Strategic Payments in Financial Networks

In their seminal work on systemic risk in financial markets, Eisenberg and Noe [Larry Eisenberg and Thomas Noe, 2001] proposed and studied a model with n firms embedded into a network of debt relations. We analyze this model from a game-theoretic point of view. Every firm is a rational agent in a directed graph that has an incentive to allocate payments in order to clear as much of its debt as possible. Each edge is weighted and describes a liability between the firms. We consider several variants of the game that differ in the permissible payment strategies. We study the existence and computational complexity of pure Nash and strong equilibria, and we provide bounds on the (strong) prices of anarchy and stability for a natural notion of social welfare. Our results highlight the power of financial regulation - if payments of insolvent firms can be centrally assigned, a socially optimal strong equilibrium can be found in polynomial time. In contrast, worst-case strong equilibria can be a factor of ?(n) away from optimal, and, in general, computing a best response is an NP-hard problem. For less permissible sets of strategies, we show that pure equilibria might not exist, and deciding their existence as well as computing them if they exist constitute NP-hard problems

(Pre)kernel catchers for cooperative games

The paper provides a new (pre)kernel catcher in that the relevant set always contains the (pre)kernel. This new (pre)kernel catcher gives rise to a better lower bound ɛ*** such that the kernel is included in strong ɛ-cores for all real numbers ɛ not smaller than the relevant bound ɛ***

On bargaining sets of supplier-firm-buyer games

We study a special three-sided matching game, the so-called supplier-firm-buyer game, in which buyers and sellers (suppliers) trade indirectly through middlemen (firms). Stuart (1997) showed that all supplier-firm-buyer games have non-empty core. We show that for these games the core coincides with the classical bargaining set (Davis and Maschler, 1967), and also with the Mas-Colell bargaining set (Mas-Colell, 1989)

On Cost Allocation in Networks with Threshold Based Discounting

We study network design in which each pair of nodes can communicate via a direct link and the communication flow can be delivered through any path in the network. The cost of flow through each link is discounted if and only if the amount of flow exceeds certain threshold. This exploitation of economies of scale encourages the concentration of flows and use of relatively small number of links. Applications include telecommunications, airline traffic flow, and mail delivery networks. The cost of services delivered through such a network is distributed among its users who may be individuals or organizations with possibly conflicting interests. The cooperation between these users is essential for the exploitation of economies of scale. Consequently, there is a need to ensure a fair distribution of the cost of providing the service among network users. In order to describe this cost allocation problem we formulate the associated cooperative game, to be referred to as the threshold game. We then demonstrate that certain cost allocation solution (the core of the threshold game) can be efficiently applied to relatively ’large’ networks with threshold-based discounting

Network cost allocation games based on threshold discounting

Consider networks in which each pair of nodes needs to communicate. The communication flow between any pair of nodes can be delivered through a direct link or via some connecting path in the network. By discounting the cost of flow through links for which the high flow volume is anticipated, network designers exploit economies of scale. This approach encourages the concentration of flows and use of relatively small number of links. This led to the design of well known hub networks and more recently hub-like networks. Applications include telecommunications, airline traffic flow, and mail delivery networks. The cost of services delivered through such networks is distributed among its users who may be individuals or organizations with possibly conflicting interests. The cooperation of these users is essential for the exploitation of economies of scale. Consequently, there is a need to find a fair distribution of the cost of providing the service among network users. In this paper, we present a survey of some recent results in the development of cooperative game theory based mechanisms to efficiently characterize cost allocation solutions for hub and hub-like networks. Specifically, we formulate the associated hub and hub-like network cost allocation games. Then, while paying special attention to users\u27 contribution to economies of scale, we demonstrate that some attractive cost allocation solutions, which provide users with the incentive to cooperate, can be efficiently computed

Network Connectivity Game

We investigate the cost allocation strategy associated with the problem of providing service /communication between all pairs of network nodes. There is a cost associated with each link and the communication between any pair of nodes can be delivered via paths connecting those nodes. The example of a cost efficient solution which could provide service for all node pairs is a (non-rooted) minimum cost spanning tree. The cost of such a solution should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Network Connectivity (NC) game, and develop a stable and efficient cost allocation scheme. The NC game is related to the Minimum Cost Spanning Tree games and to the Shortest Path games. The profound difference is that in those games the service is delivered from some common source node to the rest of the network, while in the NC game there is no source and the service is established through the two-way interaction among all pairs of participating nodes. We formulate Network Connectivity (NC) game and construct an efficient cost allocation algorithm which finds some points in the core of the NC game. Finally, we discuss the Egalitarian Network Cost Allocation (ENCA) rule and demonstrate that it finds an additional core point

The kernel is in the least core for permutation games

Permutation games are totally balanced transferable utility cooperative games arising from certain sequencing and re-assignment optimization problems. It is known that for permutation games the bargaining set and the core coincide, consequently, the kernel is a subset of the core. We prove that for permutation games the kernel is contained in the least core, even if the latter is a lower dimensional subset of the core. By means of a 5-player permutation game we demonstrate that, in sense of the lexicographic center procedure leading to the nucleolus, this inclusion result can not be strengthened. Our 5-player permutation game is also an example (of minimum size) for a game with a non-convex kernel

Uniform-price assignment markets

Uniform-price assignment games are introduced as those assignment markets with the core reduced to a segment. In these games, for all active agents, competitive prices are uniform although products may be non-homogeneous. A characterization in terms of the assignment matrix is given. The only assignment markets where all submarkets are uniform are the Bohm-Bawerk horse markets. We prove that for uniform-price assignment games the kernel, or set of symmetrically-pairwise bargained allocations, either coincides with the core or reduces to the nucleolusEls jocs d'assignació amb preu uniforme són aquells mercats d'assignació on el core es redueix a un segment. En aquests casos, per a tots els agents actius en el mercat, els preus competitius varien de forma uniforme, tot i que els productes poden ser no homogenis. En aquest treball es dona una caracterització dels mercats amb preu uniforme a partir de la matriu d'assignació Els únics mercats on tots els subjocs són de preu uniforme són els mercats de cavalls de Bohm-Bawerk. Finalment,provem que en aquests mercats de preu uniforme el kernel, o conjunt de pagaments que s'obtenen a partir d'un procés de negociació bilateral i simètric, o bé coincideix amb tot core o es redueix al seu punt mig que és el nucleolus

Insights into the nucleolus of the assignment game

We show that the family of assignment matrices which give rise to the same nucleolus form a compact join-semilattice with one maximal element, which is always a valuation. -see p.43, Topkis, 1998-. We give an explicit form of this valuation matrix. The above family is in general not a convex set, but path-connected, and we construct minimal elements of this family. We also analyze the conditions to ensure that a given vector is the nucleolus of some assignment game