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Game Theory 101 (#9): How NOT to Write a Mixed Strategy Nash Equilibrium
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Game Theory 101 (#9): How NOT to Write a Mixed Strategy Nash Equilibrium

Introduction

In this section, William Spaniel introduces the topic of how not to write a mixed strategy Nash equilibrium in game theory. He highlights a common mistake made by beginners and emphasizes the importance of understanding this concept to avoid losing easy points in homework and exams.

How Not to Write a Mixed Strategy Nash Equilibrium

  • Writing fractions as decimals can lead to incorrect solutions.
  • For example, writing 1/3 as 0.33 is technically wrong.
  • The derived equations for player two's mixed strategy will not hold if decimals are used instead of fractions.
  • Substituting 0.33 for Sigma left in the equations results in inconsistent expected utilities for player one.
  • Player one's expected utility for up is -0.35, while the expected utility for down is -0.33, indicating that down is slightly better.
  • As player one has a profitable deviation by always choosing down instead of using the mixed strategy, this cannot be a Nash equilibrium.

Conclusion

The key takeaway from this lesson is to write mixed strategies as fractions rather than decimals. Using fractions avoids rounding issues and ensures accurate analysis without losing points due to incorrect conclusions or arguments.

Video description

Game Theory 101: The Complete Textbook on Amazon: https://www.amazon.com/Game-Theory-101-Complete-Textbook/dp/1492728152/ http://gametheory101.com/courses/game-theory-101/ For a player to be willing to mix between two strategies, he must be indifferent between them. Put differently, he must expect to earn exact the same amount by choosing either strategy in expectation. As a result, we must be very careful when we write mixed strategies. Although we commonly write 1/3 as .33, those two numbers are not equal to one another. This lecture shows why using decimals can be problematic. In general, you should always play it safe and write numbers as fractions, not decimals.