The Game Where You Guess What They're Guessing
The Infinite Mirror of Tic-Tac-Toe
You are playing tic-tac-toe against your best friend. You put an X in a corner. She puts an O in the center. You already know you have to block her next move, but she knows you have to block her. So you try to set a trap two moves ahead. But she sees the trap, so she ruins it.
This back-and-forth of “I think that she thinks that I think” is at the heart of a field called epistemic game theory. It is the study of what happens when you have to guess the plans of someone who is guessing your plans. It is a game played not just with cards or tokens, but with thoughts. And understanding it can help us make better decisions, build smarter computers, and even figure out how trust works.
The Map and the Mind
To understand the trick, we first need a map. Philosophers and economists call any situation where you have to make a choice a game. A game has three parts: the players, the choices (or strategies) they can make, and the payoffs (the points, money, or satisfaction you get from the outcome).
In the 1960s, a thinker named Thomas Schelling (1921–2016) noticed something crucial about games. He called them “interdependent decision problems.” That is a fancy way of saying your best move depends entirely on what the other person does. If your friend runs left, you want to throw the ball to the left. But if she runs right, you have to throw right.
Epistemic game theory adds a new layer to this map. It asks two huge questions that go beyond the rules of the board. The first is strategic uncertainty: What will the other player actually do? The second, trickier question is higher-order information: What do they believe I will do? And what do they believe that I believe they will do? It creates a ladder of beliefs that stretches up forever.
Never Choose Broccoli
Imagine you have to pick a dessert, but your friend is picking a topping, and you hate broccoli. The options are cake or broccoli. There is absolutely no topping your friend could pick that would make broccoli taste better than cake. Choosing broccoli would be a huge mistake.
In game theory, this is called a strictly dominated strategy. It is a choice that is always strictly worse than another option, no matter what your opponent does. A basic rule of epistemic game theory is that a rational player—someone who always acts to get the best possible result given what they believe—will never choose a strictly dominated strategy.
Now, this rule creates a chain reaction. If I know you are rational, I know you will scratch broccoli off your list. Suddenly, the game is smaller. In this new, smaller game, maybe a topping that was only bad because of broccoli is now the clear winner. This process of scratching out bad strategies, then scratching out the strategies that become bad in the new game, is called iterative elimination of strictly dominated strategies (IESDS). Logicians Robert Bernheim and David Pearce showed in the 1980s that if we assume everyone is rational, and everyone believes everyone is rational, and everyone believes that, and so on forever—what we call common belief in rationality—the only strategies left will be the ones that survive this scratching-out process.
The Coin Flip in Your Brain
But what if being predictable is the worst thing you can do? Consider the game “Matching Pennies.” You and an opponent each secretly flip a coin to show Heads or Tails. If the coins match, you win a dollar. If they don’t match, you lose a dollar.
If your opponent can predict your choice, they will win every time simply by mimicking you. The only way to protect yourself is to become completely unpredictable—even to yourself. You must use a mixed strategy, where you randomize your choice. Maybe you flip a coin in your head, deciding to show Heads exactly 50% of the time.
This idea is deeply controversial among philosophers. Ariel Rubinstein questioned in 1991 whether humans can ever be truly random, or if a mixed strategy just represents the opponent’s uncertainty about our secret plan. If you have a surefire plan to win, can you force yourself to flip a coin and possibly lose? Most philosophers agree that for the math to work in these high-stakes games, we have to act as if we are randomizing, even if deep down we might not be.
The Broken Mirror of Assumptions
If you think too deeply about thinking, you can fall into a trap. In 2006, Adam Brandenburger and H. Jerome Keisler discovered a brain-twisting paradox that shows the limits of common knowledge.
Imagine two players, Ann and Bob. Ann believes that Bob assumes that Ann thinks Bob’s assumption is wrong. An assumption is your strongest possible belief. Now ask: Does Ann think Bob is wrong?
If Ann thinks Bob is wrong, then Bob’s assumption (whatever it is) must be false. But if Bob’s assumption is that Ann thinks he is wrong, and Ann does think he is wrong, then Bob is actually right. So if Ann thinks Bob is wrong, Bob is right. But if a very smart Ann realizes this, she will see that thinking Bob is wrong leads to a contradiction. So she must think Bob is right. But if she thinks Bob is right, then Bob’s assumption is true. But wait, we just assumed Ann thought Bob was wrong at the start!
This logical loop proves something profound: You cannot build a perfect map of everyone’s beliefs that includes all possibilities. There will always be a statement—like Ann’s belief about Bob’s assumption—that breaks the system. Just like looking into two mirrors facing each other, the image eventually dissolves into blurry chaos. Our reasoning has fundamental limits.
Why We Still Play
Sometimes this all seems like an abstract puzzle about coins and chessboards. But these rules actually run the modern world. When computer scientists build an AI to play poker—where bluffing and mind-reading are everything—they use epistemic logic. The AI doesn’t just calculate odds; it builds a model of what you think it is holding, and what you think it thinks you are holding.
Economists use this to design auctions and trade deals. They have to figure out what buyers believe about sellers and what sellers believe about those beliefs. And in our daily life, we make choices based on our “theory of mind.” When you cross the street, you’re not just guessing if the driver sees you; you’re guessing whether the driver assumes you will stop, and whether the driver assumes you assume they will speed up.
The math of epistemic game theory shows us that pure logic sometimes demands we act randomly to survive, and that a perfectly clear view of another person’s thoughts is mathematically impossible. Understanding this doesn’t just make you better at games. It teaches us that our choices are always wrapped up in the mysterious, tangled, and infinite task of guessing what other people believe.
Think about it
- If a computer can always pick the strictly rational move, but a human sometimes picks a risky one because they have a “bad feeling,” who is playing the game better?
- You’re in a staring contest. You want to blink, but you don’t want them to know you want to blink. How does your strategy change if you know they are thinking the exact same thing?
- Can you ever prove that a choice you made was completely random, or does your brain always have a secret pattern that can be predicted?
