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.
@cite_7
1596450573 2135737379 2141368734 2010404638 2083596217 2002373723 2073060670 1971467559 2136241552 2069363808 2015596743 1608583290 2008587708 2046633098 2811269531 2558809543 2015247315 2151123669 2952499036 2139774323
In the generalized Russian cards problem, we have a card deck @math of @math cards and three participants, Alice, Bob, and Cathy, dealt @math , @math , and @math cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (weak @math -security) to not gaining any probabilistic advantage in guessing the fate of some set of @math cards (perfect @math -security). As we demonstrate, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly @math -secure strategies and @math -designs on @math points with block size @math , when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect @math -security against Cathy when @math . We leverage a known combinatorial design to construct a strategy with @math , @math , and @math that is perfectly @math -secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak @math -security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of @math cards. Alternatively, the construction yields solutions for arbitrary @math , @math and any @math . 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. 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. In the generalized Russian cards problem, the three players Alice, Bob and Cath draw @math a , b and @math c cards, respectively, from a deck of @math a + b + c cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. For a natural number @math k , the communication is said to be @math k -safe if Cath does not learn whether or not Alice holds any given set of at most @math k cards that are not Cath's, a notion originally introduced as weak @math k -security by Swanson and Stinson. An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces rather than projective planes, and call it the geometric protocol'. Given arbitrary @math c , k > 0 , this protocol gives an informative and @math k -safe solution to the generalized Russian cards problem for infinitely many values of @math ( a , b , c ) with @math b = O ( a c ) . This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees @math k -safety when Cath has more than one card. Consider the following probabilistic one-player game: The board is a graph with @math vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of @math available colors to each edge immediately, where @math is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph @math has been created, and Painter's goal is to “survive” for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are chosen not randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number @math , the ratio of edges to vertices in all subgraphs of the evolving board ne... In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof. 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. Two players A and B are randomly placed on a line. The distribution of the distance between them is unknown except that the expected initial distance of the (two) players does not exceed some constant @math The players can move with maximal velocity 1 and would like to meet one another as soon as possible. Most of the paper deals with the asymmetric rendezvous in which each player can use a different trajectory. We find rendezvous trajectories which are efficient against all probability distributions in the above class. (It turns out that our trajectories do not depend on the value of @math ) We also obtain the minimax trajectory of player A if player B just waits for him. This trajectory oscillates with a geometrically increasing amplitude. It guarantees an expected meeting time not exceeding @math We show that, if player B also moves, then the expected meeting time can be reduced to @math The expected meeting time can be further reduced if the players use mixed strategies. We show that if player B rests, then the optimal strategy of player A is a mixture of geometric trajectories. It guarantees an expected meeting time not exceeding @math This value can be reduced even more (below @math ) if player B also moves according to a (correlated) mixed strategy. We also obtain a bound for the expected meeting time of the corresponding symmetric rendezvous problem. 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. Motivated by the problem of searching in the presence of adversarial errors, we consider a version of the game “Twenty Questions” played on the set 0,…,N-1 where the player giving answers may lie in her answers. The questioner is allowed Q questions and the responder may lie in upto [ rQ ] of the answers, for some fixed and previously known fraction r . Under three different models of this game and for two different question classes, we give precise conditions (i.e. tight bounds on r and, in most cases, optimal bounds on Q under which the questioner has a winning strategy in the game. 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. 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. We construct an uncoupled randomized strategy of repeated play such that, if every player follows such a strategy, then the joint mixed strategy profiles converge, almost surely, to a Nash equilibrium of the one-shot game. The procedure requires very little in terms of players' information about the game. In fact, players' actions are based only on their own past payoffs and, in a variant of the strategy, players need not even know that their payoffs are determined through other players' actions. The procedure works for general finite games and is based on appropriate modifications of a simple stochastic learning rule introduced by Foster and Young. We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(log2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open. The @math -online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge at each step and Painter immediately paints the edge red or blue. Builder's goal is to force Painter to create either a red @math or a blue @math using as few edges as possible. The online Ramsey number @math is the minimum number of edges Builder needs to guarantee a win in the @math -online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement [ r (n,n) 2^ (2- 2 )n + O(1) ] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement [ r (m,n) n^ (2- 2 )m + O(1) ] for the off-diagonal case, where @math is fixed and @math . Using a different randomized Painter strategy and the Lopsided Lov 'asz Local Lemma, we prove that @math , determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for @math . In connection with the online Ramsey game with random Painter, we study the problem of finding a copy of a target graph @math in a sufficiently large unknown Erd o s--R ' e nyi random graph @math using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem. We introduce GuessWhat?!, a two-player guessing game as a testbed for research on the interplay of computer vision and dialogue systems. The goal of the game is to locate an unknown object in a rich image scene by asking a sequence of questions. Higher-level image understanding, like spatial reasoning and language grounding, is required to solve the proposed task. Our key contribution is the collection of a large-scale dataset consisting of 150K human-played games with a total of 800K visual question-answer pairs on 66K images. We explain our design decisions in collecting the dataset and introduce the oracle and questioner tasks that are associated with the two players of the game. We prototyped deep learning models to establish initial baselines of the introduced tasks. A simple game is a pair consisting of a finite set N of players and a set of winning coalitions. (Vector-) weighted majority games ((V) WMG) are a special case of simple games, in which an integer (vector) weight can be assigned to each player and there is a quota which a coalition has to achieve in order to win. Binary decision diagrams (BDDs) are used as compact representations for Boolean functions and sets of subsets. This paper shows, how a quasi-reduced and ordered BDD (QOBDD) of the winning coalitions of a (V) WMG can be build, how one can compute the minimal winning coalitions and how one can easily compute the Banzhaf, Shapley-Shubik, Holler-Packel and Deegan-Packel indices of the players. E.g. in case of weighted majority games it is shown that the Banzhaf and Holler-Packel indices of all players can be computed in expected time and in general, the Banzhaf indices can be computed in time linear in the size of the QOBDD representation of the winning coalitions. Other running times are proven as well. The algorithms were tested on some real world games, e.g. the International Monetary Fund and the EU Treaty of Nice. Games are used to evaluate and advance Multiagent and Artificial Intelligence techniques. Most of these games are deterministic with perfect information (e.g. Chess and Checkers). A deterministic game has no chance element and in a perfect information game, all information is visible to all players. However, many real-world scenarios with competing agents are stochastic (non-deterministic) with imperfect information. For two-player zero-sum perfect recall games, a recent technique called Counterfactual Regret Minimization (CFR) computes strategies that are provably convergent to an e-Nash equilibrium. A Nash equilibrium strategy is useful in two-player games since it maximizes its utility against a worst-case opponent. However, for multiplayer (three or more player) games, we lose all theoretical guarantees for CFR. However, we believe that CFR-generated agents may perform well in multiplayer games. To test this hypothesis, we used this technique to create several 3-player limit Texas Hold'em poker agents and two of them placed first and second in the 3-player event of the 2009 AAAI IJCAI Computer Poker Competition. We also demonstrate that good strategies can be obtained by grafting sets of two-player subgame strategies to a 3-player base strategy after one of the players is eliminated. 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. (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."