Flow Allocation Games

TL;DR

This paper introduces a game-theoretic model of flow allocation in networks, analyzing equilibrium existence, complexity, and efficiency, with implications for financial network incentives.

Contribution

It provides new insights into equilibrium properties and computational complexity in flow allocation games, highlighting differences between ranking strategies.

Findings

Pure Nash and strong equilibria existence results

Complexity classifications for equilibrium computation

Bounds on prices of anarchy and stability

Abstract

We study a game-theoretic variant of the maximum circulation problem. In a flow allocation game, we are given a directed flow network. Each node is a rational agent and can strategically allocate any incoming flow to the outgoing edges. Given the strategy choices of all agents, a maximal circulation that adheres to the chosen allocation strategies evolves in the network. Each agent wants to maximize the amount of flow through her node. Flow allocation games can be used to express strategic incentives of clearing in financial networks. We provide a cumulative set of results on the existence and computational complexity of pure Nash and strong equilibria, as well as tight bounds on the (strong) prices of anarchy and stability. Our results show an interesting dichotomy: Ranking strategies over individual flow units allow to obtain optimal strong equilibria for many objective functions. In contrast, more intuitive ranking strategies over edges can give rise to unfavorable incentive properties.

Equations

$$c_v^{+}=\sum _{e\in E^{+}(v)}{c}_e.$$

$$0\le a_e(y)\le c_e,$$

$$\sum _{e\in E^{+}(v)}{a}_e(y)\le y,$$

$$\sum _{e\in E^{+}(v)}{a}_e(y)=\min \{c_v^{+},y\}.$$

$$f_{(v,w)}=a_{(v,w)}(f_v).\text{(fixed point constraint)}$$

$$v_2\text{ plays }$$

$$v_1\text{ plays }$$

$$a_{(v_1,v_3)}(y)=y-a_{(v_1,v_2)}(y)\text{ for }y\ge 0.$$

$$a_e(y)\ge a_e(y^{\prime }).\text{(monotonicity constraint)}$$

$$g(\mathbf{f}{)}_{(u,v)}=a_{(u,v)}\left(\sum _{e^{\prime }\in E^{-}(u)}{f}_{e^{\prime }}+b_u\right)\text{ for every }(u,v)\in E$$

$$u_v(\mathbf{a})=u_v(\mathbf{a},\hat{\mathbf{f}})=\sum _{e\in E^{+}(v)}{\hat{f}}_e.$$

$$\underset{\mathbf{a}\in {\mathcal{A}}_{\text{Eq}}(\Gamma )}{\max }\frac{\text{textsc}Q({\mathbf{a}}^{\ast })}{\text{textsc}Q(\mathbf{a})}=\frac{\text{textsc}Q({\mathbf{a}}^{\ast })}{{\min }_{\mathbf{a}\in {\mathcal{A}}_{\text{Eq}}(\Gamma )}\text{textsc}Q(\mathbf{a})}.$$

$$\text{textsc}Flow(\mathbf{a})=\sum _{e\in E}{\hat{f}}_e=\sum _{v\in V}\sum _{e\in E^{+}(v)}{\hat{f}}_e=\sum _{v\in V}{u}_v(\mathbf{a}),$$

$$\sum _{e\in E^{+}(v)}{a}_e(f_v)=\min \{f_v,c_v^{+}\}.$$

$$\sum _{v\in V}{b}_v$$

$$=\sum _{v\in V}\left(f_v-\sum _{e\in E^{+}(v)}{a}_e(f_v)\right)=\sum _{v\in V}\max \{0,f_v-c_v^{+}\}.$$

$$o(C)=\{v\in V\mid \exists \text{ v-u-path of active edges, for some }u\in C\},$$

$$\sum _{e\in E^{\prime }}{f}_e^{\ast }$$

$$=2\sum _{v\in V}{b}_v+\text{textsc}Flow({\mathbf{f}}^{\ast })\ge 2\sum _{v\in V}{b}_v+\text{textsc}Flow(\mathbf{a}).$$

$$d=\underset{{\mathbf{f}}^{\ast },\mathcal{C}({\mathbf{f}}^{\ast })}{\min }\underset{C\in \mathcal{C}({\mathbf{f}}^{\ast })}{\max }|C_i|$$

$$\text{textsc}Flow({\mathbf{a}}^{\ast })=\sum _{v\in V}\sum _{e\in E^{+}(v)}{f}_e^{\ast }=\sum _{C_i\in \mathcal{C}({\mathbf{f}}^{\ast })}|C_i|-2\sum _{v\in V}{b}_v\le \sum _{C_i\in \mathcal{C}({\mathbf{f}}^{\ast })}d.$$

$$\text{textsc}Flow(\mathbf{a})\ge \sum _{C_i\in \mathcal{C}({\mathbf{f}}^{\ast })}1$$

Full text

We provide a cumulative set of results on the existence and computational complexity of pure Nash and strong equilibria, as well as tight bounds on the (strong) prices of anarchy and stability. Our results show an interesting dichotomy: Ranking strategies over individual flow units allow to obtain optimal strong equilibria for many objective functions. In contrast, more intuitive ranking strategies over edges can give rise to unfavorable incentive properties.

1 Introduction.

Flows and circulations in networks are a classic problem domain in combinatorial optimization. A network flow is called a feasible circulation in a graph if it maintains flow conservation at all nodes, and some given lower and maybe upper bounds on the flow on edges are fulfilled. Dinitz [Dinitz70] and Edmonds and Karp [EdmondsK72] provided strongly polynomial-time algorithms for solving the existence problem of a circulation in a network. Tardos85 even showed that the minimum-cost circulation problem can be solved in strongly polynomial time. There is a vast number of applications of flow problems, and flows give rise to beautiful and favorable mathematical and algorithmic properties. As a consequence, variants of flow and circulation problems have been investigated for decades.

In this paper, we explore a novel game-theoretic model for circulations in networks. In our model, each node belongs to a player who aims to maximize the flow through the node. We assume that players can strategically allocate the flow entering their node to outgoing edges. Given the strategy choices of all players, a maximal circulation that adheres to the chosen allocation strategies evolves in the network.

Flows and circulations have a broad range of applications to different areas. A fascinating area, where circulations play a key role, is in the analysis of financial networks and systemic risks. A very popular approach to model financial networks has emerged from the seminal work by EisenbergN01. In their model, the financial market can be seen as a directed graph $G=(V,E)$. The node set $V$ corresponds to the set of institutions (or firms). The set of directed edges $E$ expresses the debt relations among firms. Each edge $e=(u,v)\in E$ has a capacity $c_{e}$ that corresponds to the nominal liability of firm $u$ to firm $v$. In addition, each firm $v$ has a non-negative supply $b_{v}$ that corresponds to external assets. Given this networked scenario, the goal is to understand the properties of clearing, i.e., the resulting payments when firms have to clear their debt and “pay their bills” to their creditors. For this clearing task, we view the graph as a flow network, where payments constitute a flow of funds or assets (which we will call money). Each node has a non-negative supply and potentially receives additional money over its’ incoming edges. It then can use all this money to allocate flow towards its’ outgoing edges. As a consequence, money starts to circulate in the network. Eventually, all edges of a node become tight (and all debt is cleared), or the node runs out of funds.

In the majority of the literature, the circulation flow is governed by a static proportional (also called pro-rata) strategy for each node. The node must allocate the entire outgoing flow in proportion to the capacities of its outgoing edges. More recently, interest has emerged in more general, decentralized, and monotone allocation strategies. Notably, in an influential work, CsokaH18 consider flows based on an arbitrary integral, monotone allocation strategy for each node (which we term unit ranking below). This extension leads to a variety of interesting questions for the resulting money circulations.

In this work, our interest lies in the decentralized and, more concretely, incentive and stability properties of the circulation problem. In our model, nodes can individually choose the flow allocation strategy for the outgoing edges. Flow allocation games of this kind have been of interest recently – in particular, GuhaKV19 explore games, in which each agent controls the strategies of one or more nodes and wants to maximize the flow routed to an agent-specific sink in the network. Our game is closely related, with the difference that each agent corresponds to a single node, but it might neither be a source nor a sink. Instead, the goal of the node in our game is to maximize the flow circulating through the node. In a financial context, this is equivalent to the natural goal of maximizing the equity (total assets minus total liabilities).

Each node can choose as strategy an allocation function that yields for each amount of available flow an assignment of this flow to the outgoing edges. Similar to CsokaH18 our interest lies in monotone, ranking-based strategies. Game-theoretic variants of maximum flows, even based on rankings, have been of recent interest in economics. Fleiner2014 introduced stable flows, which have been further developed by KiralyP13, Cseh2013, and CsehM19. In stable flows, each node has an inherent preference ranking over the edges. A node $v$ strives to maximize the amount of flow on its preferred edges. Thus, a stable flow can be seen as a flow that is immune to coalitional deviations of players, i.e., nodes of a non-saturated walk can jointly decide to add flow on this walk. In our work, the ranking is not externally given, but represents a strategic decision of the players.

To the best of our knowledge, flow allocation games studied in this paper have not been addressed before. We provide a cumulative analysis of the properties of equilibria in these games. We focus on pure Nash and strong equilibria. In these equilibria, nodes have no unilateral (pure Nash) or coalitional incentives (strong) to deviate from their chosen strategies. Depending on the set of strategies, the resulting games have different properties. If we assume that strategies are restricted to priority orderings over edges, existence of a pure Nash or a strong equilibrium is not guaranteed and becomes strongly NP-hard to decide. Instead, if nodes can assign each unit of available flow in an arbitrary monotone fashion, a strong equilibrium always exists and can be computed in strongly polynomial time. Moreover, this strong equilibrium maximizes the total amount of flow circulating in the network (and, as such, Pareto optimizes the utility of all nodes). In addition, we show that for a diverse set of objective functions, there is a strong equilibrium that is optimal w.r.t. this objective (such as, e.g., minimizing the number of nodes or firms that are in default and unable to pay their liabilities).

This interesting technical dichotomy between games with different payment strategies (restricted edge-based vs. arbitrary integral and monotone) offers insights into the properties of financial networks. Our results show that a benevolent designer could realize a clearing mechanism with monotone unit-based payment strategies that leads to a socially optimal state, for many different notions of “social optimum”. It comes with the additional guarantee of giving no coalition of firms an incentive to pay their debts differently. In contrast, if clearing payments are determined in a decentralized fashion resulting in some arbitrary Nash or strong equilibrium, the total amount of flow in the system can deteriorate drastically (and similarly the social quality for many objectives).

Similar problems arise if a centralized mechanism is restricted to edge-based priorities. This can lead to non-existence of pure equilibria in the resulting games. Even if pure equilibria exist, they can be undesirable since, e.g., the total amount of circulating flow can be very small. This shows a marked contrast between centralized and decentralized approaches and highlights how the structure of permissible strategies impacts the structural properties of the resulting flows.

1.1 Contribution and outline.

In Section 2 we introduce our formal model of a flow allocation game. We focus on natural classes of ranking-based payment strategies for the nodes as introduced by CsokaH18. For an edge-ranking strategy, a node ranks its outgoing edges and assigns its incoming flow in the order of the ranking222Csóka and Herings called them priority rules.. As a superset of strategies, we consider unit-ranking strategies, where flow is considered in units. Instead of edges, each node ranks single units of each edge capacity. Edge- and unit-ranking strategies both are classes of monotone strategies, where the mapping of the available flow of a node to every outgoing edge is an arbitrary monotone function.

There can be several feasible flows for a given strategy profile. In fact, the feasible flows form a lattice with a partial order based on the total outgoing flow of each node [CsokaH18]. There is a unique feasible flow that forms the supremum of the lattice – it pointwise maximizes the outgoing flow to each node (for the given profile of monotone strategies). We assume that this supremum flow is the clearing flow and defines the utility of each node in the game. After discussing some structural insights on the feasible flows in Section 3.1, we show in Section 3.2 that the supremum flow can be computed in strongly polynomial time (Proposition 8) for profiles of unit-ranking strategies.

In Section 4 we study unit-ranking games, in which all nodes use unit-ranking strategies. Our interest lies in the existence, computational complexity, and social quality of equilibria. We show that in every such game there exists a strategy profile that represents a strong equilibrium, in which no coalition of nodes has an incentive to deviate (Theorem 9). Furthermore, there are even strong equilibria that represent a state that maximizes the circulation flow. Moreover, for a large variety of natural notions of social objective functions (such as the sum, the minimum, or the geometric mean of all utilities, the number of fully saturated nodes, etc.) there is a strong equilibrium that maximizes the objective over all possible circulations (Corollary 12).

For simplicity, in the remainder of the paper we then focus on one standard objective in circulation problems, the total amount of flow circulating in the network. A strong equilibrium that maximizes the circulation can be computed in strongly polynomial time (Theorem 9). It can even be represented using a number of bits polynomial in the size of the input (i.e., the size of the graph and the size of all numbers in logarithmic encoding), even though every strategy shall rank all single flow units that a node might have available, and their number could be pseudo-polynomial (i.e., linear in the unary encoding size of the input). In contrast, it is strongly NP-hard to find a best-response strategy for a single node in a given arbitrary strategy profile of a unit-ranking game (Theorem 14).

For worst-case equilibria and the strong price of anarchy, we show that the deterioration of the circulation in a strong equilibrium compared to a max-circulation is tightly characterized by the min-max length of cycles in any max-circulation (Theorem 16). This implies that in networks, in which an optimal circulation is composed of small cycles, we see a small inefficiency in strong equilibria. In contrast, the circulation of a worst-case Nash equilibrium, which is stable only against unilateral deviations, can be arbitrarily worse than in an optimum, even in simple games with a constant number of nodes (Proposition 15).

In Section LABEL:sec:edge we study a natural and interesting restriction on the strategies and analyze edge-ranking games, in which all nodes are restricted to edge-ranking strategies. Restricting the strategy space to rankings over edges can have devastating consequences for the existence of equilibria and the amount of circulating flow in an equilibrium. In edge-ranking games, pure Nash and strong equilibria can be absent, and deciding their existence is strongly NP-hard (Theorem LABEL:thm:edgeExistNPC). The same hardness applies for computing an optimal strategy profile, and for computing a pure Nash or strong equilibrium when it is guaranteed to exist. Even the best strong equilibrium can be a factor of $\Omega(n)$ worse than an optimum in terms of the circulating flow (Proposition LABEL:prop:poa). For pure Nash equilibria, even the best one can be arbitrarily worse than an optimum (Proposition LABEL:prop:posExtrnalInflow).

We conclude in Section LABEL:sec:conclude with a summary of the main findings, a discussion of our results, and directions for future work.

1.2 Related work.

Flow allocation games are based on circulations in financial network models that emerged from [EisenbergN01]. Rather than proportional payments, CsokaH18 analyze edge- and unit-ranking strategies as well as arbitrary monotone strategies. They analyze the structure of clearing flows and show that they constitute a complete lattice. For completeness, in Section 2 we recapitulate these findings using our notation.

Flow games.

Our game-theoretic approach is related to a number of existing game-theoretic models based on flows in networks. In cooperative game theory, there are several notions of flow games based on a directed flow network. Existing variants include games, where edges are players [DubeyS84, KalaiZ82, KalaiZ82a, GranotG92, DengIN99, BachrachR09], or each player owns a source-sink pair [Papadimitriou01, MarkakisS05]. The total value of a coalition $C$ is the profit from a maximum (multi-commodity) flow that can be routed through the network if only the players in $C$ are present. There is a rich set of results on structural characterizations and computability of solutions in the core, as well as other solution concepts for cooperative games. In contrast to our work, these games are non-strategic. We consider each player as a single node with a strategic decision about flow allocation.

More recently, a class of strategic flow games has been proposed in [GuhaKV19, KupfermanVV17]. There is a capacitated flow network with a set of source nodes. At each source node, a given amount of flow enters the network. Each node of the network is owned by a single player. Each player always owns a designated sink node, as well as one or more additional nodes from the network. A player can choose a flow strategy for each of her nodes. The flow strategy specifies, for every node $v$ and every $x\geq 0$, how an incoming flow of $x$ at $v$ is distributed onto the outgoing edges (if any). Each flow strategy needs to fulfill flow conservation constraints at every node, subject to capacity on the outgoing edges. Each player aims to maximize the incoming flow at its sink node.

For these games there exist a number of $\Sigma_{2}^{p}$-completeness results for, e.g., determining the value of a game in a two-player Stackelberg variant, or determining the existence of a pure Nash equilibrium in a multi-player variant. In the latter game, computing a best response can also be NP-hard. Our approach is related to these games. However, motivated by financial networks we assume each firm is a single node. The firm optimizes the incoming flow at its node (without it being a designated sink node). We study the computational complexity and social quality of equilibria. Moreover, strategic incentives arise mainly from cycles in the network (see Section 2.4 below) – a condition absent in the existing work on max-flow games [GuhaKV19, KupfermanVV17] where the network is assumed to be acyclic.

The problem of computing a clearing state for a given strategy profile in our games is closely related to the notion of a stable flow studied in [Fleiner2014, CsehM19]. In the stable flow problem, each node is equipped with an intrinsic preference order over both incoming and outgoing arcs. The goal is to route as much flow as possible over most preferred arcs. There always exists a stable flow, where no group of agents all can benefit from rerouting the flow along a walk, and such a flow can be computed in polynomial time. The set of stable flows forms a lattice. The model has been extended to an over time setting [Cseh2013] and to a multi-commodity variant [KiralyP13].

Financial networks.

We consider issues of strategic choice and computational complexity in flow allocation games. Flow allocation games have a strong connection to clearings in financial networks. There have been works addressing computational complexity of diverse issues in financial networks, such as pricing options with [AroraBBG11] and without information asymmetry [BravermanP14], finding clearing payments with credit default swaps [SchuldenzuckerSB17], or estimating the number of defaults when providing a shock in the financial system [HemenwayK16].

In addition, many extensions to the model by Eisenberg and Noe have been proposed in the literature on financial networks. However, even models including cross-holdings of equity [Suzuki02], default costs [RogersV13], or debt contracts of different seniorities [Fischer14] follow the idea of the basic approach that all contracts have to be cleared consistently, i.e., clearing payments locally adhere to the rather mechanical clearing rule and constitute a fixed point solution globally. Indeed, BaruccaBCDVBC20 have shown that many of the above models can be unified in terms of self-consistent network valuations. A well-known result of such models is the “robust-yet-fragile” property exhibited by financial networks, i.e., contagion arises in an all-or-nothing fashion akin to the formation of a giant connected component in random graph models [GaiS10]. This provides important insights into systemic risk and advises the need for macro-prudential regulation.

An extended abstract of this paper has appeared in the proceedings of the ITCS 2020 conference [BertschingerHS20]. Subsequent to publication of the extended abstract, there has been a significant interest in extending our results and analyzing closely related computational problems in the context of financial networks. KanellopoulosKZ21 study flow allocation games with credit default swaps (CDS). They obtain a number of NP-hardness results for equilibrium existence. Instead of CDSes, HoeferW22 consider extensions to seniorities (expressed by a class of threshold strategies) along with minimal clearing based on the infimum (rather than the supremum) of feasible flows. For endogeneous seniorities, they extend our main existence result for strong equilibria; for exogenenous seniorities, they observe NP-hardness results for equilibrium existence.

More fundamentally, Ioannidis et al. [IoannidisKV22ICALP, IoannidisKV23] prove FIXP-completeness results for computing a strong approximation of clearing states with debts and CDSes, contrasting PPAD-completeness for weaker notions of approximation [SchuldenzuckerSB17]. Further hardness results for computing optimal clearing states are provided by PappW21WINE. They also study the question of computing the set of banks that default in such networks [PappW21]. Interestingly, for networks with CDSes, the point in time at which a bank announces to default plays an important role. The same authors analyze incentives for adjusting the network structure to optimize individual funds in the clearing state by deleting an incoming edge or gifting some money to other banks [PappW20a]. Similar questions of optimally removing debt contracts or bail-outs are analyzed by KanellopoulosKZ22. In a related spirit, the effects of debt swaps of incoming payment obligations by two banks are analyzed in [PappW21EC, FroeseHW23].

In a broader context, strategic and financial aspects of networks have received substantial attention over the last decade. A related, yet orthogonal, body of work considers trading networks [HatfieldKNOW13, Ostrovsky08]. These models are closely related to two-sided matching under preferences and the study of competitive equilibrium. Rather than circulation effects, agents strive to establish profitable trades with their neighbors. Depending on the model variant, this results in agents matching into pairs, exchanging goods, or establishing upstream/downstream relations with suppliers and customers. The analysis of these networks usually addresses similar issues as the ones we consider here, such as existence, structure, and computational complexity of equilibria. Equilibrium computation in trading networks has recently been studied in, e.g., [CandoganEV21, FleinerJST23]. These works also contain a good overview of related work and pointers into the existing literature.

2 Flow allocation games.

2.1 Network model, monotone strategies, and utilities.

Network model.

In a flow allocation game $\Gamma=(G,(b_{v})_{v\in V},(c_{e})_{e\in E})$ we are given a graph $G=(V,E)$ with a node set $V$ and a set of directed edges $E$. Each node $v\in V$ corresponds to a player and has a fixed supply $b_{v}\geq 0$. The capacity $c_{e}\geq 0$ of an edge $e=(u,v)$ is the maximum amount of flow that $u$ can forward to $v$. In terms of financial networks, $b_{v}$ is the amount of external assets of $v$, and $e$ represents a liability of value $c_{e}$ of firm $u$ to firm $v$. We follow standard notation in graph theory and denote by $E^{+}(v)=\{(v,u)\in E\}$ and by $E^{-}(v)=\{(u,v)\in E\}$ the set of outgoing and incoming edges of $v\in V$, respectively. The saturating output $c^{+}_{v}$ of a node $v$ is the maximum amount of flow $v$ can send to other players, specified by the weighted outdegree

[TABLE]

We strive to analyze issues of computational complexity. As such, we will assume that all numbers in the input, i.e., all $b_{v}$ and $c_{e}$, are integer numbers in binary encoding.

Allocation strategies.

We analyze allocation strategies in flow allocation games. Each player $v\in V$ strategically allocates its total supply, i.e., the fixed supply $b_{v}$ plus the total amount of incoming flow, to the outgoing edges $E^{+}(v)$. More formally, each player $v\in V$ chooses as a strategy a parametrized flow allocation function $a_{e}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ for every outgoing edge $e\in E^{+}(v)$. This function specifies an amount of flow $a_{e}(y)$ that is forwarded to the edge $e$, for every $y\in\mathbb{R}_{\geq 0}$. Here, $y$ represents a possible value of total supply. Intuitively, the strategy specifies for every possible value $y\geq 0$ of total flow that $v$ might have available, how $v$ will allocate this flow to the outgoing edges. The strategy $\mathbf{a}_{v}=(a_{e})_{e\in E^{+}(v)}$ of player $v\in V$ must satisfy for every $y\geq 0$ and $e\in E^{+}(v)$

[TABLE]

The capacity constraint ensures that no edge is overused, and the weak-flow conservation constraint ensures that $v$ cannot generate additional flow. The no-fraud constraint ensures that each player forwards as much of its total supply as possible. No-fraud strategies are desirable in the application of financial networks, since they do not allow financial firms to hide or malversate assets while having unpaid debt (hence the name “no-fraud”)333For example, EisenbergN01 restrict the choice of each player to a specific no-fraud strategy with pro-rata payments: Each node $v$ distributes its total supply in proportion to the edge capacities. Formally, for every edge $e\in E^{+}(v)$ the strategy is fixed to $a_{e}(y)=\min\left\{c_{e},y\cdot\frac{c_{e}}{c^{+}_{v}}\right\}$.. We assume this property in the strategies for simplicity. Note that even if we drop the no-fraud condition as a constraint, then in flow allocation games (with utilities resulting from monotone strategies defined below) it turns out that every player always has a best response that satisfies the no-fraud condition (see Section 3.1 below).

A flow is a vector of edge valuations $\mathbf{f}=(f_{e})_{e\in E}$. Given some flow $\mathbf{f}$, we slightly abuse notation and denote the total supply of $v$ by $f_{v}=b_{v}+\sum_{e\in E^{-}(v)}f_{e}$. For a strategy profile $\mathbf{a}=(\mathbf{a}_{e})_{e\in E}$, a flow is called feasible if for all edges $(v,w)\in E$ it holds

[TABLE]

A node is called fully saturated if the feasible flow saturates all outgoing edges, i.e., $f_{e}=c_{e}$ for all $e\in E^{+}(v)$, or equivalently $f_{v}=c^{+}_{v}$. Given some feasible flow $\mathbf{f}$ for a strategy profile $\mathbf{a}$, the utility of player $v$ is defined by $u_{v}(\mathbf{a},\mathbf{f})=\sum_{e\in E^{+}(v)}f_{e}$, i.e., $v$’s goal is to choose a strategy to maximize the total outgoing flow. Capacity and weak flow conservation constraints are not sufficient for a consistent definition of utility. In particular, even if a strategy profile satisfies capacity and weak flow conservation constraints, it may not allow a feasible flow.

Example 1**.**

Consider a graph depicted in Fig. 1 with three nodes $V=\{v_{1},v_{2},v_{3}\}$ and edges $(v_{1},v_{2})$, $(v_{2},v_{1})$ and $(v_{1},v_{3})$. Let $c_{e}=2$ for all edges $e\in E$. For the strategy profile $\mathbf{a}$, we assume

[TABLE]

Informally, $v_{1}$ forwards the total supply to $v_{2}$ if it is at most 1. Otherwise, $v_{1}$ forwards all supply to $v_{3}$. The fixed supply is $b_{v_{1}}=1$ and 0 for the other nodes.

There is no feasible flow for strategy profile $\mathbf{a}$. Suppose the total supply of $v_{1}$ is 1, then $v_{1}$ routes all supply to $v_{2}$, who forwards it back to $v_{1}$. The total supply of $v_{1}$ must be at least 2, a contradiction. Suppose the total supply of $v_{1}$ is more than 1, then $v_{1}$ forwards all supply to $v_{3}$, so $v_{1}$ only has the fixed supply. The total supply of $v_{1}$ must be 1, a contradiction. $\blacksquare$

The main problem with feasible flows in the example is that the strategies are not monotone.

Monotonicity.

Monotonicity is a natural condition for payment strategies. In a flow allocation game with monotone strategies, each player $v\in V$ forwards the flow in a monotone fashion. Monotone strategies are characterized by capacity and weak-flow conservation constraints, and, for every $y,y^{\prime}\in\mathbb{R}_{\geq 0}$ with $y\geq y^{\prime}$

[TABLE]

Monotone strategies have been proposed and studied before by CsokaH18. In the following theorem, we recapitulate their main structural insight on feasible flows. Consider a flow-allocation game and a strategy profile $\mathbf{a}$ of monotone strategies. Let $\mathcal{F}$ be the set of feasible flows for $\mathbf{a}$. We observe that $\mathcal{F}$ is non-empty, i.e., a feasible flow always exists. Moreover, $(\mathcal{F},\leq)$ forms a lattice with the coordinate-wise comparison. Formally, $\mathbf{f}\leq\mathbf{f}^{\prime}$ iff $f_{e}\leq f^{\prime}_{e}$ for all $e\in E$; and $\mathbf{f}<\mathbf{f}^{\prime}$ iff $f_{e}\leq f^{\prime}_{e}$ for all $e$ and $f_{e}<f^{\prime}_{e}$ for at least one edge $e$.

Theorem 2** ([CsokaH18]).**

For every strategy profile $\mathbf{a}$ in a flow-allocation game with monotone strategies, the pair $(\mathcal{F},\leq)$ is a non-empty complete lattice.

Csoka and Herings state the result only for integral monotone strategy profiles $\mathbf{a}$. We here reiterate the proof to show that it works for all monotone ones. This theorem motivates the study of flow allocation games with monotone strategies. It follows using the Knaster-Tarski theorem.

Theorem 3** ([tarski1955lattice]).**

Let $(L,\leq)$ be any complete lattice. Suppose $g:L\to L$ is order-preserving, i.e., for all $x,y\in L$ we have that $x\leq y$ implies $g(x)\leq g(y)$. Then the set of all fixed points of $g$ is a non-empty, complete lattice with respect to $\leq$.

Proof of Theorem 2..

Consider $F=\left\{\mathbf{f}\mathrel{}\middle|\mathrel{}0\leq f_{e}\leq c_{e},\hskip 4.26773pt\forall e\in E\right\}$, a compact superset of all possible flow vectors. Obviously, $(F,\leq)$ forms a complete lattice with the coordinate-wise comparison defined above. For a given strategy profile $\mathbf{a}$, the map $g:F\to F$ with

[TABLE]

is an order-preserving function for every edge $e\in E$, since the strategies are monotone. Obviously, the set of feasible flows $\mathcal{F}$ is the set of fixed-points of $g$. The result follows by applying the Knaster-Tarski theorem. ∎

Clearing states and utilities.

We consider feasible flows that arise due to the strategic flow allocation decisions. We determine the utility $u_{v}(\mathbf{a},\mathbf{f})$ of a player $v$ by using a feasible flow $\mathbf{f}$ for the given strategy profile $\mathbf{a}$. We focus on monotone strategies in order to guarantee the existence of at least one feasible flow. This is a necessary condition to make the game well-defined. In many cases, for a fixed strategy profile $\mathbf{a}$, there is a unique feasible flow $\mathbf{f}$. However, even in very special cases, there might be infinitely many feasible flows for the same strategy profile444For example, consider a simple cycle with two nodes $v$ and $w$. The capacities $c_{(v,w)}=c_{(w,v)}=1$, the fixed supplies $b_{v}=b_{w}=0$, and the strategies $a_{(v,w)}(y)=a_{(w,v)}(y)=y$. Every flow with $f_{(v,w)}=f_{(w,v)}\in[0,1]$ is a feasible flow. $\mathbf{a}$.

Based on these properties, and similarly to the vast majority of the literature, we concentrate on the supremum of the lattice. For every profile $\mathbf{a}$, we focus on the unique feasible flow that maximizes the total flow in the network. We denote this feasible flow by $\mathbf{\hat{f}}$ and call it the clearing state of strategy profile $\mathbf{a}$. The clearing state $\mathbf{\hat{f}}$ determines the utility of every player in $\mathbf{a}$, i.e.,

[TABLE]

Clearly, the clearing state $\mathbf{\hat{f}}$ and the resulting utilities significantly depend on the strategy choices of the nodes in the profile $\mathbf{a}$.

2.2 Ranking-based strategies.

In this paper we are interested in classes of intuitive, expressive, and meaningful monotone strategies. We concentrate on strategies that can be derived via rankings [CsokaH18].

Unit-ranking strategies.

In unit-ranking games, we rely on integrality of all values for $c_{e}$ and $b_{v}$, and choose the strategies such that the feasible flows will be integral. We define the parametrized flow functions $a_{e}(y)$ for every outgoing edge $e\in E^{+}(v)$ on the non-negative integer numbers $a_{e}(y):\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$. Note that Theorem 2 can also be shown for $a_{e}(y):\mathbb{N}_{0}\rightarrow\mathbb{N}_{0}$. The proof is analogous, where we define $F$ on integrals and replace compact by finite. This ensures that unit-ranking strategies are well-defined in our model. We interpret the flow as being discretized into unsplittable “units” or “particles” of size 1.

Unit-ranking strategies are relevant for the application of flow-allocation games to financial networks. Usually, currencies have some smallest indivisible amount of money. In this way, unit-ranking strategies provide a rich and powerful class of strategies that can be used to express payments strategies in this context.

Edge-ranking strategies.

In a flow-allocation game with edge-ranking strategies, each player $v\in V$ forwards its total supply according to a strict and total order over $E^{+}(v)$, which we represent by a permutation $\pi_{v}=(e_{1},e_{2},\ldots)$. Player $v$ first allocates the maximum possible flow to edge $e_{1}=\pi_{v}(1)$, then $e_{2}=\pi_{v}(2)$, etc. until all edges are at their capacity or $v$ has no supply left. Formally, $a_{e_{i}}(y)=\min\{c_{e_{i}},\max\{0,y-\sum_{j<i}c_{e_{j}}\}\}$. The edge-ranking strategy555These strategies have also been termed singleton liability priority lists in [IoannidisKV23]. of $v$ is fully described by the ranking $\pi_{v}$, hence we denote a strategy profile in edge-ranking games by $\bm{\pi}=(\pi_{v})_{v\in V}$.

Edge-ranking strategies are a special case of unit-ranking strategies in the sense that every edge-ranking strategy can be written as a unit-ranking strategy. Maybe counterintuitively, every unit-ranking game is also a special edge-ranking game – replacing each edge $e$ with capacity $c_{e}$ many multi-edges of unit capacity expands a unit-ranking game into an equivalent edge-ranking game. There is a one-to-one correspondence between unit-ranking strategies in the original game and edge-ranking strategies in the expanded game. Intuitively, for a unit-ranking strategy in the original game, a player $v$ assigns the first particle of flow to the multi-edge $\pi_{v}(1)$, the second particle to $\pi_{v}(2)$, etc. in the expanded edge-ranking game until all outgoing edges are saturated or $v$ runs out of supply. The expansion of the game implies a pseudo-polynomial blowup in representation size. Nevertheless, the structural equivalence turns out to be very useful for characterizing and analyzing feasible flows and equilibria in unit-ranking games.

Note that the trivial representation of a unit-ranking strategy might be pseudo-polynomial in the size of the original non-expanded game, since edge capacities are given in binary encoding. We will address this issue briefly in Theorem 9 when we discuss polynomial-time computation of equilibria. It turns out that there always exist equilibria with a representation that is polynomial in the input size of the game.

2.3 Equilibria and Social Quality

Equilibrium concepts.

We study pure Nash and strong equilibria of flow allocation games. A (pure) Nash equilibrium in a flow allocation game is a strategy profile $\mathbf{a}$ such that no player $v$ has an incentive to unilaterally deviate from the strategy $\mathbf{a}_{v}$. More formally, in a pure Nash equilibrium $\mathbf{a}$ we have $u_{v}(\mathbf{a})\geq u_{v}(\mathbf{a}^{\prime}_{v},\mathbf{a}_{-v})$ for every player $v\in V$ and every strategy $\mathbf{a}^{\prime}_{v}$. Here $\mathbf{a}_{-v}$ denotes the reduced profile composed of all entries of $\mathbf{a}$ except the entries for player $v$.

For the definition of a strong equilibrium, we first define the notion of a profitable deviation of a coalition. A coalition $C\subseteq V$ of nodes has a profitable deviation $\mathbf{a}^{\prime}_{C}=(\mathbf{a}^{\prime}_{v})_{v\in C}$ if upon joint deviation of $C$ to $\mathbf{a}^{\prime}_{C}$, the resulting utility in the new profile $(\mathbf{a}^{\prime}_{C},\mathbf{a}_{-C})$ is strictly better for every player in $C$, i.e., $u_{v}(\mathbf{a}^{\prime}_{C},\mathbf{a}_{-C})>u_{v}(\mathbf{a})$ for every $v\in C$. Here $\mathbf{a}_{-C}$ denotes the reduced profile composed of all entries of $\mathbf{a}$ except the ones for players $v\in C$. A strategy profile $\mathbf{a}$ is a strong equilibrium if no coalition $C\subseteq V$ has any profitable deviation. Thus, by definition, a strong equilibrium is a Nash equilibrium.

In general, pure Nash or strong equilibria might not exist in a flow allocation game. If they are guaranteed to exist, they might not be unique.

Prices of anarchy and stability.

In addition to existence and computational complexity, we also quantify the performance of a feasible flow in equilibrium in terms of natural notions of social quality. For any non-negative objective function $\textsc{Q}(\mathbf{a})$ measuring the quality of a strategy profile, we rely on standard notions of price of anarchy and price of stability to relate the quality in equilibrium to the one that could be obtained in a strategy profile that maximizes Q.

The price of anarchy for an equilibrium concept in a game $\Gamma$ is given by the ratio

[TABLE]

Here $\mathcal{A}_{\text{Eq}}(\Gamma)$ is a set of equilibria (e.g., the set of all Nash equilibria, or the set of all strong equilibria) of the game $\Gamma$, and $\mathbf{a}^{*}$ is a strategy profile maximzing Q. The price of anarchy for a class of games is the largest price of anarchy of any game in the class. The price of stability for an equilibrium concept in a game $\Gamma$ is defined by replacing $\max$ with $\min$ and vice versa in (7). The price of stability for a class of games is the largest price of stability in any game in the class. Note that both prices of anarchy and stability are at least 1.

Intuitively, an upper bound of $\rho$ on the price of anarchy implies that every equilibrium has a quality of at least $\textsc{Q}(\mathbf{a}^{*})/\rho$ in every game of the class. A lower bound of $\rho$ implies that for some game there is some equilibrium in that game with quality at most $\textsc{Q}(\mathbf{a}^{*})/\rho$. Similarly, an upper bound of $\rho$ on the price of stability implies that at least one equilibrium has a quality of at least $\textsc{Q}(\mathbf{a}^{*})/\rho$ in every game of the class. A lower bound of $\rho$ implies that for some game it holds that every equilibrium in that game with quality at most $\textsc{Q}(\mathbf{a}^{*})/\rho$.

Our main result for unit-ranking games shows the existence of optimal strong equilibria. More in detail, the price of stability is 1 for pure Nash and strong equilibria for every quality function from a large class of so-called flow-monotone functions Q (for details see Corollary 12 below).

For the remaining results on prices of anarchy and stability in this paper, we concentrate on the total amount of flow, which is a standard objective for circulation and flow allocation problems. Formally, the total (amount of) flow for a strategy profile $\mathbf{a}$ is

[TABLE]

since the clearing state $\mathbf{\hat{f}}$ determines the utilities for all players. Hence, $\textsc{Flow}(\mathbf{a})$ also represents the utilitarian welfare.

2.4 Games on DAGs

Flow allocation games are designed to analyze incentives in networks with circulation flows. In contrast to previous work [GuhaKV19, KupfermanVV17], incentives in our games are inherently connected to cycles. To see this, we briefly discuss games on directed acyclic graphs (DAGs).

Proposition 4**.**

In a flow allocation game on a DAG $G$, every strategy profile is a strong equilibrium.

Proof.

We prove the statement inductively. Consider an arbitrary profile $\mathbf{a}$. We first discuss a natural algorithm to construct a feasible flow. A DAG contains a node $v$ without incoming edges. Clearly, the incoming assets of $v$ are fixed to $b_{v}$. Thus, every strategy $\mathbf{a}_{v}$ allocates an amount of $\min\{c_{v}^{+},b_{v}\}$ to the outgoing edges. This is independent of the behavior of $v$ or any other player, so $\mathbf{a}_{v}$ represents a best response. The algorithm then constructs an equivalent network for the remaining players by (1) removing $v$ and all its outgoing edges, and (2) increasing $b_{w}$ by $a_{(v,w)}(b_{v})$, for every $(v,w)\in E^{+}(v)$. The new network is again a DAG and contains a node without incoming edges. In this way, the algorithm proceeds until the remaining network contains no edges.

To show that $\mathbf{a}$ is a strong equilibrium, consider any coalition $C$ and the first player $v_{1}\in C$ that was processed by our algorithm above. $v_{1}$ and all players being processed after $v_{1}$ cannot increase the incoming assets of $v_{1}$ by changing their strategies. As such, $v_{1}$ has no incentive to deviate from $\mathbf{a}_{v_{1}}$. $C$ has no profitable deviation. Therefore, $\mathbf{a}$ is a strong equilibrium. ∎

Our arguments do not rely on monotonicity – the algorithm constructs a feasible flow even for non-monotone strategies $\mathbf{a}$. Moreover, by the same induction, for any given strategy profile $\mathbf{a}$ the feasible flow is unique. The proposition holds even for flow allocation games with non-monotone strategies. Since every strategy profile is a strong equilibrium, it is stable against arbitrary deviations of coalitions. Hence, the result continues to hold even when we restrict to games on DAGs with monotone unit- or edge-ranking strategies.

3 Properties of feasible flows.

We observe a useful circulation representation of feasible flows in flow allocation games and some preliminary results that will be used in the subsequent sections. For edge-ranking games, the circulation representation can be used to describe a polynomial time algorithm that computes the clearing state $\mathbf{\hat{f}}$ in polynomial time.

3.1 Preliminaries on feasible flows and clearing states.

Circulation structure.

Given a game $\Gamma$ on $G$ with strategies $\mathbf{a}$, we build a circulation network $G^{\prime}$ with strategies $\mathbf{a}^{\prime}$ and consider an extended game in $G^{\prime}$ as follows. We add to $G$ an auxiliary node $s$. For every $v\in V$, we add an auxiliary edge $(v,s)$ with capacity $c_{(v,s)}=\infty$. For every $v\in V$ with $b_{v}>0$ we add an auxiliary edge $(s,v)$ with $c_{(s,v)}=b_{v}$, and choose $b_{v}^{\prime}=0$ in the new game. In this way, the supply of $v$ becomes an incoming flow to $v$ on edge $(s,v)$. The strategy $\mathbf{a}^{\prime}_{s}$ of player $s$ is arbitrary. For every strategy profile $\mathbf{a}$ of the original game, we modify $\mathbf{a}_{v}$ such that flow not forwarded by some player $v$ will now be a flow on edge $(v,s)$ under $\mathbf{a}^{\prime}_{v}$.

Proposition 5**.**

For every feasible flow $\mathbf{f}$ for a strategy profile $\mathbf{a}$, the corresponding flow for the modified strategy profile $\mathbf{a}^{\prime}$ in the extended game $G^{\prime}$ can be decomposed and represented as a circulation. The auxiliary node $s$ has an incoming flow of $\sum_{v\in V}b_{v}$, and all auxiliary edges $(s,v)$ are saturated.

Proof.

The proposition is a simple consequence of fixed point constraint and no-fraud constraint. Non-forwarded flow at firm $v\in V$ in the original game exists only if $v$ saturates all outgoing edges

[TABLE]

Moreover, the total sum of non-forwarded flow in the original game is exactly the sum of all fixed supply:

[TABLE]

The non-forwarded flow in the original game gets routed to the auxiliary node $s$ in the extended game. This constitutes the incoming flow of $s$, i.e., $f_{s}=\sum_{v\in V}\max\{0,f_{v}-c^{+}_{v}\}=\sum_{v\in V}b_{v}$, and all auxiliary edges $(s,v)$ are saturated. Overall, by routing the non-forwarded flow in the original game to the auxiliary node $s$ in the extended game, we obtain exact flow conservation at every node. As such, the flow can be decomposed and represented as a circulation. ∎

Fully saturated nodes.

In flow allocation games $\Gamma$, it turns out that the clearing state $\mathbf{\hat{f}}$ is fixed as long as all nodes that are not fully saturated stick to their strategies. Since every strategy satisfies capacity and no-fraud constraints, the forwarded flow of fully saturated nodes remains the same if they have the same total supply, and vice versa. Consequently, strategies of fully saturated nodes have no impact on the clearing state $\mathbf{\hat{f}}$. For any fully saturated node $v$, every strategy constitutes a best response.

Proposition 6**.**

For a given flow allocation game, consider any profile $\mathbf{a}$, the corresponding clearing state $\mathbf{\hat{f}}$, and any fully saturated node $v$ with $\hat{f}_{v}\geq c_{v}^{+}$. Every strategy $\mathbf{a}^{\prime}_{v}$ is a best response for $v$ against the other strategies $\mathbf{a}_{-v}$ and results in the same clearing state $\mathbf{\hat{f}}$.

Proof.

Firm $v$ is fully saturated under $\mathbf{\hat{f}}$, thus $\hat{f}_{v}\geq\sum_{e\in E^{+}(v)}c_{e}$ and $\hat{f}_{e}=c_{e}$ for all $e\in E^{+}(v)$. Consider a deviation $\mathbf{a}^{\prime}_{v}$, the resulting profile $\mathbf{a}^{\prime}=(\mathbf{a}^{\prime}_{v},\mathbf{a}_{-v})$. It suffices to show that $\mathbf{\hat{f}}$ is feasible under $\mathbf{a}^{\prime}$. Note that capacity, weak flow conservation and no-fraud constraints ensure that $a^{\prime}_{e}(\hat{f}_{v})=c_{e}$ for all $e\in E^{+}(v)$, which immediately implies the feasibility of $\mathbf{\hat{f}}$ under $\mathbf{a}^{\prime}$. ∎

No-Fraud Property.

We observe that violating the no-fraud constraint is never in the interest of any player in any game.

Proposition 7**.**

Suppose $\mathbf{a}$ is a strategy profile of monotone strategies that do not necessarily fulfill the no-fraud constraint (3), and strategy $\mathbf{a}_{v}$ of player $v$ is such that for a value $y$ of total supply we have $\sum_{e\in E^{+}(v)}a_{e}(y)<\min\{c^{+}_{v},y\}$. There is a no-fraud strategy $\mathbf{a}^{\prime}_{v}$ such that $u_{w}(\mathbf{a})\leq u_{w}(\mathbf{a}^{\prime}_{v},\mathbf{a}_{-v})$ for every player $w\in V$.

Proof.

Consider any no-fraud strategy $\mathbf{a}_{v}^{\prime}$ arising from $\mathbf{a}_{v}$ by increasing the functions $a_{e}$ arbitrarily such that Condition (3) holds. Consider the clearing states $\mathbf{\hat{f}}$ for $\mathbf{a}$ and $\mathbf{\hat{f}}^{\prime}$ for $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$. For the same total supply $y$ of $v$ we know $a_{e}(y)\leq a^{\prime}_{e}(y)$ for every $e\in E^{+}(v)$. Now consider $\mathbf{\hat{f}}$. If $\mathbf{\hat{f}}$ is a feasible flow for $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$, then $\mathbf{\hat{f}}^{\prime}\geq\mathbf{\hat{f}}$, so the utility of every player in $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$ is weakly improved and we are done.

Otherwise, $\mathbf{\hat{f}}$ is not a feasible flow for $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$. We apply the straightforward monotone fixed-point iteration based on the map $g$ in Equation (6). Since the space $F$ of possible flow vectors is compact, the iteration converges to a feasible flow for $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$ that is coordinate-wise at least $\mathbf{\hat{f}}$. Hence $\mathbf{\hat{f}}^{\prime}\geq\mathbf{\hat{f}}$, so the utility of every player in $(\mathbf{a}_{v}^{\prime},\mathbf{a}_{-v})$ is weakly improved. ∎

3.2 Structure and computation of feasible flows in games with edge-ranking strategies.

For edge-ranking games, we provide a more detailed analysis of the structure of feasible flows in a strategy profile $\bm{\pi}$. We observed above that every feasible flow $\mathbf{f}\in\mathcal{F}$ is in a one-to-one correspondence to a circulation in the circulation network. Here, we observe that the circulation follows a partial order. We use this structural insight to show that the clearing state $\mathbf{\hat{f}}$ can be computed in polynomial time.

Before we describe the algorithm, we will define the notion of an active edge. For a given strategy profile $\bm{\pi}$, consider the circulation network $G^{\prime}=(V^{\prime},E^{\prime})$ with auxiliary node $s$ and the corresponding auxiliary edges described in the previous section. We also extend $\bm{\pi}$ to $\bm{\pi}^{\prime}$ and define the auxiliary edge $(v,s)$ to be the least preferred edge by node $v$. For completeness, we treat $s$ as a node with a fixed strategy $\pi^{\prime}_{s}$ over its outgoing auxiliary edges. Due to Proposition 5, $s$ exactly saturates all outgoing edges in every feasible flow. As such, the strategy $\pi^{\prime}_{s}$ has no impact on the set of feasible flows.

Let $\mathbf{f}$ be a feasible flow in a circulation network $G^{\prime}$. In games with ranking-based strategies, we can observe the following. Independent of the total supply $f_{v}$ of some node $v$, there is a uniquely defined outgoing edge for the next unit of $v$’s supply. We call this edge the active edge of $v$. If $\mathbf{f}\equiv\mathbf{0}$, the active edge of a node $v$ is $\pi^{\prime}_{v}(1)$. If $\sum_{i=1}^{k-1}{c_{\pi^{\prime}_{v}(i)}}\leq f_{v}<\sum_{i=1}^{k}{c_{\pi^{\prime}_{v}(i)}}$ for some $k$, the active edge is $\pi^{\prime}_{v}(k)$.

In the following, we will describe the TopCycleIncrease algorithm. For a formal description see Algorithm 1. In the description we distinguish necessary and optional cycles. This classification builds on the lattice structure of feasible flows (c.f. Theorem 2). Since the algorithm computes the supremum of the lattice, it raises flow along all – necessary and optional – cycles.

Necessary cycles.

Consider the set of all active edges for $\mathbf{f}\equiv\mathbf{0}$. Every node $v\neq s$ has exactly one outgoing active edge $\pi^{\prime}_{v}(1)$. The set of active edges form disjoint cycles with attached trees. Each tree attached to a cycle is rooted in a node from the cycle, and directed towards the cycle $C$. We define the orbit of $C$ by

[TABLE]

i.e., the set of nodes $v$ from which we can reach $C$ over active edges. Now, consider the auxiliary node $s$. Note that $\mathbf{f}\equiv 0$ is a feasible circulation in $G^{\prime}$, but $\sum_{v}f_{(v,s)}=c^{+}_{s}=\sum_{v}b_{v}$ is a necessary constraint to ensure that $\mathbf{f}$ represents a feasible flow in the original graph $G$. However, as long as $\sum_{v}f_{(v,s)}<\sum_{v}b_{v}$ there is an active outgoing edge of $s$ and all nodes (including $s$) belong to some orbit. Hence, there is a cycle $C$ with $s\in o(C)$. Flow conservation and the monotonicity of strategies implies that some amount of flow of $s$ must eventually reach $C$. Due to flow conservation in $C$, a flow of at least $\delta_{C}=\min\{c_{e}\mid e\in C\}$ must thus be present on every edge of $C$. This is a necessary condition in every feasible flow $\mathbf{f}\in\mathcal{F}$.

It is straightforward to inductively apply this argument, thereby obtaining a sequence of necessary cycles $C$ that must be filled with flow $\delta_{C}=\min\{c_{e}-f_{e}\mid e\in C\}$. In particular, when a flow of $f_{e}=\delta_{C}$ has been assigned to every edge $e\in C$, the active edge of at least one of the vertices in $C$ changes. This implies that the orbits change, i.e., the orbit $o(C)$ partitions into new suborbits, or parts that get attached to other orbits. Note that once a vertex $v$ becomes fully saturated and all regular outgoing edges are filled, the active edge becomes $(v,s)$.

Orbits present at the same time are always mutually disjoint. Thus, for two existing orbits $o(C_{1})$ and $o(C_{2})$, pushing flow along $C_{1}$ can never change the active edges of $o(C_{2})$. Hence, it is necessary that all the cycles $C$, where $s$ eventually appears in the orbit, must get assigned a flow increase $\delta_{C}$ in order to reach $\sum_{v}f_{(v,s)}=\sum_{v}b_{v}$. Once we reach a flow $\mathbf{f}$ where $\sum_{v}f_{(v,s)}=\sum_{v}b_{v}$, an “orbit” $o(s)$ emerges composed of a tree rooted in $s$. At this point, we have indeed constructed a feasible flow, which by induction is the unique minimal circulation in $G^{\prime}$ that represents a feasible flow in $G$.

Optional cycles.

In the following, we characterize the structure of all other feasible flows $\mathbf{f}\in\mathcal{F}$ by applying similar observations. Fix some flow $\mathbf{f}$ in the circulation network that is also feasible in $G$, i.e., $\sum_{v}f_{(s,v)}=\sum_{v}b_{v}$ and consider the set of all active edges.

Suppose there is a cycle $C$ with some orbit $o(C)$, i.e., a set of nodes $v$ that are not attached to the tree rooted in $s$. In this case, one can push a non-zero amount of flow along $C$, i.e., increase $f_{v}$ by a strictly positive amount for every $v\in C$. This obviously yields a new feasible clearing state. When an edge becomes saturated, the set of active edges changes and the orbit $o(C)$ disappears, i.e., gets split up as explained above (new suborbits, parts attached to other orbits, parts attached to the tree rooted in $s$).

Note that there might exist multiple orbits at the same time and, thus, multiple possibilities to extend $\mathbf{f}$ by increasing flow along a cycle. However, as observed above, orbits present at the same time are always mutually disjoint, and pushing flow along cycle $C_{1}$ can never change the active edges in an orbit $o(C_{2})$ present at that time. Now consider some vertex $v\in o(C)$. In order to create some feasible flow $\mathbf{f}^{\prime}$ with $f^{\prime}_{v}>f_{v}$ it is necessary to push flow along $C$ until $f_{v}$ is reached (if $v\in o(C)\cap C$) or $o(C)$ disappears and splits up (if $v\in o(C)\setminus C$). Since the flow adjustments monotonically increase all flow values, there is a one-to-one correspondence between sets of cycles with flow increase and the feasible flows.

Note that the cycles chosen for flow increase form a partial order: A cycle $C^{\prime}$ might not be present in the beginning – there might be predecessor-cycles $C$ that have to be filled up to $\delta_{C}$ to break an existing orbit $o(C)$, change some of the active edges, and make $C^{\prime}$ appear. In the argumentation above, it can be seen that the set of predecessor cycles $pred(C^{\prime})$ for some cycle $C^{\prime}$ is uniquely defined resulting from the ranking of edges in $\bm{\pi}$.

The arguments above imply a natural algorithm to compute $\mathbf{\hat{f}}$, which is similar in spirit to the classic Top-Trading-Cycles algorithm for house allocation [ShapleyS74]. The TopCycleIncrease algorithm iteratively raises flow along cycles among the active edges in the circulation network $G^{\prime}$. Thereby it computes the unique maximal feasible flow $\mathbf{\hat{f}}$ from the lattice. The algorithm runs in polynomial time – in every round it increases the flow along a cycle $C$ by $\delta_{C}$. At this point at least one edge (from $G$ or auxiliary) becomes saturated. In terms of the original network $G=(V,E)$, the algorithm needs at most $O(|V|+|E|)$ rounds. Each round can easily be implemented in strongly polynomial time.

Proposition 8**.**

In every edge-ranking game, the TopCycleIncrease algorithm computes the clearing state $\mathbf{\hat{f}}$ in strongly polynomial time.

Since unit-ranking games can be cast as edge-ranking games with unit-capacity multi-edges, the algorithm can be applied in unit-ranking games as well. Note that the running time does not necessarily remain polynomial due to the pseudo-polynomial blowup in representation size.

4 Unit-ranking games.

4.1 Existence and computation of equilibria.

In this section we consider equilibria in unit-ranking games. Our first result is that in every unit-ranking game there is a strong equilibrium that can be computed in polynomial time. Moreover, for a large class of quality functions Q, we can guarantee the existence of a strong equilibrium that is optimal with respect to Q. In particular, every such equilibrium profile of unit-ranking strategies can be represented compactly. In the next theorem, we prove the existence and representation result, and we show that the price of stability is 1 for Flow. The extension to more general quality functions Q is discussed subsequently.

Consider an arbitrary flow allocation game and a circulation that maximizes the total flow in the circulation network $G^{\prime}$. We show that this circulation can be expressed as a clearing state of a strong equilibrium in unit-ranking strategies.

Theorem 9**.**

For every unit-ranking game, there is a strong equilibrium that maximizes the total amount of flow. The strong equilibrium can be computed in polynomial time, and the equilibrium profile can be represented in polynomial space.

Proof.

Consider the circulation network $G^{\prime}=(V,E^{\prime})$. For a moment, assume this is a standard flow network without strategic flow allocation. Consider an optimal circulation $\mathbf{f}^{*}$ that maximizes the total flow value, i.e., it maximizes the sum of flow on all edges. This implies, in particular, that it saturates all outgoing auxiliary edges from $s$. Clearly, $\mathbf{f}^{*}$ yields an upper bound on the achievable total amount of flow (denoted by $\textsc{Flow}(\mathbf{f}^{*})$) in any strategy profile $\mathbf{a}$ of the game

[TABLE]

$\mathbf{f}^{*}$ can be computed in strongly polynomial time [Tardos85]. Since all edge capacities are integral, we can assume all $f_{e}^{*}$ are integral.

We now turn this circulation into a clearing state for a carefully chosen strategy profile $\mathbf{a}^{*}$ of unit-ranking strategies. We will choose $\mathbf{a}^{*}$ such that it can be compactly represented by threshold-ranking strategies.

In a threshold-ranking strategy, every firm $v$ chooses a permutation $\pi_{v}$ over $E^{+}(v)$ and thresholds $\tau_{e}$. The interpretation of threshold-ranking strategies is that node $v$ first assigns $\tau_{e}$ particles to every edge $e\in E^{+}(v)$, sequentially in the order given by $\pi_{v}$. Then, it assigns the remaining $c_{e}-\tau_{e}$ particles to every edge in the order given by $\pi_{v}$. That is, $v$ first considers edge $\pi_{v}(1)$ and forwards the first $\tau_{\pi_{v}(1)}$ particles to this edge. The next $\tau_{\pi_{v}(2)}$ particles are forwarded to edge $\pi_{v}(2)$ etc. until $\sum_{j=1}^{|E^{+}(v)|}\tau_{\pi_{v}(j)}$ particles are sent to the edges (or $v$ runs out of flow). Then, the remaining $c_{\pi_{v}(1)}-\tau_{\pi_{v}(1)}$ particles are forwarded to edge $\pi_{v}(1)$, then the next $c_{\pi_{v}(2)}-\tau_{\pi_{v}(2)}$ particles to $\pi_{v}(2)$ etc. Clearly, threshold-ranking strategies are more general than edge-ranking strategies. They constitute a special class of unit-ranking strategies with compact representation.

We choose $\mathbf{a}^{*}$ as follows. Every firm $v$ chooses an arbitrary permutation $\pi_{v}$ over $E^{+}(v)$ and sets $\tau_{e}=f^{*}_{e}$. It is easy to see that in $\mathbf{a}^{*}$, the optimal circulation $\mathbf{f}^{*}$ corresponds to the clearing state. Let us prove that $\mathbf{a}^{*}$ is a strong equilibrium, i.e., that no coalition $C\subseteq V$ has a profitable deviation.

Suppose for contradiction that there is a coalition $C$ with a profitable deviation. Examine the new profile $(\mathbf{a}^{\prime}_{C},\mathbf{a}^{*}_{-C})$ and assume $\mathbf{\hat{f}}^{\prime}$ is the clearing state. Consider a node $v\in C$. Since $u_{v}(\mathbf{a}^{\prime}_{C},\mathbf{a}^{*}_{-C})>u_{v}(\mathbf{a}^{*})$, there must be strictly more outgoing flow from $v$ in the new profile. Due to the no-fraud condition, this can only happen if $v$ also has strictly more incoming flow in the new profile. Hence, there is an incoming edge $e=(w,v)\in E^{-}(v)$ with $\hat{f}^{\prime}_{e}>f^{*}_{e}$. Now consider node $w$. If $w\in C$, then $u_{w}(\mathbf{a}^{\prime}_{C},\mathbf{a}^{*}_{-C})>u_{w}(\mathbf{a}^{*})$, so by the same reasoning there is again some incoming edge in $E^{-}(w)$ that has strictly more flow in $\mathbf{\hat{f}}^{\prime}$. Otherwise, if $w\not\in C$, then $w$ still plays the strategy $\mathbf{a}_{w}^{*}$. Due to monotonicity, a higher flow on $(w,v)$ can only occur if $w$ has larger total supply. Thus, there is again some incoming edge in $E^{-}(w)$ that has strictly more flow in $\mathbf{\hat{f}}^{\prime}$.

We can repeat this argument indefinitely. As such, there must be a cycle of edges that all have more flow in $\mathbf{\hat{f}}^{\prime}$ than in $\mathbf{f}^{*}$. Such a cycle can be used to increase the circulation, which contradicts that $\mathbf{f}^{*}$ is an optimal circulation in $G^{\prime}$. ∎

Remark 10**.**

For the profitable deviation, we can even allow arbitrary continuous strategies and any choice of clearing state for the deviation profile. This applies even for games with non-monotone and fraud strategies, as long as a feasible flow exists and the clearing state is chosen arbitrarily among the feasible flows that are not weakly dominated in terms of coordinate-wise comparison.

Remark 11**.**

If we consider deviations that weakly improve the coalition (i.e., $u_{v}(\mathbf{a}^{\prime}_{C},\mathbf{a}_{-C})\geq u_{v}(\mathbf{a})$ for all $v\in C$ and $u_{w}(\mathbf{a}^{\prime}_{C},\mathbf{a}_{-C})>u_{w}(\mathbf{a})$ for at least one $w\in C$), it is a simple exercise to see that there are unit-ranking games, in which no such (often termed “super-strong”) equilibrium exists.

As mentioned before, the result can be generalized quite substantially beyond Flow to a large class containing various quality functions $\textsc{Q}(\mathbf{a})$. Let $\mathcal{A}$ be the set of all possible strategy profiles of a unit-ranking game. Let $\mathbf{a},\mathbf{a}^{\prime}\in\mathcal{A}$ be two strategy profiles and $\mathbf{\hat{f}}$ and $\mathbf{\hat{f}}^{\prime}$ the clearing states in $\mathbf{a}$ and $\mathbf{a}^{\prime}$, respectively. Recall the coordinate-wise comparison of flows used in Theorem 2. The function $\textsc{Q}:\mathcal{A}\to\mathbb{R}$ is flow-monotone if $\mathbf{\hat{f}}>\mathbf{\hat{f}}^{\prime}$ implies $\textsc{Q}(\mathbf{a})\geq\textsc{Q}(\mathbf{a}^{\prime})$.

Corollary 12**.**

For every unit-ranking game and every flow-monotone social welfare function Q, there is a strong equilibrium that maximizes Q. The equilibrium profile can be represented in polynomial space.

Proof.

The proof uses the same argument as above. Consider the optimal value $\textsc{Q}_{\max}=\max_{\mathbf{a}\in\mathcal{A}}\textsc{Q}(\mathbf{a})$ and the set $\mathcal{A}_{\max}=\{\mathbf{a}\mid\textsc{Q}(\mathbf{a})=\textsc{Q}_{\max}\}$ of optimal strategy profiles w.r.t. Q. Consider a profile $\mathbf{a}^{*}\in\mathcal{A}_{\max}$ such that the clearing state $\mathbf{\hat{f}}^{*}$ is not coordinate-wise dominated by a clearing state of another optimal profile, i.e., there is no $\mathbf{a}^{\prime}\in\mathcal{A}_{\max}$ with $\mathbf{\hat{f}}^{\prime}>\mathbf{\hat{f}}^{*}$. We show that $\mathbf{a}^{*}$ is a strong equilibrium.

For contradiction, suppose there is a coalitional deviation in $\mathbf{a}^{*}$. By the same arguments as in the proof above, the deviation implies that there is a cycle that allows to increase the flow. Thus, there is a circulation $\mathbf{f}>\mathbf{\hat{f}}^{*}$ in the flow network $G^{\prime}$. By constructing a strategy profile $\mathbf{a}$ using threshold-ranking strategies, $\mathbf{f}$ can be turned into the clearing state of $\mathbf{a}$. Since Q is flow-monotone, $\textsc{Q}_{\max}=\textsc{Q}(\mathbf{a}^{*})\leq\textsc{Q}(\mathbf{a})$. Hence, $\mathbf{a}\in\mathcal{A}_{\max}$ and $\mathbf{f}>\mathbf{\hat{f}}^{*}$, a contradiction to the choice of $\mathbf{a}^{*}$.

Finally, by using threshold-ranking strategies the equilibrium profile has a compact representation. ∎

The corollary implies that the price of stability for strong equilibria is 1 for a very wide range of natural quality functions. For example, instead of $\textsc{Flow}(\mathbf{a})$ (representing utilitarian welfare (8)) we might prefer to express the quality of a strategy profile by

• •

egalitarian welfare $\textsc{EW}(\mathbf{a})=\min_{v\in V}u_{v}(\mathbf{a})$, i.e., the minimum utility of any player in the network, or

• •

Nash social welfare $\textsc{NSW}(\mathbf{a})=\left(\prod_{v\in V}u_{v}(\mathbf{a})\right)^{1/n}$, i.e., the geometric mean of player utilities, or

• •

the number of fully saturated nodes $\textsc{FSN}(\mathbf{a})=|\{v\in V\mid\hat{f}_{v}=c_{v}^{+}\}|$, which in financial networks corresponds to the number of solvent firms, or

• •

any monotone transformation or combination of the above functions.

Corollary 12 implies the existence of an optimal strong equilibrium for all these functions.

While we can compute any strong equilibrium in polynomial time, for some functions Q an optimal strong equilibrium can be NP-hard to compute. A simple reduction shows that for each of the objective functions in the list above it is NP-hard to compute an optimal strategy profile and, consequently, also an optimal strong equilibrium.

Theorem 13**.**

In unit-ranking games, it is strongly NP-hard to compute a strategy profile that maximizes EW, NSW, or FSN.

Proof.

We start by considering the problem of optimizing FSN. This task can be at least as hard as deciding Exact Cover by 3-Sets (X3C). In an instance of X3C, we have a set $R$ of $3k$ elements for an integer $k\geq 1$ and a set $\mathcal{S}\subseteq 2^{E}$ of $m$ triplets (i.e., $|S|=3$ for each $S\in\mathcal{S}$). The goal is to decide if there are $k$ non-overlapping sets in $\mathcal{S}$.

For the reduction, we construct a game containing a source node $v$ with $b_{v}=3k$. Each $S\in\mathcal{S}$ is a node, and for each $r\in R$ there are three nodes $r^{1},r^{2},r^{3}$. We have edges $(v,S)$ of capacity $3k+1$, for every $S\in\mathcal{S}$, edges $(S,r^{1})$ of capacity 1, for every $r\in S$, and two edges $(r^{1},r^{2}),(r^{2},r^{3})$ of capacity 1 for every $r\in R$. Note that all $r^{3}$ are fully saturated, and $v$ can never be fully saturated. By routing the flow from $v$ via the sets to the elements, we can fully saturate at least the $9k$ nodes corresponding to elements in $R$. If nodes of overlapping sets are fully saturated, we route a flow of $x\geq 2$ to some element node $r^{1}$, which implies that $2(x-1)$ nodes for $x-1$ other elements $r^{\prime}\in R$ are not fully saturated. This does not happen in an optimal profile if (and only if) the X3C instance is a yes-instance. More formally, an optimal profile saturates at least $9k+k$ nodes for elements and sets if and only if there are $k$ non-overlapping sets in $\mathcal{S}$.

To prove the result for objectives EW and NSW, we introduce two auxiliary nodes $S_{a}$ and $r_{a}$. For each $S\in\mathcal{S}$, we add edges $(S,S_{a})$ and $(S_{a},S)$ with capacity 1. W.l.o.g. we can assume that all nodes $S$ give first priority to the edge $(S,S_{a})$. Consequently, each $S$ has a positive utility of at least 1. $S_{a}$ has utility $|\mathcal{S}|$. Similarly, for each node $r^{3}$, we add edges $(r^{3},r_{a})$ and $(r_{a},r^{3})$ with capacity 1. As such, all nodes $r^{3}$ can be assumed to have a utility of at least 1. $r_{a}$ has utility $3k$.

Now the problem of optimizing EW or NSW reduces to deciding whether all nodes $r^{1},r^{2}$ can simultaneously obtain positive utility, since otherwise the objective value will be 0. This property, however, is equivalent to choosing $k$ non-overlapping sets to route the total demand of $3k$ from $v$ to the $3k$ nodes $r^{1}$. As such, a positive value for EW or NSW can be obtained if and only if the X3C instance is a yes-instance. ∎

Irrespective of the quality function, even computing a best-response strategy for a single node $v$ in a strategy profile can be strongly NP-hard, since best responses can provide answers to computationally hard decision problems. This holds even in games without fixed supply and with edge capacities in $\{0,1\}$.

Theorem 14**.**

For a given strategy profile $\mathbf{a}$ of a unit-ranking game with $b_{v}=0$ for all $v\in V$ and $c_{e}\in\{0,1\}$ for all $e\in E$, deciding whether a given node $v$ has a best response resulting in utility at least $k$ is strongly NP-complete.

Proof.

For unit-ranking games with edge capacities in $\{0,1\}$ and fixed supplies equal to [math], the decision problem is obviously contained in NP: We can represent every unit-ranking strategy as a ranking over edges. Then, our TopCycleIncrease algorithm to compute $\mathbf{\hat{f}}$ discussed in Section 3.2 runs in polynomial time and we can efficiently verify the utilities for all players.

For strong NP-hardness, suppose we are given an instance $I$ of Satisfiability in conjunctive normal form with $n$ variables and $m$ clauses. We construct a unit-ranking game in edge-ranking representation with a node $v$ and a strategy profile $\bm{\pi}_{-v}$ for the other players such that the following holds: There is a strategy $\pi_{v}$ with utility $u_{v}(\pi_{v},\bm{\pi}_{-v})\geq k^{\prime}+n$ if and only if $I$ has a variable assignment that fulfills at least $k^{\prime}$ clauses.

We construct the game as follows. We denote the variables of $I$ by $x_{1},\ldots,x_{n}$ and the clauses by $C_{1},\ldots,C_{m}$. For each variable $x_{i}$ we create nodes $x_{i,j,0}$ and $x_{i,j,1}$ for all $j\in\{1,\dots,m\}$, as well as a node $z_{i}$. For each clause $C_{j}$ we add a clause node $c_{j}$. In addition, there is a separate node $v$, for which we strive to find a best response.

For each clause $C_{j}$, we add a unit-capacity edge from $x_{i,j,0}$ to $c_{j}$ if $x_{i}$ appears as $\neg x_{i}$ in $C_{j}$ and from $x_{i,j,1}$ to $c_{j}$ if it appears as $x_{i}$ in $C_{j}$. Flow incoming to $c_{j}$ will eventually indicate a literal that fulfills the clause $C_{j}$. There is an edge $(c_{j},v)$ for all $j\in\{1,\dots,m\}$. We will show below that this edge ensures that satisfying clause $C_{j}$ adds exactly one unit to the total supply of $v$.

For each variable $x_{i}$, we add a variable gadget. It consists of nodes $v,x_{i,j,0}$ and $x_{i,j,1}$ for all $j\in\{1,\dots,m\}$, as well as auxiliary nodes $z_{i}$, $z_{i,0}$ and $z_{i,1}$. There are unit-capacity edges $(v,x_{i,j,0})$ and $(v,x_{i,j,1})$ for all $j\in\{1,\dots,m\}$, edges $(x_{i,j,0},x_{i,j+1,0})$, $(x_{i,j,1},x_{i,j+1,1})$ for $j\in\{1,\dots,m-1\}$, and edges $(v,z_{i,0})$, $(z_{i,0},x_{i,1,0})$ and $(v,z_{i,1})$, $(z_{i,1},x_{i,1,1})$. Firm $z_{i}$ has edges $(x_{i,m,0},z_{i})$, $(x_{i,m,1},z_{i})$, and $(z_{i},v)$. An example for the gadget that is constructed for a variable $x_{i}$ and $m=4$ is depicted in Fig. 2(a). Note that for every strategy of $v$, there is at most one cycle emerging in this gadget, since all cycles must include the outgoing edge of $z_{i}$.

In Fig. 2(b), we show an example of the network without the variable gadgets for the Satisfiability instance $I=(x_{1}\vee x_{2}\vee\neg x_{3})\wedge(x_{1}\vee\neg x_{2}\vee x_{4})\wedge(x_{3}\neg x_{4})\wedge(x_{2}\vee\neg x_{3}\vee x_{4}\vee x_{5})$.

We construct a strategy profile $\bm{\pi}_{-v}$ as follows. Observe that nodes $c_{1},\ldots,c_{m}$ and $x_{1},\ldots,x_{n}$ each have a single outgoing edge, their strategies are trivial. If a node $x_{i,j,0}$ or $x_{i,j,1}$ has multiple outgoing edges, it always prioritizes the edge to nodes $x_{i,j+1,0}$ and $x_{i,j+1,1}$, respectively, or to $z_{i}$ if $j=m$.

In the following, we argue that there is a best response of $v$ with utility $k^{\prime}+n$ if and only if there is a variable assignment such that $k^{\prime}$ clauses are fulfilled in $I$. Suppose there is a variable assignment such that $k^{\prime}$ clauses are fulfilled. Fix this assignment, and for every satisfied clause $c$ choose a single literal $l_{c}$ that evaluates to true in the clause. Let $v$ choose the following strategy: First, prioritize edges $(v,x_{i,1,0})$ if $x_{a}=\text{false}$ in the assignment and $(v,x_{i,1,1})$ if $x_{a}=\text{true}$ in the assignment. All these edges will close a cycle $(v,x_{i,1,0},x_{i,2,0},\dots,x_{i,m,0},x_{i},v)$ or $(v,x_{i,1,1},x_{i,2,1},\dots,x_{i,m,1},x_{i},v)$. After that, for all clause-fulfilling literals $l_{c}$ prioritize the edges $(v,x_{i,j,0})$ if $c=C_{j}$ and $l_{c}=\neg x_{i}$ in any order. Prioritize the edges $(v,x_{i,j,1})$ if $c=C_{j}$ and $l_{c}=x_{i}$. All these edges close a cycle via the clause node $c_{j}$, leading to a total inflow of $k^{\prime}+n$.

For showing the other direction, we observe the following structural property for all variables $x_{i}$. If there is some flow on an edge $(x_{i,m,0},z_{i})$ there cannot be any flow on edge $(x_{i,m,1},z_{i})$, and vice versa. We conclude that flow on some edge $(x_{i,j,0},c_{j})$ implies flow on edge $(x_{i,j,0},x_{i,j+1,0})$ (since it has a higher priority), and $(x_{i,m,0},z_{i})$, and thus no flow on $(x_{i,j^{\prime},1},c_{j^{\prime}})$ for all $j^{\prime}$. Analogously, we observe that any flow on some edge $(x_{i,j,0},c_{j})$ implies no flow on edges $(x_{i,j^{\prime},1},c_{j^{\prime}})$.

We observe that if there is a best response of node $v$ with inflow equal to $k^{\prime}+n$, the node has $k^{\prime}+n$ incoming edges that carry flow. At most $n$ of these edges can be $(z_{i},v)$-edges, so there are at least $k^{\prime}$ clause-edges $(c_{j},v)$ that carry flow. Thus, all these $k^{\prime}$ clause nodes receive incoming flow. If this flow for some clause $c_{j}$ comes from a node $x_{i,j,0}$, we know by the observation above that no edge $x_{i,j^{\prime},1}$ carries flow. Thus we can set the variable $x_{i}$ to false, which fulfills clause $C_{j}$. Applying this and analogous operations for flow on an edge $(x_{i,j,1},v)$ yields a variable assignment which fulfills at least $k^{\prime}$ clauses. ∎

4.2 Total flow in equilibrium.

In this section, we analyze the quality of pure Nash and strong equilibria in unit-ranking games, mostly in terms of the Flow objective. In the last section we observed that the prices of stability for Nash and strong equilibria in unit-ranking games are both 1. We here bound the prices of anarchy for Nash and strong equilibria. The total flow depends crucially on the emergence of cycles in the strategy profile. This requires an effort that is inherently coalitional. As such, it might be unsurprising that there are games in which the worst Nash equilibrium may fail to provide any reasonable fraction of the optimal total flow.

Proposition 15**.**

The price of anarchy for Nash equilibria in terms of Flow is unbounded, even in unit-ranking games without fixed supplies.

Proof.

Consider the game depicted in Fig. 3. All edges have capacity 1, fixed supplies are $b_{v}=0$ for all nodes $v$. Consider $\bm{\pi}$ with $\pi_{1}=(e_{1},e_{3})$ and $\pi_{2}=(e_{2},e_{4})$. It is a pure Nash equilibrium with $\textsc{Flow}(\bm{\pi})=0$. No unilateral deviation can close a cycle and increase the value of $\mathbf{\hat{f}}$. The optimal solution $\bm{\pi}^{*}$ with $\pi^{*}_{1}=(e_{3},e_{1})$ and $\pi^{*}_{2}=(e_{4},e_{2})$ has $\textsc{Flow}(\mathbf{a}^{*})=2$. ∎

To analyze the quality of strong equilibria, we again consider the unit-ranking game in the form of unit-capacity multi-edges. Consider an optimal circulation $\mathbf{f}^{*}$ of maximum social welfare in the circulation network $G^{\prime}$. Since we have unit-capacity edges, we can assume that the optimal circulation has binary flows on each edge. Let $\mathcal{C}(\mathbf{f}^{*})=\{C_{1},\ldots,C_{k}\}$ be a decomposition of $\mathbf{f}^{*}$ into cycles of unit flow. We denote by

[TABLE]

the min-max size of any cycle, in any decomposition $\mathcal{C}(\mathbf{f}^{*})$ of any optimal circulation $\mathbf{f}^{*}$.

Theorem 16**.**

In unit-ranking games, the price of anarchy for strong equilibria in terms of Flow is at most $d$.

Proof.

Consider an optimal circulation $\mathbf{f}^{*}$ and a decomposition $\mathcal{C}(\mathbf{f}^{*})$ such that all flow cycles $C_{i}\in\mathcal{C}(\mathbf{f}^{*})$ have size at most $|C_{i}|\leq d$. As observed in the proof of Theorem 9, this circulation yields the total flow of an optimal strategy profile $\mathbf{a}^{*}$, i.e.,

[TABLE]

Now consider any strong equilibrium $\mathbf{a}$ in the unit-ranking game with clearing state $\mathbf{\hat{f}}$. The flow $\mathbf{\hat{f}}$ can be assumed to have binary edge flows. Suppose there is a cycle $C_{i}\in\mathcal{C}(\mathbf{f}^{*})$ such that $a_{e}(\mathbf{\hat{f}}_{v})=0$ for all $e=(v,w)\in C_{i}$. Then the nodes in this cycle have an incentive to jointly deviate and place the edges of $C_{i}$ on the first position in their ranking. Then the clearing state $\mathbf{\hat{f}}$ will emerge as before, adding a flow of 1 along the cycle $C_{i}$. This is a profitable deviation for the nodes of $C_{i}$.

Consequently, for every cycle $C_{i}\in\mathcal{C}(\mathbf{f}^{*})$ there must be at least one edge $e=(u,v)\in C_{i}$ such that $a_{e}(\mathbf{\hat{f}}_{u})=1$. Thus, the total flow in the strong equilibrium $\mathbf{a}$ is

[TABLE]

and, hence, the ratio is at most $d$. ∎