Can war be mutually beneficial? Because the costs and benefits of war can be complex and dependent on the environment, you might think that a simple example couldn't hold the answers to cost-benefit analyses used in real situations. Think again: This lesson by William Spaniel provides a no-frills, theoretical example involving Colombia, Venezuela, and $80 billion worth of oil that confirms war inefficiency through simple mathematics.
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Want to compare, add, or subtract fractions? To complete any of these operations, you need to find a common denominator. In this quick and easy math lesson by Eric Buffington, learn how to find the least common multiple (LCM) and put two fractions on a level playing field. Finally, test your skills with several practice problems.
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Adding and Subtracting Fractions is the most difficult of the operations. In this lesson we go over, in a visual way, the basics of adding or subtracting fractions, then practice practice practice.
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Tackle the ‘Battle of the Sexes’ game from a new angle and learn to solve for pure and mixed strategy Nash equilibria using variables instead of numbers. This Game Theory 101 lesson explains how to find the expected payoff of certain strategies using the mixed strategy algorithm and probability distributions - even when some factors in the game are unknown. Use the approach covered in this lesson to understand and solve a variety of generalized games.
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You already know how to multiply fractions; what about division? In this basic math lesson, Mr. Buffington shows you a quick trick with reciprocals to help you divide fractions without a calculator. You’ll also work with negative fractions to practice your understanding of operators and their effect on a number’s positive or negative value.
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Mixed strategies, pure strategies, Nash equilibria... why can't they all get along? In this lesson from William Spaniel's Game Theory 101 series, learn the rules to determine if a pure strategy is in support of a mixed strategy Nash equilibrium. You’ll work on calculating probability and expected utility, but this time will apply your skills to a 3 by 3 rather than the usual 2 by 2 game.
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Everyone loves having choices in life. In this lesson, learn two methods for solving equivalent fractions with variables and choose the method you like best! After you pick your favorite method, Mr. Buffington provides a few practice problems so that you can cement your knowledge of this important skill.
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Learn how not to write a subgame perfect equilibrium with this lesson from William Spaniel's Game Theory 101 series. Avoid the classic blunders that can trip you up and lose you points on an exam: remember that a subgame perfect equilibrium is a complete and contingent plan of action, and must state what happens on as well as off the equilibrium path of play. This lesson includes a handy trick to check your work by comparing the number of strategies you list with the number of game nodes.
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In part two of a three-part series, Eric Buffington breaks down several practice problems for adding and subtracting fractions and reviews reducing fractions to lowest terms. Whether you have never worked with fractions before or you want to brush up on old skills, this lesson has got you covered. Note: you’ll need to know how to reduce fractions to lowest terms and find the lowest common denominator.
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Have you ever struggled to know what gets done first in a math problem? This lesson outlines the correct order of operations, and many of the common challenges and mistakes people make when trying to solve math problems. Tackle your next math problem with the right strategy - and you'll have an easy time knowing when to add, subtract, multiply, and divide for the correct answer.
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You’ve already learned how to add and subtract fractions; but can you apply your skills to word problems? This lesson, the final part of a three-part series, breaks down three word problems that require you to find the least common multiple (LCM), add or subtract fractions, and reduce the answer to lowest terms. Each step is clearly explained and demonstrated so that you can solve similar questions on your own.
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In this lesson by Eric Buffington, learn how to solve algebraic equations that involve fractions. Whether you need to add, subtract, multiply, or divide, the process is just a combination of basic math skills. Get started with a few practice problems. As a bonus, you’ll learn about the property of equality and improve your understanding of equations.
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Have you ever lost points on a game theory assignment or test because you came to a slightly wrong conclusion? In this installment of his series on Game Theory, William Spaniel reveals one of the most frequent mistakes game theory students make: expressing mixed strategy Nash equilibria as decimals, not fractions. William demonstrates why 1/3 is not the same as .33, supporting his claim that when solving game theory equations, its always safe to stick with fractions.
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Do we have free will, or do we just boil down to particles following the laws of physics? Professor Massimo Pigliucci from Plato Footnote leads us into a world where science and philosophy collide by discussing the concept of reductionism. This lesson also includes an explanation of the difference between ontological and epistemological reductionism, and how these debates come to bear on ideas of emergent properties, morality, and free will.
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Never struggle with multiplying fractions again. In this quick and easy math lesson, Eric Buffington explains how to multiply several different kinds of fractions and get a simplified answer by reducing to lowest terms. You’ll also learn how to work with variables and negative fractions and cement your knowledge with several practice problems.
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The conclusion set out in "War's Inefficiency Puzzle" was that war is inefficient; this was proven using a theoretical example involving Colombia, Venezuela, and $80 billion worth of oil. However, what if we want a purer mathematical model with which to draw conclusions? This lesson from William Spaniel demonstrates how to create an algebraic model of war, which reinforces the conclusion that war is inefficient.
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Game theory fans, meet comparative statics. This lesson from William Spaniel's Game Theory 101 series expands on the previous Soccer Penalty Kicks scenario to explore the world of comparative statics. Discover how a game's outputs change as a function of the game's inputs – in this case, a change in the probability of the kicker scoring a goal when he kicks in a certain direction. Learn and practice the four steps to calculating comparative statics.
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Science clearly progresses. All the time, new discoveries are made, advancements in medicine are discovered, and theories are refined. But what does philosophy achieve? When asked about philosophers do you think of wise long-bearded men who sit and talk in circles, never reaching any conclusions or useful agreements? While philosophy may not answer the same questions that science does, it doesn’t aim to. So learn what philosophy has to offer, and what progress in the field actually achieves.
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You and your best friend are offered free money, and so long as you both choose to accept it in a blind vote, you both get richer! What could go wrong? Learn about the Free Money game theory scenario, and why there's a possible outcome where everyone ends up poorer (but is still a Nash equilibrium). The Free Money game is an exception to the Odd Rule, which states that almost all games have an odd number of equilibria.
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