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Various other algorithms return EF1 allocations that are also Pareto-efficient; see Efficient approximately fair item allocation. For two agents with arbitrary monotone valuations, or three agents with additive valuations, an EF1 allocation can be computed using a number of queries logarithmic in the number of items.
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.
Similarly, slice( sum = sum + i + w, i) only contains "for(i = 1; i < N; ++i) {" and slice( sum = sum + i + w, w) only contains the statement "int w = 7". When we union all of those statements, we do not have executable code, so to make the slice an executable slice we merely add the end brace for the for loop and the declaration of i.
A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. [5] A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as = where = (,, …,) is the vector of goods, both for all i.
Both are guaranteed to return an allocation with no envy-cycles. However, the allocation is not guaranteed to be Pareto-efficient. The Approximate-CEEI mechanism returns a partial EF1 allocation for arbitrary preference relations. It is PE w.r.t. the allocated objects, but not PE w.r.t. all objects (since some objects may remain unallocated). [3]
The algorithm also is designed for two parties only; when there are three or more parties, there may be no allocation that is simultaneously envy-free, equitable, and Pareto-optimal. This can be shown by the following example, constructed by J.H.Reijnierse, [ 2 ] : 82–83 involving three parties and their valuations:
In object-oriented computer programming, a null object is an object with no referenced value or with defined neutral (null) behavior.The null object design pattern, which describes the uses of such objects and their behavior (or lack thereof), was first published as "Void Value" [1] and later in the Pattern Languages of Program Design book series as "Null Object".
An envy-free cake-cutting is a kind of fair cake-cutting.It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.