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8 of 13

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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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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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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Game theory needn't mix you up! In this final lesson of William Spaniel's Game Theory 101 series, learn how to solve game theory scenarios with three possible strategies. Expand the mixed strategy algorithm and apply it to games with more than two strategies. Figure out the mixed strategy Nash equilibrium for a modified version of Rock, Paper, Scissors. Solve for a player's indifference and calculate expected utilities as a function of a given mixed strategy.

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6 of 13

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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5 of 13

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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7:12
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You've tackled Game Theory 101, so what's next? William Spaniel gives you a sneak preview of more advanced Game Theory concepts such as repeated games and incomplete information. What the heck is a Bayesian Nash equilibrium? Who gets a better deal in a negotiation, and why? How and when is war inefficient? Why do people vote? All these questions and more can be answered with the magic—um, science—of game theory!

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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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7:47
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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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7:23
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13 of 13

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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7 of 13

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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9 of 13

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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3 of 13

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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4 of 13

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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6:32
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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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13:01
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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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7:58
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Slice into the precarious world of knife-edge equilibria - a kind of game theory equilibrium that exists for a single and exact payoff value. If this value varies in even the slightest way, the game matrix equilibrium is destroyed. Follow along with William Spaniel and his Game Theory 101 lesson as he explores this rare occurrence that is frequently ignored in game theory studies.

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8:52
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You know about backward induction in game theory, which assumes that all future play will be rational. But what about the opposite technique? In this lesson from William Spaniel's Game Theory 101 series, you'll be introduced to forward induction, which assumes that all past play was rational. Using a version of the Stag Hunt game, learn how Player 2 can look at payoffs and use forward induction to infer Player 1's move.

Assignment:
7:51
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Discover the Golden Rule - of Probability, that is - for solving game theory puzzles! Follow along with William Spaniel as he introduces advanced strategic form games and probability distributions using ‘Battle of the Sexes.’ This lesson will teach you an efficient way of solving a game by introducing payoff variables, along with handy rules and formulas - a great alternative to repeatedly calculating outcomes with the mixed strategy algorithm!

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