Mathematical optimization | Fairness criteria | Apportionment method criteria

Optimal apportionment

Optimal apportionment is an approach to apportionment that is based on mathematical optimization. In a problem of apportionment, there is a resource to allocate, denoted by . For example, it can be an integer representing the number of seats in a house of representatives. The resource should be allocated between some agents. For example, these can be federal states or political parties. The agents have different entitlements, denoted by a vector of fractions with a sum of 1. For example, ti can be the fraction of votes won by party i. The goal is to find allocation - a vector with . The ideal share for agent i is his/her quota, defined as . If it is possible to give each agent his/her quota, then the allocation is maximally fair. However, exact fairness is usually unattainable, since the quotas are not integers and the allocations must be integers. There are various approaches to cope with this difficulty (see mathematics of apportionment). The optimization-based approach aims to attain, for eacn instance, an allocation that is "as fair as possible" for this instance. An allocation is "fair" if for all agents i, that is, each agent's allocation is exactly proportional to his/her entitlement. in this case, we say that the "unfairness" of the allocation is 0. If this equality must be violated, one can define a measure of "total unfairness", and try to minimize it. (Wikipedia).

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Maximum and Minimum

Maximum and Minimum of a set In this video, I define the maximum and minimum of a set, and show that they don't always exist. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh

From playlist Real Numbers

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Limit of sqrt(9x^6 - x)/(x^3 + 1) as x approaches infinity

Limit of sqrt(9x^6 - x)/(x^3 + 1) as x approaches infinity. We divide the numerator and denominator by the sqruare root of x^6 to do this. Note sqrt(x^6) = x^3 in this problem because x is approaching infinity. If you enjoyed this video please consider liking, sharing, and subscribing. U

From playlist Limits at Infinity

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Limit of (4u^4 + 5)/((u^2 - 2)(2u^2 - 1)) as u approaches infinity

Limit of (4u^4 + 5)/((u^2 - 2)(2u^2 - 1)) as u approaches infinity. This is a calculus problem where we find a limit as u approaches infinity. In this case we have a rational function and the numerator and denominator have the same growth rate, so the limit is the ratio of the leading coef

From playlist Limits at Infinity

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Easiest Way To Solve an Absolute Value Equation and Determine when Only One Solution

Learn how to solve absolute value equations with extraneous solutions. Absolute value of a number is the positive value of the number. For instance, the absolute value of 2 is 2 and the absolute value of -2 is also 2. To solve an absolute value problem, we first isolate the absolute value

From playlist Solve Absolute Value Equations

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Using parent graphs to understand the left and right hand limits

👉 Learn how to evaluate the limit of an absolute value function. The limit of a function as the input variable of the function tends to a number/value is the number/value which the function approaches at that time. The absolute value function is a function which only takes the positive val

From playlist Evaluate Limits of Absolute Value

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Optimal Silo

Optimization In this video, I present a very interesting optimization problem, namely I calculate the dimensions of an optimal silo that minimize the cost of building the silo. The catch is that the top dome is more expensive than the lateral part. Enjoy this hour-packed calculus adventur

From playlist Calculus

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Introduction to Apportionment

This video provides an introduction to apportionment. Site: http://mathispower4u.com

From playlist Apportionment

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Apportionment: The Alabama Paradox

This video explains and provides an example of the Alabama paradox. Site: http://mathispower4u.com

From playlist Apportionment

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Apportionment: Hamilton's Method

This video explains and provides an example of the Hamilton's method of apportionment.. Site: http://mathispower4u.com

From playlist Apportionment

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Solving Absolute value equations with a variable on both sides

Learn how to solve absolute value equations with extraneous solutions. Absolute value of a number is the positive value of the number. For instance, the absolute value of 2 is 2 and the absolute value of -2 is also 2. To solve an absolute value problem, we first isolate the absolute value

From playlist Solve Absolute Value Equations

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Apportionment: Huntington-Hill Method

This video explains and provides an example of the Huntington-Hill method of apportionment. Site: http://mathispower4u.com

From playlist Apportionment

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How To Solve an Absolute Value Equation when There is Only One Solution

Learn how to solve absolute value equations with extraneous solutions. Absolute value of a number is the positive value of the number. For instance, the absolute value of 2 is 2 and the absolute value of -2 is also 2. To solve an absolute value problem, we first isolate the absolute value

From playlist Solve Absolute Value Equations

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Apportionment: Webster's Method

This video explains Webster's method of apportionment. Site: http://mathispower4u.com

From playlist Apportionment

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Apportionment: Jefferson's Method

This video explains and provides an example of Jefferson's method of apportionment.. Site: http://mathispower4u.com

From playlist Apportionment

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Math for Liberal Studies: Apportionment -- Introduction

In this video, we learn the basic concepts we use to solve apportionment problems. Once you have watched this video, you're ready to learn about how to actually solve an apportionment problem using one of the methods below: Hamilton's Method: Jefferson's Method: Adams' Method: Webster's

From playlist Math for Liberal Studies

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Apportionment: The New States Paradox

The video explains the new states paradox and provides an example of the population paradox. Site: http://mathispower4u.com

From playlist Apportionment

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Evaluate the limit for a value of a function

👉 Learn how to evaluate the limit of an absolute value function. The limit of a function as the input variable of the function tends to a number/value is the number/value which the function approaches at that time. The absolute value function is a function which only takes the positive val

From playlist Evaluate Limits of Absolute Value

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Math for Liberal Studies - Lecture 2.7.1 The Apportionment Problem

This is the first video lecture for Math for Liberal Studies Section 2.7: Apportionment. In this video, I give an overview of the apportionment problem: assigning representatives to states based on their population.

From playlist Math for Liberal Studies Lectures

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Solving Absolute Value Equations Using Multi Steps

Learn how to solve absolute value equations with extraneous solutions. Absolute value of a number is the positive value of the number. For instance, the absolute value of 2 is 2 and the absolute value of -2 is also 2. To solve an absolute value problem, we first isolate the absolute value

From playlist Solve Absolute Value Equations

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Math for Liberal Studies - Lecture 2.7.2 Hamilton's Method and Apportionment Paradoxes

This is the second video lecture for Math for Liberal Studies Section 2.7: Apportionment. In this video, we learn how to use Hamilton's Method to assign a whole number of seats to each state based on their population. We also discuss several "paradoxes" that result from this method.

From playlist Math for Liberal Studies Lectures

Related pages

Utilitarian rule | Mathematics of apportionment | Webster/Sainte-Laguë method | Egalitarian rule | Mathematical optimization | Leximin order