Stabilizing Weighted Graphs

An edge-weighted graph G=(V,E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G

The Complexity of Matching Games: A Survey

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an overview of matching games and extensions, such as $b$-matching games and partitioned matching games; the latter originating from the emerging area of international kidney exchange. In this survey we focus on computational complexity aspects of various game-theoretical solution concepts, such as the core, nucleolus and Shapley value, when the input is restricted to some (generalized) matching game

Stabilization of Capacitated Matching Games

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G. In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case. Finally, in unit-capacity graphs there is an equivalence between the stability of a graph, existence of a stable solution for network bargaining games, and existence of a stable solution for cooperative matching games. We show that this equivalence does not extend to the capacitated case.Comment: 14 pages, 3 figure

Vertex Stabilizers for Network Bargaining Games

Network bargaining games form a prominent class of examples of game theory problems defined on graphs, where vertices represent players, and edges represent their possible interactions. An instance of a \emph{network bargaining game} is given by a graph $G = (V, E)$ with edge weights $w \in \mathbb{R}^E_{+}$ and vertex capacities $c \in \mathbb{Z}^V_+$. A \emph{solution} to an instance $(G, w, c)$ of a network bargaining game is data $(M, z)$, where $M \subset E$ is a $c$-matching, and $z \in \mathbb{R}^{2E}_{\geq 0}$ is a vector which assigns each edge $uv \in E$ a pair of values $z_{uv}$ and $z_{vu}$, such that $z_{uv} + z_{vu} = w_{uv}$ if $uv \in M$, and $z_{uv} = z_{vu} = 0$ otherwise. An instance $(G, w, c)$ is said to be \emph{stable} if it admits a so-called `stable' solution, which represents a solution where a player has no incentive to deviate. Not all instances of a network bargaining game have a stable solution, and this naturally motivates the problem of how to modify the underlying graph such that the instance becomes stable. In recent years, researchers have investigated various modifications, typically by adding or removing edges or vertices. The natural algorithmic question which stems from this is whether these modifications can be performed efficiently. The answer varies, depending on the modification in question, and on whether the edges/vertices have been restricted to be unit weight/capacity. In this work, we consider the vertex-removal setting for a general instance $(G, w, c)$ of a network bargaining game. A set of vertices whose deletion from $G$ results in a stable instance of the induced network bargaining game is called a \emph{vertex stabilizer}. We demonstrate in this work that the algorithmic problem of finding a minimum cardinality vertex stabilizer is $NP$-complete, and give an efficient $2$-approximation algorithm. Further, we show that no efficient $(2 - \epsilon)$-approximation for this problem exists for any $\epsilon > 0$, assuming the Unique Games Conjecture holds. These results hold even in the case when all edges are of unit-weight. In contrast, if we are given an instance $(G, w, c)$ together with a maximum weight $c$-matching $M$, we show that the problem of finding a minimum cardinality vertex stabilizer that avoids $M$ can be solved efficiently. We provide a polynomial time algorithm for solving this problem

Bounds on the Cost of Stabilizing a Cooperative Game

This is the author accepted manuscript. The final version is available from the AI Access Foundation via the DOI in this record.A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core—the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.DFGEuropean Science FoundationNRF (Singapore)European Research CouncilHorizon 2020 European Research Infrastructure projectIsrael Science FoundationIsrael Ministry of Science and TechnologyGoogle Inter-University Center for Electronic Markets and AuctionsEuropean Social Fund (European Commission)Calabria Regio

Entrepreneurial Action and Entrepreneurial Rents

This dissertation is comprised of three independently standing research papers (chapters 2, 3 and 4) that come together in the common theme of investigating the relationship between entrepreneurial action and performance. The introduction chapter argues that this theme is the main agenda of an entrepreneurial approach to strategy, and provides some background and context for the core chapters. The entrepreneurial approach to strategy falls in line with an array of action-based theories of strategy that trace their economic foundations to the Austrian school of economics. Chapters 2 and 3 take a game theoretical modeling and computer simulation approach and represent one of the first attempts at formal analysis of the Austrian economic foundations of action-based strategy theory. These chapters attempt to demonstrate ways in which formal analysis can begin to approach compatibility with the central tenets of Austrian economics (i.e., subjectivism, dynamism, and methodological individualism). The simulation results shed light on our understanding of the relationship between opportunity creation and discovery, and economic rents in the process of moving towards and away from equilibrium. Chapter 4 operationalizes creation and discovery as exploration and exploitation in an empirical study using data from the Kauffman Firm Survey and highlights the trade-offs faced by start-ups in linking action to different dimensions of performance (i.e., survival, profitability, and getting acquired). Using multinomial logistic regression for competing risks analysis and random effects panel data regression, we find that high technology start-ups face a trade-off between acquisition likelihood and profitability-given-survival while low and medium technology start-ups face a trade-off between survival and profitability-given-survival. Chapter 5 concludes the dissertation by highlighting some of the overall contributions and suggesting avenues for future research

Does foreign aid delay stabilization

This paper addresses the question how the expectation of unconditional external shocks like foreign aid increase or decrease saving and, thus, accelerate or delay macroeconomic stabilizations. We build on Casella and Eichengreen (1996) who first used Alesina and Drazen's (1991) war-of-attrition model to investigate the consequences of anticipation of foreign aid on the expected date of stabilization. Casella and Eichengreen's main result, namely that aid that is announced or disbursed after critical dates will delay stabilization, will be shown to be based on an invalidly modified equilibrium condition. A correct incorporation of anticipated foreign aid in the war-of-attrition model yields the result that aid unambiguously accelerates stabilization.

Instituciones políticas, procesos de diseño de políticas y resultados de las políticas en Chile

Este articulo caracteriza los rasgos principales del proceso de diseño de políticas en Chile. El articulo resalta la influencia de las instituciones políticas en dicho proceso y examina la conexión entre el diseño de políticas y el resultado final. Los rasgos principales del proceso de diseño Chileno de políticas son el sistema electoral y el sistema de partidos asociativo, caracterizados por dos coaliciones establecidas, un Ejecutivo poderoso con el control sobre la agenda política, una magistratura independiente, una burocracia que es relativamente libre de corrupción juzgada por los criterios de la OECD, y una serie de vetos en el proceso de diseño de políticas que permiten a facciones afectadas bloquear el cambio de políticas. En consistencia con la estructura teórica de Spiller y Tommasi (2003), el número de actores que interactúan repetidamente y la predictabilidad de la implementación de políticas y una aplicación legal conducen a un proceso de diseño de políticas en el cual los costos de operación son bajos y los intercambios políticos ínter temporales son creíbles. Los grupos de oposición que ejercen su derecho a veto le dan a estos intercambios ínter temporales su credibilidad, aunque también pueden bloquear las reformas. Analizando las políticas desde una perspectiva transversal, encontramos que las políticas en las cuales los intereses de los políticos son mejor representados y las cuales llevan a un cambio exógeno rápido, están asociadas con mas éxito hacia la reforma. En contraste, las políticas que no comparten ningún interés con la rama ejecutiva y con los varios grupos de oposición, tienden a estancarse.