Several different "hat games" have recently received a fair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players' hat colours. Some versions of these games have been motivated by research in complexity theory and have ties to well-known research problems in coding theory, and some variations have led to interesting new research. In this paper, we review Ebert's Hat Game, which garnered a considerable amount of publicity in the late 90's and early 00's, and the Hats-on-a-line Game. Then we introduce a new hat game which is a "hybrid" of these two games and provide an optimal strategy for playing the new game. The optimal strategy is quite simple, but the proof involves an interesting combinatorial argument.
A few years prior to the introduction of Ebert's Hat Game, in 1994, a similar game was described by Aspnes, Beigel, Furst and Rudich @cite_7 . In their version of the game, players are not allowed to pass, and the objective is for a majority of the players to guess correctly. For the three-player game, it is easy to describe a strategy that will succeed with probability @math , just as in Ebert's game: Alice votes the opposite of Bob's hat colour; Bob votes the opposite of Charlie's hat colour; and Charlie votes the opposite of Alice's hat colour.
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In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs outputs. The entangled value ω *(G) of a game G is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. We show a parallel repetition theorem for the entangled value @math of any two-player one-round game @math where the questions @math to Alice and Bob are drawn from a product distribution on @math . We show that for the @math -fold product @math of the game @math (which represents the game @math played in parallel @math times independently), @math , where @math and @math represent the sets from which the answers of Alice and Bob are drawn. We study the following one-person game against a random graph process: the Player's goal is to @math -colour a random sequence of edges @math of a complete graph on @math vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about @math , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with @math edges, where @math is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round. In the generalized Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of size a+b+c. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the ''colouring protocol'', as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve the generalized Russian cards problem in cases where a is a power of a prime, c=O(a^2) and b=O(c^2). This improves substantially on the set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players. 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). Every day millions of users are connected through online social networks, generating a rich trove of data that allows us to study the mechanisms behind human interactions. Triadic closure has been treated as the major mechanism for creating social links: if Alice follows Bob and Bob follows Charlie, Alice will follow Charlie. Here we present an analysis of longitudinal micro-blogging data, revealing a more nuanced view of the strategies employed by users when expanding their social circles. While the network structure affects the spread of information among users, the network is in turn shaped by this communication activity. This suggests a link creation mechanism whereby Alice is more likely to follow Charlie after seeing many messages by Charlie. We characterize users with a set of parameters associated with different link creation strategies, estimated by a Maximum-Likelihood approach. Triadic closure does have a strong effect on link formation, but shortcuts based on traffic are another key factor in interpreting network evolution. However, individual strategies for following other users are highly heterogeneous. Link creation behaviors can be summarized by classifying users in different categories with distinct structural and behavioral characteristics. Users who are popular, active, and influential tend to create traffic-based shortcuts, making the information diffusion process more efficient in the network. 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. Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1 m2(F), where n2−1 m2(F) is the threshold of the corresponding offline colouring problem. Consider three players Alice, Bob and Cath who hold a, b and c cards, respectively, from a deck of d = a+b+c cards. The cards are all different and players only know their own cards. Suppose Alice and Bob wish to communicate their cards to each other without Cath learning whether Alice or Bob holds a specific card. Considering the cards as consecutive natural numbers 0, 1, . . . , we investigate general conditions for when Alice or Bob can safely announce the sum of the cards they hold modulo an appropriately chosen integer. We demonstrate that this holds whenever a, b > 2 and c = 1. Because Cath holds a single card, this also implies that Alice and Bob will learn the card deal from the other player’s announcement. Coalition formation is the process of bringing together two or more agents so as to achieve goals that individuals on their own cannot, or to achieve them more efficiently. Typically, in such situations, the agents have conflicting preferences over the set of possible joint goals. Thus, before the agents realize the benefits of cooperation, they must find a way of resolving these conflicts and reaching a consensus. In this context, cooperative game theory offers the voting game as a mechanism for agents to reach a consensus. It also offers the Shapley value as a way of measuring the influence or power a player has in determining the outcome of a voting game. Given this, the designer of a voting game wants to construct a game such that a player's Shapley value is equal to some desired value. This is called the inverse Shapley value problem. Solving this problem is necessary, for instance, to ensure fairness in the players' voting powers. However, from a computational perspective, finding a player's Shapley value for a given game is #P-complete. Consequently, the problem of verifying that a voting game does indeed yield the required powers to the agents is also #P-complete. Therefore, in order to overcome this problem we present a computationally efficient approximation algorithm for solving the inverse problem. This method is based on the technique of 'successive approximations'; it starts with some initial approximate solution and iteratively updates it such that after each iteration, the approximate gets closer to the required solution. This is an anytime algorithm and has time complexity polynomial in the number of players. We also analyze the performance of this method in terms of its approximation error and the rate of convergence of an initial solution to the required one. Specifically, we show that the former decreases after each iteration, and that the latter increases with the number of players and also with the initial approximation error. A value theory is developed for voting games in which a sizable fraction or the total vote is controlled by a few major players and the rest is distributed among a continuous infinity of individually insignificant minor players. The latter are referred to collectively as an “ocean,” to suggest the total lack of order or cohesion that is assumed. Abstract A simple game ( P, W ) can serve as a model of a voting system in which an alternative is pitted against the status quo. In what follows, we investigate the following three aspects of such games as they apply to four real-world examples of voting systems: a characterization of weighted voting systems in terms of the ways in which coalitions can gain or lose by trading players; the application of a graph-theoretic notion of dimension to simple games and voting systems; and the consideration of a way to measure the power of a player as an interval of real numbers. American Mathematical Society Classification Numbers: 90A28, 90D80, 05C65. Journal of Economic Literature Classification Number: C71. We present the first formal mathematical presentation of the generalized Russian cards problem, and provide rigorous security definitions that capture both basic and extended versions of weak and perfect security notions. In the generalized Russian cards problem, three players, Alice, Bob, and Cathy, are dealt a deck of @math cards, each given @math , @math , and @math cards, respectively. The goal is for Alice and Bob to learn each other's hands via public communication, without Cathy learning the fate of any particular card. The basic idea is that Alice announces a set of possible hands she might hold, and Bob, using knowledge of his own hand, should be able to learn Alice's cards from this announcement, but Cathy should not. Using a combinatorial approach, we are able to give a nice characterization of informative strategies (i.e., strategies allowing Bob to learn Alice's hand), having optimal communication complexity, namely the set of possible hands Alice announces must be equivalent to a large set of @math -designs, where @math . We also provide some interesting necessary conditions for certain types of deals to be simultaneously informative and secure. That is, for deals satisfying @math for some @math , where @math and the strategy is assumed to satisfy a strong version of security (namely perfect @math -security), we show that @math and hence @math . We also give a precise characterization of informative and perfectly @math -secure deals of the form @math satisfying @math involving @math -designs. Oblivious transfer is a fundamental primitive in cryptography. While perfect information theoretic security is impossible, quantum oblivious transfer protocols can limit the dishonest player's cheating. Finding the optimal security parameters in such protocols is an important open question. In this paper we show that every 1-out-of-2 oblivious transfer protocol allows a dishonest party to cheat with probability bounded below by a constant strictly larger than 1 2. Alice's cheating is defined as her probability of guessing Bob's index, and Bob's cheating is defined as his probability of guessing both input bits of Alice. In our proof, we relate these cheating probabilities to the cheating probabilities of a bit commitment protocol and conclude by using lower bounds on quantum bit commitment. Then, we present an oblivious transfer protocol with two messages and cheating probabilities at most 3 4. Last, we extend Kitaev's semidefinite programming formulation to more general primitives, where the security is against a dishonest player trying to force the outcome of the other player, and prove optimal lower and upper bounds for them. Multiplayer online battle arena has become a popular game genre. It also received increasing attention from our research community because they provide a wealth of information about human interactions and behaviors. A major problem is extracting meaningful patterns of activity from this type of data, in a way that is also easy to interpret. Here, we propose to exploit tensor decomposition techniques, and in particular Non-negative Tensor Factorization, to discover hidden correlated behavioral patterns of play in a popular game: League of Legends. We first collect the entire gaming history of a group of about one thousand players, totaling roughly @math matches. By applying our methodological framework, we then separate players into groups that exhibit similar features and playing strategies, as well as similar temporal trajectories, i.e., behavioral progressions over the course of their gaming history: this will allow us to investigate how players learn and improve their skills. We consider a well-known distributed colouring game played on a simple connected graph: initially, each vertex is coloured black or white; at each round, each vertex simultaneously recolours itself by the colour of the simple (strong) majority of its neighbours. A set of vertices M is said to be a dynamo, if starting the game with only the vertices of M coloured black, the computation eventually reaches an all-black configuration.The importance of this game follows from the fact that it models the spread of faults in point-to-point systems with majority-based voting; in particular, dynamos correspond to those sets of initial failures which will lead the entire system to fail. Investigations on dynamos have been extensive but restricted to establishing tight bounds on the size (i.e., how small a dynamic monopoly might be).In this paper we start to study dynamos systematically with respect to both the size and the time (i.e., how many rounds are needed to reach all-black configuration) in various models and topologies.We derive tight tradeoffs between the size and the time for a number of regular graphs, including rings, complete d-ary trees, tori, wrapped butterflies, cube connected cycles and hypercubes. In addition, we determine optimal size bounds of irreversible dynamos for butterflies and shuffle-exchange using simple majority and for DeBruijn using strong majority rules. Finally, we make some observations concerning irreversible versus reversible monotone models and slow complete computations from minimal dynamos. 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. Modern Massively Multiplayer Online Role-Playing Games (MMORPGs) provide lifelike virtual environments in which players can conduct a variety of activities including combat, trade, and chat with other players. While the game world and the available actions therein are inspired by their offline counterparts, the games' popularity and dedicated fan base are testaments to the allure of novel social interactions granted to people by allowing them an alternative life as a new character and persona. In this paper we investigate the phenomenon of "gender swapping," which refers to players choosing avatars of genders opposite to their natural ones. We report the behavioral patterns observed in players of Fairyland Online, a globally serviced MMORPG, during social interactions when playing as in-game avatars of their own real gender or gender-swapped. We also discuss the effect of gender role and self-image in virtual social situations and the potential of our study for improving MMORPG quality and detecting online identity frauds. 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. (This article originally appeared in Management Science, November 1967, Volume 14, Number 3, pp. 159-182, published by The Institute of Management Sciences.) The paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation, such as the payoff functions, the strategies available to various players, the information other players have about the game, etc. However, each player has a subjective probability distribution over the alternative possibilities. In most of the paper it is assumed that these probability distributions entertained by the different players are mutually "consistent," in the sense that they can be regarded as conditional probability distributions derived from a certain "basic probability distribution" over the parameters unknown to the various players. But later the theory is extended also to cases where the different players' subjective probability distributions fail to satisfy this consistency assumption. In cases where the consistency assumption holds, the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution, and the outcome of this lottery will decide which particular subgame will be played, i.e., what the actual values of the relevant parameters will be in the game. Yet, each player will receive only partial information about the outcome of the lottery, and about the values of these parameters. However, every player will know the "basic probability distribution" governing the lottery. Thus, technically, the resulting game will be a game with complete information. It is called the Bayes-equivalent of the original game. Part I of the paper describes the basic model and discusses various intuitive interpretations for the latter. Part II shows that the Nash equilibrium points of the Bayes-equivalent game yield "Bayesian equilibrium points" for the original game. Finally, Part III considers the main properties of the "basic probability distribution."