We study the inverse power index problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we study various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, O(2^(n^2) p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the inverse power index problem that runs in exponential time. This algorithm is straightforward and general: it computes the error for each game enumerated, and outputs the game that minimizes this error. By the genericity of our approach, our algorithm can be used to find a weighted voting game that optimizes any exponential time computable function. We implement our algorithm for the case of the normalized Banzhaf index, and we perform experiments in order to study performance and error convergence.
Threshold functions @cite_5 @cite_29 are of fundamental research interest in voting games, circuit complexity and neural networks. The problem of realizing Boolean threshold functions by neural networks has been extensively studied @cite_51 @cite_66 @cite_34 , where upper and lower bounds are derived on the synaptic weights for such realization. The enumeration of threshold functions is closely related to the enumeration of weighted voting games (as threshold functions are essentially weighted voting games where negative weights are allowed.). The enumeration of threshold functions up to six variables has been done in @cite_52 . Subsequently, in @cite_39 @cite_0 , all threshold functions of respectively seven and eight variables were enumerated. Krohn and Sudh "olter @cite_23 enumerated the canonical weighted voting games up to eight players, as well as the class of canonical linear games. Kurz @cite_14 was the first to enumerate all nine player canonical weighted voting games, and Freixas and Molinero @cite_6 were the first to enumerate all nine player canonical linear games.
@cite_14 @cite_29 @cite_52 @cite_6 @cite_39 @cite_0 @cite_23 @cite_5 @cite_34 @cite_51 @cite_66
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In many multiagent settings, situations arise in which agents must collectively make decisions while not every agent is supposed to have an equal amount of influence in the outcome of such a decision. Weighted voting games are often used to deal with these situations. The amount of influence that an agent has in a weighted voting game can be measured by means of various power indices. This paper studies the problem of finding a weighted voting game in which the distribution of the influence among the agents is as close as possible to a given target value. We propose a method to exactly solve this problem. This method relies on a new efficient procedure for enumerating weighted voting games of a fixed number of agents. The enumeration algorithm we propose works by exploiting the properties of a specific partial order over the class of weighted voting games. The algorithm enumerates weighted voting games of a fixed number of agents in time exponential in the number of agents, and polynomial in the number of games output. As a consequence we obtain an exact anytime algorithm for designing weighted voting games. Weighted voting games are mathematical models, used to analyse situations where voters with variable voting weight vote in favour of or against a decision. They have been applied in various political and economic organizations. Similar combinatorial models are also encountered in neuroscience, threshold logic, reliability theory and distributed systems. The calculation of voting powers of players in a weighted voting game has been extensively researched in the last few years. However, the inverse problem of designing a weighted voting game with a desirable distribution of power has received less attention. We present an elegant algorithm which uses generating functions and interpolation to compute an integer weight vector for target Banzhaf power indices. This algorithm has better performance than any other known to us. It can also be used to design egalitarian two-tier weighted voting games and a representative weighted voting game for a multiple weighted voting game. The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g., for voting games this complexity is #p-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen's multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen's. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k^2n) and the upper bound on its approximation error is O(k^2 n). Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coalitional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational incentive to leave it), and finding a division of payoffs to coalition members (an imputation) that is fair. We investigate the computational complexity of such questions for weighted threshold games. We study the core, the least core, and the nucleolus, distinguishing those problems that are polynomial-time computable from those that are NP-hard, and providing pseudopolynomial and approximation algorithms for the NP-hard problems. This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1. A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation. The ‘dollar game’ represents a kind of diffusion process on a graph. Under the rules of the game some cofigurations are both stable and recurrent, and these are known as critical cofigurations. The set of critical configurations can be given the structure of an abelian group, and it turns out that the order of the group is the tree-number of the graph. Each critical configuration can be assigned a positive weight, and the generating function that enumerates critical configurations according to weight is a partial evaluation of the Tutte polynomial of the graph. It is shown that the weight enumerator can also be interpreted as a growth function, which leads to the conclusion that the (partial) Tutte polynomial itself is a growth function. We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors. The number of threshold functions of eight variables is counted by ILLIAC II, the computer of the University of Illinois. Sets of optimum weights of majority elements realizing these functions also are investigated. Actually, canonical positive self-dual threshold functions of nine variables are investigated instead of directly investigating threshold functions of eight variables because it is easier to deal with them. The number and optimum weights of threshold functions of eight variables are easily obtained from these functions of nine variables and their realization. Simple coalitional games are a fundamental class of cooperative games and voting games which are used to model coalition formation, resource allocation and decision making in computer science, artificial intelligence and multiagent systems. Although simple coalitional games are well studied in the domain of game theory and social choice, their algorithmic and computational complexity aspects have received less attention till recently. The computational aspects of simple coalitional games are of increased importance as these games are used by computer scientists to model distributed settings. This thesis fits in the wider setting of the interplay between economics and computer science which has led to the development of algorithmic game theory and computational social choice. A unified view of the computational aspects of simple coalitional games is presented here for the first time. Certain complexity results also apply to other coalitional games such as skill games and matching games. The following issues are given special consideration: influence of players, limit and complexity of manipulations in the coalitional games and complexity of resource allocation on networks. The complexity of comparison of influence between players in simple games is characterized. The simple games considered are represented by winning coalitions, minimal winning coalitions, weighted voting games or multiple weighted voting games. A comprehensive classification of weighted voting games which can be solved in polynomial time is presented. An efficient algorithm which uses generating functions and interpolation to compute an integer weight vector for target power indices is proposed. Voting theory, especially the Penrose Square Root Law, is used to investigate the fairness of a real life voting model. Computational complexity of manipulation in social choice protocols can determine whether manipulation is computationally feasible or not. The computational complexity and bounds of manipulation are considered from various angles including control, false-name manipulation and bribery. Moreover, the computational complexity of computing various cooperative game solutions of simple games in dierent representations is studied. Certain structural results regarding least core payos extend to the general monotone cooperative game. The thesis also studies a coalitional game called the spanning connectivity game. It is proved that whereas computing the Banzhaf values and Shapley-Shubik indices of such games is #P-complete, there is a polynomial time combinatorial algorithm to compute the nucleolus. The results have interesting significance for optimal strategies for the wiretapping game which is a noncooperative game defined on a network. Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. In the following paper we offer a method for the a priori evaluation of the division of power among the various bodies and members of a legislature or committee system. The method is based on a technique of the mathematical theory of games, applied to what are known there as “simple games” and “weighted majority games.” We apply it here to a number of illustrative cases, including the United States Congress, and discuss some of its formal properties. The designing of the size and type of a legislative body is a process that may continue for many years, with frequent revisions and modifications aimed at reflecting changes in the social structure of the country; we may cite the role of the House of Lords in England as an example. The effect of a revision usually cannot be gauged in advance except in the roughest terms; it can easily happen that the mathematical structure of a voting system conceals a bias in power distribution unsuspected and unintended by the authors of the revision. How, for example, is one to predict the degree of protection which a proposed system affords to minority interests? Can a consistent criterion for “fair representation” be found? It is difficult even to describe the net effect of a double representation system such as is found in the U. S. Congress (i.e., by states and by population), without attempting to deduce it a priori . The method of measuring “power” which we present in this paper is intended as a first step in the attack on these problems. In a document network such as a citation network of scientific documents, web-logs, etc., the content produced by authors exhibits their interest in certain topics. In addition some authors influence other authors' interests. In this work, we propose to model the influence of cited authors along with the interests of citing authors. Moreover, we hypothesize that apart from the citations present in documents, the context surrounding the citation mention provides extra topical information about the cited authors. However, associating terms in the context to the cited authors remains an open problem. We propose novel document generation schemes that incorporate the context while simultaneously modeling the interests of citing authors and influence of the cited authors. Our experiments show significant improvements over baseline models for various evaluation criteria such as link prediction between document and cited author, and quantitatively explaining unseen text. An important aspect of mechanism design in social choice protocols and multiagent systems is to discourage insincere and manipulative behaviour. We examine the computational complexity of false-name manipulation in weighted voting games which are an important class of coalitional voting games. Weighted voting games have received increased interest in the multiagent community due to their compact representation and ability to model coalitional formation scenarios. Bachrach and Elkind in their AAMAS 2008 paper examined divide and conquer false-name manipulation in weighted voting games from the point of view of Shapley-Shubik index. We analyse the corresponding case of the Banzhaf index and check how much the Banzhaf index of a player increases or decreases if it splits up into sub-players. A pseudo-polynomial algorithm to find the optimal split is also provided. Bachrach and Elkind also mentioned manipulation via merging as an open problem. In the paper, we examine the cases where a player annexes other players or merges with them to increase their Banzhaf index or Shapley-Shubik index payoff. We characterize the computational complexity of such manipulations and provide limits to the manipulation. The annexation non-monotonicity paradox is also discovered in the case of the Banzhaf index. The results give insight into coalition formation and manipulation. Let ( F ) be a family of subsets of the ground set [n] = 1, 2, . . . , n . For each ( i [n] ) we let ( p( F ,i) ) be the number of pairs of subsets that differ in the element i and exactly one of them is in ( F ). We interpret ( p( F ,i) ) as the influence of that element. The normalized Banzhaf vector of ( F ), denoted ( B( F ) ), is the vector ( (B( F ,1), ,B( F ,n)) ), where ( B( F ,i)= p( F ,i) p( F ) ) and ( p( F ) ) is the sum of all ( p( F ,i) ). The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n − k dummy voters. Abstract Documents in computer-readable form can be used to provide information about other documents, i.e. those they cite. To do this efficiently requires procedures for computer recognition of citing statements. This is not easy, especially for multi-sentence citing statements. Computer recognition procedures have been developed which are accurate to the following extent: 73 of the words in statements selected by computer procedures as being citing statements are words which are correctly attributable to the corresponding documents. The retrieval effectiveness of computer-recognized citing statements was tested in the following way. First, for eight retrieval requests in inorganiic chemistry, average recall by search of Chemical Abstracts Service indexing and Chemical Abstracts abstract text words was found to be 50 . Words from citing statements referring to the papers to be retrieved were then added to the index terms and abstract words as additional access points, and searching was repeated. Average recall increased to 70 . Only words from citing statements published within a year of the cited papers were used. The retrieval effect of citing statement words alone (published within a year) without index or abstract terms was the following: average recall was 40 . When just the words of the titles of the cited papers were added to those citing statement words, average recall increased to 50 . In the online F-avoidance edge-coloring game with r colors, a graph on n vertices is generated by randomly adding a new edge at each stage. The player must color each new edge as it appears; the goal is to avoid a monochromatic copy of F. Let N"0(F,r,n) be the threshold function for the number of edges that the player is asymptotically almost surely able to paint before he she loses. Even when F=K"3, the order of magnitude of N"0(F,r,n) is unknown for r>=3. In particular, the only known upper bound is the threshold function for the number of edges in the offline version of the problem, in which an entire random graph on n vertices with m edges is presented to the player to be r edge-colored. We improve the upper bound for the online triangle-avoidance game with r colors, providing the first result that separates the online threshold function from the offline bound for r>=3. This supports a conjecture of Marciniszyn, Spohel, and Steger that the known lower bound is tight for cliques and cycles for all r. We examine agent failures in weighted voting games. In our cooperative game model, R-WVG, each agent has a weight and a survival probability, and the value of an agent coalition is the probability that its surviving members would have a total weight exceeding a threshold. We propose algorithms for computing the value of a coalition, finding stable payoff allocations, and estimating the power of agents. We provide simulation results showing that on average the stability level of a game increases as the failure probabilities of the agents increase. This conforms to several recent results showing that failures increase stability in cooperative games. We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds.