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Single-peaked preferences are a class of preference relations. A group has single-peaked preferences over a set of outcomes if the outcomes can be ordered along a line such that: Each agent has a "best outcome" in the set, and; For each agent, outcomes that are further from his or her best outcome are preferred less.
In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal".
A rule is anonymous and strategyproof for all single-peaked preferences if it is equivalent to a median rule with at most n+1 phantoms. A rule is strategyproof for all single-peaked preferences iff it is equivalent to a minmax rule of the following form. There are 2 n parameters, b S for any subset S of voters.
Black proved that by replacing unrestricted domain with single-peaked preferences in Arrow's theorem removes the impossibility: there are Pareto-efficient non-dictatorships that satisfy the "independence of irrelevant alternatives" criterion. However, Black's 1948 proof was published before Arrow's impossibility theorem was published in 1950 ...
The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
Whenever the distribution of voters has a unique median in all directions, and voters rank candidates in order of proximity, the median voter theorem applies: the candidate closest to the median will have a majority preference over all his or her rivals, and will be elected by any voting method satisfying the median voter property in one dimension.
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The one-dimensional case is the special case in which there are only two issues, e.g. defense and education. In this case, distributions can be represented by a single parameter: the allocation to issue #1 (the allocation to issue #2 is simply the total budget minus the allocation to issue #1).