Testing Stability Properties in Graphical Hedonic Games

In hedonic games, players form coalitions based on individual preferences over the group of players they belong to. Several concepts to describe the stability of coalition structures in a game have been proposed and analyzed. However, prior research focuses on algorithms with time complexity that is at least linear in the input size. In the light of very large games that arise from, e.g., social networks and advertising, we initiate the study of sublinear time property testing algorithms for existence and verification problems under several notions of coalition stability in a model of hedonic games represented by graphs with bounded degree. In graph property testing, one shall decide whether a given input has a property (e.g., a game admits a stable coalition structure) or is far from it, i.e., one has to modify at least an $\epsilon$-fraction of the input (e.g., the game's preferences) to make it have the property. In particular, we consider verification of perfection, individual rationality, Nash stability, (contractual) individual stability, and core stability. Furthermore, we show that while there is always a Nash-stable coalition (which also implies individually stable coalitions), the existence of a perfect coalition can be tested. All our testers have one-sided error and time complexity that is independent of the input size

Forming Probably Stable Communities with Limited Interactions

A community needs to be partitioned into disjoint groups; each community member has an underlying preference over the groups that they would want to be a member of. We are interested in finding a stable community structure: one where no subset of members $S$ wants to deviate from the current structure. We model this setting as a hedonic game, where players are connected by an underlying interaction network, and can only consider joining groups that are connected subgraphs of the underlying graph. We analyze the relation between network structure, and one's capability to infer statistically stable (also known as PAC stable) player partitions from data. We show that when the interaction network is a forest, one can efficiently infer PAC stable coalition structures. Furthermore, when the underlying interaction graph is not a forest, efficient PAC stabilizability is no longer achievable. Thus, our results completely characterize when one can leverage the underlying graph structure in order to compute PAC stable outcomes for hedonic games. Finally, given an unknown underlying interaction network, we show that it is NP-hard to decide whether there exists a forest consistent with data samples from the network.Comment: 11 pages, full version of accepted AAAI-19 pape

Testing Core Membership in Public Goods Economies

This paper develops a recent line of economic theory seeking to understand public goods economies using methods of topological analysis. Our first main result is a very clean characterization of the economy's core (the standard solution concept in public goods). Specifically, we prove that a point is in the core iff it is Pareto efficient, individually rational, and the set of points it dominates is path connected. While this structural theorem has a few interesting implications in economic theory, the main focus of the second part of this paper is on a particular algorithmic application that demonstrates its utility. Since the 1960s, economists have looked for an efficient computational process that decides whether or not a given point is in the core. All known algorithms so far run in exponential time (except in some artificially restricted settings). By heavily exploiting our new structure, we propose a new algorithm for testing core membership whose computational bottleneck is the solution of $O(n)$ convex optimization problems on the utility function governing the economy. It is fairly natural to assume that convex optimization should be feasible, as it is needed even for very basic economic computational tasks such as testing Pareto efficiency. Nevertheless, even without this assumption, our work implies for the first time that core membership can be efficiently tested on (e.g.) utility functions that admit "nice" analytic expressions, or that appropriately defined $\varepsilon$-approximate versions of the problem are tractable (by using modern black-box $\varepsilon$-approximate convex optimization algorithms).Comment: To appear in ICALP 201

Hedonic Games and Treewidth Revisited

We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree ?. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2^{O(??t)}. We present an algorithm with parameter dependence (? t)^{O(? t)}, significantly improving upon the parameter dependence on ? given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence t^{o(t)} for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on ? and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH. We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided

Novel Hedonic Games and Stability Notions

We present here work on matching problems, namely hedonic games, also known as coalition formation games. We introduce two classes of hedonic games, Super Altruistic Hedonic Games (SAHGs) and Anchored Team Formation Games (ATFGs), and investigate the computational complexity of finding optimal partitions of agents into coalitions, or finding - or determining the existence of - stable coalition structures. We introduce a new stability notion for hedonic games and examine its relation to core and Nash stability for several classes of hedonic games

Humans and the Core Partition: An Agent-Based Modeling Experiment

Although strategic coalition formation is traditionally modeled using cooperative game theory, behavioral game theorists have repeatedly shown that outcomes predicted by game theory are different from those generated by actual human behavior. To further explore these differences, in a cooperative game theory context, we experiment to compare the outcomes resulting from human participants’ behavior to those generated by a cooperative game theory solution mechanism called the core partition. Our experiment uses an interactive simulation of a glove game, a particular type of cooperative game, to collect the participant’s decision choices and their resultant outcomes. Two different glove games are considered, and the outputs from 62 trial games are analyzed. The experiment’s outcomes show that core coalitions, which are coalitions in a core partition, are found in about 42% of games. Though this number may seem low, a trial’s outcome is more complex than whether the human player finds a core coalition or not. Finding the core coalition depends on factors such as the other possible feasible solutions and the payoffs available from these solutions. These factors, and the complexity they generate, are discussed in the paper

Testing Core Membership in Public Goods Economies

This paper develops a recent line of economic theory seeking to understand public goods economies using methods of topological analysis. Our first main result is a very clean characterization of the economy\u27s core (the standard solution concept in public goods). Specifically, we prove that a point is in the core iff it is Pareto efficient, individually rational, and the set of points it dominates is path connected. While this structural theorem has a few interesting implications in economic theory, the main focus of the second part of this paper is on a particular algorithmic application that demonstrates its utility. Since the 1960s, economists have looked for an efficient computational process that decides whether or not a given point is in the core. All known algorithms so far run in exponential time (except in some artificially restricted settings). By heavily exploiting our new structure, we propose a new algorithm for testing core membership whose computational bottleneck is the solution of O(n) convex optimization problems on the utility function governing the economy. It is fairly natural to assume that convex optimization should be feasible, as it is needed even for very basic economic computational tasks such as testing Pareto efficiency. Nevertheless, even without this assumption, our work implies for the first time that core membership can be efficiently tested on (e.g.) utility functions that admit ``nice\u27\u27 analytic expressions, or that appropriately defined epsilon-approximate versions of the problem are tractable (by using modern black-box epsilon-approximate convex optimization algorithms)

Does Hazardous Waste Matter? Evidence from the Housing Market and the Superfund Program

Approximately $30 billion (2000$) has been spent on Superfund clean-ups of hazardous waste sites, and remediation efforts are incomplete at roughly half of the 1,500 Superfund sites. This study estimates the effect of Superfund clean-ups on local housing price appreciation. We compare housing price growth in the areas surrounding the first 400 hazardous waste sites to be cleaned up through the Superfund program to the areas surrounding the 290 sites that narrowly missed qualifying for these clean-ups. We cannot reject that the clean-ups had no effect on local housing price growth, nearly two decades after these sites became eligible for them. This finding is robust to a series of specification checks, including the application of a quasi-experimental regression discontinuity design based on knowledge of the selection rule. Overall, the preferred estimates suggest that the benefits of Superfund clean-ups as measured through the housing market are substantially lower than the $43 million mean cost of Superfund clean-ups.Valuation of environmental goods, Hazardous waste sites, Environmental regulation, Regression discontinuity, Superfound, Externalities

Coalition Formation For Distributed Constraint Optimization Problems

This dissertation presents our research on coalition formation for Distributed Constraint Optimization Problems (DCOP). In a DCOP, a problem is broken up into many disjoint sub-problems, each controlled by an autonomous agent and together the system of agents have a joint goal of maximizing a global utility function. In particular, we study the use of coalitions for solving distributed k-coloring problems using iterative approximate algorithms, which do not guarantee optimal results, but provide fast and economic solutions in resource constrained environments. The challenge in forming coalitions using iterative approximate algorithms is in identifying constraint dependencies between agents that allow for effective coalitions to form. We first present the Virtual Structure Reduction (VSR) Algorithm and its integration with a modified version of an iterative approximate solver. The VSR algorithm is the first distributed approach for finding structural relationships, called strict frozen pairs, between agents that allows for effective coalition formation. Using coalition structures allows for both more efficient search and higher overall utility in the solutions. Secondly, we relax the assumption of strict frozen pairs and allow coalitions to form under a probabilistic relationship. We identify probabilistic frozen pairs by calculating the propensity between two agents, or the joint probability of two agents in a k-coloring problem having the same value in all satisfiable instances. Using propensity, we form coalitions in sparse graphs where strict frozen pairs may not exist, but there is still benefit to forming coalitions. Lastly, we present a cooperative game theoretic approach where agents search for Nash stable coalitions under the conditions of additively separable and symmetric value functions