracts are shown as blue arrows from debtor to creditor, and CDSsare shown as brown arrows from debtor to creditor, with a dotted line specifying the reference entity.
While the principle of proportionality is a simple and natural assumption, financial systemsoften have more complex payment rules in practice. Thus we also introduce a more generalmodel of payments with priorities .That is, we assume that there is a constant number of priority classes P , and eachcontract belongs to one of these priority classes. If a node v is in default, then it first spendsall its assets to fulfill its liabilities in the highest priority class. If v does not have enoughassets to fulfill all such obligation, it spends all its assets on the payments for these edges,proportionally to the amount of liabilities. On the other hand, if v has more assets thanhighest-priority liabilities, then v pays for all the liabilities in this highest priority level, andcontinues using the rest of its assets for the lower-priority liabilities in a similar fashion.More formally, in our modified model, each contract in the network receives another priority parameter (besides its weight), which is an integer in { , ..., P } . The value 1 denotesthe highest priority (i.e. liabilities that have to be paid first), while class P denotes thelowermost priority level.Given a clearing vector r , for each node v , let l ( ρ ) v denote the total amount of liabilitiesof v due to edges on priority level ρ . Let us also introduce the notation l ( ≤ ρ ) v = P ρi =1 l ( i ) v .Assume that v has total assets of a v , and a liability of l v,u on priority level ρ towards anothernode u . Then the payment of v to u is defined as p v,u = , if a v ≤ l ( ≤ ρ − va v − l ( ≤ ρ − v l ( ρ ) v · l v,u , if a v ∈ (cid:16) l ( ≤ ρ − v , l ( ≤ ρ ) v (cid:17) l v,u , if a v ≥ l ( ≤ ρ ) .v For an example, consider a modified version of the network in Figure 1. Assume wenow have 2 priority levels: the debt from u to w is on the higher level, while the other twocontracts are on the lower level. For the case of u , this still means l u = 4, a u = 2 and r u = as before. However, now u uses its 2 units of assets to pay its full liability to w , since thiscontract has higher priority than the debt to v . Hence p u,v = 0 and p u,w = 2, resulting in a w = 2. Since r u = still implies l w = 1 for the CDS, the rest of the payments and recoveryrates remain unchanged: we still have p w,v = 1 and r w = r v = 1. However, the payoffs ofthe banks in the system are now q u = 0, q w = 1 and q v = 2.The main motivation for introducing payment priorities is that in many cases, it is veryclose to what happens in real-world financial systems. In many countries, economic lawsprovide a specific priority list for companies to follow when paying their debts in case of a Network-Aware Strategies in Financial Systems default. This might start with salaries and other payments to the employees of the companyfirst, then specific kind of debt contracts, and so on.Another advantage of priorities is that we can use them to replace so-called default costs .Default costs (also studied in [21, 22]) are an extension of the original model, assumingthat when banks go into default, they immediately lose a specific portion of their assets.This represents the fact that in practice, once a company goes into default, it has a rangeof immediate payment obligations (e.g. employees’ wages) before it can make paymentsto other banks in the system. If we instead represent the bank’s employees as a separatenode in the network, and model this payment obligation with a high-priority edge, then thisallows us to describe the phenomenon without the use of default costs.This observation is crucial because the introduction of default costs comes at a significantprice: intuitively speaking, default costs introduce a point of discontinuity into the updatefunction, and as a result, some financial systems do not have a solution at all [21]. Incontrast to this, without default costs, systems always have at least one solution, as shownby a fixed-point argument in [21]. We point out that the same fixed-point theorem proofalso applies in our model with payment priorities: even though the functions p u,v ( r ) and a v ( r ) become significantly more complicated, they are still continuous.This shows that by introducing priorities, we obtain a model that is significantly morerealistic on one hand, but also ensures the existence of a solution at the same time. ◮ Theorem 1.
Every financial system with payment priorities has at least one solution.
Proof (sketch).
The proof of this claim is identical to the same proof in the original financialsystem model, described in the results of [21]. The main idea of the proof is to apply the fixed-point theorem of Kakutani [16], which ensures the existence of a fixed point of the updatefunction f , and thus a solution. This proof can still be applied after the introduction ofpriorities, since both a v ( r ) and l v ( r ) still remains a continuous function of r , and so does theupdate function f v ( r ) = min( a v ( r ) l v ( r ) , l v ( r ) >
0. The technicalpart of the proof is slightly more complicated, since one has to consider the l v ( r ) = 0 caseseparately. For more details on this proof, we refer the reader to the work of [21]. ◭ We now discuss a wide range of actions that a bank can execute in order to obtain a morefavorable outcome in the system. Note that except for Section 5.3 which explicitly studiesreadjusting priorities, all the results also hold in the base model without priorities.
One of the most natural actions for a bank v would be to simply cancel a debt contractin which v is a creditor. Since the creditor is considered the beneficiary of a debt, in somefinancial/legal frameworks, the regulations may indeed allow a bank to nullify an incomingdebt contract. However, in case of a financial system with short positions, it is actuallypossible that in the end, this indirectly increases the payoff of v . ◮ Theorem 2.
Removing an incoming debt of v can increase the payoff of v . More precisely, our claim is as follows: there exists a financial system S such that (i) S has only one solution r , in which v has payoff q v and an incoming debt contract, and (ii)in the modified financial system S ′ obtained by removing this debt, there is again only onesolution r ′ , in which the payoff q ′ v satisfies q ′ v > q v . .A. Papp and R. Wattenhofer 7 wu v Figure 2
Example for removing anincoming debt /γ
211 1 wu v /γ − γ Figure 3
Removing γ portion of anincoming debt Proof.
Consider the network in Figure 2. Note that the unlabeled nodes in this system canalways pay all their liabilities, so their recovery rate is always 1. Originally, the system has a u = 1 and l u = 2, thus r u = in any case. This implies a w = 2 · (1 − ) = 1, and thus r w = 1. With r w = 1, v obtains no payment from its incoming CDS at all, so the payoff of v in this only solution is q v = p u,v = r u · .One the other hand, consider the system obtained by removing the debt contract from u to v . In this case, a u = l u = 1, and thus r u = 1. This means that w receives no incomingpayments at all, and with a w = 0, we have r w = 0. As a result, v obtains a payment of2 · (1 − r w ) = 2 from its incoming CDS, so we have q v = 2. ◭ The proof shows that releasing an outgoing debt increases the recovery rate of u , whichindirectly yields an extra payoff for v . Note that v could also achieve this result by donatingfunds to u , i.e. by increasing e u by 1. This is even more realistic in a legal framework: theowner(s) of bank v can simply donate a specific amount to bank u , who would accept it inhope of avoiding default. Naturally, this is only a favorable step to v if by donating x unitsof money, it can increase its own payoff by more than x . ◮ Theorem 3.
Donating external assets to another node u can be a favorable step. More precisely, there is a system S such that (i) S has only one solution r , in whichnode v has payoff q v , and (ii) in the system S ′ obtained by replacing external funds of u by e ′ u := e u + x , there is again only one solution r ′ which satisfies q ′ v > q v + x .The proof of the theorem is identical to that of Theorem 2: if v increases e u by x = 1in Figure 2, then again r u = 1, which ultimately provides a payoff of q v = 3 (as opposed tothe original ). Note that in general, this action may allow banks to improve their positionby affecting a bank that is arbitrarily far in the topology of the network.Finally, if v can increase its payoff by releasing an incoming debt, it is natural to wonderif it is always optimal for v to erase the entire debt, or whether it could be beneficial to onlyreduce the amount in some cases. We show that reducing a debt to a given portion γ of itsoriginal weight can also be an optimal strategy. ◮ Theorem 4.
For each constant γ ∈ [0 , , there is a financial system where bank v achievesits maximal payoff by reducing an incoming debt to a γ portion of its original weight. Proof.
Consider a modified version of our previous systems, as shown in Figure 3. We showthat for any γ parameter, the optimal action of v in this system is to let go of γ portionof the incoming debt from u , i.e. to reduce its weight to 1 − γ .Assume that v reduces the incoming debt by a γ portion for some γ ∈ [0 , v as a function of γ . Note that a choice of γ = γ implies that Network-Aware Strategies in Financial Systems a u = l u exactly, and thus r u = 1, r w = 0 and q v = (1 − γ ) + 2 = 3 − γ as a result. Hencewe have to show that q v < − γ in any other case.First consider the case when γ < γ . Since u has l u = 1 + (1 − γ ) = 2 − γ > − γ , u isin default. Then r u = − γ − γ , and thus w receives an incoming payment of a w = 2 γ · (cid:18) − − γ − γ (cid:19) = 2 · ( γ − γ ) γ · (2 − γ ) . This is a decreasing function in γ , and it equals 1 exactly for γ = 0, so a w < γ > w is in default with r w = a w . Then the amount v receives from the CDS is2 · (1 − r w ) = 2 · (cid:18) − · ( γ − γ ) γ · (2 − γ ) (cid:19) = 2 · γ · (2 − γ ) γ · (2 − γ ) . Since q v = (1 − γ ) · r u + 2 · (1 − r w ), we need to show that3 − γ > (1 − γ ) · − γ − γ + 2 · γ · (2 − γ ) γ · (2 − γ ) . After multiplying this by γ · (2 − γ ), expanding the brackets and removing terms that cancelout, we are left with γ · (4 − γ ) > γ · (4 − γ ), which naturally holds since γ < γ .On the other hand, if γ > γ , then a u > l u , and thus r u = 1. This means r w = 0, so v receives an amount of 2 from the CDS, and has a total payoff of (1 − γ ) + 2 = 3 − γ , whichis again less than 3 − γ . Thus selecting γ = γ is indeed the best option for v . ◭ In light of Theorem 3, it is natural to ask if v can also increase its payoff by injecting furtherfunds into its own external assets. That is, if increasing e v by x would allow v to increaseits payoff by more than x in the only solution, then the owner of bank v would be motivatedto invest these extra funds into the bank.However, somewhat surprisingly, it turns out that this is not possible in the same wayas in previous cases: we cannot increase the payoff of v by more than x in the only solutionof the system. More specifically, if a vector r ′ is a solution to the new system and providesa payoff of q ′ v , then r ′ was already a solution of the original system with a payoff of q ′ v − x . ◮ Theorem 5.
Assume that every solution of system S provides a payoff of at most q v for v . Then setting e ′ v = e v + x cannot introduce a new solution r ′ with q ′ v > q v + x . Proof.
Assume that such a new solution r ′ is introduced. Since payoff is always nonnegative, q v ≥
0, and thus q ′ v > x in r ′ . This means that we have a ′ v > x + l ′ v in r ′ . Hence, even if e ′ v was reduced by x (back to its original value e v ), then v could still pay all of its liabilities;thus the same recovery vector r ′ and the same payments on each edge also provide a solutionin the original system S . The payoff of v in this solution is q ′ v − x , which is larger than q v by assumption. This contradicts the fact that q v was the maximal payoff for v in S . ◭ Naturally, if v is in default, then recovery rate of v can indeed be increased in the onlysolution by injecting extra funds. However, an increase of r v does not translate to an increasein payoff, so it is a waste for the owners of v to invest resources for this.On the other hand, while its not possible to produce a new, more favorable solution for v , it is possible to invalidate solutions that are unfavorable to v . That is, if the originalfinancial system had multiple solutions with different payoffs for v , and v is unsure which .A. Papp and R. Wattenhofer 9 wu v Figure 4
A bank v increasing itsown external assets δ wu v δ Figure 5
Readjusting the priority ofoutgoing contracts of these solutions will be implemented by a financial authority, then it is possible that v can inject extra funds to remove a solution where its payoff is much smaller than in othersolutions. This may allow v to increase its worst-case payoff, or its payoff in expectation (incase of a randomized choice of solution). ◮ Theorem 6.
Given a financial system S with two solutions, it is possible that setting e ′ v = e v + x removes the solution which is unfavorable to v . More precisely, there is a system S such that (i) S has two solutions r and r , withsolution r satisfying q v = 0, and (ii) in the system S ′ obtained by setting e ′ v := e v + x , theonly solution is r ′ = r , satisfying q ′ v > x . Proof.
Consider the system in Figure 4, which has two solutions. The design of the systemensures r u = r v and r w = 1 − r u . If r v = 1, then this implies r u = 1 and r w = 0, in whichcase v has a v = 100, giving a solution with q v = 99. On the other hand, if r v <
1, then ithas to satisfy r v = 100 · (1 − r w )1 = 100 · r u = 100 · r v . This is only satisfied if r v = 0, so this is the only other solution, providing q v = 0.Now assume that v invests x = 1 extra funds to have e v = 1. In this case, the systemalways has r v = 1, hence r u = 1 and r w = 0. This implies that v obtains a payment of100 in the CDS, resulting in a payoff of q v = 100. Even if we subtract the extra x = 1investment, v has an extra payoff of 99, and thus it has indeed increased its worst-casepayoff significantly. ◭ Assuming payments with priorities as discussed in Section 4, it is also interesting to knowif a node can improve its situation by readjusting the priorities of its outgoing edges. Thatis, in a more flexible regulation framework, banks may be allowed to choose to some extentthe order in which they fulfill their payment obligations. However, we show that similarlyto the previous case, readjusting the priorities of outgoing edges cannot introduce a bettersolution. ◮ Theorem 7.
Assume that every solution of system S provides a payoff of at most q v for v . Then redefining v ’s outgoing priorities cannot introduce a new solution r ′ with q ′ v > q v . Proof.
Assume that such a new solution r ′ is introduced. Payoff is nonnegative, so q v ≥ q ′ v >
0. This implies that a ′ v > l ′ v in r ′ , i.e. v is able to pay all of its liabilities inevery outgoing contract. However, in this case, the priorities on the outgoing edges do notmatter; hence r ′ is a solution of S ′ regardless of how the priorities of outgoing contracts arechosen. In particular, r ′ is already a solution of the initial system S before the prioritieswere reorganized, giving the same payoff q ′ v in S . This contradicts the fact that q v was themaximal payoff for v in S . ◭ However, it is again possible that v can increase its recovery rate by readjusting priorities.Recall that in the previous case of increasing the bank’s own external assets, we did notexplore this possibility, since it required the bank v to invest extra funds while not yielding(the same amount of) extra payoff. However, readjusting priorities is an action that v mightbe able to execute free of charge. Thus if we define the recovery rate as the secondaryobjective function of a bank (i.e. even if v is in default and thus has 0 payoff, it is notoblivious to the outcome, and prefers a higher recovery rate), then redefining priorities mayallow v to achieve a more preferred outcome without having to invest any extra funds. ◮ Theorem 8.
Redefining v ’s outgoing priorities can increase the recovery rate of v . Proof.
Consider the system in Figure 5 with a choice of δ = . Originally, each contractis in the same (lower) priority class. Bank v never has enough assets to pay its liabilities,hence u is also in default. In this case, we have r u = r v and r w = 2 − · r u , so v receives δ · (1 − r w ) = δ · (2 · r v −
1) funds from the CDS. This means that r v = δ · (2 · r v −
1) + 12 , which, after reorganization, gives δ − · ( δ − · r v , and thus r v = . This is the onlysolution of the system if δ = 1.Now assume that v is able to raise the debt towards u to the higher priority level. Inthis new system, v first fulfills its payment obligation to u , which is always possible from itsexternal assets. Hence r u = 1 in this case, implying r w = 0 and thus a payment of to v inthe CDS. This implies r v = in the only solution of the new system. ◭ Finally, we show that redefining priorities can allow v to remove an unfavorable solution,and thus increase its worst-case or expected payoff as in the previous subsection. ◮ Theorem 9.
Given a financial system S with two solutions, redefining v ’s outgoing pri-orities can remove the solution which is unfavorable to v . Proof.
Consider the system in Figure 5 with a choice of δ = 100. As discussed in the proofof Theorem 8, if r v <
1, then the only solution is r v = . However, the large δ value nowallows another solution in the original system: if r v = 1, then r u = 1 and r w = 0, ensuringthat v indeed has enough funds to pay its liabilities. The two solutions come with payoffsof q v = 0 and q v = 98, respectively.Now if v raises its debt towards u to the higher priority level, then r u = 1 is alwaysguaranteed, so r w = 0 and thus v indeed has a payoff of 98 in the only solution. ◭ Finally, we briefly show that the attempts of banks to influence the system can also easilylead to situations that can be described by classical game-theoretic settings. .A. Papp and R. Wattenhofer 11 wu v v Figure 6
Prisoner’s dilemma infinancial systems
11 11 δδ vu u ′ v ′ δ
00 000
Figure 7
Dollar auction game infinancial systems
We first show an example where if two nodes simultaneously try to influence the systemto their advantage, then the resulting situation is essentially identical to the well-knownprisoner’s dilemma [19].Consider the financial system in Figure 6, where banks v and v want to influence thesystem to achieve a better outcome. Assume that in the current legal framework, the onlystep available to these banks is to completely remove their incoming debt contract from u (as in Theorem 2); both banks can decide whether to execute this step or not. Note thatcanceling a debt increases the recovery rate of u , which indirectly implies a larger payment onthe CDS for both v and v , and thus can be beneficial for both banks. Applying prisoner’sdilemma terminology, we also refer to the step of canceling the debt as cooperation , and thestep of not canceling the debt as defection .Now let us analyze the payoff of v and v in each strategy profile. Note that r w = 1 − r u ,so the payment on the CDSs for both v and v is 3 · (1 − r w ) = 3 · r u in any case.If both of the nodes cooperate (i.e. both debts are removed), then u can pay its remainingliabilities, thus r u = 1. This implies a payment of 3 on the CDS, which is the only asset ofthe acting nodes in this case; hence q v = q v = 3.If both of the nodes defect (no debt is removed), then we only have r u = , resultingin a payment of 1 from the CDS. However, in this case, both v and v also get a directpayment of 5 · r u = from the defaulting u , which adds up to a total payoff of = 2 . ˙6.Finally, assume that only one of the nodes cooperate (say, v ). With only one of theoutgoing debts removed, u will have a recovery rate of r u = . This results in a paymentof on the CDS for both nodes. However, note that v still has an incoming debt contractfrom u , and receives a payment of 5 · r u = on this contract. This implies q v = = 1 . q v = 4 for the strategy profile. The symmetric case yields q v = 4 and q v = 1 . v , defection yields a payoff of 4(instead of only 3) if v cooperates, and it yields a payoff of 2 . ˙6 (instead of only 1 .
5) if v defects. Thus the Nash-Equilibrium of the game is obtained when both players defect, with q v = q v = 2 . ˙6. However, both players would be better off in the social optimum of mutualcooperation, which gives q v = q v = 3.Some other well-known two-player games, e.g. the chicken or stag hunt game [19] can alsoeasily occur in financial networks in a similar fashion. We outline some example financialsystems that correspond to these further games in Appendix A. We also show an example of the dollar auction game [23] in financial systems. Considerthe system in Figure 7, and assume that banks u ′ and v ′ want to influence this system bydonating extra funds to banks u or v (as in Theorem 3). Note that the payoff of u ′ and v ′ depends on the recovery rates of u and v , respectively, which in turn have a recovery ratedepending on each other. Due to the design of the system, u ′ prefers bank u to be in default,and thus it wants to increase e v ; similarly, v ′ prefers bank v to be in default, so it wants toincrease e u . We assume that 1 unit of money is a very high amount in our context, and thus u ′ and v ′ cannot donate enough to ensure that u or v pays its debt entirely from externalassets; i.e. we assume that e u , e v < ǫ of fundsthat u ′ or v ′ can donate in one step. In our example, we choose a δ value in the magnitudeof this ǫ , e.g. δ = 6 ǫ .Let us now analyze the recovery rates of u and v in the solutions of the system.The vector r u = r v = 1 cannot be a solution, since it would imply no payment on theincoming CDSs, and thus these recovery rates would only be possible if e u , e v ≥ r u = 1, r v < v receives no incoming payments, wemust have r v = e v = e v . Thus bank u has assets of e u + 1 − e v , which has to be at least1 for r u = 1 to hold. Hence this is only a solution if e u + 1 − e v ≥
1, i.e. e u ≥ e v . In asymmetric manner, r v = 1, r u = e u is only a solution if e v ≥ e u .If r u < r v < r u = e u + 1 − r v and r v = e v + 1 − r u must hold.This implies e u = e v , and r u + r v = 1 + e u . Hence if e u = e v , then any r u , r v with r u + r v = 1 + e u provides a solution.Thus as long as e u , e v <
1, the behavior of the system is as follows:If e u < e v , then the only solution is r u = e u , r v = 1. This means q u ′ = δ · (1 − e u ) and q v ′ = 0.If e u > e v , then the only solution is r u = 1, r v = e v . This implies q u ′ = 0 and q v ′ = δ · (1 − e v ).If e u = e v , then any r u , r v ≤ r u + r v = 1 + e u is a solution of the system. In thegeneral case, q u ′ = δ · (1 − r u ) and q v ′ = δ · (1 − r v ).This describes a setting that is very similar to a dollar auction. In the beginning, with e u = e v = 0, we have a range of different solutions, and a choice among these depends ona financial authority. One of the banks (say, bank u ′ ) decides to donate a small ǫ amountof funds to v ; then with e v = ǫ > e u = 0, bank u ′ receives a payment of δ · (1 −
0) in theonly resulting solution. At this point, the payoff of v ′ is 0; however, at the cost of donating2 · ǫ funds to u , it could achieve e u = 2 ǫ > e v = ǫ , thus resulting in a single solution witha payoff of q v ′ = δ · (1 − ǫ ). Since this increases the payoff of v ′ by δ · (1 − ǫ ) at the cost ofonly 2 ǫ , this is indeed a rational step for the appropriate δ and ǫ values. However, then u ′ is again motivated to donate 2 ǫ more funds to increase e v over e u again, and so on.Assuming that both u ′ and v ′ has at most funds to donate, we always have e u , e v ∈ [0 , ]. This shows that e.g. if we have e u > e v , then the payoff of bank v ′ is always within q v ′ = δ · (1 − e v ) ∈ [ δ/ , δ ] = [3 ǫ , ǫ ] . Hence in every step, it is indeed rational for v ′ to donate another 2 ǫ funds, since it increasesits payoff from 0 to at least 3 ǫ . After a couple of rounds, u ′ and v ′ will have both donatedsignificantly more money than their payoff of at most 6 ǫ . However, the banks are still alwaystempted to execute the next donation step to mitigate their losses. .A. Papp and R. Wattenhofer 13 References Daron Acemoglu, Vasco M Carvalho, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. Thenetwork origins of aggregate fluctuations.
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Appendices
A Further two-player games in financial systems
We now show some further examples of financial system that represent other well-knowntwo-player-two-strategy games, similarly to the case of the prisoner’s dilemma.
A.1 Stag Hunt
We first analyze the financial system in Figure 8, which represents the coordination gameknown as stag hunt [19]. We again assume that the two acting nodes v and v can onlyexecute the action of completely removing their incoming debt contract from u and u ,respectively. As before, we refer to the decisions of removing and not removing the debt ascooperation and defection, respectively.Recall that canceling an incoming debt and donating funds to another bank are verysimilar operations in some sense. With a slight modification to our system, we could alsopresent the same example game in a setting where the acting banks must decide to donateor not donate a specific amount of funds to a bank. For our example systems, we select theaction that allows a simpler presentation.Let us analyze the payoffs in the different strategy profiles. If both players cooperate,then both u and u will only have a liability of 2, which implies r u = r u = 1. In thiscase, w receives no payment from either of the CDSs, resulting in r w = 0. This means thatboth v and v get a payment of 3 from their incoming CDSs. With their debt contractscanceled, we get q v = q v = 3.If both players defect and keep their debt contract, then both u and u will have arecovery rate of only . This implies a payment of 1 to w on both CDSs, so w avoids defaultwith r w = 1. This means that the acting nodes will not receive any payment on the CDS.On their debt contracts, they both receive ·
2, i.e. q v = q v = 1.Finally, assume that v cooperates but v defects. In this case, we end up with recoveryrates of r u = 1 and r u = . Thus w only receives payment on the CDS that is in referenceto u . However, this payment of · w to fulfill its liabilities, andhence r w = 1. Again, v and v do not receive any payment on the CDS. However, v stillhas an incoming debt contract that ensures a payment of · v has no assets atall. Thus the solution provides q v = 0 and q v = 1. In a symmetric manner, the case when v cooperates and v defects incurs q v = 1, q v = 0.Thus the system represents a game where the players are incentivized to coordinate theirstrategies. Both the case when both banks cooperate and when both banks defect is a pureNash-Equilibrium, with mutual cooperation being the social optimum. However, if a bankis unsure whether the other bank will cooperate, it might be motivated to defect in order toavoid the risk of getting no payoff at all.
22 3 32 22 2 1 wu u v v Figure 8
Stag hunt game in afinancial system wu v v Figure 9
Chicken game in afinancial system
A.2 Chicken game
Finally, we also provide an example of the chicken game (also known as the hawk-dove game[19]) when the pure Nash-Equilibria are obtained in the asymmetric strategy profiles.Consider the financial system in Figure 9, and assume the acting banks v and v nowhave the options to either donate 1 unit or money to another bank, or do not donate moneyat all. Due to the structure of the network, the nodes are motivated to donate this 1 unitof money to u , since this results in a payment on their incoming CDS contract. We againrefer to donating a unit of money to u as cooperation, and not donating as defection.If both nodes defect, then u still has no assets at all, implying r u = 0. This results in r w = 1, and hence the acting nodes receive no incoming payment, so q v = q v = 0.If both nodes cooperate, then u has more than enough assets to pay its liabilities, res-ulting in r u = 1 and r w = 0. This means that both nodes get a payment of 3 in the CDS.After subtracting the amount they have donated, we get q v = q v = 2.However, to ensure that u does not go into default, it is enough if only one of the twonodes make a donation. I.e. if v cooperates but v defects, then u still has 1 asset, whichalready implies r u = 1, r w = 0 and a payment of 3 to both v and v on their incomingCDS. After subtracting the donated funds, this gives q v = 2 and q v = 3. Similarly, if v cooperates and v defects, we obtain q v = 3, q v = 2.The payoffs show that there is no dominant strategy in the game: if v cooperates, thenthe best response of v is to defect, while if v defects, then the best response of v is tocooperate. This implies that the two pure Nash-Equilibria are obtained in the strategyprofiles when the banks choose the opposite strategies.Note that we can easily generalize this setting to the case of more than 2 acting nodes, res-ulting in the so-called volunteer’s dilemma. For any k , we can add distinct banks v , v , ..., v k that are all connected to the financial network in the same way (through an incoming CDSof weight 3 in reference to w ), and all have the same two options of either donating 1 unitof money to u or not acting at all. Note that we also have to ensure that the (currentlyunlabeled) debtor of the CDSs to these acting nodes has enough resources to make paymentson these CDSs in any case, i.e. it must have external assets of at least 3 · k .In this case, we obtain a game where again only one volunteer bank v i is required tomake a donation to u , and this already ensures a payoff of 3 for every other bank (and apayoff of 2 for v ii