Nash-Equilibrium

Mixed Strategies: Why You Should Be Unpredictable You’ve learned about dominant strategies and Nash equilibria in pure strategies. But what happens when there’s no pure strategy Nash equilibrium? What if being predictable is your worst enemy? Welcome to the world of mixed strategies — where randomness becomes your most powerful weapon. The Problem with Being Predictable Imagine you’re a goalkeeper facing a penalty kick. You can dive left or right. The striker can shoot left or right. If you both go the same way, the striker scores. If you guess correctly, you save. ...

John Nash won the Nobel Prize for an idea so simple you can explain it in 5 minutes. Yet this idea revolutionized economics, predicted Cold War outcomes, explains why you’re stuck in traffic, and even helps explain evolution. Let’s understand Nash Equilibrium—the most important concept in game theory. The Core Idea (In One Sentence) A Nash Equilibrium is a situation where no player can improve their outcome by changing their strategy alone—everyone is doing the best they can given what everyone else is doing. ...

When most people hear the word “game,” they think of fun activities like chess, poker, or video games. But in mathematics, a “game” has a much broader and more precise meaning. Game theory studies strategic situations where the outcome depends not just on your choices, but on the choices of others too. What Makes Something a “Game”? In game theory, a game is any situation where: Multiple decision-makers (called “players”) interact Each player has choices (called “strategies”) they can make Each combination of choices leads to an outcome with specific rewards or costs (called “payoffs”) Players care about these outcomes and try to make rational decisions The beauty of this definition is that it applies to far more than board games. Business competition, political negotiations, evolutionary biology, and even social dilemmas are all “games” in the mathematical sense. ...

If you want to master game theory, you need to master payoff matrices. They’re the single most important tool for analyzing two-player games, and once you understand them, you’ll see strategic situations everywhere. A payoff matrix is simply a table that shows every possible outcome of a game and what each player gets in each scenario. But this simple visualization unlocks powerful insights about human behavior, business strategy, and why rational people sometimes make seemingly irrational choices. ...

Here’s one of the most unsettling discoveries in mathematics: perfectly rational players, each acting in their own self-interest, can all end up worse off than if they had acted irrationally. This isn’t a flaw in game theory—it’s a feature of reality that game theory reveals. This paradox explains traffic jams, arms races, overfishing, climate change negotiations, and why businesses sometimes engage in destructive price wars. Understanding it will change how you see human cooperation (and its failures). ...