Constrained Games: The Impact of the Attitude to Adversary's Constraints

We consider properties of constrained games, where the strategy set available to a player depends on the choice of strategies made by other players. We show that the utilities of each player associated with that player's own performance and constraints are not sufficient to model a constrained game and to define equilibria; for the latter, one also needs to model how a player values the fact that other players meet their constraints. We study three different approaches to other players' constraints, and show that they exhibit completely different equilibrium behaviors. Further, we study a general class of stochastic games with partial information, and focus on the case where the players are indifferent to whether the constraints of other players hold.     

MMORPG Player actions: Network performance, session patterns and latency requirements analysis

Providing Massively Multiplayer Online Role-Playing Games (MMORPGs) is a big challenge for future mobile, IP-based networks. Understanding how the players’ actions affect the network parameters, the game platform, and the overall perceived quality is highly relevant for the purposes of game design, as well as for the networking infrastructure and network support for games. We break player actions down into discrete categories, and show that each category is distinct in terms of several key metrics. We discuss which categories of actions could be supported on current mobile devices, and present evidence in form of a user survey demonstrating the demand for such services. The starting points into the discussion include the networking, session and latency requirements for particular player actions on one side, and the players’ interest on the other. The Blizzard Entertainment’s World of Warcraft (WoW) is used as a case study.     

Game-theoretic analysis of an ancient Chinese horse race problem

This paper analyzes a legendary Chinese horse race problem involving the King of Qi and General Tianji which took place more than 2000 years ago. In this problem each player owns three horses of different speed classes and must choose the sequence of horses to compete against each other. Depending on the payoffs received by the players as a result of the horse races, we analyze two groups of constant-sum games. In each group, we consider three separate cases where the outcomes of the races are (i) deterministic, (ii) probabilistic within the same class, and (iii) probabilistic across classes. In the first group, the player who wins the majority of races receives a one-unit payoff. For this group we show analytically that the three different games with non-singular payoff matrices have the same solution where each player has a unique optimal mixed strategy with equal probabilities. For the second group of games where the payoff to a player is the total number of races his horses have won, we use linear programming with non-numeric data to show that the solution of the three games are mixed strategies given as a convex combination of two extreme points. We invoke results from information theory to prove that to maximize the opponent's “entropy” the players should use the equal probability mixed strategy that was found for the one-unit games.     

基于重复博弈参与者有权重的秘密共享方案

在大多数参与者有权重的秘密共享方案中,各参与者子秘密份额数量的不同会导致秘密重构阶段产生不公平问题。为此,提出一个基于重复博弈的理性秘密共享方案。在参与者原有份额的基础上,为其构造数量差不超过1的有效子秘密份额,利用重复博弈使每个参与者可以获得其他参与者的全部份额,进而重构出秘密。分析结果表明,该方案可以使理性参与者始终遵守协议,完成秘密重构,且具有较高的安全性和良好的可扩展性。     

Equilibria problems on games: Complexity versus succinctness

We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others, the problems of deciding the existence of a pure Nash equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete, when the number of players is large and the number of actions for each player is constant, and that the problem is -complete when the number of players is a constant and the size of the action sets is exponential in the size of the game representation. Again when the game is described explicitly the problem is decidable in polynomial time.     

On Dynamics and Nash Equilibriums of Networked Games

Networked noncooperative games are investigated, where each player (or agent) plays with all other players in its neighborhood. Assume the evolution is based on the fact that each player uses its neighbors' current information to decide its next strategy. By using sub-neighborhood, the dynamics of the evolution is obtained. Then a method for calculating Nash equilibriums from mixed strategies of multi-players is proposed. The relationship between local Nash equilibriums based on individual neighborhoods and global Nash equilibriums of overall network is revealed. Then a technique is proposed to construct Nash equilibriums of an evolutionary game from its one step static Nash equilibriums. The basic tool of this approach is the semi-tensor product of matrices, which converts strategies into logical matrices and payoffs into pseudo-Boolean functions, then networked evolutionary games become discrete time dynamic systems.      

The partial migration of game state and dynamic server selection to reduce latency

Massively multi-player online games (MMOGs) have stringent latency requirements and must support large numbers of concurrent players. To handle these conflicting requirements, it is common to divide the virtual environment into virtual regions. As MMOGs are world-spanning games, it is plausible to disperse these regions on geographically distributed servers. Core selection can then be applied to locate an optimal server for placing a region, based on player latencies. Functionality for migrating objects supports this objective, with a distributed name server ensuring that references to the moved objects are maintained. As a result we anticipate a decrease in the aggregate latency for the affected players. The core selection relies on a set of servers and measurements of the interacting players latencies. Measuring these latencies by actively probing the network is not scalable for a large number of players. We therefore explore the use of latency estimation techniques to gather this information.     

Symmetries and the complexity of pure Nash equilibrium

Strategic games may exhibit symmetries in a variety of ways. A characteristic feature, enabling the compact representation of games even when the number of players is unbounded, is that players cannot, or need not, distinguish between the other players. We investigate the computational complexity of pure Nash equilibria in four classes of symmetric games obtained by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. In contrast to other types of succinctly representable multi-player games, the pure equilibrium problem is tractable in all four classes when only a constant number of actions is available to each player. Identical payoff functions make the difference between TC⁰-completeness and membership in AC⁰, while a growing number of actions renders the equilibrium problem NP-hard for three of the classes and PLS-hard for the most restricted class for which the existence of a pure equilibrium is guaranteed. Our results also extend to larger classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.     

Interactive strategy sets in multiple payoff games

In game theory, it is usually assumed that each player has only one payoff function and the strategy set of the game is composed of the topological product of individual players’ strategy sets. In real business and system design or control problems, however, players’ strategy sets may be interactive and each player may have more than one payoff function. This paper, investigates the more general situation of multiple payoff and multiple person games in a normal form. In this paper, each player has several payoff functions which are dominated by certain convex cones, and the feasible strategy set of each player may be interactive with those of the other players. This new model is applied to a classical example without requiring variational and quasi-variational inequalities, or point-to-set mappings.     

Price of Stability in Survivable Network Design

We study the survivable version of the game theoretic network formation model known as the Connection Game, originally introduced in Anshelevich et al. (Proc. 35th ACM Symposium on Theory of Computing, 2003). In this model, players attempt to connect to a common source node in a network by purchasing edges, and sharing their costs with other players. We introduce the survivable version of this game, where each player desires 2 edge-disjoint connections between her pair of nodes instead of just a single connecting path, and analyze the quality of exact and approximate Nash equilibria. This version is significantly different from the original Connection Game and have more complications than the existing literature on arbitrary cost-sharing games since we consider the formation of networks that involve many cycles.     

Collusion in Atomic Splittable Routing Games

We study how collusion affects the social cost in atomic splittable routing games. Suppose that players form coalitions and each coalition behaves as if it were a single player controlling all the flows of its participants. We investigate the following question: under what conditions would the social cost of the post-collusion equilibrium be bounded by the social cost of the pre-collusion equilibrium? We show that if (i) the network is “well-designed” (satisfying a natural condition), and (ii) the delay functions are affine, then collusion is always beneficial for the social cost in the equilibrium flows. On the other hand, if either of the above conditions is unsatisfied, collusion can worsen the social cost. Our main technique is a novel flow-augmenting algorithm to build equilibrium flows. Our positive result for collusion is obtained by applying this algorithm simultaneously to two different flow value profiles of players and observing the difference in the derivatives of their social costs. Moreover, for a non-trivial subclass of selfish routing games, this algorithm finds the exact equilibrium flows in polynomial time.     

What's in a name? Ages and names predict the valence of social interactions in a massive online game

Multi-player online battle arena games (MOBAs) are large virtual environments requiring complex problem-solving and social interaction. We asked whether these games generate psychologically interesting data about the players themselves. Specifically, we asked whether user names, which are chosen by players outside of the game itself, predicted in-game behaviour. To examine this, we analysed a large anonymized dataset from a popular MOBA (‘League of Legends’) – by some measures the most popular game in the world.We find that user names contain two pieces of information that correlate with in-game social behaviour. Both player age (estimated from numerical sequences within name) and the presence of highly anti-social words are correlated with the valences of player/player interactions within the game.Our findings suggest that players' real-world characteristics influence behaviour and interpersonal interactions within online games. Anonymized statistics derived from such games may therefore be a valuable tool for studying psychological traits across global populations.     

Multi-player non-zero-sum games: Online adaptive learning solution of coupled Hamilton–Jacobi equations

In this paper we present an online adaptive control algorithm based on policy iteration reinforcement learning techniques to solve the continuous-time (CT) multi player non-zero-sum (NZS) game with infinite horizon for linear and nonlinear systems. NZS games allow for players to have a cooperative team component and an individual selfish component of strategy. The adaptive algorithm learns online the solution of coupled Riccati equations and coupled Hamilton–Jacobi equations for linear and nonlinear systems respectively. This adaptive control method finds in real-time approximations of the optimal value and the NZS Nash-equilibrium, while also guaranteeing closed-loop stability. The optimal-adaptive algorithm is implemented as a separate actor/critic parametric network approximator structure for every player, and involves simultaneous continuous-time adaptation of the actor/critic networks. A persistence of excitation condition is shown to guarantee convergence of every critic to the actual optimal value function for that player. A detailed mathematical analysis is done for 2-player NZS games. Novel tuning algorithms are given for the actor/critic networks. The convergence to the Nash equilibrium is proven and stability of the system is also guaranteed. This provides optimal adaptive control solutions for both non-zero-sum games and their special case, the zero-sum games. Simulation examples show the effectiveness of the new algorithm.     

Two-terminal routing games with unknown active players

We analyze 2-terminal routing games with linear cost functions and with unknown number of active players. We deal with both splittable and unsplittable models. We prove the existence and uniqueness of a symmetric safety-level equilibrium in such games and show that in many cases every player benefits from the common ignorance about the number of players. Furthermore, we prove new theorems on existence and uniqueness of equilibrium in 2-terminal convex routing games with complete information.     

Influence of head-up displays’ characteristics on user experience in video games

Head-up displays (HUD) are important parts of visual interfaces of virtual environments such as video games. However, few studies have investigated their role in player–video game interactions. Two experiments were designed to investigate the influence of HUDs on player experience according to player expertise and game genre. Experiment 1 used eye-tracking and interviews to understand how and to what extent players use and experience HUDs in two types of commercial games: first-person shooter and real-time strategy games. Results showed that displaying a permanent HUD within the visual interface may improve the understanding of this environment by players. They also revealed that two HUD characteristics, namely composition and spatial organization, have particular influence on player experience. These critical characteristics were manipulated in experiment 2 to study more precisely the influence of HUD design choices on player experience. Results showed that manipulation of design of these HUD characteristics influences player experience in different ways according to player expertise and game genre. For games with HUDs that are perceived as very useful, the higher player expertise is, the more player experience is influenced. Recommendations for video game design based on these results are proposed.     

EcoTRADE – A multi-player network game of a tradable permit market for biodiversity credits

EcoTRADE is a multi-player network game of a virtual biodiversity credit market. Each player controls the land use of a certain amount of parcels on a virtual landscape. The biodiversity credits of a particular parcel depend on neighboring parcels, which may be owned by other players. The game can be used to study the strategies of players in experiments or classroom games and as a communication tool for stakeholders participating in credit markets that include spatially interdependent credits.     

Improving player experience in Computer Games by using players' behavior analysis and linguistic descriptions

Understanding player behavior has an interest to computer games researchers and developers since it allows them to improve the design and implementation of computer games and also to ensure that players have the expected experiences. Currently this knowledge is not usually reported to players as feedback, although sometimes it is already used as an analysis tool. This paper presents a novel technology for automatically generating linguistic reports and immediate feedback from actions performed by players during play sessions. These reports allow developers to provide players with a more complete and personalized feedback about their behaviors, abilities, attitudes, skills or movements. In order to show and explore the possibilities of this new technology, we have incorporated it in the core of a computer game. We have evaluated positively that the incorporation of this kind of feedback into the core of YADY computer game allows us to improve the overall player experience.     

Online calibrated forecasts: Memory efficiency versus universality for learning in games

We provide a simple learning process that enables an agent to forecast a sequence of outcomes. Our forecasting scheme, termed tracking forecast, is based on tracking the past observations while emphasizing recent outcomes. As opposed to other forecasting schemes, we sacrifice universality in favor of a significantly reduced memory requirements. We show that if the sequence of outcomes has certain properties—it has some internal (hidden) state that does not change too rapidly—then the tracking forecast is weakly calibrated so that the forecast appears to be correct most of the time. For binary outcomes, this result holds without any internal state assumptions. We consider learning in a repeated strategic game where each player attempts to compute some forecast of the opponent actions and play a best response to it. We show that if one of the players uses a tracking forecast, while the other player uses a standard learning algorithm (such as exponential regret matching or smooth fictitious play), then the player using the tracking forecast obtains the best response to the actual play of the other players. We further show that if both players use tracking forecast, then under certain conditions on the game matrix, convergence to a Nash equilibrium is possible with positive probability for a larger class of games than the class of games for which smooth fictitious play converges to a Nash equilibrium.     

Iterative ADP learning algorithms for discrete-time multi-player games

Adaptive dynamic programming (ADP) is an important branch of reinforcement learning to solve various optimal control issues. Most practical nonlinear systems are controlled by more than one controller. Each controller is a player, and to make a tradeoff between cooperation and conflict of these players can be viewed as a game. Multi-player games are divided into two main categories: zero-sum game and non-zero-sum game. To obtain the optimal control policy for each player, one needs to solve Hamilton–Jacobi–Isaacs equations for zero-sum games and a set of coupled Hamilton–Jacobi equations for non-zero-sum games. Unfortunately, these equations are generally difficult or even impossible to be solved analytically. To overcome this bottleneck, two ADP methods, including a modified gradient-descent-based online algorithm and a novel iterative offline learning approach, are proposed in this paper. Furthermore, to implement the proposed methods, we employ single-network structure, which obviously reduces computation burden compared with traditional multiple-network architecture. Simulation results demonstrate the effectiveness of our schemes.     

Personalise your massively multiplayer online game (MMOG) with Artemis

Better mobile computing, broadband and devices have contributed to the fast growth and popularity of massively multiplayer online games (MMOGs). Players are now expecting a more personalised gaming experience as personalisation has begun to filter into most games and not just MMOGs. In this paper, we explain how players of MMOGs can enjoy a ubiquitous and personalised gaming experience anywhere, on any device, and on any network with Artemis. MPEG-7 is not traditionally used for modelling games; however, Artemis adopts MPEG-7 for modelling the player, their device and the game, in conjunction with genetic algorithms for personalising the game, where possible.