Game Tree Research Articles

Minimum-cost spanning tree games: Bird rule revisited

Minimum-cost spanning tree (MCST) game is a classical model in the theory of cooperative games. The MCST game has been drawing continuous attention since it was raised by Claus and Kleitman in 1973. The MCST game is closely related to the MCST problem, a fundamental model in graph theory and combinatorial optimization, and has been widely applied in cost allocations for constructions of water networks, power networks, road networks and railway networks, etc. Due to its simplicity and intuitiveness, the Bird rule is one of the most famous solutions in the MCST game and has been extensively studied in the literature. Based on the linear programming formulation of the MCST problem by Edmonds, we provide a new equivalent closed-form formula for the Bird rule. We also study a generalization of the MCST game with weighted nodes. We show that the core is nonempty in the generalized model, and in particular, a modified Bird rule is proved to be still applicable.

Computing the Subgame Perfect Nash Equilibriums in Parallel Allocation Indivisible Items

Multi-Agent resource allocation has also become a research hotspot in the field of artificial intelligence in recent years. A lot of related work aims to design a procedural resource allocation system with execution efficiency. This paper Studied a parallel allocation method for allocating indivisible items between agents in which every agent asks for an object among those that re-main in every round, and every reported item will be allocated randomly to an agent reporting it. In order to depicts the entire distribution process, the authors propose a method of constructing a game tree of parallel allocation. Study the corresponding relationship between the total number of branches of the game tree and items. In the last part, prof the parallel allocation mechanism is dynamic game and show that the subgame perfect Nash equilibrium (SPNE) can be found by backward introduction the game trees. Provides a method to compute the unique subgame perfect Nash equilibrium in linear time.

ДА ПОСТРОИМ ЗАЕДНО КЪЩА – ЗАНИМАНИЕ ПО ИНТЕРЕСИ В ГРУПА ЗА ЦЕЛОДНЕВНА ОРГАНИЗАЦИЯ НА УЧЕБНИЯ ДЕН

The presented report offers development of activity by interests, conducted in a group for all-day organization of the educational day, 5th grade, First primary school „St. Cyril and St. Methodius“, Gotse Delchev. The topic „How to build a house together – ours group, school, society – where all the differences are being respected and feels like home“ selection is based on the specific features of the school group. The group aim is to focus the students attention to themselves and further through precised class organization provided by the teacher to reach realization of the goals is being used game working approach. „The Plot“ of the student’s group is compared to building a house. Playing consequently tree games, the students build their „house“. In the game process attention is given to the respect and tolerance based in the personality of the students. Moreover, the process of skill development of the students is being provoked through discussion regarding „National symbols“ where the aim is to build a common sense related to the social norms of behavior. Game working approach is being used as a reliable tool in order to increase the students activity in all organization forms of education and socialization.

TASKS FOR PROGRAMMING OF THE INTELLECTUAL OLYMPIAD OF 2020 OF THE VOLGA FEDERAL DISTRICT AMONG SCHOOLCHILDREN

The article presents the results of the annual Intellectual Olympiad for programming of the Volga Federal District among schoolchildren, in which representatives of fourteen regions participated. Each team consisted of two schoolchildren who were selected in regional contests. The author's solutions of the Olympiad tasks are considered. The set of tasks is balanced by subject and difficulty. The topics of the tasks are related to the organization of rational data structures, bracket sequences, optimization in graphs, depth-first and breadth-first search, analysis of areas on a checkered field, construction and evaluation of the game tree. Important conditions for a successful performance were good teamwork and a correct estimation of the computational complexity of the algorithms. The winner of the Olympiad was the team of the Samara region, which solved all eight of the proposed tasks. The article is intended for schoolchildren, students, trainers, leading classes in Olympiad programming.

Characterizing the Nature of Probability-Based Proof Number Search: A Case Study in the Othello and Connect Four Games

Variants of best-first search algorithms and their expansions have continuously been introduced to solve challenging problems. The probability-based proof number search (PPNS) is a best-first search algorithm that can be used to solve positions in AND/OR game tree structures. It combines information from explored (based on winning status) and unexplored (through Monte Carlo simulation) nodes from a game tree using an indicator called the probability-based proof number (PPN). In this study, PPNS is employed to solve randomly generated positions in Connect Four and Othello, in which the results are compared with the two well-known best-first search algorithms (proof number search (PNS) and Monte Carlo proof number search). Adopting a simple improvement parameter in PPNS reduces the number of nodes that need to be explored by up to 57%. Moreover, further observation showed the varying importance of information from explored and unexplored nodes in which PPNS relies critically on the combination of such information in earlier stages of the Othello game. Discussion and insights from these findings are provided where the potential future works are briefly described.

Design and improvement of the pruning algorithm of the Chinese chess in the computer games

Computer games have always been considered to be the most challenging task in the field of artificial intelligence specially the Chinese chess as an example. The core technology of the computer games is the search. This work is studied to research an improved pruning strategy in order to achieve a deeper level search of the game tree in a limited time. On the basis of the traditional alpha–beta search algorithm, the worthless nodes are discarded to be never searched by introducing the deep iterative, history table and other auxiliary methods of pruning. The number of nodes searched is effectively reduced to make the pruning earlier to shorten the search time. At the same time, its search depth is higher than the original search algorithm. The advanced and modified algorithm is proved to be practical and applicative by experimentations and tests of computer games system provided in this study.

An efficient characterization of submodular spanning tree games

Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

Market Trading Model of Urban Energy Internet Based on Tripartite Game Theory

As an important driving force to promote the energy revolution, the emergence of the energy internet has provided new ideas for the marketization and flexibility of multi-energy transactions. How to realize multi-energy joint trading is a key issue in the development of the energy market. An urban energy internet market trading model among energy suppliers, energy service providers and the large users in the urban area, based on tripartite game theory, is established in this paper. Considering the cost–income function of each market entity and the basic market trading mechanism, a new game-tree search method is proposed to solve the Nash equilibria for the game model. The Nash equilibria of the tripartite game can be obtained, and the market transaction status corresponding to the Nash equilibria is analyzed from the perspective of the market transactions. The multi-energy joint transaction and market equilibria can be easily implemented for the bids and offers of the multiple energy entities in the urban energy internet market.

Limited lookahead in imperfect-information games

Limited lookahead has been studied for decades in perfect-information games. We initiate a new direction via two simultaneous deviation points: generalization to imperfect-information games and a game-theoretic approach. We study how one should act when facing an opponent whose lookahead is limited. We study this for opponents that differ based on their lookahead depth, based on whether they, too, have imperfect information, and based on how they break ties. We characterize the hardness of finding a Nash equilibrium or an optimal commitment strategy for either player, showing that in some of these variations the problem can be solved in polynomial time while in others it is PPAD-hard, NP-hard, or inapproximable. We proceed to design algorithms for computing optimal commitment strategies—for when the opponent breaks ties favorably, according to a fixed rule, or adversarially. We then experimentally investigate the impact of limited lookahead. The limited-lookahead player often obtains the value of the game if she knows the expected values of nodes in the game tree for some equilibrium—but we prove this is not sufficient in general. Finally, we study the impact of noise in those estimates and different lookahead depths.

Parallel Computing of PVS Node based GTS Algorithm on GPU

The developments in the field of computer architecture, it allows humans to play games like Chess, Tic-tac-toe, Go, etc. with computer machines using AI technology. In AI, Game tree search (GTS) is an important approach and is directed toward finding the finest choice of move for computer games. Using the traditional GTS algorithm the computer could not win a human. So there is a need for enhancing algorithms using dynamic parallelism of GPU. The block of thread set is organized on GPU is either one or two or three dimensional for parallel computations. On this, each thread designated by an inimitable mixture of indices. In this paper, parallel computing of node-based Principal Variation Search (PVS) GTS algorithm presented, which runs on GPU using libraries of CUDA. this experiment tested on chess games with different depths and results are compared with the threads on CPU and GPU. The results proved that GPU improves the performance of speedup up to 80 percent on the checkers game. Parallel computing greatly increasing the efficient use of CPU and improves the performances of the PVS-GTS algorithm on GPU to search deeper layers and find the optimal moves for the current players for two-player computer games.

Automated construction of bounded-loss imperfect-recall abstractions in extensive-form games

Extensive-form games (EFGs) model finite sequential interactions between players. The amount of memory required to represent these games is the main bottleneck of algorithms for computing optimal strategies and the size of these strategies is often impractical for real-world applications. A common approach to tackle the memory bottleneck is to use information abstraction that removes parts of information available to players thus reducing the number of decision points in the game. However, existing information-abstraction techniques are either specific for a particular domain, they do not provide any quality guarantees, or they are applicable to very small subclasses of EFGs. We present domain-independent abstraction methods for creating imperfect recall abstractions in extensive-form games that allow computing strategies that are (near) optimal in the original game. To this end, we introduce two novel algorithms, FPIRA and CFR+IRA, based on fictitious play and counterfactual regret minimization. These algorithms can start with an arbitrary domain specific, or the coarsest possible, abstraction of the original game. The algorithms iteratively detect the missing information they require for computing a strategy for the abstract game that is (near) optimal in the original game. This information is then included back into the abstract game. Moreover, our algorithms are able to exploit imperfect-recall abstractions that allow players to forget even history of their own actions. However, the algorithms require traversing the complete unabstracted game tree. We experimentally show that our algorithms can closely approximate Nash equilibrium of large games using abstraction with as little as 0.9% of information sets of the original game. Moreover, the results suggest that memory savings increase with the increasing size of the original games.

A Game Theory Based Clustering Protocol to Support Multicast Routing in Cognitive Radio Mobile Ad Hoc Networks

The frequent movement of nodes in cognitive radio mobile ad hoc networks (CRAHNs) causes challenges in scalability, stability, channel sensing and channel access problems that can be solved by using clustering technique. Game theory is a feasible approach to solve such problems by casting clustering problems as distributed optimization problems. The main contributions of this article are as follows. Firstly, we propose a minimum connected weighted inner edge spanning tree (MWIEST) game to find an approximate solution of a MWIEST problem in CRAHNs. In this game, a link-weight function of each link is designed based on a combination of link-stability and link-connectivity ratio functions. Secondly, we prove that the MWIEST game is an exact potential game that exists at least one Nash equilibrium ( NE ) point which is an approximate solution of the MWIEST problem. Besides, we also prove that best responses ( BR s) of the game converge to a NE in finite iterations. Thirdly, based on the MWIEST game, we propose four algorithms including the node information exchange (NIE), the best response selection (BRS), the intermediate nodes selection (INS) and the forming cluster (FC). Specifically, the algorithms NIE, BRS and INS provide a set of intermediate nodes (SetIN) which supports the FC algorithm to form clusters. Finally, we propose the game theory based clustering (GBC) protocol which is combination of the FC algorithm and the proposed cluster maintenance algorithm to construct high stable clusters supporting multicast routing in CRAHNs. Moreover, each obtained cluster includes most members having the same receiving channel which avoids the affected regions of licensed channels. For the performance evaluation, we implement the GBC protocol in OMNET++ platform to demonstrate its performance improvement over the state-of-the-art protocols in terms of network stability and control overheads.

Generating 2D Lego Compatible Puzzles Using Reinforcement Learning

We present a framework that generates a 2D Lego-compatible puzzle layout of greater than thousands pieces of bricks using a reinforcement learning technique. Many existing 2D legorization strategies have limitations in producing a Lego layout, which is composed of more than thousands of pieces. We attack this problem by employing a reinforcement learning technique, which accelerates the progress of various game strategies. We represent the legorization process as a game tree search problem, where each leaf node of the tree corresponds to a Lego layout. The goal of legorization is to find an optimal Lego layout that achieves maximum reward. To efficiently find a leaf node for the maximum reward layout, we reduce the search space using a dueling deep Q-Network (DQN), which is a widely used reinforcement learning model. Our framework is composed of a learning stage and a legorization stage. In the learning stage, we design a dueling DQN model and train this model using three heuristics for legorization strategies. In the legorization stage, we efficiently generate a large-scaled 2D Lego-compatible puzzle layout by reducing the search space using the trained dueling DQN. This approach enables us to produce a puzzle layout of more than a thousand of pieces, which has not been feasible for existing legorization schemes.

The dynamic Nash bargaining solution for 2-stage cost sharing game

The problem of constructing the Dynamic Nash Bargaining Solution in a 2-stage game is studied. In each stage, a minimum cost spanning tree game is played, all players select strategy profiles to construct graphs in the stage game. At the second stage, players may change the graph using strategy profiles with transition probabilities, which decided by players in the first stage. The players' cooperative behavior is considered. As solution the Dynamic Nash Bargaining Solution is proposed. A theorem is proved to allow the Dynamic Nash Bargaining Solution to be time-consistent.

Network Attack and Defense Game Theory Based on Bayes-Nash Equilibrium

In the process of constructing the traditional offensive and defensive game theory model, these are some shortages for considering the dynamic change of security risk problem. By analysing the critical indicators of the incomplete information game theory model, incomplete information attack and defense game theory model and the mathematical engineering method for solving Bayes-Nash equilibrium, the risk-averse income function for information assets is summarized as the problem of maximising the return of the equilibrium point. To obtain the functional relationship between the optimal strategy combination of the offense and defense and the information asset security probability and risk probability. At the same time, the offensive and defensive examples are used to visually analyse and demonstrate the incomplete information game and the Harsanyi conversion method. First, the incomplete information game and the Harsanyi conversion problem is discussed through the attack and defense examples and using the game tree. Then the strategy expression of incomplete information static game and the engineering mathematics method of Bayes-Nash equilibrium are given. After that, it focuses on the offensive and defensive game problem of unsafe information network based on risk aversion. The problem of attack and defense is obtained by the issue of maximizing utility, and then the Bayes-Nash equilibrium of offense and defense game is carried out around the security risk of assets. Finally, the application model in network security penetration and defense is analyzed by designing a simulation example of attack and defense penetration. The analysis results show that the constructed income function model is feasible and practical.

Simulation of NLH DON game strategies for identifying violations of online gambling rules by clients

One of the most serious types of violation by a client of a user agreement with an operator providing online gambling services in the segment of card games is «team play» – a conspiracy between two or more participants to play in a co-op and receive an unfair advantage. The effective detection of user agreement violations is an important task in the field of production activities of the information security service of a gambling platform. Various types of game strategies are considered: Nash, Sklansky-Chubukov equilibrium, team play strategies, hybrid strategies for which mathematical models and algorithms for software implementation based on decision trees are built. The game tree expresses features of the conflict of interests of players through actions that take place in the game format under consideration. When forming game strategies, a model of independent chips is used. As a result of the work, the calculation was performed on model situations, the areas of the game situations space were clarified, on which it is possible to effectively differentiate the strategy chosen by the client as a strategy that implements the forbidden «team play» methods, thereby flagrantly breaking the rules. This result will help support the adoption of an earlier and more effective decision by the information security service of the online gambling operator to detect «team play» in the gaming environment. This is an important element of the information security policy declared and implemented by gambling operators – leaders of the online gambling industry: to guarantee strict compliance with the rules by other participants for each client, thereby ensuring positive dynamics in changing their reputation and rating, and, accordingly, game and financial traffic

An LP Approach for Solving Two-Player Zero-Sum Repeated Bayesian Games

This paper studies two-player zero-sum repeated Bayesian games in which every player has a private type that is unknown to the other player, and the initial probability of the type of every player is publicly known. The types of players are independently chosen according to the initial probabilities, and are kept the same all through the game. At every stage, players simultaneously choose actions, and announce their actions publicly. For finite horizon cases, an explicit linear program is provided to compute players’ security strategies. Moreover, this paper shows that a player's sufficient statistics, which is independent of the strategy of the other player, consists of the belief over the player's own type, the regret over the other player's type, and the stage. Explicit linear programs, whose size is linear in the size of the game tree, are provided to compute the initial regrets, and the security strategies that only depends on the sufficient statistics. For discounted cases, following the same idea in the finite horizon, this paper shows that a player's sufficient statistics consists of the belief of the player's own type and the antidiscounted regret with respect to the other player's type. Besides, an approximated security strategy depending on the sufficient statistics is provided, and an explicit linear program to compute the approximated security strategy is given. This paper also obtains a bound on the performance difference between the approximated security strategy and the security strategy, and shows that the bound converges to 0 exponentially fast.

Simplicial Complexes are Game Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

Improving RTS Game AI by Supervised Policy Learning, Tactical Search, and Deep Reinforcement Learning

Constructing strong AI systems for video games is difficult due to enormous state and action spaces and the lack of good state evaluation functions and high-level action abstractions. In spite of recent research progress in popular video game genres such as Atari 2600 console games and multiplayer online battle arena (MOBA) games, to this day strong human players can still defeat the best AI systems in adversarial video games. In this paper, we propose to use a deep Convolutional Neural Network (CNN) to select among a limited set of abstract action choices in Real-Time Strategy (RTS) games, and to utilize the remaining computation time for game tree search to improve low-level tactics. The CNN is trained by supervised learning on game states labeled by Puppet Search, a strategic search algorithm that uses action abstractions. Replacing Puppet Search by a CNN frees up time that can be used for improving units' tactical behavior while executing the strategic plan. Experiments in the μRTS game show that the combined algorithm results in higher win-rates than either of its two independent components and other state-of-the-art μRTS agents. We then present a case-study that investigates how deep Reinforcement Learning (RL) can be used in modern video games, such as Total War: Warhammer, to improve tactical multi-agent AI modules. We use popular RL algorithms such as Deep-Q Networks (DQN) and Asynchronous AdvantageActor Critic (A3C), basic network architectures and minimal hyper-parameter tuning to learn complex cooperative behaviors that defeat the highest difficulty built-in AI in mediumscale scenarios.

A Modeling Approach for Getting to Win-Win in Industrial Collaboration under Strategic Coopetition

Interorganizational coopetition describes a relationship in which two or more organizations cooperate and compete simultaneously. Actors under coopetition cooperate to achieve collective objectives and compete to maximize their individual benefits. Such relationships are based on the logic of win-win strategies that necessitate decision-makers in coopeting organizations to develop relationships that yield favorable outcomes for each actor. We follow a strategic modeling approach that combines i* goal-modeling to explore strategic alternatives of actors with Game Tree decision-modeling to evaluate the actions and payoffs of those players. In this article, we elaborate on the method, illustrating one particular pathway towards a positive-sum outcome – through the introduction of an intermediary actor. This article demonstrates the activation of one component in this guided approach of systematically searching for alternatives to generate a new win-win strategy. We also present a meta-model for relating i* models and Game Trees. A hypothetical industrial scenario focusing on the Industrial Data Space, which is a platform that can help organizations to overcome obstacles to data sharing in a coopetitive ecosystem, is used to explain this approach.