Game Theory

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Meet Alice and Bob

In math and computer science, when you need two people for an example, you use Alice and Bob. Always. It started with a 1978 paper on cryptography and just stuck. Every textbook, every paper, every thought experiment—Alice and Bob.

Why does this matter? Because game theory revolves heavily around thought exercises, and Alice and Bob are the most common names used. If you go read a paper on cryptography, economics, network theory—you'll see Alice and Bob everywhere. Now you know why. You're in on it.

But more importantly: game theory isn't really about games. It's about thought. It's about modeling what happens before it happens. Alice doesn't move. Bob doesn't move. They THINK about moving. They simulate each other's possible actions inside their own heads. The entire field runs on thought experiments because the power IS the thought.

The game is never played. The game is computed.

What's a Game?

In game theory, a "game" is any situation where:

1. There are multiple players (at least two—Alice and Bob).

2. Each player has choices (strategies).

3. The outcome depends on everyone's choices combined.

4. Each player has preferences about outcomes (payoffs).

That's it. A game. By this definition, chess is a game, but so is negotiating rent with your landlord. So is merging onto the highway. So is deciding whether to study for an exam when your grade depends on the curve—which depends on whether OTHER students study.

Any time your outcome depends on someone else's decision AND their outcome depends on yours—you're in a game. And game theory is the math of figuring out what to do.

The Prisoner's Dilemma: The Most Famous Game

Alice and Bob committed a crime together. They're arrested and put in separate rooms. Each is offered a deal:

If you rat on the other person and they stay silent: you go free, they get 10 years.

If you both stay silent: you each get 1 year.

If you both rat: you each get 5 years.

What do you do?

Here's the math. If Bob stays silent, Alice's best move is to rat (0 years beats 1 year). If Bob rats, Alice's best move is ALSO to rat (5 years beats 10 years). No matter what Bob does, ratting is Alice's best individual strategy. Same logic for Bob.

So they both rat. They both get 5 years. Even though if they'd BOTH stayed silent, they'd only get 1 year each.

That's the dilemma. Individual rationality leads to collective irrationality. The best move for each person, independently, produces the worst outcome for both. The selfish path is a trap.

This isn't just a thought experiment. This is climate change (everyone's better off polluting individually, everyone's worse off when everyone pollutes). This is arms races. This is why competitors cut prices until nobody profits. This is why trust matters—not as a moral luxury, but as a mathematical advantage.

Nash Equilibrium: Where Nobody Wants to Move

John Nash. Beautiful mind. Actual beautiful mind, not just the movie. In 1950 he proved that every finite game has at least one Nash equilibrium—a set of strategies where no player can improve their outcome by changing strategy alone.

In the Prisoner's Dilemma, both ratting IS the Nash equilibrium. Neither Alice nor Bob can do better by unilaterally changing their choice. If Alice switches to silent while Bob rats, she goes from 5 years to 10. Worse. So she stays put. That's equilibrium—not because it's optimal, but because nobody can individually improve.

Nash equilibria aren't necessarily good outcomes. They're stable outcomes. The distinction matters. A lot of terrible situations in the world are Nash equilibria—everyone's stuck because nobody can improve things alone. Breaking out requires coordination. Trust. Communication. Cooperation.

This is why game theory matters beyond math. It explains WHY cooperation requires effort and defection is the default. It's not that people are bad. It's that the structure of many situations makes selfishness the equilibrium. Changing the outcome means changing the game—adding communication, adding repeated interactions, adding consequences.

The Trampoline: Competition and Cooperation

Think of a trampoline. You compete with the force of gravity going up—you push against it, you fight it, you launch yourself skyward. Then you cooperate with it going down—you let it pull you, you use its force to compress the springs, to store energy. Both of these are required to maximize the bounce.

That's the deep insight of game theory. Competition and cooperation aren't opposites. They're two phases of the same motion. You compete to cooperate and cooperate to compete.

In a market, companies compete for customers. But they cooperate with standards, with supply chains, with the basic rules of trade. Without the cooperation, the competition collapses. Without the competition, the cooperation stagnates.

In evolution, organisms compete for resources. But cells cooperate within organisms. Species cooperate in ecosystems. The competition happens WITHIN a cooperative framework, and the cooperative framework exists BECAUSE of competition.

Game theory shows this mathematically. The games where everyone does best aren't zero-sum (your gain is my loss). They're positive-sum—the total grows when players cooperate, even while competing within that cooperation. The trampoline analogy is real. You push against gravity to go up, you work WITH gravity to come down, and the cycle amplifies the motion. Fighting it all the time gets you nowhere. Going limp gets you nowhere. The maximum motion requires both.

Zero-Sum vs Positive-Sum: The Casino

Remember the preface? Life is a casino. Now let's be precise about what kind.

A zero-sum game is one where the total payoff is fixed. What you win, I lose. Poker is zero-sum—the money on the table doesn't grow or shrink, it just moves between players.

A positive-sum game is one where the total payoff can grow. Trade is positive-sum—I have apples, you have oranges, we both prefer variety, so after trading we're both better off. The total happiness increased.

Most of life is positive-sum. Most of the things that feel zero-sum actually aren't—we just treat them that way because we're stuck in the wrong game theory. Job markets feel zero-sum (if you get the job, I don't) but they're actually positive-sum (more skilled workers create more opportunities for everyone).

The casino is an interesting case. The house has an edge—that's guaranteed. So player versus house is negative-sum for the player. But the entertainment value, the social experience, the thrill—that's value the player gets even when losing money. Whether the casino is a good deal depends on what you're counting as payoff.

Game theory forces you to be explicit about what you value. Money? Status? Time? Fun? Relationships? The math works no matter what you plug in. But the answers change dramatically based on what you're optimizing for.

Repeated Games: Why Nice Guys Finish First

The Prisoner's Dilemma is tragic when played once. But what if you play it over and over, with the same person, and you remember what they did last time?

In 1984, Robert Axelrod ran a tournament. He invited game theorists to submit strategies for repeated Prisoner's Dilemma. The strategies were run against each other, hundreds of rounds each.

The winner was the simplest strategy submitted. It was called Tit for Tat:

Start by cooperating. Then do whatever the other player did last round.

That's it. Cooperate first. If they cooperate, keep cooperating. If they betray, betray back once. If they return to cooperation, forgive immediately.

Tit for Tat won because it has four properties: it's nice (never betrays first), it's retaliatory (punishes betrayal immediately), it's forgiving (returns to cooperation as soon as the other player does), and it's clear (the other player can easily figure out what you're doing).

This is boundaries. Mathematical boundaries. Be kind by default. Respond to aggression. Forgive when aggression stops. Be predictable enough that the other player can learn your pattern and choose cooperation.

Over repeated interactions, cooperation emerges not because players are altruistic but because betrayal has consequences and cooperation has rewards. The STRUCTURE of repeated games—the fact that there's a future where your reputation matters—transforms the equilibrium from mutual betrayal to mutual cooperation.

This is why community works. Why relationships work. Why trust works. Not because of feelings. Because of math. Repeated games with the same players change the optimal strategy from selfish to cooperative.

The Power of Thought

Here's what I want you to take from this whole chapter, and honestly from this whole book.

Alice and Bob never move. They think. They model. They simulate. They consider what the other person might do, what they'd do in response, what the other person would do in response to THAT. The game is played in the mind before it's played in the world.

That's what math is. It's the power to simulate reality before experiencing it. Arithmetic lets you predict how quantities combine without counting physical objects. Algebra lets you reason about unknowns. Geometry lets you understand space without physically measuring it. Calculus lets you predict motion. Statistics lets you assess paths under uncertainty. Linear algebra lets you handle systems. Boolean logic lets you reason precisely. Topology lets you understand shape without measurement.

And game theory lets you understand what happens when you're not the only one thinking.

You walked into this book and I said life is a big ass casino. Now you know the math of the casino. You know probability—the measure of available paths. You know linear algebra—the degrees of freedom. You know Boolean logic—the binary choices underlying every decision. You know topology—the shape of the possibility space. And now you know game theory—the math of what happens when every player at the table is doing the same calculations you are.

The game is never played. The game is computed. The computation is the game. And now you can compute.

Welcome to the casino.