48,141 research outputs found

    Aggregation, non-contradiction ans excluded-middle

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    This paper investigates the satisfaction of the Non-Contradiction (NC) and Excluded-Middle (EM) laws within the domain of aggregation operators. It provides characterizations both for those aggregation operators that satisfy NC/EM with respect to (w.r.t.) some given strong negation, as well as for those satisfying them w.r.t. any strong negation. The results obtained are applied to some of the most important known classes of aggregation operators

    Aggregation of Votes with Multiple Positions on Each Issue

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    We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We require the aggregation to be supportive, i.e. for every issue jj the corresponding component fjf_j of every aggregator on every issue should satisfy fj(x1,,,xn){x1,,,xn}f_j(x_1, ,\ldots, x_n) \in \{x_1, ,\ldots, x_n\}. We prove that, in such a set-up, non-dictatorial aggregation of votes in a society of some size is possible if and only if either non-dictatorial aggregation is possible in a society of only two members or a ternary aggregator exists that either on every issue jj is a majority operation, i.e. the corresponding component satisfies fj(x,x,y)=fj(x,y,x)=fj(y,x,x)=x,x,yf_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =x, \forall x,y, or on every issue is a minority operation, i.e. the corresponding component satisfies fj(x,x,y)=fj(x,y,x)=fj(y,x,x)=y,x,y.f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =y, \forall x,y. We then introduce a notion of uniformly non-dictatorial aggregator, which is defined to be an aggregator that on every issue, and when restricted to an arbitrary two-element subset of the votes for that issue, differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniformly non-dictatorial aggregator. Then making use of Bulatov's dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with combinatorial complexity by proving that if a set of feasible voting patterns XX has a uniformly non-dictatorial aggregator of some arity then the multi-sorted conservative constraint satisfaction problem on XX, in the sense introduced by Bulatov and Jeavons, with each issue representing a sort, is tractable; otherwise it is NP-complete

    Judgment aggregation in search for the truth

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    We analyze the problem of aggregating judgments over multiple issues from the perspective of whether aggregate judgments manage to efficiently use all voters' private information. While new in judgment aggregation theory, this perspective is familiar in a different body of literature about voting between two alternatives where voters' disagreements stem from conflicts of information rather than of interest. Combining the two bodies of literature, we consider a simple judgment aggregation problem and model the private information underlying voters' judgments. Assuming that voters share a preference for true collective judgments, we analyze the resulting strategic incentives and determine which voting rules efficiently use all private information. We find that in certain, but not all cases a quota rule should be used, which decides on each issue according to whether the proportion of ‘yes’ votes exceeds a particular quota

    A pooling approach to judgment aggregation

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    The literature has focused on a particular way of aggregating judgments: Given a set of yes or no questions or issues, the individuals’ judgments are then aggregated separately, issue by issue. Applied in this way, the majority method does not guarantee the logical consistency of the set of judgments obtained. This fact has been the focus of critiques of the majority method and similar procedures. This paper focuses on another way of aggregating judgments. The main difference is that aggregation is made en bloc on all the issues at stake. The main consequence is that the majority method applied in this way does always guarantee the logical consistency of the collective judgments. Since it satisfies a large set of attractive properties, it should provide the basis for more positive assessment if applied using the proposed pooling approach than if used separately. The paper extends the analysis to the pooling supermajority and plurality rules, with similar result

    Aggregative movement and front propagation for bi-stable population models

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    Front propagation for the aggregation-diffusion-reaction equation is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation

    The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions

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    We study the multidimensional aggregation equation u_t+\Div(uv)=0, v=Kuv=-\nabla K*u with initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d). We prove that with biological relevant potential K(x)=xK(x)=|x|, the equation is ill-posed in the critical Lebesgue space L_{d/(d-1)}(\bR^d) in the sense that there exists initial data in \cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d) such that the unique measure-valued solution leaves L_{d/(d-1)}(\bR^d) immediately. We also extend this result to more general power-law kernels K(x)=xαK(x)=|x|^\alpha, 0<α<20<\alpha<2 for p=ps:=d/(d+α2)p=p_s:=d/(d+\alpha-2), and prove a conjecture in Bertozzi, Laurent and Rosado [5] about instantaneous mass concentration for initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d) with p<psp<p_s. Finally, we classify all the "first kind" radially symmetric similarity solutions in dimension greater than two.Comment: typos corrected, 18 pages, to appear in Comm. Math. Phy
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