3
Part I · The rules of the game

Nobody wants to switch

John Nash asked a simple question about any game: after everyone has chosen, does anyone wish they had chosen differently? When the answer is no for every player at once, the game has settled. That settled place carries his name. Every game has one, though not always where you expect, and sometimes not in the grid at all.

Here is the test. Pick any box in a payoff grid. Cover Robo's choice with your hand and ask: "Knowing Robo will do that, would I rather be in a different row?" Then cover your own choice and ask: "Knowing I will do this, would Robo rather be in a different column?" If both answers are no, the box is a Nash equilibrium (say: NASH ee-kwi-LIB-ree-um). Nobody wants to switch. The game has come to rest.

Chapter 2 gave this a name: each player is playing a best response to the other. An equilibrium is a box where both best responses land at once.

Five games. Click boxes to test them. One game has one equilibrium, some have two, and one has none at all. Solve all five for a star.

Equilibrium hunt

Five games, five different answers.

Arrows find it for you

There is a faster way than testing every box. In each column, draw a red arrow from every box toward your best response in that column. In each row, draw a blue arrow toward Robo's best response. A box that no arrow leaves is an equilibrium: nobody has anywhere better to go.

Choose a game below and the arrows appear. Then press Random game and, before the arrows show, predict how many equilibria the grid has. Get five in a row right for a star.

Best-response arrows

Red arrows: where you would move. Blue arrows: where Robo would move.
Game
Streak0

"Let them play" drops a token on a random box and lets you and Robo take turns moving to your best responses. Sometimes the token settles. Sometimes it chases its tail forever.

Design your own

Type any payoffs you like. The board counts the equilibria as you type. Can you make a 2 by 2 game with exactly three? Think about what has to happen at the fourth box.

Equilibrium counter

Red numbers are yours, blue are Robo's. Ties count.
Equilibria0
Math corner

The definition, properly. Write your payoff as u(a, b) and Robo's as v(a, b), where a is your row and b is Robo's column. A pair (a*, b*) is a Nash equilibrium when

u(a*, b*) ≥ u(a, b*) for every row a
v(a*, b*) ≥ v(a*, b) for every column b

The first line says no row beats a* against b*. The second says no column beats b* against a*. Notice the ≥: a tie counts. If switching gets you exactly the same payoff, you do not want to switch.

Nash's theorem (1950). Every game with a finite number of players and a finite number of strategies has at least one equilibrium, as long as players are allowed to choose at random with chosen odds. Matching Pennies has no equilibrium among the four boxes, but it has one if each player flips a coin. That is Chapter 4.

Big idea

A Nash equilibrium is a box where each player is already playing a best response to the other, so nobody wants to switch. It is where a game comes to rest. It is not always good for the players, it is not always fair, and there is not always exactly one, but there is always at least one once dice are allowed.

Try it on paper

1. Rows Up and Down, columns Left and Right. Payoffs (yours, Robo's): Up-Left (2, 1), Up-Right (0, 0), Down-Left (0, 0), Down-Right (1, 2). Find every equilibrium.

Answer

Two: Up-Left and Down-Right. You prefer Up-Left, Robo prefers Down-Right, but both are settled. This game is sometimes called the Battle of the Sexes; a better name is "which movie shall we see".

2. A dominant strategy equilibrium (like Grab-Grab in the Cookie Game) is always a Nash equilibrium. Explain why in one sentence.

Answer

A dominant strategy is a best response to everything, so it is certainly a best response to whatever the other player is doing.

3. Can a game have an equilibrium where one player is playing a dominated strategy?

Answer

Not a strictly dominated one. If some other row is strictly better in every column, it is strictly better in the equilibrium column too, so the player would want to switch. Weakly dominated strategies can appear in equilibria, because ties count.

Challenge

Use the equilibrium counter to build a 2 by 2 game with exactly three equilibria. Then explain why exactly three is possible but requires a tie somewhere.

Answer

One way: make three boxes pay (1, 1) and the fourth pay (0, 0). Every (1, 1) box is settled because moving away either keeps you at 1 (a tie, no reason to move) or drops you to 0. The (0, 0) box is not settled. Why a tie is needed: with no ties, each column has exactly one red arrow target and each row exactly one blue target, so at most one box per column can be an equilibrium, which caps a 2 by 2 game at two.

True story

John Nash proved his theorem in 1950 as a 21-year-old graduate student at Princeton, in a PhD thesis shorter than many school reports. The idea was so simple that the great John von Neumann, when Nash described it to him, reportedly said "that's trivial, you know, that's just a fixed point theorem." It was not trivial. It became the foundation of modern economics, and in 1994 Nash shared the Nobel Prize for it. In between, he spent decades fighting a serious mental illness, a story told in the film A Beautiful Mind. He died in 2015, at 86, on the way home from receiving another of mathematics' top prizes.

Stars in this chapter

Earn them by doing the clever thing, not by clicking around.