Showing papers on "Combinatorial game theory published in 1997"

An Introduction to Applicable Game Theory

Abstract: This paper offers an introduction to game theory for applied economists I try to give simple definitions and intuitive examples of the basic kinds of games and their solution concepts There are four kinds of games: static or dynamic, and complete or incomplete information ( Complete information means there is no private information) The corresponding solution concepts are: Nash equilibrium in static games of complete information; backwards induction (or subgame-perfect Nash equilibrium) in dynamic games of complete information; Bayesian Nash equilibrium in static games with incomplete information; and perfect Bayesian (or sequential) equilibrium in dynamic games with incomplete information The main theme of the paper is that these solution concepts are closely linked As we consider progressively richer games, we progressively strengthen the solution concept, to rule out implausible equilibria in the richer games that would survive if we applied solution concepts available for simpler games In each case, the stronger solution concept differs from the weaker concept only for the richer games, not for the simpler games

Cooperative Game Theory and Applications: Cooperative Games Arising from Combinatorial Optimization Problems

Abstract: Preface. 1. Cooperative Games and Solution Concepts. 2. Linear Programming Games. 3. Assignment Games and Permutation Games. 4. Sequencing Games and Generalizations. 5. Travelling Salesman Games and Routing Games. 6. Minimum Cost Spanning Tree Games. 7. Location Games. References. Index.

Classics in game theory

Abstract: Classics in Game Theory assembles in one sourcebook the basic contributions to the field that followed on the publication of Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern (Princeton, 1944). The theory of games, first given a rigorous formulation by von Neumann in a in 1928, is a subfield of mathematics and economics that models situations in which individuals compete and cooperate with each other. In the "heroic era" of research that began in the late 1940s, the foundations of the current theory were laid; it is these fundamental contributions that are collected in this volume. In the last fifteen years, game theory has become the dominant model in economic theory and has made significant contributions to political science, biology, and international security studies. The central role of game theory in economic theory was recognized by the award of the Nobel Memorial Prize in Economic Science in 1994 to the pioneering game theorists John C. Harsanyi, John Nash, and Reinhard Selten. The fundamental works for which they were honored are all included in this volume.Harold Kuhn, himself a major contributor to game theory for his reformulation of extensive games, has chosen eighteen essays that constitute the core of game theory as it exists today. Drawn from a variety of sources, they will be an invaluable tool for researchers in game theory and for a broad group of students of economics, political science, and biology.

Personal computer adventure games: their structure, principles, and applicability for training

Abstract: Personal computer adventure games, in which the player assumes the role of a fantasy character to pursue an adventure, have enjoyed enormous popularity and commercial success. Beyond their entertainment value, these games also have an educational value, training users to become better problem solvers in the game domain and probably beyond.In order to understand better this type of game and determine its potential use for managerial training, we analyzed adventure games with respect to three issues. First, what makes a computer simulation an adventure game; second, what makes such a game enjoyable and challenging; and finally, how well does such a game facilitate learning? In search of answers, we analyzed a number of adventure games to determine game structures, and completed a small empirical study in which subjects played a game and had to answer both content and preference related questions.Overall, our analysis reconfirmed people's liking of adventure games. Players seem to enjoy the games' story and interface, and are challenged by the game tasks. Players also learn from gaming; yet levels of learning vary widely, based on the type of knowledge to be conveyed. Fact learning proved to be easiest; plan learning and learning of negative knowledge, the hardest.

Games and structures

Abstract: Frequently applications of game theory assume, but do not show, that games are contained in social structures. The new analysis offered here uncovers games embedded in structures by attributing strategies to some positions and deriving the pay-off matrices for others. As structures vary so do the games embedded in them. All strong power structures contain prisoners' dilemma games for at least some range of pay-offs while some contain a chain of prisoners' dilemma games linked by defections. As a result, the development of interpersonal power in strong power structures is produced by free-riding of those low in power. Examples of other types of structures are given and other games are found that do not contain defection chains. Issues of dynamics including rates of change of power and coalition formation as a condition of countervailing power are addressed. New experiments offer support for central formulations.

Algorithms for Finding Repeated Game Equilibria

Abstract: This paper describes computational techniques for finding all equilibria in infinitely repeated games with discounting and perfect monitoring. It illustrates these techniques with a three player Cournot game. This is the first infinitely repeated three player game ever solved. The paper also presents the solution for the set of equilibria in a two country tariff war. In both games the set of equilibria is large even when the players are not patient.

Different ways to represent weighted majority games

Abstract: It is well known that every simple game is the intersection of weighted majority games. the aim of this paper is to gather together various ways of expressing weighted majority games and, for each game of this type, to give the simplest way to define it. Normalized representations, the parameters of a simple game and the characteristic invariants of a complete game merit special attention.

A Relationship Among Gentzen's Proof-Reduction, Kirby-Paris' Hydra Game and Buchholz's Hydra Game

Abstract: We first note that Gentzen's proof-reduction for his consistency proof of PA can be directly interpreted as moves of Kirby-Paris' Hydra Game, which implies a direct independence proof of the game (Section 1 and Appendix). Buchholz's Hydra Game for labeled hydras is known to be much stronger than PA. However, we show that the one-dimensional version of Buchholz's Game can be exactly identified to Kirby-Paris' Game (which is two-dimensional but without labels), by a simple and natural interpretation (Section 2). Jervell proposed another type of a combinatorial game, by abstracting Gentzen's proof-reductions and showed that his game is independent of PA. We show (Section 3) that this Jervell's game is actually much stronger than PA, by showing that the critical ordinal of Jervell's game is φω (0) (while that of PA or of Kirby-Paris' Game is φ1 (0) = ϵ0) in the Veblen hierarchy of ordinals.

On the infiltration game

Abstract: An infiltrator tries to go through a graph ofn arcs, within a time limit, without being caught by a guard. The latter is allowed a restricted number of tentatives to catch the infiltrator. This paper describes optimal strategies and gives the value of this discrete zero-sum infiltration game.

Adaptive critic design in learning to play game of Go

Abstract: This paper examines the performance of an HDP-type adaptive critic design (ACD) of the game Go. The game Go is an ideal problem domain for exploring machine learning; it has simple rules but requires complex strategies to play well. All current commercial Go programs are knowledge based implementations; they utilize input feature and pattern matching along with minimax type search techniques. But the extremely high branching factor puts a limit on their capabilities, and they are very weak compared to the relative strengths of other game programs like chess. In this paper, the Go-playing ACD consists of a critic network and an action network. The HDP type critic network learns to predict the cumulative utility function of the current board position from training games, and, the action network chooses a next move which maximizes critics next step cost-to-go. After about 6000 different training games against a public domain program, WALLY, the network (playing WHITE) began to win in some of the games and showed slow but steady improvements on test games.

Multi-agent reinforcement learning in Markov games

Abstract: Game playing has been a popular problem area for research in artificial intelligence and machine learning for many years. In almost every study of game playing and machine learning, the focus has been on games with a finite set of states and a finite set of actions. Further, most of this research has focused on a single player or team learning how to play against another player or team that is applying a fixed strategy for playing the game. In this dissertation, we provide and evaluate algorithms for learning strategies in two player games with large state and action spaces. First, we focus on the class of differential games in which the state space and the action space are both continuous. We model these games as discrete Markov games and provide methods for representing the state and actions spaces at varying levels of resolution. Second, we explore multi-agent learning and develop algorithms for "co-learning" in which all players attempt to learn their optimal strategies simultaneously. Specifically, in this dissertation we compare several algorithms for a single player to learn an optimal strategy against a fixed opponent. Next we combine the results of one algorithm--a genetic algorithm--with a second algorithm--a memory-based learning algorithm--to yield performance exceeding the capabilities of either algorithm alone. Then we explore two approaches to co-learning in which both players learn simultaneously. We demonstrate strong performance by a memory-based reinforcement learner and comparable but faster performance with a tree-based reinforcement learner. In addition to the experimental results, we also provide an overview of machine learning and game playing as well as an overview of differential and Markov games.

Sums of hot and tepid combinatorial games

Abstract: In this dissertation, combinatorial games are the main objects of study. Games can be classified into hot, cold and tepid games according to their temperatures. Roughly speeking, the temperature of a game is a measure of the size of the next move in the game. The information about the temperatures of the games in a sum can be used to find out good moves in the sum. In the first part of this dissertation, we study hot games. A new method for calculating the mean/temperature of a game is presented. This new method improves the classical approach by ignoring unnecessary searches in the game tree. Two new game tree pruning techniques called M-cut and T-cut, analog to alpha-beta cut, are introduced. In the second part, we study tepid games. Tepid games are those games where the players fight for the last move, instead of points. The values of these games are called infinitesimals. We introduce several new classes of infinitesimals which are completely solved. As byproducts of the result, there are five games introduced in this disseration. Four of them are completely solved, the only unsolved one is an NP-hard game. The application of the result to computer Go is also discussed.

Cooperative Theory with Incomplete Information

Abstract: This paper surveys cooperative game theory when players have incomplete or asymmetric information, especially when the TU and NTU games are derived from economic models. First some results relating balanced games and markets are summarized, including theorems guaranteeing that the core is nonempty. Then the basic pure exchange economy is extended to include asymmetric information. The possibilities for such models to generate cooperative games are examined. Here the core is emphasized as a solution, and criteria are given for its nonemptiness. Finally, an alternative approach is explored based on Harsanyi’s formulation of games with incomplete information.

Comparing the Power of Games on Graphs

Abstract: The descriptive complexity of a problem is the complexity of describing the problem in some logical formalism. One of the few techniques for proving separation results in descriptive complexity is to make use of games on graphs played between two players, called the spoiler and the duplicator. There are two types of these games, which differ in the order in which the spoiler and duplicator make various moves. In one of these games, the rules seem to be tilted towards favoring the duplicator. These seemingly more favorable rules make it easier to prove separation results, since separation results are proven by showing that the duplicator has a winning strategy. In this paper, the relationship between these games is investigated. It is shown that in one sense, the two games are equivalent. Specifically, each family of graphs used in one game (the game with the seemingly more favorable rules for the duplicator) to prove a separation result can in principle be used in the other game to prove the same result. This answers an open question of Ajtai and the author from 1989. It is also shown that in another sense, the games are not equivalent, in that there are situations where the spoiler requires strictly more resources to win one game than the other game. This makes formal the informal statement that one game is easier for the duplicator to win.

Decisions in games: why there should be a special exemption from Bayesian rationality

Abstract: I examine the Bayesian foundations of game theory and advance three main theses: (i) that if Bayesianism is to be used in game theory, then simpler versions of it are methodologically preferable to the more sophisticated ones; (ii) that it is dubious whether the Savage (or Anscombe and Aumann) axiom system can yield an axiomatic justification for Bayesianism in game theory; and (iii) that there exist other foundational frameworks for games which are at least as convincing as the Bayesian one.

Consistency and Egalitarianism: The Egalitarian Set

Abstract: In this paper we extend an egalitarian solution for two-person games to n-person cooperative games so that the estalished standard is observed for every two agents of the game

Zero-Sum Games

Abstract: We introduce here zero-sum games, which represent the strategic interactions where two agents have totally opposed interests. The concept of rational solution for such interactions is then defined by the notions of value and optimal strategies. These quantities always exist if the information is perfect (there is no uncertainty) and the number of stages of the interaction is bounded.

An Iterated Hawk-and-Dove Game

Abstract: A fundamental problem in multi agent systems is conflict resolution. A conflict occurs in general when the agents have to deal with inconsistent goals, such as a demand for shared resources. We investigate some theoretical game approaches as efficient methods to examine a class of conflicts in multi agent systems.

Stationary strategies for recursive games

Abstract: We study two-person, zero-sum recursive matrix games framing them in the more general context of nonleavable games. Our aim is to extend a result of Orkin Orkin, M. 1972. Recursive matrix games. J. Appl. Probab.9 813--820. by proving that a uniformly e-optimal stationary strategy is available to player I player II in any recursive game such that the set where the value of the game is strictly greater less than its utility is finite. The proof exploits some new connections between nonleavable and leavable games. In particular we find sufficient conditions for the existence of e-optimal stationary strategies for player I in a leavable game and we use these as a basis for constructing uniformly e-optimal stationary strategies for the recursive games with which we are concerned.

Four way chess game

Abstract: The present invention is chess game played by up to four players. The players may play against each other or in combination. The game consists of a standard set of chess game pieces controlled by standard chess rules. The standard chess game board is made wider by two rows and columns and 3×8 square appendages are added to each edge. The game may be played with two to four players. The players may be arranged in teams of two each.

Checkers for teams

Abstract: A method of playing a two to four person checkerboard game utilizing a traditional game board consisting of 64 equally sized squares in an 8 by 8 matrix known as the battleground. An 8 by 3 matrix, known as a players home base, is attached to each of the four sides of the main game board, providing an additional area in which each players' game pieces are placed. Four sets of 12 game pieces in different colors are provided, allowing two to four players, or teams of players, each to play in a single game. The game utilizes a novel set of rules for kinging and jumping checkers.

Games of Incomplete Information: The Inconsistent Case

Abstract: In his well known series of papers Harsanyi [1967–68] develops the theory of noncooperative games of incomplete information, in which such games are replaced by games of complete information involving chance. He emphasizes the consistent case, in which the various types of players in a game of incomplete information Γ, as generated from the various beliefs, can be thought as being derived from a joint probability matrix. In this case Harsanyi constructs a game G of perfect recall and in extensive form, whose players represent the various types of the players in Γ. They are called agents and G itself is called the agent-form game representing Γ.