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.     

Matrix games with missing,interval, and ambiguous lottery payoffs of pure strategy profiles and compound strategy profiles

In a matrix game, the interactions among players are based on the assumption that each player has accurate information about the payoffs of their interactions and the other players are rationally self‐interested. As a result, the players should definitely take Nash equilibrium strategies. However, in real‐life, when choosing their optimal strategies, sometimes the players have to face missing, imprecise (i.e., interval), ambiguous lottery payoffs of pure strategy profiles and even compound strategy profile, which means that it is hard to determine a Nash equilibrium. To address this issue, in this paper we introduce a new solution concept, called ambiguous Nash equilibrium, which extends the concept of Nash equilibrium to the one that can handle these types of ambiguous payoff. Moreover, we will reveal some properties of matrix games of this kind. In particular, we show that a Nash equilibrium is a special case of ambiguous Nash equilibrium if the players have accurate information of each player's payoff sets. Finally, we give an example to illustrate how our approach deals with real‐life game theory problems.     

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.     

Evolutionarily stable sets in quantum penny flip games

In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also a strict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players’ mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players’ mixed classical strategies were invaded by quantum strategies, a new quantum ES set was emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.     

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.     

Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg’s quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.     

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     

Coalition-hierarchical game under uncertainty conditions

An optimization problem in a coalition-hierarchical game under uncertainty conditions is formulated. In the game, information assumptions are that the player of the high hierarchical level (controlling Center) and each low-level coalition estimates uncertainty in its own way. The Center constructs its strategy from the maximum condition for its own payoff function and its minimum in uncertainty. The relationships between coalitions are built upon the guaranteeing absolute active equilibrium understood in the sense of providing the players with guaranteed payoff under the actual uncertainty. The guaranteed uncertainty is obtained with the help of Slater principle. The total equilibrium in the game is called CH-equilibrium. For a quadratic game version, sufficient optimality conditions are obtained. A numerical procedure for solving the game is described and an example is given.     

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.     

On engineering game theory with its application in power systems

Due to its capability of solving decision-making problems involving multiple entities and objectives, as well as complex action sequences, game theory has been a basic mathematical tool of economists, politicians, and sociologists for decades. It helps them understand how strategic interactions impact rational decisions of individual players in competitive and uncertain environment, if each player aims to get the best payoff. This situation is ubiquitous in engineering practices. This paper streamlines the foundations of engineering game theory, which uses concepts, theories and methodologies to guide the resolution of engineering design, operation, and control problems in a more canonical and systematic way. An overview of its application in smart grid technologies and power systems related topics is presented, and intriguing research directions are also envisioned.     

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.     

A game theoretical perspective on the quantum probabilities associated with a GHZ state

In the standard approach to quantum games, players’ strategic moves are local unitary transformations on an entangled state that is subsequently measured. Players’ payoffs are then obtained as expected values of the entries in the payoff matrix of the classical game on a set of quantum probabilities obtained from the quantum measurement. In this paper, we approach quantum games from a diametrically opposite perspective. We consider a classical three-player symmetric game along with a known expression for a set of quantum probabilities relevant to a tripartite Einstein–Podolsky–Rosen (EPR) experiment that depends on three players’ directional choices in the experiment. We define the players’ strategic moves as their directional choices in an EPR setting and then express their payoff relations in the resulting quantum game in terms of their directional choices, the entries of the payoff matrix, and the quantum probability distribution relevant to the tripartite EPR experiment.     

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.      

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.     

Fuzzy signaling game of deception between ant-inspired deceptive robots with interactive learning

In this study the robotic deception phenomenon is raised in the framework of a signaling game which utilizes fuzzy logic and game theory along with inspirations from nature. Accomplishing the fuzzy signaling strategy set for deceptive players serves as a great part of our contribution and on this aim, hierarchical fuzzy inference systems support receiver’s actions and sender’s ant-inspired deceptive signals (track and pheromone). In addition, special deceptive robots and visually-supported experimental environment are also provided. The fuzzy behavior of robots defines the strategy type of players. The final result of deception process depends on this strategy type which leads to proposing a payoff matrix in which each cell of mutual costs is defined with special supporting logic related to our deception game with pursuit–evasion applications. Furthermore, motivated by animal signaling, through applying mixed strategies on deceiver’s honesty level and rival’s trust level, the corresponding learning dynamics are investigated and the conceptual discussion put forward serves as a proof to the smart human-like behavior that occurs between the robots: the interactive learning. Simulation results show that robots are capable of interactive learning within deceptive interaction and finally change their strategies to adopt themselves to new situation occurred due to opponent’s strategy change. Because of repetitive change in strategies as a result of learning, the conditions of a persistent deception without breakdown holds for this game where deceiver can frequently benefit from deception without leaving rival to lose its trust totally. The change in strategy will happen after a short time needed to learn the new situation. In rival’s learning process, this short time, which we call the ignorance time, exactly is the period that deceiver can benefit from deception while its evil intends are still concealed. Moreover, in this study an algorithm is given for the proposed signaling game of deception and an illustrative experiment in the introduced experimental environment demonstrates the process of a successful deception. The paper also gives solution to the proposed game by analyzing mixed Nash equilibrium which turns out to be the interior center fixed point of the learning dynamics.     

Multiple capture in Pontryagin’s almost periodic example

This paper considers Pontryagin’s generalized nonstationary example with several participants under the same dynamic and inertial capabilities of the players, in which the set of admissible control actions is a convex compact set and the terminal sets are convex compact sets. We obtain sufficient conditions for the multiple capture of one evader by a group of pursuers under the assumption that some functions associated with the initial data and game parameters are almost periodic. Each pursuer cannot make a capture more than once before being eliminated from the game. Such a situation may happen when the evader must be “terminated” but contact between the pursuer and the evader does not guarantee termination.     

Toward a quizmaster robot for speech-based multiparty interaction

This paper presents an interactive quizmaster robot that can manage a multiparty speech-based quiz game. The basic flow of the quiz game is that (1) the robot reads a question, (2) one or more players answer it, and (3) the robot judges the correctness of the answers. We categorize such speech-based quiz games into school-type interaction and auction-type interaction. The former asks players to say ‘Yes’ to get the right to answer before answering a question and the latter allows players to directly answer a question without any advance notice. To realize such interaction, the robot needs the capability of recognizing utterances from multiple people using its own microphones (i.e. ears), even if those utterances are made simultaneously. To cope with such situations, the robot estimates which player made the fastest utterance and recognizes it by localizing and separating a mixture of audio signals. Experiments were conducted to evaluate the success rates of the fastest player identification and speech recognition. The results showed that the robot could identify the fastest speakers with a success rate of 90.0% more accurately than humans when only one speaker slightly preceded the other speakers. We found that although the success rate of speech recognition for the fastest speakers did not reach that of humans, the robot attains amusing quiz game interaction.     

The Impact of Payoff Function and Local Interaction on the N-Player Iterated Prisoner's Dilemma

The N-player iterated prisoner's dilemma (NIPD) game has been widely used to study the evolution of cooperation in social, economic and biological systems. This paper studies the impact of different payoff functions and local interactions on the NIPD game. The evolutionary approach is used to evolve game-playing strategies starting from a population of random strategies. The different payoff functions used in our study describe different behaviors of cooperation and defection among a group of players. Local interaction introduces neighborhoods into the NIPD game. A player does not play against every other player in a group any more. He only interacts with his neighbors. We investigate the impact of neighborhood size on the evolution of cooperation in the NIPD game and the generalization ability of evolved strategies. Received 18 August 1999 / Revised 27 February 2000 / Accepted 15 May 2000     

Controlling the Bit Rate of Multi-Object Videos With Noncooperative Game Theory

This paper proposes an object-level rate control algorithm to jointly controlling the bit rates of multiple video objects. Utilizing noncooperative game theory, the proposed rate control algorithm mimics the behaviors of players representing video objects. Each player competes for available bits to optimize its visual quality. The algorithm finds an “optimal solution” in that it conforms to the mixed strategy Nash equilibrium, which is the probability distribution of the actions carried by the players that maximizes their expected payoffs (the number of bits). The game is played iteratively, and the expected payoff of each play is accumulated. The game terminates when all of the available bits for the specific time instant have been distributed to video object planes (VOPs). The advantage of the proposed scheme is that the bidding objects divide the bits among themselves automatically and fairly, according to their encoding complexity, and with an overall solution that is strategically optimal under the given circumstances. To minimize buffer fluctuation and avoid buffer overflow and underflow, a proportional-integral-derivative (PID) control based buffer policy is utilized.      

Game-theoretic Evolutionary Algorithm Based on Behavioral Expectation and its Performance Analysis

Game theory has been widely recognized as an important tool in many fields, which provides general mathematical techniques for analyzing situations in which two or more individuals make decisions that will influence one another’s welfare. This paper presents a game-theoretic evolutionary algorithm based on behavioral expectation, which is a type of optimization approach based on game theory. A formulation to estimate the payoffs expectation is given, which is a mechanism of trying to master the player’s information so as to facilitate the player becoming the rational decision maker. GameEA has one population (players set), and generates new offspring only by the imitation operator and the belief learning operator. The imitation operator is used to learn strategies and actions from other players to improve its competitiveness and applies it into the future game, namely that one player updates its chromosome by strategically copying some segments of gene sequences from the competitor. Belief learning refers to models in which a player adjusts its own strategies, behavior or chromosome by analyzing current history information with respect to an improvement of solution quality. The experimental results on various classes of problems using real-valued representation show that GameEA outperforms not only the standard genetic algorithm (GA) but also other GAs having additional mechanisms of accuracy enhancement. Finally, we compare the convergence of GameEA with different numbers of players to determine whether this parameter has a significant effect on convergence. The statistical results show that at the 0.05 significance level, the number of players has a crucial impact on GameEA's performance. The results suggest that 50 or 100 players will provide good results with unimodal functions, while 200 players will provide good results for multimodal functions.