Aggregation of monotone reciprocal relations with application to group decision making

This paper is composed of two complementary parts. The first part is a formal investigation into the interplay of properties of reciprocal relations, how monotonicity relates to some natural and intuitive properties, including stochastic transitivity. The goal is to aggregate monotone reciprocal relations on a given set of alternatives. Monotonicity is expressed w.r.t. a linear order on the set of alternatives. The second part is a practical protocol to both determine the best fitting linear order underlying the alternatives, and construct a reciprocal relation monotone w.r.t. it. We formulate the problem as an optimization problem, where the aggregated linear order is that for which the implied stochastic monotonicity conditions are closest to being satisfied by the distribution of the input monotone reciprocal relations. We show that if stochastic monotonicity conditions are satisfied, a monotone reciprocal relation is easily found on the basis of the (possibly constructed) stochastically monotone reciprocal distributional relation.     

Priority theory and utility theory

The first axiom of the Analytic Hierarchy Process (AHP) states that the pairwise comparison judgements are reciprocal. Here we explore the relationships between the reciprocal property and preference relations with and without the axiom of transitivity and point out some important distinctions between utility theory and the AHP.     

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems. The authors like to thank the comments of the anonymous referees for their remarks, which greatly improved the presentation of this paper.     

A simple randomised algorithm for convex optimisation

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.     

Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications

Kernel methods and rough sets are two general pursuits in the domain of machine learning and intelligent systems. Kernel methods map data into a higher dimensional feature space, where the resulting structure of the classification task is linearly separable; while rough sets granulate the universe with the use of relations and employ the induced knowledge granules to approximate arbitrary concepts existing in the problem at hand. Although it seems there is no connection between these two methodologies, both kernel methods and rough sets explicitly or implicitly dwell on relation matrices to represent the structure of sample information. Based on this observation, we combine these methodologies by incorporating Gaussian kernel with fuzzy rough sets and propose a Gaussian kernel approximation based fuzzy rough set model. Fuzzy T-equivalence relations constitute the fundamentals of most fuzzy rough set models. It is proven that fuzzy relations with Gaussian kernel are reflexive, symmetric and transitive. Gaussian kernels are introduced to acquire fuzzy relations between samples described by fuzzy or numeric attributes in order to carry out fuzzy rough data analysis. Moreover, we discuss information entropy to evaluate the kernel matrix and calculate the uncertainty of the approximation. Several functions are constructed for evaluating the significance of features based on kernel approximation and fuzzy entropy. Algorithms for feature ranking and reduction based on the proposed functions are designed. Results of experimental analysis are included to quantify the effectiveness of the proposed methods.     

一致性互反Fuzzy判断矩阵的传递性

研究矩阵元素为Fuzzy数的互反判断矩阵的传递性质.首先得到了判断两个Fuzzy数近似相等的等价条件,并得到了揭示Fuzzy数的核之间关系的一个充要条件.在此基础上,进一步证明了一致性互反Fuzzy判断矩阵具有传递性的两个结论.这两个结论说明:在层次分析法中,用一致性互反Fuzzy判断矩阵表示一组方案在同一目标下的两两重要性比较是符合理性决策的思维特征的.     

Consistency analysis of triangular fuzzy reciprocal preference relations

In order to simulate the uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. Based on the reciprocity property, this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations. The new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by Buckley (1985). The properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail. In addition, the shortcomings of the proof procedure of the proposition given by Wang and Chen (2008) are pointed out. And the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations. Finally, using the (n − 1) restricted comparison ratios, a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed, and an algorithm is shown to make a consistent decision ranking. Numerical results are further calculated to illustrate the new definition and the obtained algorithm.     

Fictitious play with incomplete learning

In this paper, we consider a case that a game is played repeatedly in an incomplete learning process where each player updates his belief only in the learning periods rather than all the stages. For fictitious play process with incomplete learning, we discuss the absorbability of Nash equilibriums and the consistency of utilities in a finite game and discuss the convergence in a 2×2 game with an identical learning-period set. The main results for incomplete learning models are that, if it is uniformly played, a strict Nash equilibrium is absorbing in a fictitious play process; a fictitious play has the property of utility consistency if it exhibits infrequent switches and players learn frequently enough; a 2×2 game with an identical learning-period set has fictitious play property that any fictitious process for the game converges to equilibrium provided that players learn frequently enough.     

Learning Rates of Least-Square Regularized Regression

This paper considers the regularized learning algorithm associated with the least-square loss and reproducing kernel Hilbert spaces. The target is the error analysis for the regression problem in learning theory. A novel regularization approach is presented, which yields satisfactory learning rates. The rates depend on the approximation property and on the capacity of the reproducing kernel Hilbert space measured by covering numbers. When the kernel is C^∞ and the regression function lies in the corresponding reproducing kernel Hilbert space, the rate is m^ζ with ζ arbitrarily close to 1, regardless of the variance of the bounded probability distribution.     

Evolutionary multiobjective optimization in noisy problem environments

This paper presents a multiobjective evolutionary algorithm (MOEA) capable of handling stochastic objective functions. We extend a previously developed approach to solve multiple objective optimization problems in deterministic environments by incorporating a stochastic nondomination-based solution ranking procedure. In this study, concepts of stochastic dominance and significant dominance are introduced in order to better discriminate among competing solutions. The MOEA is applied to a number of published test problems to assess its robustness and to evaluate its performance relative to NSGA-II. Moreover, a new stopping criterion is proposed, which is based on the convergence velocity of any MOEA to the true Pareto optimal front, even if the exact location of the true front is unknown. This stopping criterion is especially useful in real-world problems, where finding an appropriate point to terminate the search is crucial.     

About the non-convex optimization problem induced by non-positive semidefinite kernel learning

During the last years, kernel based methods proved to be very successful for many real-world learning problems. One of the main reasons for this success is the efficiency on large data sets which is a result of the fact that kernel methods like support vector machines (SVM) are based on a convex optimization problem. Solving a new learning problem can now often be reduced to the choice of an appropriate kernel function and kernel parameters. However, it can be shown that even the most powerful kernel methods can still fail on quite simple data sets in cases where the inherent feature space induced by the used kernel function is not sufficient. In these cases, an explicit feature space transformation or detection of latent variables proved to be more successful. Since such an explicit feature construction is often not feasible for large data sets, the ultimate goal for efficient kernel learning would be the adaptive creation of new and appropriate kernel functions. It can, however, not be guaranteed that such a kernel function still leads to a convex optimization problem for Support Vector Machines. Therefore, we have to enhance the optimization core of the learning method itself before we can use it with arbitrary, i.e., non-positive semidefinite, kernel functions. This article motivates the usage of appropriate feature spaces and discusses the possible consequences leading to non-convex optimization problems. We will show that these new non-convex optimization SVM are at least as accurate as their quadratic programming counterparts on eight real-world benchmark data sets in terms of the generalization performance. They always outperform traditional approaches in terms of the original optimization problem. Additionally, the proposed algorithm is more generic than existing traditional solutions since it will also work for non-positive semidefinite or indefinite kernel functions.     

The principal-agent problem with evolutionary learning

We extend agency theory to incorporate bounded rationality of both principals and agents. In this study we define a simple version of the principal-agent game and examine it using object-oriented computer simulation. Player learning is simulated with a genetic algorithm model. Our results show that players of incentive games in highly uncertain environments may take on defensive strategies. These defensive strategies lead to equilibria which are inferior to Nash equilibria. If agents are risk averse, the principal may not be able to provide enough monetary compensation to encourage them to take risks. But principals may be able to improve system performance by identifying good performers and facilitating information exchange among agents.The authors would like to thank the anonymous referees for their helpful suggestions.     

Stochastic rearrangement inequalities

We develop a unified theory for obtaining stochastic rearrangement inequalities and show how this theory may be applied in statistical contexts such as ranking problems, hypothesis testing, contamination models, and optimal assembly of systems.     

Methods for comparison of coalition influence on games in characteristic function form and their interrelationships

This paper extends two existent methods, called the blockability relation and the viability relation, for simple games to compare influence of coalitions, to those for games in characteristic function form, and shows that the newly defined relations satisfy transitivity and completeness. It is shown in this paper that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship.     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

Optimal learning for sequential sampling with non-parametric beliefs

We propose a sequential learning policy for ranking and selection problems, where we use a non-parametric procedure for estimating the value of a policy. Our estimation approach aggregates over a set of kernel functions in order to achieve a more consistent estimator. Each element in the kernel estimation set uses a different bandwidth to achieve better aggregation. The final estimate uses a weighting scheme with the inverse mean square errors of the kernel estimators as weights. This weighting scheme is shown to be optimal under independent kernel estimators. For choosing the measurement, we employ the knowledge gradient policy that relies on predictive distributions to calculate the optimal sampling point. Our method allows a setting where the beliefs are expected to be correlated but the correlation structure is unknown beforehand. Moreover, the proposed policy is shown to be asymptotically optimal.     

Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation

The purpose of the present work is to implement well-known statistical decision and game theory strategies into multiobjective stochastic control problems of fluid dynamics. Such goal is first justified by the fact that deterministic (either singleobjective or multiobjective) control problems that are obtained without taking into account the uncertainty of the model are usually unreliable. Second, in most real-world problems, several goals must be satisfied simultaneously to obtain an optimal solution and, as a consequence, a multiobjective control approach is more appropriate. Therefore, we develop a multiobjective stochastic control algorithm for general fluid dynamics applications, based on the Bayes decision, adjoint formulation and the Nash equilibrium strategies. The algorithm is exemplified by the multiobjective stochastic control of a periodic Burgers equation.     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

DYNAMIC SUPPLY FROM COMMON PROPERTY RESOURCE: WATER DIVERSIONS FROM THE GREAT LAKES

For many purposes, the five Great Lakes can be classified as an open access resource since there is no well-defined system of property rights governing these water uses in the lakes. Yet the Great Lakes Charter and the Water Resources Act of 1966 establish regulations concerning the diversion of water out of the basin. Thus, for interbasin water diversions the Great Lakes is more like a common property resource with rules governing such transfers. Moreover, water diversion decisions are interconnected since withdrawing water from one point may affect water levels in the entire lake system. This, in turn, can adversely affect hydropower production and commercial navigation. Contributing to the complexity of the problem are the eight U.S. states, two Canadian provinces and two federal governments which are involved in Great Lakes management. Game theory is used to describe this situation. Several games are constructed to describe different market structures. Of particular interest is the number of players who participate in the game, as well as the expectations they hold. Open-loop (where players commit themselves to future actions) and closed-loop (where players do not commit themselves to future actions) are compared for the ten players game (eight states and two provinces), two players game (U.S. and Canada) and one player game (a social planner's solution). The open-loop game is shown to ignore part of the externalities involved, and thus can underestimate the social loss caused by diversions from the lakes.     

Fuzzy relations for the analysis of traders’ preferences in an information market game

In this paper we focus on preference and decision data gathered during a computer-supported information market game in which 35 students participated during seven consecutive trading sessions. The participants’ individual preferences on the market shares are collected to calculate a collective preference ranking using the Borda social choice method. Comparing this preference ranking to the shares’ actual market ranking resulting from the participants’ trading, we find a statistically significant difference between both rankings. As the preferences established by market behavior cannot be adequately explained through a social choice rule, we propose an alternative explanation based on the herd behavior phenomenon where traders imitate the most successful trader in the market. Using a decision analysis technique based on fuzzy relations, we study the participants’ rankings of the best share in the market during 7 weeks and compare the most successful trader to the other traders. The results from our analysis show that a substantial number of traders is indeed following the market leader.