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Decision Making Beyond Arrow’s Impossibility Theorem
 International Journal of Intelligent Systems
, 2009
"... Abstract — In 1951, K. J. Arrow proved that, under certain assumptions, it is impossible to have group decision making rules which satisfy reasonable conditions like symmetry. This Impossibility Theorem is often cited as a proof that reasonable group decision making is impossible. We start our paper ..."
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Cited by 9 (8 self)
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Abstract — In 1951, K. J. Arrow proved that, under certain assumptions, it is impossible to have group decision making rules which satisfy reasonable conditions like symmetry. This Impossibility Theorem is often cited as a proof that reasonable group decision making is impossible. We start our paper by remarking that Arrow’s result only covers the situations when the only information we have about individual preferences is their binary preferences between the alternatives. If we follow the main ideas of modern decision making and game theory and also collect information about the preferences between lotteries (i.e., collect the utility values of different alternatives), then reasonable decision making rules are possible: e.g., Nash’s rule in which we select an alternative for which the product of utilities is the largest possible. We also deal with two related issues: how we can detect individual preferences if all we have is preferences of a subgroup, and how we take into account mutual attraction between participants.
Level Sets and Minimum Volume Sets of Probability Density Functions
 Approximate Reasoning
, 2003
"... Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse propositio ..."
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Cited by 7 (1 self)
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Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduced. It provides a di#erent visualization of the distribution of the probability for the random variable in question. It is also very useful to prove the above proposition. The volume # of the minimum volume sets varies according to its probability #: smaller volume implies lower probability and vice versa and larger volume implies larger probability and vice versa. In this context, 1 # is the error of an erroneously classification of a new observation inside of the minimum volume set or corresponding level set. To decide the volume and/or the error of the level set that will serve to summarize the support of the random variable, a # # plot is defined. We also study the relation of the minimum volume set approach with random set theory when # is a random variable and extend the most specific probabilitypossibility transformation proposed in [DPS93] to continuous piecewise strictly monotone probability density functions. Keywords: Minimum volume set, level set, random set, probabilitypossibility transformation.
A New Differential Formalism for IntervalValued Functions and Its Potential Use in . . .
 ITS POTENTIAL USE IN DETECTING 1D LANDSCAPE FEATURES, PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON INFORMATION TECHNOLOGY INTECH’03, CHIANG MAI
, 2003
"... In many practical problems, it is important to know the slope (derivative) dy=dx of one quantity y with respect to some other quantity x. For example, different 1D landscape features can be characterized by different values of the derivative dy=dx, where y is an altitude, and x is a horizontal ..."
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Cited by 2 (1 self)
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In many practical problems, it is important to know the slope (derivative) dy=dx of one quantity y with respect to some other quantity x. For example, different 1D landscape features can be characterized by different values of the derivative dy=dx, where y is an altitude, and x is a horizontal coordinate. In practice, we often know the values of y(x) for different x with interval uncertainty. How can we then find the set of possible values of the slope? In this paper, we formulate this problem of differentiating intervalvalues functions in precise terms, and we describe an (asymptotically) optimal algorithm for computing the corresponding derivative.
How to Avoid Gerrymandering: A New Algorithmic Solution
"... Abstract—Subdividing an area into voting districts is often a very controversial issue. If we divide purely geographically, then minority groups may not be properly represented. If we start changing the borders of the districts to accommodate different population groups, we may end up with very arti ..."
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Cited by 1 (0 self)
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Abstract—Subdividing an area into voting districts is often a very controversial issue. If we divide purely geographically, then minority groups may not be properly represented. If we start changing the borders of the districts to accommodate different population groups, we may end up with very arti cial borders – borders which are often to set up in such a way as to give an unfair advantage to incumbents. In this paper, we describe redistricting as a precise optimization problem, and we propose a new algorithm for solving this problem. I. FORMULATION OF THE PRACTICAL PROBLEM The notion of electoral districts. In the USA and in many other countries, voting is done by electoral districts: • in elections to the US House of Representative, every federal voting district elects one representative; • in elections to the state legislature, each state district
Intermediate Degrees are Needed for the World to be Cognizable: Towards a New Justification for Fuzzy Logic Ideas
"... Summary. Most traditional examples of fuzziness come from the analysis of commonsense reasoning. When we reason, we use words from natural language like “young”, “well”. In many practical situations, these words do not have a precise trueorfalse meaning, they are fuzzy. One may therefore be left w ..."
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Summary. Most traditional examples of fuzziness come from the analysis of commonsense reasoning. When we reason, we use words from natural language like “young”, “well”. In many practical situations, these words do not have a precise trueorfalse meaning, they are fuzzy. One may therefore be left with an impression that fuzziness is a subjective characteristic, it is caused by the specific way our brains work. However, the fact that that we are the result of billions of years of successful adjustingtotheenvironment evolution makes us conclude that everything about us humans is not accidental. In particular, the way we reason is not accidental, this way must reflect some reallife phenomena – otherwise, this feature of our reasoning would have been useless and would not have been abandoned long ago. In other words, the fuzziness in our reasoning must have an objective explanation – in fuzziness of the real world. In this paper, we first give examples of objective realworld fuzziness. After these example, we provide an explanation of this fuzziness – in terms of cognizability of the world. 1
I. EVERYTHING IS A MATTER OF DEGREE: ONE OF THE MAIN IDEAS BEHIND FUZZY LOGIC
"... Abstract—One of the main ideas behind fuzzy logic and its applications is that everything is a matter of degree. We are often accustomed to think that every statement about a physical world is true or false – that an object is either a particle or a wave, that a person is either young or not, either ..."
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Abstract—One of the main ideas behind fuzzy logic and its applications is that everything is a matter of degree. We are often accustomed to think that every statement about a physical world is true or false – that an object is either a particle or a wave, that a person is either young or not, either well or ill – but in reality, we sometimes encounter intermediate situations. In this paper, we show that the existence of such intermediate situations can be theoretically explained – by a natural assumption that the real world is cognizable.
Application of OrderPreserving Functions to the Modeling of Computational Mechanics Problems with Uncertainty ∗
"... In many engineering problems the shape of the structure is not exactly known. In that situation it is possible to consider a family of shapes which belong to the interval set Ω ∈ [Ω,Ω]. In order to find upper and lower bound of the solution, which depend on uncertain (interval) shape, it is possible ..."
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In many engineering problems the shape of the structure is not exactly known. In that situation it is possible to consider a family of shapes which belong to the interval set Ω ∈ [Ω,Ω]. In order to find upper and lower bound of the solution, which depend on uncertain (interval) shape, it is possible to use properties of topological derivative or differentials which are positive definite (in many cases positive definite differentials are simply positive). Numerical examples are related to the heat transfer, imprecise probability, and integration on manifolds.
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