Before doing the following exercises you should compute the following quantities: c_1 and c2, respectively the.alphabetical index.associated with the first and second character in your first name (or in the first name you are signing this homework with). Example: If your name is Anaximander then c_1 = 1,c_2 = 14 since A and N are respectively the 1st and 14th letter of the English alphabet. (x_1, y_1) = (c_1% 3, c_1% 5) (a% b stands for \"remainder of a when divided by b); (x_2, y_2) = (c_2%3,c_2%5); N_v = (c_1%5) + 4 In the following exercise we will focus on pairs (tuples of two items) and explore ways to compare them. All pairs (x, y) are in the set R^2. The following definition is often used in game theory, AI and economics. It is related to a concept called Pareto Domination. We say that (x, y) (x\', y\') iff x lessthanorequalto x\' and y lessthanorequalto y\'. Draw four points in the plane showing which ones are to another; two of the points should be (x_1, y_2) and (x_2,y_2). Prove that it is a partial order. Is it total? How about this other definition though? (x, y) (x\',y\') iff lessthanorequalto or y lessthanorequalto y\'. Draw four points in the plane showing which ones are: to another; two of the points should be (xi,yi) and (x^2,y^2). Prove it is not ant symmetric and thus it is not a partial order. Is it even a preorder? (A preorder is a reflexive and transitive relation). 2. Assume mi undirected, simple graph G has exactly 17 edges mid.ill its nodes have degree greater or equal than 3. What is the maximum number of nodes G may have? Draw an undirected acyclic graph that is not a tree having exactly Nv nodes.
Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off. The term is named after Vilfredo Pareto (1848–1923), an Italian engineer and economist who used the concept in his studies ofeconomic efficiency and income distribution. The concept has applications in academic fields such as economics, engineering, and the life sciences.
Pareto improvement is defined to be a change to a different allocation that makes at least one individual better off without making any other individual worse off, given a certain initial allocation of goods among a set of individuals. An allocation is defined as \"Pareto efficient\" or \"Pareto optimal\" when no further Pareto improvements can be made.
Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society.[1][2] The notion of Pareto efficiency can also be applied to the selection of alternatives in engineering and similar fields. Each option is first assessed under multiple criteria and then a subset of options is identified with the property that no other option can categorically outperform any of its members.
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