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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

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Lesson Transcript

Instructor:
*Michael Quist*

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

There are many ways to approach game-like challenges, especially those that involve unknowns or other human players. In this lesson, we'll discuss the 'dominant' strategy and look at examples of how you can use it.

Most people like games. Whether it's chess, football, or Jeopardy, it's fun to participate in a simulated event where the excitement is real, yet the risk is limited. **Game theory** is the science of strategy, the study of math and logic behind conflict and cooperation. Everyone uses some sort of **strategy** (the algorithm, or logical set of steps, that determines how they make their choices) to approach life. Some strategies are effective for winning, others for maintaining an **equilibrium** where no one gets ahead, and some create losses.

For example, every move you make on a chessboard carries consequences and opportunities. If you advance your king's pawn two squares, you have also created a set of risks and possibilities for your opponent. His strategy will determine what he does with those risks and possibilities. Game theory is especially applicable in situations where models can be useful, such as economics or politics, and it can be helpful in games like poker or bridge.

A **dominant strategy** is one that will have the absolute best effects, no matter what your opponents or partners do. It's a set of choices that creates the highest balance of reward/risk ratios. Sometimes, there is no truly dominant strategy, which leaves you in an **intransitive** situation, where your strategy must adapt to the actions of others. For example, chess is an extremely complex game and is purely adversarial (one player must lose for the other to win), so it's almost impossible to create a dominant strategy. You and your opponent are constantly adapting using intransitive strategies.

The **dominated strategy** produces the absolute worst result, regardless of the choices made by the other players. Going back to our chess example, some players will randomly advance their pieces, heedless of the risks or disadvantages they create for themselves. This strategy causes the players to lose, regardless of how any normal opponent plays.

Most people are familiar with the rock-paper-scissors game. You and your opponent each hide one of your hands and then form scissors, rock, or paper with the other. Each choice wins against one of the others, loses against another, and ties if the opponent makes the same choice. Rock wins over scissors (smashes them), scissors wins over paper (cuts it), and paper wins over rock (covers it).

At first glance, this game would seem to be fairly intransitive. No matter what choice you make, there is a choice your opponent can make that will defeat you and another choice that will cause you to win. Is there a dominant strategy for this simple game? People who study game theory would say that there is a dominant strategy against a certain opponent given the moves he or she historically tends to make. The following steps might be part of your analysis and strategy:

- If you know that almost every time you begin a rock-paper-scissors game with a certain person, he starts with rock, then your dominant strategy would include using the correct counter-move and throw paper.
- If you observe that whenever your opponent loses a turn, he tends to pick the choice that defeated him for the next turn, then your dominant strategy can include a counter for that move.
- If you observe that your opponent tends to play the same move over and over again, you can counter that in your strategy until he chooses to change.

Using analysis like these steps, game theorists can establish frighteningly effective strategies, until it almost feels like they're reading your mind. The dominant strategy is not always apparent, though, and often depends on the sum of everyone else's choices and consequences.

For example, game theory can be used in economics. Imagine that there are only two companies that make cell phones. The two phones are virtually identical, so the only real reason for the customer to choose one over the other is price. To keep it simple, we'll assume the following:

- The phones cost $30 to make.
- The companies can charge either $50 ($20 profit) or $100 ($70 profit) for the phone.
- There are a million willing customers.
- If one company's price is higher than the others, that company will only get 10% of the market. If the two prices are the same, then each company will get 50%.

The possible situations and payoffs are shown in this figure. The first number in each pair is Company 1's profit, while the second number is what Company 2 will get.

Okay, notice what happens. If both companies go for the low price to make sure they get at least their market share, they will each earn $10 million. If one tries to go high, while the other stays low, the ratio shifts, with the high-priced company making only $7 million while the lower-priced company makes $18 million. Here's the key point. What if the companies collaborate? If they're working together, they can agree to both set their prices high, which will produce the most profit for both companies. In some situations, such as negotiation, diplomacy, arbitration, and similar situations, cooperation can offer the dominant strategy.

**Game theory** is the science of strategy and the study of how math and logic apply to matters of conflict and cooperation. A **strategy** is the set of steps that you use to make decisions. A **dominant strategy** is the one that produces the best results, regardless of other choices made in the situation. An **intransitive** strategy is one that depends upon the strategies chosen by others. The **dominated strategy** is the one where the results are the worst, regardless of how the other participants react. The study of games and how they're played is serious business.

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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

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