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Hisao Ishibuchi - One of the best experts on this subject based on the ideXlab platform.

  • a decomposition based large scale multi modal multi objective optimization algorithm
    Congress on Evolutionary Computation, 2020
    Co-Authors: Yiming Peng, Hisao Ishibuchi
    Abstract:

    A multi-modal multi-objective optimization problem is a special kind of multi-objective optimization problem with multiple Pareto subsets. In this paper, we propose an efficient multi-modal multi-objective optimization algorithm based on the widely used MOEA/D algorithm. In our proposed algorithm, each weight vector has its own sub-population. With a clearing mechanism and a greedy removal strategy, our proposed algorithm can effectively preserve equivalent Pareto optimal solutions (i.e., different Pareto optimal solutions with same objective values). Experimental results show that our proposed algorithm can effectively preserve the diversity of solutions in the Decision Space when handling large-scale multi-modal multi-objective optimization problems.

  • a double niched evolutionary algorithm and its behavior on polygon based problems
    Parallel Problem Solving from Nature, 2018
    Co-Authors: Yiping Liu, Hisao Ishibuchi, Yusuke Nojima, Naoki Masuyama, Ke Shang
    Abstract:

    Multi-modal multi-objective optimization problems are commonly seen in real-world applications. However, most existing researches focus on solving multi-objective optimization problems without multi-modal property or multi-modal optimization problems with single objective. In this paper, we propose a double-niched evolutionary algorithm for multi-modal multi-objective optimization. The proposed algorithm employs a niche sharing method to diversify the solution set in both the objective and Decision Spaces. We examine the behaviors of the proposed algorithm and its two variants as well as three other existing evolutionary optimizers on three types of polygon-based problems. Our experimental results suggest that the proposed algorithm is able to find multiple Pareto optimal solution sets in the Decision Space, even if the diversity requirements in the objective and Decision Spaces are inconsistent or there exist local optimal areas in the Decision Space.

  • Many-objective and many-variable test problems for visual examination of multiobjective search
    2013 IEEE Congress on Evolutionary Computation, 2013
    Co-Authors: Hisao Ishibuchi, Masakazu Yamane, Naoya Akedo, Yusuke Nojima
    Abstract:

    In the development of evolutionary multiobjective optimization (EMO) algorithms, it is important to implement a good balancing mechanism between the convergence of solutions towards the Pareto front and their diversity over the Pareto front. When an EMO algorithm is applied to a two-objective problem, the balance can be easily visualized by showing all solutions at each generation in the two-dimensional objective Space. However, such a visual examination of the multiobjective search is difficult for many-objective problems with four or more objectives. The use of many-objective test problems with two Decision variables has been proposed in some studies to visually examine the search behavior of EMO algorithms. Such test problems are defined by a number of points in a two-dimensional Decision Space where the distance minimization from each point is an objective. Thus the number of objectives is the same as the number of points. The search behavior of EMO algorithms can be visually examined in the two-dimensional Decision Space. In this paper, we propose the use of many-objective test problems for visual examination of the search behavior in a high-dimensional Decision Space. More specifically, our m-objective test problem with n variables is generated by specifying m points on a plane in an n-dimensional Decision Space. We examine the behavior of EMO algorithms through computational experiments on such an m-objective n-variable test problem. Our experimental results show that the number of variables has a large effect on the search behavior of EMO algorithms with respect to the diversity of solutions.

  • relation between neighborhood size and moea d performance on many objective problems
    International Conference on Evolutionary Multi-criterion Optimization, 2013
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    MOEA/D is a simple but powerful scalarizing function-based EMO algorithm. Its high search ability has been demonstrated for a wide variety of multiobjective problems. MOEA/D can be viewed as a cellular algorithm. Each cell has a different weight vector and a single solution. A certain number of the nearest cells are defined for each cell as its neighbors based on the Euclidean distance between weight vectors. A new solution is generated for each cell from current solutions in its neighboring cells. The generated solution is compared with the current solutions in the neighboring cells for solution replacement. In this paper, we examine the relation between the neighborhood size and the performance of MOEA/D. In order to examine the effect of local mating and local replacement separately, we use a variant of MOEA/D with two different neighborhoods: One is for local mating and the other is for local replacement. The performance of MOEA/D with various combinations of two neighborhoods is examined using the hypervolume in the objective Space and a diversity measure in the Decision Space for many-objective problems. Experimental results show that MOEA/D with a large replacement neighborhood has high search ability in the objective Space. However, it is also shown that small replacement and mating neighborhoods are beneficial for diversity maintenance in the Decision Space. It is also shown that the appropriate specification of two neighborhoods strongly depends on the problem.

  • a many objective test problem for visually examining diversity maintenance behavior in a Decision Space
    Genetic and Evolutionary Computation Conference, 2011
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    Recently distance minimization problems in a two-dimensional Decision Space have been utilized as many-objective test problems to visually examine the behavior of evolutionary multi-objective optimization (EMO) algorithms. Such a test problem is usually defined by a single polygon where the distance from a solution to each vertex is minimized in the Decision Space. We can easily generate different test problems from different polygons. We can also easily generate test problems with multiple equivalent Pareto optimal regions using multiple polygons of the same shape and the same size. Whereas these test problems have a number of advantages, they have no clear relevance to real-world situations since they are artificially generated unrealistic test problems. In this paper, we generate a distance minimization problem from a real-world map. Our test problem has four objectives, which are to minimize the distances to the nearest elementary school, junior high school, railway station, and convenience store. Using our test problem, we examine the behavior of well-known and frequently-used EMO algorithms in terms of their diversity maintenance ability in the two-dimensional Decision Space.

Yusuke Nojima - One of the best experts on this subject based on the ideXlab platform.

  • a double niched evolutionary algorithm and its behavior on polygon based problems
    Parallel Problem Solving from Nature, 2018
    Co-Authors: Yiping Liu, Hisao Ishibuchi, Yusuke Nojima, Naoki Masuyama, Ke Shang
    Abstract:

    Multi-modal multi-objective optimization problems are commonly seen in real-world applications. However, most existing researches focus on solving multi-objective optimization problems without multi-modal property or multi-modal optimization problems with single objective. In this paper, we propose a double-niched evolutionary algorithm for multi-modal multi-objective optimization. The proposed algorithm employs a niche sharing method to diversify the solution set in both the objective and Decision Spaces. We examine the behaviors of the proposed algorithm and its two variants as well as three other existing evolutionary optimizers on three types of polygon-based problems. Our experimental results suggest that the proposed algorithm is able to find multiple Pareto optimal solution sets in the Decision Space, even if the diversity requirements in the objective and Decision Spaces are inconsistent or there exist local optimal areas in the Decision Space.

  • Many-objective and many-variable test problems for visual examination of multiobjective search
    2013 IEEE Congress on Evolutionary Computation, 2013
    Co-Authors: Hisao Ishibuchi, Masakazu Yamane, Naoya Akedo, Yusuke Nojima
    Abstract:

    In the development of evolutionary multiobjective optimization (EMO) algorithms, it is important to implement a good balancing mechanism between the convergence of solutions towards the Pareto front and their diversity over the Pareto front. When an EMO algorithm is applied to a two-objective problem, the balance can be easily visualized by showing all solutions at each generation in the two-dimensional objective Space. However, such a visual examination of the multiobjective search is difficult for many-objective problems with four or more objectives. The use of many-objective test problems with two Decision variables has been proposed in some studies to visually examine the search behavior of EMO algorithms. Such test problems are defined by a number of points in a two-dimensional Decision Space where the distance minimization from each point is an objective. Thus the number of objectives is the same as the number of points. The search behavior of EMO algorithms can be visually examined in the two-dimensional Decision Space. In this paper, we propose the use of many-objective test problems for visual examination of the search behavior in a high-dimensional Decision Space. More specifically, our m-objective test problem with n variables is generated by specifying m points on a plane in an n-dimensional Decision Space. We examine the behavior of EMO algorithms through computational experiments on such an m-objective n-variable test problem. Our experimental results show that the number of variables has a large effect on the search behavior of EMO algorithms with respect to the diversity of solutions.

  • relation between neighborhood size and moea d performance on many objective problems
    International Conference on Evolutionary Multi-criterion Optimization, 2013
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    MOEA/D is a simple but powerful scalarizing function-based EMO algorithm. Its high search ability has been demonstrated for a wide variety of multiobjective problems. MOEA/D can be viewed as a cellular algorithm. Each cell has a different weight vector and a single solution. A certain number of the nearest cells are defined for each cell as its neighbors based on the Euclidean distance between weight vectors. A new solution is generated for each cell from current solutions in its neighboring cells. The generated solution is compared with the current solutions in the neighboring cells for solution replacement. In this paper, we examine the relation between the neighborhood size and the performance of MOEA/D. In order to examine the effect of local mating and local replacement separately, we use a variant of MOEA/D with two different neighborhoods: One is for local mating and the other is for local replacement. The performance of MOEA/D with various combinations of two neighborhoods is examined using the hypervolume in the objective Space and a diversity measure in the Decision Space for many-objective problems. Experimental results show that MOEA/D with a large replacement neighborhood has high search ability in the objective Space. However, it is also shown that small replacement and mating neighborhoods are beneficial for diversity maintenance in the Decision Space. It is also shown that the appropriate specification of two neighborhoods strongly depends on the problem.

  • a many objective test problem for visually examining diversity maintenance behavior in a Decision Space
    Genetic and Evolutionary Computation Conference, 2011
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    Recently distance minimization problems in a two-dimensional Decision Space have been utilized as many-objective test problems to visually examine the behavior of evolutionary multi-objective optimization (EMO) algorithms. Such a test problem is usually defined by a single polygon where the distance from a solution to each vertex is minimized in the Decision Space. We can easily generate different test problems from different polygons. We can also easily generate test problems with multiple equivalent Pareto optimal regions using multiple polygons of the same shape and the same size. Whereas these test problems have a number of advantages, they have no clear relevance to real-world situations since they are artificially generated unrealistic test problems. In this paper, we generate a distance minimization problem from a real-world map. Our test problem has four objectives, which are to minimize the distances to the nearest elementary school, junior high school, railway station, and convenience store. Using our test problem, we examine the behavior of well-known and frequently-used EMO algorithms in terms of their diversity maintenance ability in the two-dimensional Decision Space.

  • many objective test problems to visually examine the behavior of multiobjective evolution in a Decision Space
    Parallel Problem Solving from Nature, 2010
    Co-Authors: Hisao Ishibuchi, Yasuhiro Hitotsuyanagi, Noritaka Tsukamoto, Yusuke Nojima
    Abstract:

    Many-objective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are non-dominated with each other in many-objective EMO algorithms, we may need a different fitness evaluation scheme from the case of two and three objectives. One difficulty in the design of many-objective EMO algorithms is that we cannot visually observe the behavior of multiobjective evolution in the objective Space with four or more objectives. In this paper, we propose the use of many-objective test problems in a two- or three-dimensional Decision Space to visually examine the behavior of multiobjective evolution. Such a visual examination helps us to understand the characteristic features of EMO algorithms for many-objective optimization. Good understanding of existing EMO algorithms may facilitates their modification and the development of new EMO algorithms for many-objective optimization.

Thomas J Bossert - One of the best experts on this subject based on the ideXlab platform.

  • Decision Space and capacities in the decentralization of health services in fiji comment on decentralisation of health services in fiji a Decision Space analysis
    International journal of health policy and management, 2016
    Co-Authors: Thomas J Bossert
    Abstract:

    The study of decentralization in Fiji shows that increasing capacities is not necessarily related to increasing Decision Space of local officials, which is in contrast with earlier studies in Pakistan. Future studies should address the relationship among Decision Space, capacities, and health system performance.

  • health sector decentralization and local Decision making Decision Space institutional capacities and accountability in pakistan
    Social Science & Medicine, 2011
    Co-Authors: Thomas J Bossert, Andrew D Mitchell
    Abstract:

    Abstract Health sector decentralization has been widely adopted to improve delivery of health services. While many argue that institutional capacities and mechanisms of accountability required to transform decentralized Decision-making into improvements in local health systems are lacking, few empirical studies exist which measure or relate together these concepts. Based on research instruments administered to a sample of 91 health sector Decision-makers in 17 districts of Pakistan, this study analyzes relationships between three dimensions of decentralization: decentralized authority (referred to as “Decision Space”), institutional capacities, and accountability to local officials. Composite quantitative indicators of these three dimensions were constructed within four broad health functions (strategic and operational planning, budgeting, human resources management, and service organization/delivery) and on an overall/cross-function basis. Three main findings emerged. First, district-level respondents report varying degrees of each dimension despite being under a single decentralization regime and facing similar rules across provinces. Second, within dimensions of decentralization—particularly Decision Space and capacities—synergies exist between levels reported by respondents in one function and those reported in other functions (statistically significant coefficients of correlation ranging from ρ  = 0.22 to ρ  = 0.43). Third, synergies exist across dimensions of decentralization, particularly in terms of an overall indicator of institutional capacities (significantly correlated with both overall Decision Space ( ρ  = 0.39) and accountability ( ρ  = 0.23)). This study demonstrates that decentralization is a varied experience—with some district-level officials making greater use of Decision Space than others and that those who do so also tend to have more capacity to make Decisions and are held more accountable to elected local officials for such choices. These findings suggest that Pakistan’s decentralization policy should focus on synergies among dimensions of decentralization to encouraging more use of de jure Decision Space, work toward more uniform institutional capacity, and encourage greater accountability to local elected officials.

  • decentralization of health systems in ghana zambia uganda and the philippines a comparative analysis of Decision Space
    Health Policy and Planning, 2002
    Co-Authors: Thomas J Bossert, Joel C Beauvais
    Abstract:

    : This study reviews the experience of decentralization in four developing countries: Ghana, Uganda, Zambia and the Philippines. It uses two analytical frameworks to describe and compare the types and degrees of decentralization in each country. The first framework specifies three types of decentralization: deconcentration, delegation and devolution. The second framework uses a principal agent approach and innovative maps of 'Decision Space' to define the range of choice for different functions that is transferred from the centre to the periphery of the system. The analysis finds a variety of different types and degrees of decentralization, with the Philippines demonstrating the widest range of choice over many functions that were devolved to local government units. The least choice was transferred through delegation to an autonomous health service in Ghana. Uganda and Zambia display variations between these extremes. There was insufficient evidence of the impact of decentralization to assess how these differences in 'Decision Space' influenced the performance of each health system. The authors suggest that this is a major area for future research.

  • analyzing the decentralization of health systems in developing countries Decision Space innovation and performance
    Social Science & Medicine, 1998
    Co-Authors: Thomas J Bossert
    Abstract:

    Decentralization has long been advocated as a desirable process for improving health sys- tems. Nevertheless, we still lack a suAcient analytical framework for systematically studying how decen- tralization can achieve this objective. We do not have adequate means of analyzing the three key elements of decentralization: (1) the amount of choice that is transferred from central institutions to in- stitutions at the periphery of health systems, (2) what choices local oAcials make with their increased discretion and (3) what eAect these choices have on the performance of the health system. This article proposes a framework of analysis that can be used to design and evaluate the decentralization of health systems. It starts from the assumption that decentralization is not an end in itself but rather should be designed and evaluated for its ability to achieve broader objectives of health reform: equity, eAciency, quality and financial soundness. Using a ''principal agent'' approach as the basic framework, but incor- porating insights from public administration, local public choice and social capital approaches, the article presents a Decision Space approach which defines decentralization in terms of the set of functions and degrees of choice that formally are transferred to local oAcials. The approach also evaluates the incentives that central government can oAer to local Decision-makers to encourage them to achieve health objectives. It evaluates the local government characteristics that also influence Decision-making and implementation at the local level. Then it determines whether local oAcials innovate by making choices that are diAerent from those directed by central authorities. Finally, it evaluates whether the local choices have improved the performance of the local health system in achieving the broader health objectives. Examples from Colombia are used to illustrate the approach. The framework will be used to analyze the experience of decentralization in a series of empirical studies in Latin America. The results of these studies should suggest policy recommendations for adjusting Decision Space and incentives so that localities make Decisions that achieve the objectives of health reform. # 1998 Elsevier Science Ltd. All rights reserved.

  • analyzing the decentralization of health systems in developing countries Decision Space innovation and performance
    Social Science & Medicine, 1998
    Co-Authors: Thomas J Bossert
    Abstract:

    Decentralization has long been advocated as a desirable process for improving health sys- tems. Nevertheless, we still lack a suAcient analytical framework for systematically studying how decen- tralization can achieve this objective. We do not have adequate means of analyzing the three key elements of decentralization: (1) the amount of choice that is transferred from central institutions to in- stitutions at the periphery of health systems, (2) what choices local oAcials make with their increased discretion and (3) what eAect these choices have on the performance of the health system. This article proposes a framework of analysis that can be used to design and evaluate the decentralization of health systems. It starts from the assumption that decentralization is not an end in itself but rather should be designed and evaluated for its ability to achieve broader objectives of health reform: equity, eAciency, quality and financial soundness. Using a ''principal agent'' approach as the basic framework, but incor- porating insights from public administration, local public choice and social capital approaches, the article presents a Decision Space approach which defines decentralization in terms of the set of functions and degrees of choice that formally are transferred to local oAcials. The approach also evaluates the incentives that central government can oAer to local Decision-makers to encourage them to achieve health objectives. It evaluates the local government characteristics that also influence Decision-making and implementation at the local level. Then it determines whether local oAcials innovate by making choices that are diAerent from those directed by central authorities. Finally, it evaluates whether the local choices have improved the performance of the local health system in achieving the broader health objectives. Examples from Colombia are used to illustrate the approach. The framework will be used to analyze the experience of decentralization in a series of empirical studies in Latin America. The results of these studies should suggest policy recommendations for adjusting Decision Space and incentives so that localities make Decisions that achieve the objectives of health reform. # 1998 Elsevier Science Ltd. All rights reserved.

Naoya Akedo - One of the best experts on this subject based on the ideXlab platform.

  • Many-objective and many-variable test problems for visual examination of multiobjective search
    2013 IEEE Congress on Evolutionary Computation, 2013
    Co-Authors: Hisao Ishibuchi, Masakazu Yamane, Naoya Akedo, Yusuke Nojima
    Abstract:

    In the development of evolutionary multiobjective optimization (EMO) algorithms, it is important to implement a good balancing mechanism between the convergence of solutions towards the Pareto front and their diversity over the Pareto front. When an EMO algorithm is applied to a two-objective problem, the balance can be easily visualized by showing all solutions at each generation in the two-dimensional objective Space. However, such a visual examination of the multiobjective search is difficult for many-objective problems with four or more objectives. The use of many-objective test problems with two Decision variables has been proposed in some studies to visually examine the search behavior of EMO algorithms. Such test problems are defined by a number of points in a two-dimensional Decision Space where the distance minimization from each point is an objective. Thus the number of objectives is the same as the number of points. The search behavior of EMO algorithms can be visually examined in the two-dimensional Decision Space. In this paper, we propose the use of many-objective test problems for visual examination of the search behavior in a high-dimensional Decision Space. More specifically, our m-objective test problem with n variables is generated by specifying m points on a plane in an n-dimensional Decision Space. We examine the behavior of EMO algorithms through computational experiments on such an m-objective n-variable test problem. Our experimental results show that the number of variables has a large effect on the search behavior of EMO algorithms with respect to the diversity of solutions.

  • relation between neighborhood size and moea d performance on many objective problems
    International Conference on Evolutionary Multi-criterion Optimization, 2013
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    MOEA/D is a simple but powerful scalarizing function-based EMO algorithm. Its high search ability has been demonstrated for a wide variety of multiobjective problems. MOEA/D can be viewed as a cellular algorithm. Each cell has a different weight vector and a single solution. A certain number of the nearest cells are defined for each cell as its neighbors based on the Euclidean distance between weight vectors. A new solution is generated for each cell from current solutions in its neighboring cells. The generated solution is compared with the current solutions in the neighboring cells for solution replacement. In this paper, we examine the relation between the neighborhood size and the performance of MOEA/D. In order to examine the effect of local mating and local replacement separately, we use a variant of MOEA/D with two different neighborhoods: One is for local mating and the other is for local replacement. The performance of MOEA/D with various combinations of two neighborhoods is examined using the hypervolume in the objective Space and a diversity measure in the Decision Space for many-objective problems. Experimental results show that MOEA/D with a large replacement neighborhood has high search ability in the objective Space. However, it is also shown that small replacement and mating neighborhoods are beneficial for diversity maintenance in the Decision Space. It is also shown that the appropriate specification of two neighborhoods strongly depends on the problem.

  • a many objective test problem for visually examining diversity maintenance behavior in a Decision Space
    Genetic and Evolutionary Computation Conference, 2011
    Co-Authors: Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima
    Abstract:

    Recently distance minimization problems in a two-dimensional Decision Space have been utilized as many-objective test problems to visually examine the behavior of evolutionary multi-objective optimization (EMO) algorithms. Such a test problem is usually defined by a single polygon where the distance from a solution to each vertex is minimized in the Decision Space. We can easily generate different test problems from different polygons. We can also easily generate test problems with multiple equivalent Pareto optimal regions using multiple polygons of the same shape and the same size. Whereas these test problems have a number of advantages, they have no clear relevance to real-world situations since they are artificially generated unrealistic test problems. In this paper, we generate a distance minimization problem from a real-world map. Our test problem has four objectives, which are to minimize the distances to the nearest elementary school, junior high school, railway station, and convenience store. Using our test problem, we examine the behavior of well-known and frequently-used EMO algorithms in terms of their diversity maintenance ability in the two-dimensional Decision Space.

Arul Mishra - One of the best experts on this subject based on the ideXlab platform.

  • thinking outside the euclidean box riemannian geometry and inter temporal Decision making
    PLOS ONE, 2016
    Co-Authors: Himanshu Mishra, Arul Mishra
    Abstract:

    Inter-temporal Decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these Decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the Decision Space, where the Decision-maker perceives time and monetary payoffs, is a Euclidean Space. By relaxing the rigid assumption of Euclidean Space, we propose that the Decision Space is a more flexible Riemannian Space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature Space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal Decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the Decision Space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the Decision Space resembles those of the Negative Curvature Space rather than the Euclidean Space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.

  • thinking outside the euclidean box riemannian geometry and inter temporal Decision making
    2015
    Co-Authors: Himanshu Mishra, Arul Mishra
    Abstract:

    Inter-temporal Decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these Decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the Decision Space, where the Decision-maker perceives time and payoffs, is a Euclidean Space. By relaxing the rigid assumption of Euclidean Space, we propose that the Decision Space is a more flexible Riemannian Space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature Space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal Decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the Decision Space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the Decision Space resembles those of the Negative Curvature Space rather than the Euclidean Space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.