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.     

Multi-objective design method based on evolution game and its application for suspension

Through research and bionics of biology survival mode, game players with competition, cooperation and self-adaptation capacity are introduced in the multi-objective design. The dynamic behavior and bounded rationality in game processes for players are considered according to Chinese saying “In success, commit oneself to the welfare of the society; in distress, maintain one’s own integrity”. An evolution rule, Poor-Competition-Rich-Cooperation (short for PCRC), is proposed. Then, the corresponding payoff functions of competition and cooperation behavior are established and a multi-objective design method based on evolution game is proposed. The calculation steps are as follows: 1) Taking the design objectives as different game players, and calculating factors of the design variables to objective and fuzzy clustering. The design variables are divided into multiple strategy subsets owned by each game player. 2) According to the evolution rule, each player determines its behavior and payoff function in this game round. 3) In their own strategy subsets, each game player takes their payoff as mono-objective for optimization. It gives the best strategy upon other players. And so the best strategies of all players conform the group strategy in this round. The final equilibrium solution is obtained through multi-round game based on convergence criterion. The validity and reliability of this method are shown by the results of an example of a tri-objective optimization design of passive suspension parameters.     

Self-tuning leader-follower games

The optimization of a class of linear discrete-time stochastic systems with unknown parameters and multiple decision-makers or controllers each having his own single-stage quadratic objective is considered. Based on a centralized information pattern, a steady-state solution is obtained for the stochastic adaptive leader-follower game problem. It is shown that this class of leader-follower game problems can be transformed to an equivalent class of multiple-input-multiple-output self-tuning control problems with a single decision-maker. The convergence of the multiple-input-multiple-output self-tuning controller is established and applied to the leader-follower game. A numerical simulation illustrates the proposed self-tuning algorithm for an adaptive leader-follower strategy.     

多人非合作随机自适应博弈

本文考虑系数未知的离散时间线性随机系统多人非合作的自适应博弈问题,每个参与者运用最小二乘算法和"必然等价原则"来设计博弈策略组合,目的是自适应优化自身的一步超前收益函数.本文证明此自适应策略组合使得闭环系统全局稳定,并且在一定意义下是该博弈问题的渐近纳什均衡解.     

Towards solving 2-TBSG efficiently

Two-player turn-based stochastic game (2-TBSG) is a two-player game model which aims to find Nash equilibriums and is widely utilized in reinforcement learning and AI. Inspired by the fact that the simplex method for solving the deterministic discounted Markov decision processes is strongly polynomial independent of the discount factor, we are trying to answer an open problem whether there is a similar algorithm for 2-TBSG. We develop a simplex strategy iteration where one player updates its strategy with a simplex step while the other player finds an optimal counterstrategy in turn, and a modified simplex strategy iteration. Both of them belong to a class of geometrically converging algorithms. We establish the strongly polynomial property of these algorithms by considering a strategy combined from the current strategy and the equilibrium strategy. Moreover, we present a method to transform general 2-TBSGs into special 2-TBSGs where each state has exactly two actions.     

Perfect aggregation of information in two-person multistage games with fixed sequence of moves and aggregated information on partner’s choice

Two-person multistage game with fixed sequence of moves is considered, under perfect information on existing history of the game and aggregated information on the current move of player 2. Having this information at each stage i, player 1 is the first to choose his move x(in1)(.); moreover, in the beginning of the game player 1 announces his strategy x(.)=(a(in1)(.),..., x(in)(.)) for n future stages. Given information regarding the choice of player 1 and history of the game, player 2 strives to maximize his payoff function via the strategy v=(v(in1), v(in2),..., v(inn)). In this paper the sufficient conditions of perfect aggregation, involving certain results from the theory of Lie groups, are provided for the game in question.     

Altruistic behavior in a nonantagonistic positional differential game

We consider a two-person nonantagonistic positional differential game (NPDG) whose dynamics is described by an ordinary nonlinear vector differential equation. Constraints on values of players’ controls are geometric. Final time of the game is fixed. Payoff functionals of both players are terminal. The formalization of positional strategies in an NPDG is based on the formalization and results of the general theory of antagonistic positional differential games (APDGs) (see monographs by N.N. Krasovskii and A.I. Subbotin [3, 4]). Additionally, in the present paper we assume that each player, together with the usual, normal (nor), type of behavior aimed at maximizing his own functional, can use other behavior types introduced in [2, 5]. In particular, these may be altruistic (alt), aggressive (agg), and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. Using the possibility of such switches in a repeated bimatrix 2 × 2 game in [5, 6] allowed to obtain new solutions of this game. In the present paper, extension of this approach to NPDGs leads to a new formulation of the problem. In particular, of interest is the question of how players’ outcomes at Nash solutions are transformed. An urgent problem is minimizing the time of “abnormal” behavior while achieving a good result. The paper proposes a formalization of an NPDG with behavior types (NPDGwBT). It is assumed that in an NPDGwBT each player, simultaneously with choosing a positional strategy, chooses also his own indicator function defined on the whole game horizon and taking values in the set {normal, altruistic, aggressive, paradoxical}. The indicator function of a player shows the dynamics of changes in the behavior type demonstrated by the player. Thus, in this NPDGwBT each player controls the choice of a pair {positional strategy, indicator function}. We define the notion of a BT-solution of such a game. It is expected that using behavior types in the NPDGwBT which differ from the normal one (so-called abnormal types) in some cases may lead to more favorable outcomes for the players than in the NPDG. We consider two examples of an NPDGwBT with simple dynamics in the plane in each of which one player keeps to altruistic behavior type over some time period. It is shown that in the first example payoffs of both players increase on a BT-solution as compared to the game with the normal behavior type, and in the second example, the sum of players’ payoffs is increased.     

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.     

Automatic fuzzy decision making system with learning for competing and connected businesses

We study uncertainties surrounding competition on business networks and board games. We investigate these uncertainties using concepts of fuzzy logic and game theory. We investigate how the payoff of the players is affected by a number of factors. These include the level of connectivity or number of links, the number of competitors, possible constraints on the networks and on the boards, as well as choice of strategy adopted by competitors. We introduce one fuzzy player in the game. This player uses fuzzy rules to make strategic decisions. We introduce learning to train and analyze how the fuzzy player adapts over time during the game.     

A fuzzy approach to strategic games

A game is a decision-making situation with many players, each having objectives that conflict with each other. The players involved in the game usually make their decisions under conditions of risk or uncertainty. In the paper, a fuzzy approach is proposed to solve the strategic game problem in which the pure strategy set for each player is already defined. Based on the concepts of fuzzy set theory, the approach uses a multicriteria decision-making method to obtain the optimal strategy in the game, a method which shows more advantages than the classical game methods. Moreover, with this approach, some useful conclusions are reached concerning the famous “prisoner's dilemma” problem in game theory     

Games Played under Fuzzy Constraints

Psychological experiment studies reveal that human interaction behaviors are often not the same as what game theory predicts. One of important reasons is that they did not put relevant constraints into consideration when the players choose their best strategies. However, in real life, games are often played in certain contexts where players are constrained by their capabilities, law, culture, custom, and so on. For example, if someone wants to drive a car, he/she has to have a driving license. Therefore, when a human player of a game chooses a strategy, he/she should consider not only the material payoff or monetary reward from taking his/her best strategy and others' best responses but also how feasible to take the strategy in that context where the game is played. To solve such a game, this paper establishes a model of fuzzily constrained games and introduces a solution concept of constrained equilibrium for the games of this kind. Our model is consistent with psychological experiment results of ultimatum games. We also discuss what will happen if Prisoner's Dilemma and Stag Hunt are played under fuzzy constraints. In general, after putting constraints into account, our model can reflect well the human behaviors of fairness, altruism, self‐interest, and so on, and thus can predict the outcomes of some games more accurate than conventional game theory.     

Necessary and sufficient conditions for optimality in constrained general sum stochastic games

In this paper we first derive a necessary and sufficient condition for a stationary strategy to be the Nash equilibrium of discounted constrained stochastic game under certain assumptions. In this process we also develop a nonlinear (non-convex) optimization problem for a discounted constrained stochastic game. We use the linear best response functions of every player and complementary slackness theorem for linear programs to derive both the optimization problem and the equivalent condition. We then extend this result to average reward constrained stochastic games. Finally, we present a heuristic algorithm motivated by our necessary and sufficient conditions for a discounted cost constrained stochastic game. We numerically observe the convergence of this algorithm to Nash equilibrium.     

Bayesian Coalition Game for Contention-Aware Reliable Data Forwarding in Vehicular Mobile Cloud

The exponential growth in the demands of users to access various resources during mobility has led to the popularity of Vehicular Mobile Cloud. Vehicular users may access various resources on road from the cloud which acts as a service provider for them. Most of the existing proposals on vehicular cloud use unicast sender-based data forwarding, which results in an overall performance degradation with respect to the metrics such as packet delivery ratio, end-to-end delay, and reliable data transmission. Most of the applications for vehicular cloud have tight upper bounds with respect to reliable transmission. In view of the above, in this paper, we formulate the problem of reliable data forwarding as a Bayesian Coalition Game (BCG) using Learning Automata concepts. Learning Automata (LA) are assumed as the players in the game stationed on the vehicles. For taking adaptive decisions about reliable data forwarding, each player observes the moves of the other players in the game. For this purpose, a coalition game is formulated among the players of the game for taking adaptive decisions. For each action taken by a player in the game, it gets a reward or a penalty from the environment, and accordingly, it updates its action probability vector. An adaptive Learning Automata based Contention Aware Data Forwarding (LACADF) is also proposed. The proposed scheme is evaluated in different network scenarios with respect to parameters such as message overhead, throughput, and delay by varying the density and mobility of the vehicles. The results obtained show that the proposed scheme is better than the other conventional schemes with respect to the above metrics.     

A Nash equilibrium solution in an oligopoly market: The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

The present analysis applies continuous time replicator dynamics to the analysis of oligopoly markets. In the present paper, we discuss continuous game problems in which decision-making variables for each player are bounded on a simplex by equalities and non-negative constraints. Several types of problems are considered under conditions of normalized constraints and non-negative constraints. These problems can be classified into two types based on their constraints. For one type, the simplex constraint applies to the variables for each player independently, such as in a product allocation problem. For the other type, the simplex constraint applies to interference among all players, creating a market share problem. In the present paper, we consider a game problem under the constraints of allocation of product and market share simultaneously. We assume that a Nash equilibrium solution can be applied and derive the gradient system dynamics that attain the Nash equilibrium solution without violating the simplex constraints. Models assume that three or more firms exist in a market. Firms behave to maximize their profits, as defined by the difference between their sales and cost functions with conjectural variations. The effectiveness of the derived dynamics is demonstrated using simple data. The present approach facilitates understanding the process of attaining equilibrium in an oligopoly market.     

Concurrent reachability games

We consider concurrent two-player games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zero-sum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type-1 states, player 1 has a deterministic strategy to always reach the target. From type-2 states, player 1 has a randomized strategy to reach the target with probability 1. From type-3 states, player 1 has for every real ε>0$\epsilon >0$ a randomized strategy to reach the target with probability greater than 1−ε$1-\epsilon$. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, and type-2 and type-3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.     

Evolution of iterated prisoner's dilemma game strategies in structured demes under random pairing in game playing

We discuss the evolution of strategies in a spatial iterated prisoner's dilemma (IPD) game in which each player is located in a cell of a two-dimensional grid-world. Following the concept of structured demes, two neighborhood structures are used. One is for the interaction among players through the IPD game. A player in each cell plays against its neighbors defined by this neighborhood structure. The other is for mating strategies by genetic operations. A new strategy for a player is generated by genetic operations from a pair of parent strings, which are selected from its neighboring cells defined by the second neighborhood structure. After examining the effect of the two neighborhood structures on the evolution of cooperative behavior with standard pairing in game-playing, we introduce a random pairing scheme in which each player plays against a different randomly chosen neighbor at every round (i.e., every iteration) of the game. Through computer simulations, we demonstrate that small neighborhood structures facilitate the evolution of cooperative behavior under random pairing in game-playing.     

Road Pricing Design Based on Game Theory and Multi-agent Consensus

Consensus theory and noncooperative game theory respectively deal with cooperative and noncooperative interactions among multiple players/agents. They provide a natural framework for road pricing design, since each motorist may myopically optimize his or her own utility as a function of road price and collectively communicate with his or her friends and neighbors on traffic situation at the same time. This paper considers the road pricing design by using game theory and consensus theory. For the case where a system supervisor broadcasts information on the overall system to each agent, we present a variant of standard fictitious play called average strategy fictitious play (ASFP) for large-scale repeated congestion games. Only a weighted running average of all other players' actions is assumed to be available to each player. The ASFP reduces the burden of both information gathering and information processing for each player. Compared to the joint strategy fictitious play (JSFP) studied in the literature, the updating process of utility functions for each player is avoided. We prove that there exists at least one pure strategy Nash equilibrium for the congestion game under investigation, and the players' actions generated by the ASFP with inertia (players' reluctance to change their previous actions) converge to a Nash equilibrium almost surely. For the case without broadcasting, a consensus protocol is introduced for individual agents to estimate the percentage of players choosing each resource, and the convergence property of players' action profile is still ensured. The results are applied to road pricing design to achieve socially local optimal trip timing. Simulation results are provided based on the real traffic data for the Singapore case study.      

Multistage stochastic linear differential game with decision-time constraint

In this work, we consider a class of multistage stochastic linear differential game with quadratic cost function subject to a post initial measurement decision constraint, Secure strategies, in the sense of minimax and maximin, are derived for each player by applying dynamic programming. Both of the minimax and maximin controls are shown to be of linear feedback form. It is demonstrated that both the certainty equivalence principle and the separation theorem from stochastic control theory hold. The derived algorithm, even for the multistage and stochastic nature of the problem, is of closed form and can be employed in real-time applications     

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.     

Mixed scheduling with heterogeneous delay constraints in cyber-physical systems

The physical space and the cyber space are deeply coupled in Cyber-Physical Systems (CPS). The traffic flows are constrained by heterogeneous delay constraints. In order to provide real-time and predictable communication, the paper combines the distributed scheduling algorithm with game theory. A non-cooperative game is proposed to form the scheduling set in the contention-based multiple-access scenario. In the game, each player only has its delay knowledge and makes decision without the information of other competing players. The payoff function is designed to encourage players to give the transmission chance to the player with urgent packets. Simulation results demonstrate that the game-theoretic scheduling approach can improve the real-time performance compared with the existing scheduling algorithms under different scenarios.