Game Tree Research Articles

Book Reviews

Andrés Perea of Maastricht University reviews “The Theory of Extensive Form Games,” by Carlos Alós-Ferrer and Klaus Ritzberger. The Econlit abstract of this book begins: “Extends the theory of extensive-form games to the general case without any finiteness and synthesizes different models that have been proposed for extensive-form games. Discusses game trees; pseudotrees and order theory; extensive decision problems; extensive forms; discrete extensive forms; and equilibrium.”

On the Core of Cost-Revenue Games: Minimum Cost Spanning Tree Games with Revenues

In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net profit that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a non-empty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a non-empty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).

Combinatorial multi-armed bandits for real-time strategy games

Games with large branching factors pose a significant challenge for game tree search algorithms. In this paper, we address this problem with a sampling strategy for Monte Carlo Tree Search (MCTS) a...

Kuhn's Theorem for Extensive Games with Unawareness

We extend Kuhn's Theorem to extensive games with unawareness. This extension is not entirely obvious: First, extensive games with non-trivial unawareness involve a forest of partially ordered game trees rather than just one game tree. An information set at a history in one tree may consist of histories in a less expressive tree. Consequently, perfect recall takes a more complicated form as players may also become aware of new actions during the play. Second, strategies can only be partially an object of ex-ante choice in games with unawareness. Finally, histories that a player may expect to reach with a strategy profile may not be the histories that actually occur with this strategy profile, requiring us to define appropriate notions of equivalence of strategies.

A game-theoretic model and analysis of data exchange protocols for Internet of Things in clouds

Big data, Internet of things (IoT), and cloud computing have been recognized a family of technologies for a connected world. Besides hailed hope for the future, there are also challenges to security due to complexity and unpredictability of the Internet, clouds, and data. One of the challenges is information and data exchange, for example, identifying untrustworthy cloud users and analyzing abnormal user behavior during information exchange. This paper addresses exchange mechanism, which is a useful theoretic basis to make secure electronic commerce and electronic business transactions possible. To ensure and verify the property of fairness, a crucial property of exchange mechanism, this paper proposes a specific model for behavior analysis based on the extensive game with imperfect information. Rationality and fairness properties are built in the corresponding game and the game tree. To verify the properties, a tree analysis method is proposed, and a linear time algorithm is given. As a case study, some flaws of the ASW protocol are found.

Game-Theory-Based Active Defense for Intrusion Detection in Cyber-Physical Embedded Systems

Cyber-Physical Embedded Systems (CPESs) are distributed embedded systems integrated with various actuators and sensors. When it comes to the issue of CPES security, the most significant problem is the security of Embedded Sensor Networks (ESNs). With the continuous growth of ESNs, the security of transferring data from sensors to their destinations has become an important research area. Due to the limitations in power, storage, and processing capabilities, existing security mechanisms for wired or wireless networks cannot apply directly to ESNs. Meanwhile, ESNs are likely to be attacked by different kinds of attacks in industrial scenarios. Therefore, there is a need to develop new techniques or modify the current security mechanisms to overcome these problems. In this article, we focus on Intrusion Detection (ID) techniques and propose a new attack-defense game model to detect malicious nodes using a repeated game approach. As a direct consequence of the game model, attackers and defenders make different strategies to achieve optimal payoffs. Importantly, error detection and missing detection are taken into consideration in Intrusion Detection Systems (IDSs), where a game tree model is introduced to solve this problem. In addition, we analyze and prove the existence of pure Nash equilibrium and mixed Nash equilibrium. Simulations show that the proposed model can both reduce energy consumption by up to 50% compared with the existing All Monitor (AM) model and improve the detection rate by up to 10% to 15% compared with the existing Cluster Head (CH) monitor model.

Novel threat-based AI strategies that incorporate adaptive data structures for multi-player board games

This paper considers the problem of designing novel techniques for multi-player game playing, in a range of board games and configurations. Compared to the well-known case of two-player game playing, multi-player game playing is a more complex problem with unique requirements. To address the unique challenges of this domain, we examine the potential of employing techniques inspired by Adaptive Data Structures (ADSs) to rank opponents based on their relative threats, and using this information to achieve gains in move ordering and tree pruning. We name our new technique the Threat-ADS heuristic. We examine the Threat-ADS’ performance within a range of game models, employing a number of different, well-understood update mechanisms for ADSs. We then extend our analysis to specifically consider intermediate board states, which are more interesting than the initial board state, as we do not assume the availability of “Opening book” moves, and where substantial variation can exist, in terms of available moves and threatening opponents. We expand this analysis to include an exploration of the Threat-ADS heuristic’s performance in deeper ply game trees, to confirm that it maintains its benefits even when lookahead is greater, and with an expanded examination of how the number of players present in the game impacts the performance of the Threat-ADS heuristic. We find that in nearly all environments, the Threat-ADS heuristic is able to produce meaningful, statistically significant improvements in tree pruning, demonstrating that it serves as a very reliable move ordering heuristic for multi-player game playing under a wide range of configurations, thus motivating the use of ADS-based techniques within the field of game playing.

Deploying Game Theory in Analyzing the Conflict in Afghanistan-Pakistan Bilateral Relations

The objective of this thesis is to develop a game theoretical model for analysis of the conflict in the bilateral relations of Afghanistan and Pakistan. The development of the model is preceded by historical and qualitative analysis of Afghanistan, its internal politics and consequent trajectory of relationship between Afghanistan and Pakistan. The model has been developed keeping in view the existence of a National Unity Government in Afghanistan and the conflict between Afghanistan and Pakistan on the issues of the Durand line and Taliban. The game theoretical model hence developed is structured as a two-level game which has the domestic game in Afghanistan at the level one and an inter-state game at the level two. Both games have different structures with the level one being an extensive game tree whereas for the level two of the game hyper-gaming has been deployed to account for the different perception the players may have about the same conflict. Although this thesis avoids using the formal mathematical notation, knowledge of basic game theory is a prerequisite.

On the core and nucleolus of directed acyclic graph games

We introduce directed acyclic graph (DAG) games, a generalization of standard tree games, to study cost sharing on networks. This structure has not been previously analyzed from a cooperative game theoretic perspective. Every monotonic and subadditive cost game--including monotonic minimum cost spanning tree games--can be modeled as a DAG-game. We provide an efficiently verifiable condition satisfied by a large class of directed acyclic graphs that is sufficient for the balancedness of the associated DAG-game. We introduce a network canonization process and prove various structural results for the core of canonized DAG-games. In particular, we characterize classes of coalitions that have a constant payoff in the core. In addition, we identify a subset of the coalitions that is sufficient to determine the core. This result also guarantees that the nucleolus can be found in polynomial time for a large class of DAG-games.

From reasonable preferences, via argumentation, to logic

This article demonstrates that typical restrictions which are imposed in dialogical logic in order to recover first-order logical consequence from a fragment of natural language argumentation are also forthcoming from preference profiles of boundedly rational players, provided that these players instantiate a specific player type and compute partial strategies. We present two structural rules, which are formulated similarly to closure rules for tableaux proofs that restrict players' strategies to a mapping between games in extensive forms (i.e., game trees) and proof trees. Both rules are motivated from players' preferences and limitations; they can therefore be viewed as being player-self-imposable. First-order logical consequence is thus shown to result from playing a specific type of argumentation game. The alignment of such games with the normative model of the Pragma-dialectical theory of argumentation is positively evaluated. But explicit rules to guarantee that the argumentation game instantiates first-order logical consequence have now become gratuitous, since their normative content arises directly from players' preferences and limitations. A similar naturalization for non-classical logics is discussed.

A Game Theoretic Model for the Optimal Location of Integrated Air Defense System Missile Batteries

We examine the optimal location of Integrated Air Defense System (IADS) missile batteries to protect a country’s assets, formulated as a Defender-Attacker-Defender three-stage sequential, perfect information, zero-sum game between two opponents. We formulate a trilevel nonlinear integer program for this Defender-Attacker-Defender model and seek a subgame perfect Nash equilibrium (i.e., a set of attacker and defender strategies from which neither player has an incentive to deviate). Such a trilevel formulation is not solvable via conventional optimization software, and an exhaustive enumeration of the game tree based on the discrete set of strategies is only tractable for small instances. We develop and test a customized heuristic over a set of small instances having deliberate parametric variations in a designed experiment, comparing its performance to an exhaustive enumeration algorithm. Testing results indicate the enumeration approach to be severely limited for realistically sized instances, so we demonstrate the heuristic on a larger instance from the literature for which it maintains computational efficiency.

LinUCB applied to Monte Carlo tree search

UCT is a standard method of Monte Carlo tree search (MCTS) algorithms, which have been applied to various domains and have achieved remarkable success. This study proposes a family of LinUCT algorithms that incorporate LinUCB into MCTS algorithms. LinUCB is a recently developed method that generalizes past episodes by ridge regression with feature vectors and rewards. LinUCB outperforms UCB1 in contextual multi-armed bandit problems. We introduce a straightforward application of LinUCB, LinUCTPLAIN by substituting UCB1 with LinUCB in UCT. We show that it does not work well owing to the minimax structure of game trees. To better handle such tree structures, we present LinUCTRAVE and LinUCTFP by further incorporating two existing techniques, rapid action value estimation (RAVE) and feature propagation, which recursively propagates the feature vector of a node to that of its parent. Experiments were conducted with a synthetic model, which is an extension of the standard incremental random tree model in which each node has a feature vector that represents the characteristics of the corresponding position, and Finnsson's shock step game which is used to empirically analyze the performance of UCT with respect to the distribution of suboptimal moves. The experiments results indicate that LinUCTRAVE and LinUCTFP outperform UCT, especially when the branching factor is relatively large.

A Branch-and-cut-and-price algorithm for the Stackelberg Minimum Spanning Tree Game

The Stackelberg Minimum Spanning Tree Game (StackMST) is defined in terms of a graph G=(V,B∪R), with two disjoint sets of edges, blue B and red R, and costs {ce≥0:e∈R} defined for the red edges. Once the leader of the game defines prices {pe:e∈B} to the blue edges, the follower chooses a minimum weight spanning tree (V,ET), at cost ∑e∈B∩ETpe+∑e∈R∩ETce. The goal is to find prices to maximize the revenue ∑e∈B∩ETpe collected by the leader. We introduce a reformulation and a Branch-and-cut-and-price algorithm for StackMST. The reformulation is obtained after applying KKT optimality conditions to a StackMST non-compact Bilevel Linear Programming formulation and is strengthened with a partial rank-1 RLT and with valid inequalities from the literature. We also implemented a Branch-and-cut algorithm for an extended formulation derived from another in the literature. A preliminary computational study comparing both methods is also presented.

Comparable characterizations of four solutions for permission tree games

In the field of cooperative games, there is an extensive literature that studies situations of restricted cooperation. In a communication graph game, players can only cooperate if they are connected in an undirected graph representing the communication possibilities. The Myerson value of such a game is obtained by taking the Shapley value of the corresponding restricted game. For the special case that the graph is cycle-free and connected, for each player the corresponding hierarchical outcome yields an alternative solution. In a permission tree game, the player set is enriched with a rooted directed graph (or tree) on the set of players. A coalition is said to be feasible, if for every player in the coalition, except the top (root) player, also its predecessor belong(s) to the coalition. The permission value is obtained by taking the Shapley value of the associated restricted game. In this paper, we modify the Myerson value and hierarchical outcome that are defined for (undirected) communication graph games to a value for permission tree games. We also define a new solution that assigns all payoff to the unique top player in the hierarchy. Then comparable characterizations are given of these three solutions and the known permission value.

Research on collision-preventing decisions of oncoming vehicles

An emergency collision-preventing model is proposed for two vehicles driving from opposite directions. To study the impact of drivers’ strategies on traffic safety in this model, the relevant knowledge of game theory has been introduced. Based on the drivers’ strategies and payoff of every strategy, a game tree and a payoff matrix have been constructed. Mathematical method is applied to identify the pure-strategy Nash equilibrium and the mixed strategies Nash equilibrium. The occurrence probability of accidents under the Nash equilibrium is discussed here. The results suggest that under the pure-strategy Nash equilibrium, one driver must swerve. Furthermore, the probability of keeping original route is greater when drivers adopt the mixed strategies Nash equilibrium. The results presented in this article will provide some reference for drivers under similar conditions.

Strategy-Based Warm Starting for Regret Minimization in Games

Counterfactual Regret Minimization (CFR) is a popular iterative algorithm for approximating Nash equilibria in imperfect-information multi-step two-player zero-sum games. We introduce the first general, principled method for warm starting CFR. Our approach requires only a strategy for each player, and accomplishes the warm start at the cost of a single traversal of the game tree. The method provably warm starts CFR to as many iterations as it would have taken to reach a strategy profile of the same quality as the input strategies, and does not alter the convergence bounds of the algorithms. Unlike prior approaches to warm starting, ours can be applied in all cases. Our method is agnostic to the origins of the input strategies. For example, they can be based on human domain knowledge, the observed strategy of a strong agent, the solution of a coarser abstraction, or the output of some algorithm that converges rapidly at first but slowly as it gets closer to an equilibrium. Experiments demonstrate that one can improve overall convergence in a game by first running CFR on a smaller, coarser abstraction of the game and then using the strategy in the abstract game to warm start CFR in the full game.

Refining Subgames in Large Imperfect Information Games

The leading approach to solving large imperfect information games is to pre-calculate an approximate solution using a simplified abstraction of the full game; that solution is then used to play the original, full-scale game. The abstraction step is necessitated by the size of the game tree. However, as the original game progresses, the remaining portion of the tree (the subgame) becomes smaller. An appealing idea is to use the simplified abstraction to play the early parts of the game and then, once the subgame becomes tractable, to calculate a solution using a finer-grained abstraction in real time, creating a combined final strategy. While this approach is straightforward for perfect information games, it is a much more complex problem for imperfect information games. If the subgame is solved locally, the opponent can alter his play in prior to this subgame to exploit our combined strategy. To prevent this, we introduce the notion of subgame margin, a simple value with appealing properties. If any best response reaches the subgame, the improvement of exploitability of the combined strategy is (at least) proportional to the subgame margin. This motivates subgame refinements resulting in large positive margins. Unfortunately, current techniques either neglect subgame margin (potentially leading to a large negative subgame margin and drastically more exploitable strategies), or guarantee only non-negative subgame margin (possibly producing the original, unrefined strategy, even if much stronger strategies are possible). Our technique remedies this problem by maximizing the subgame margin and is guaranteed to find the optimal solution. We evaluate our technique using one of the top participants of the AAAI-14 Computer Poker Competition, the leading playground for agents in imperfect information setting

Turn-Based War Chess Model and Its Search Algorithm per Turn

War chess gaming has so far received insufficient attention but is a significant component of turn-based strategy games (TBS) and is studied in this paper. First, a common game model is proposed through various existing war chess types. Based on the model, we propose a theory frame involving combinational optimization on the one hand and game tree search on the other hand. We also discuss a key problem, namely, that the number of the branching factors of each turn in the game tree is huge. Then, we propose two algorithms for searching in one turn to solve the problem: (1) enumeration by order; (2) enumeration by recursion. The main difference between these two is the permutation method used: the former uses the dictionary sequence method, while the latter uses the recursive permutation method. Finally, we prove that both of these algorithms are optimal, and we analyze the difference between their efficiencies. An important factor is the total time taken for the unit to expand until it achieves its reachable position. The factor, which is the total number of expansions that each unit makes in its reachable position, is set. The conclusion proposed is in terms of this factor: Enumeration by recursion is better than enumeration by order in all situations.

A Novel Game Theoretical Approach for Formative Knowledge Assessment: Feasibility Analysis

This study explored Extensive Game Theory and Concept Map (CM) for possible integration of both to develop a novel formative knowledge assessment (FKA) approach. A modified extensive game theory (MEGT) has been proposed and defined. Further, to implement this, a Concept Tree Game (CTG) has been proposed and an algorithm for the game has been developed. The outcomes of this study are promising towards facilitating the students for an exciting learning assessment task.

Characterizing existence of equilibrium for large extensive form games: a necessity result

What is the minimal structure that is needed to perform equilibrium analysis in large extensive form games? To answer this question, this paper provides conditions that are simultaneously necessary and sufficient for the existence of a subgame perfect equilibrium in any well-behaved perfect information game defined on a large game tree. In particular, the set of plays needs to be endowed with a topology satisfying two conditions. (a) Nodes are closed as sets of plays; and (b) the immediate predecessor function is an open map.