A short note on the axiomatic requirements of uncertainty measure

In this note, we argue that the axiomatic requirement of range to the measure of aggregated total uncertainty (ATU) in Dempster-Shafer theory is not reasonable.Comment: 4 pages, 1 figur

Transformation of basic probability assignments to probabilities based on a new entropy measure

Dempster-Shafer evidence theory is an efficient mathematical tool to deal with uncertain information. In that theory, basic probability assignment (BPA) is the basic element for the expression and inference of uncertainty. Decision-making based on BPA is still an open issue in Dempster-Shafer evidence theory. In this paper, a novel approach of transforming basic probability assignments to probabilities is proposed based on Deng entropy which is a new measure for the uncertainty of BPA. The principle of the proposed method is to minimize the difference of uncertainties involving in the given BPA and obtained probability distribution. Numerical examples are given to show the proposed approach.Comment: 14 page

Generalized prisoner's dilemma

Prisoner's dilemma has been heavily studied. In classical model, each player chooses to either "Cooperate" or "Defect". In this paper, we generalize the prisoner's dilemma with a new alternative which is neither defect or cooperation. The classical model is the special case under the condition that the third state is not taken into consideration.Comment: 7 pages, 2 figure

Exploring the Combination Rules of D Numbers From a Perspective of Conflict Redistribution

Dempster-Shafer theory of evidence is widely applied to uncertainty modelling and knowledge reasoning because of its advantages in dealing with uncertain information. But some conditions or requirements, such as exclusiveness hypothesis and completeness constraint, limit the development and application of that theory to a large extend. To overcome the shortcomings and enhance its capability of representing the uncertainty, a novel model, called D numbers, has been proposed recently. However, many key issues, for example how to implement the combination of D numbers, remain unsolved. In the paper, we have explored the combination of D Numbers from a perspective of conflict redistribution, and proposed two combination rules being suitable for different situations for the fusion of two D numbers. The proposed combination rules can reduce to the classical Dempster's rule in Dempster-Shafer theory under a certain conditions. Numerical examples and discussion about the proposed rules are also given in the paper.Comment: 6 pages, 4 figure

D numbers theory based game-theoretic framework in adversarial decision making under fuzzy environment

Adversarial decision making is a particular type of decision making problem where the gain a decision maker obtains as a result of his decisions is affected by the actions taken by others. Representation of alternatives' evaluations and methods to find the optimal alternative are two important aspects in the adversarial decision making. The aim of this study is to develop a general framework for solving the adversarial decision making issue under uncertain environment. By combining fuzzy set theory, game theory and D numbers theory (DNT), a DNT based game-theoretic framework for adversarial decision making under fuzzy environment is presented. Within the proposed framework or model, fuzzy set theory is used to model the uncertain evaluations of decision makers to alternatives, the non-exclusiveness among fuzzy evaluations are taken into consideration by using DNT, and the conflict of interests among decision makers is considered in a two-person non-constant sum game theory perspective. An illustrative application is given to demonstrate the effectiveness of the proposed model. This work, on one hand, has developed an effective framework for adversarial decision making under fuzzy environment; One the other hand, it has further improved the basis of DNT as a generalization of Dempster-Shafer theory for uncertainty reasoning.Comment: 59 pages, 5 figure

A quantum extension to inspection game

Quantum game theory is a new interdisciplinary field between game theory and physical research. In this paper, we extend the classical inspection game into a quantum game version by quantizing the strategy space and importing entanglement between players. Our result shows that the quantum inspection game has various Nash equilibrium depending on the initial quantum state of the game. It is also shown that quantization can respectively help each player to increase his own payoff, yet fails to bring Pareto improvement for the collective payoff in the quantum inspection game.Comment: 6 page

Distance function of D numbers

Dempster-Shafer theory is widely applied in uncertainty modelling and knowledge reasoning due to its ability of expressing uncertain information. A distance between two basic probability assignments(BPAs) presents a measure of performance for identification algorithms based on the evidential theory of Dempster-Shafer. However, some conditions lead to limitations in practical application for Dempster-Shafer theory, such as exclusiveness hypothesis and completeness constraint. To overcome these shortcomings, a novel theory called D numbers theory is proposed. A distance function of D numbers is proposed to measure the distance between two D numbers. The distance function of D numbers is an generalization of distance between two BPAs, which inherits the advantage of Dempster-Shafer theory and strengthens the capability of uncertainty modeling. An illustrative case is provided to demonstrate the effectiveness of the proposed function.Comment: 29 pages, 7 figure

Quantum games of opinion formation based on the Marinatto-Weber quantum game scheme

Quantization becomes a new way to study classical game theory since quantum strategies and quantum games have been proposed. In previous studies, many typical game models, such as prisoner's dilemma, battle of the sexes, Hawk-Dove game, have been investigated by using quantization approaches. In this paper, several game models of opinion formations have been quantized based on the Marinatto-Weber quantum game scheme, a frequently used scheme to convert classical games to quantum versions. Our results show that the quantization can change fascinatingly the properties of some classical opinion formation game models so as to generate win-win outcomes.Comment: 19 page

Basic concepts, definitions, and methods in D number theory

As a generalization of Dempster-Shafer theory, D number theory (DNT) aims to provide a framework to deal with uncertain information with non-exclusiveness and incompleteness. Although there are some advances on DNT in previous studies, however, they lack of systematicness, and many important issues have not yet been solved. In this paper, several crucial aspects in constructing a perfect and systematic framework of DNT are considered. At first the non-exclusiveness in DNT is formally defined and discussed. Secondly, a method to combine multiple D numbers is proposed by extending previous exclusive conflict redistribution (ECR) rule. Thirdly, a new pair of belief and plausibility measures for D numbers are defined and many desirable properties are satisfied by the proposed measures. Fourthly, the combination of information-incomplete D numbers is studied specially to show how to deal with the incompleteness of information in DNT. In this paper, we mainly give relative math definitions, properties, and theorems, concrete examples and applications will be considered in the future study.Comment: 28 pages, 2 figure

Belief and plausibility measures for D numbers

As a generalization of Dempster-Shafer theory, D number theory provides a framework to deal with uncertain information with non-exclusiveness and incompleteness. However, some basic concepts in D number theory are not well defined. In this note, the belief and plausibility measures for D numbers have been proposed, and basic properties of these measures have been revealed as well.Comment: 9 page