Winning a “Beauty Contest,” Or How My Professor Gambled $250 Teaching a Lesson in Game Theory

Using game theory is like playing high stakes poker. You can predict the right moves, but you can still get burned by surprises. Game theorists often gamble more than just academic integrity and pride. In one very memorable lecture, my college professor staked $250 to teach a lesson about crowd behavior.

The lecture began innocently enough. We were going to play a simple game. Here are its rules:

1. Everyone secretly submits a whole number from 0 to 20.

2. All entries will be collected, and the guesses will be averaged together.

3. The winning number will be chosen as two-thirds of the average, rounded to the closest number. For instance, if the average of all entries was 3, then the winning number would be chosen as 2. Or if the average was 4, the winning number would be 3 (rounded from 2.6666…).

4. Entries closest to the winning number get a prize of meeting with the professor over a $5 smoothie. (In the academic version of the game, multiple winners split the prize, but my professor was being generous).

Before you read on, I would like you to seriously consider what number you might pick.

Imagine you are sitting in a Stanford lecture hall and actually playing this game. You seek the glory of outsmarting 49 other Stanford kids, and you really want to meet with the professor since you find game theory fascinating. You have 10 seconds to decide before ballots are collected. Which number would you pick?

Please write down your answer on a piece of paper before reading on.

Some Guiding Logic

The game is called a “p-beauty contest.” The “p” refers to the proportion the average is multiplied by–in this case, p is two-thirds. If you’re wondering, the game has the same flavor for any value of p less than 1. Why is it called a beauty contest? It’s because the game is the numbers-analog to a beauty contest developed by John Maynard Keynes.

Here is the beauty contest that Keynes pondered. Imagine that a newspaper runs a contest to determine the prettiest face in town. Readers can vote for the prettiest face, and the face with the most votes will be the winner. Readers voting for the prettiest face will be entered in a raffle for a big prize.

How does the game play out? Keynes wanted to point out the group dynamics. The naive strategy would be to pick the face you found to be the most attractive. A better would be to picking the face that you think other people will find attractive.

The number “beauty contest” has the same kind of logic. You don’t pick a number you like. You pick a number that’s closest to two-thirds of the average of everyone else. The twist of both games is that your guess affects the average outcome. And each person is trying to outsmart everyone else.

Given the subtlety of the game, my professor was banking on paying out to only a few winners. Although it was mathematically possible for each of us to win, and he was taking that risk. In fact, he knew that if we were all rational, we would all win. He would have to pay out a $5 smoothie to 50 students–that is, he made a $250 gamble playing this game.

Why was he so confident? Let’s explore the solution to the game and see why it’s hard to be rational.

Numbers You Shouldn’t Pick

Even though it’s not possible to know what other people are guessing, this game has a solution. If everyone acts rationally, there are only two possible winning numbers. It takes some crafty thinking, but it is really based on two principles I think you will accept.

Principle 1: Don’t Play Stupid Strategies (Eliminate Dominated Strategies)

The first principle is that players should avoid writing down numbers that could never win. That sounds logical enough, but it’s not always the case. We all can agree that writing a number that could never win is just a dumb, stupid strategy. You are picking an option that’s inferior to something else, and hence is known as a dominated strategy. (For an example, see this article about a dominated strategy in real life).

Are there any dominated strategies in the beauty contest?

To start answering that question, we need to figure out which numbers will never win. A natural question is what is the highest winning number? You would never want to pick a number larger than that, unless you want to lose.

You know that the highest number anyone can pick is the number 20. If every single person picked 20, then the average would be 20. The winning number would be two-thirds of 20, which is 13 when rounded.

Should you ever find yourself submitting 20?

The answer is no–there is always a better choice, say the number 19. The only time 20 wins is precisely when everyone else picks it and everyone shares the prize. In that case, you would be better off writing 19 to win the prize unshared. Plus, by writing 19, you can possibly win in other cases, like when everyone picks 19. You are always better off writing 19 than 20. The guess of 20 is dominated–it’s dumb.

You should never choose 20. And your rational opponents should be thinking the same way. So here’s the big result: you can reason that no player ever chooses 20.

Principle 2: Trim the Game, and Apply Principle 1 Again (Iterate the Elimination Process)

By principle 1, no player will ever choose 20. Therefore, you can essentially remove 20 as a choice. The game trims to a smaller beauty contest in which everyone is picking a number between 0 and 19. The smaller game has survived one round of principle 1.

Now, repeat! Ask yourself: in the reduced game, are there any dominated strategies?

Now 19 takes the role of 20 from the last analysis. Since 19 is the highest possible average, it will never be a good idea to guess it. Applying principle 1, you can reason that 18 is always a better choice than 19. Thus, 19 is dominated and should be eliminated as a choice for every player.

The game is now trimmed to picking numbers from 0 to 18. This is the result of two iterations of principle 1.

There’s no reason to stop now. You can iterate principle 1 to successively eliminate choices of 18, 17, 16, and so on. The only numbers remaining will be 0 and 1. (This requires 19 iterations of principle 1.)

There is a name for this thought process. It’s aptly named, but a mouthful: iterated elimination of dominated strategies (IEDS). The idea is to eliminate bad moves, trim the game, and iterate the process to find the surviving moves.

These remaining strategies are considered to be rationalizable moves, that is, moves that can possibly win.

Here’s the schematic for the IEDS:

The Equilibriums

The only reasonable choices are to pick the numbers 0 and 1. Is either a better choice? This is unfortunately where IEDS cannot give insight.

It’s possible to have 0 as a winning number–imagine all players picked 0. (The winning number will be 0).

It is also possible to have 1 as a winning number–imagine all players picked 1. (The winning number is 2/3, which rounds to 1.)

The answer will depend on what people think others will be guessing. Both equilibriums–all 0 and all 1–are achievable.

Back to the Classroom

None of us in the class had this deep understanding of IEDS. We were just learning game theory–it was actually our third lecture. If you’ve read one or two of my articles, then you know more about game theory than what we knew when we faced the game. My professor was pretty sure our guesses would be all over the place.

But Stanford kids can be crafty. One student used some sharp thinking and realized that coordination would help; he asked if we could talk to each other. The professor, still feeling we were novices, confidently replied with a smile, “Sure. Go ahead.” We only had 10 seconds to write down our answers anyway.

Before the professor could change his mind, the student quickly shouted to all of us, “If we all write down 0, we all win.”

It was remarkable. He figured out the equilibrium and told us what to do! He couldn’t be tricking us because the math was clear: if we all picked 0, we would all have winning numbers.

My professor’s face seemed to drop. That’s $250 on the line. (He never let future years talk).

How Smart Are Stanford Kids?

The professor was relieved after he tallied the votes. He told us that admirably most of us wrote down the number 0 (I was among those who did). But there were larger answers too, ranging from 1 to 10.

Someone actually wrote down 10! And this was after being told the answer.

After all was said and done, the winning number turned out to be 2, and the prize was awarded to three students. Thanks to our irrationality, my professor only paid out $15.

It was even better. My professor grilled the students who wrote down larger numbers. They all squirmed, as he was physically intimidating, and explained reasons like “it was my lucky number” or “I don’t know. I wasn’t really thinking.”

The Practical Lesson

What is going on? This is a group of smart students that was told the answer to the game.

The example illustrates a flaw of IEDS. It can get you reasonable answers if you think players are reasoning out further and further in nested logic. We don’t have infinite rationality, only bounded rationality.

The practical answer to what you should write depends on the book answer plus your subjective beliefs about what other people do. It’s the combination of book smarts plus social smarts that matters.

The people who wrote down the winning numbers told the class they suspected some people would deviate for irrational reasons. And they were rewarded for not confusing theory and practice.

Additional reading

The stock market is another example of a beauty contest–prices depend on fundamentals and investor psychology. For one example, check out this article on FDA fast-tracking and the stock market.