310-2-08-Lecture01-Introduction-and-Pareto-optimality

# 310-2-08-Lecture01-Introduction-and-Pareto-optimality -...

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Unformatted text preview: Lecture 1 — Introduction and Pareto Optimality Todd Sarver Northwestern University Spring 2008 1 Notation and General Setup We begin by introducing some mathematical notation which will be used throughout the course: • Mathematical Operators: The symbol ∑ , which is called “sigma,” denotes a summation. Thus, ∑ n i =1 x i is equivalent to x 1 + ··· + x n . The symbol producttext , called “pi,” denotes a product. Thus, producttext n i =1 x i is equivalent to x 1 × ··· × x n . • Vectors: The notation ( x 1 ,... ,x n ) is an array, or vector , which enumerates the elements x 1 ,... ,x n . The term x i is referred to as the i th coordinate of this vector. Note that in vector notation, the order in which the elements are listed is important. Thus, (1 , 2) is not the same as (2 , 1). • Sets: A set is any collection of objects. Given a set X , an element x of the set is one of the objects in the set. This is denoted by x ∈ X . If x is not one of the objects in the set, then we write x / ∈ X . – If a set consists of finitely many elements, it is common to describe the set by simply listing its elements inside bracket notation. For example, { A,B } is the set containing the letters A and B , and { 1 , 2 . 5 , 7 } is the set containing the numbers 1, 2.5, and 7. – Sets can also be defined by describing the common condition that its elements satisfy. In this case, we use a colon inside the brackets and then list this necessary condition. The colon means “such that.” For example, { ( x 1 ,x 2 ) : x 1 ≤ x 2 } is the set of all 2-dimensional vectors (indicated by “( x 1 ,x 2 )”) such that (indicated by “:”) the first coordinate is less than or equal to the second coordinate (indicated by “ x 1 ≤ x 2 ”). For another example, { ( x 1 ,... ,x n ) : x i ≥ 0 and ∑ n i =1 x i = 1 } is the set of all n-dimensional vectors whose coordinates are all nonnegative and sum to 1. • Intervals: The interval [ a,b ] denotes the set of all the numbers between a and b , that is, any number x such that a ≤ x ≤ b . Using the set notation introduced above, we could write [ a,b ] = { x : a ≤ x ≤ b } . Note that [1 , 3] is not the same set as { 1 , 2 , 3 } . The interval [1 , 3] contains all points between 1 and 3, which includes numbers like 1 . 5, 4 3 , and √ 2 that are clearly not in the set { 1 , 2 , 3 } . The environment used throughout this course is described below: Lecture 1 — Introduction and Pareto Optimality 2 • Individuals: 1 , 2 , 3 ,... ,n • Alternatives: Let X denote the set of feasible alternatives, and let A,B,C,... denote elements of X . • Preferences: Each individual i has a preference followsequal i over the available alternatives....
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310-2-08-Lecture01-Introduction-and-Pareto-optimality -...

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