Not long ago, I read “The Fundamentals Are Sound” by Rusty Guinn at Epsilon Theory. What particularly intrigued me was his recounting of Dick Thaler‘s take on the Keynesian Beauty Contest (which is actually borrowed from a study by Rosemarie Nagel, which in turn was based on the “Guess 2/3 of the Average” game created by Alain Ledoux).

### The Keynesian Beauty Contest

Before covering the Thaler/Nagel/LeDoux version, however, a brief introduction to the original Keynesian Beauty Contest is in order. Mr. Guinn did as good of a job as anyone could, so I’ll quote him:

In Keynes’s version of the contest, you win by correctly picking the woman from a series of pictures in a newspaper that you think will be voted as the most beautiful by everyone participating. First-degree thinking, in Keynes’s parlance, is to pick the woman you believe is the most beautiful. Second-degree thinking is picking the woman that you believe the other participants will believe is the most beautiful. Degrees above that require thinking less about beauty or what others will think is beautiful, and more about what the contestants are likely to think about one another.

### Degrees of Thought

- Zeroth Degree: Selecting a random woman
- First Degree: Selecting the woman you think is most beautiful
- Second Degree: Selecting the woman you think everyone else will think is most beautiful
- Third Degree: Selecting the woman you think everyone else thinks everyone else will think is most beautiful

As you can see, the mental gymnastics required to move beyond the second or third degree of thought quickly becomes dizzying. You can witness Vizzini wander into this intellectual quagmire in *The Princess Bride*:

The Keynesian Beauty Contest does a great job of demonstrating the differences between degrees of thought. Are you just thinking about yourself? Are you taking into account others’ thoughts and preferences? Are you considering what they might also think about everyone else?

### The “Guess 2/3 of the Average” Game

In the beauty contest, there is no discrete knowledge of what others consider valuable, however. It is even more impossible to accurately ascertain what others could know about others’ opinions. What LeDoux and Nagel added to the thought experiment was precise value. They changed the exercise from selecting the most popular woman, to choosing 2/3 of the average number selected by a group (with a possible range from 0 to 100). Now we are able to quantify the degrees of analysis.

*Rules of the game: *Each participant must pick a number between 0 and 100. The winner will be the person who chooses the number that is closest to 2/3 of the average.

Assuming participants correctly understand the rules of the game, these are the values arrived at given the following degrees of thought:

- 0th Degree: Any value (we could also assume 50, since that would be an “average” random selection)
- 1st Degree: 33.33
- 2nd Degree: 22.22
- 3rd Degree: 14.81
- …
- 22nd Degree: 0.01
- 23rd Degree: 0.00

### Zero

As you can see, after 23 degrees we move beyond two digits of zero. In other words, as you increase the degrees of thought, you approach a limit of zero. In case you haven’t taken calculus (or purged it from your memory), a limit is a value that a function or sequence inevitably approaches from one direction, but never actually reaches. If you simply round to the nearest integer, however, the Nash Equilibrium for this game is zero.

For anyone new to the idea, see the video below for a suboptimal yet functional example of a Nash Equilibrium, from *A Beautiful Mind*:

Back to the “Guess 2/3 of the Average” game, *if* everyone understands the math behind the game, and *if* everyone participating is an objectively rational actor, then the winning answer is zero. But here’s the thing: you can’t assume either of those. Not with large groups of humans, anyway.

### Nagel’s Study

In her 1995 paper, Rosemarie Nagel carried out experiments to figure out what people’s strategies would be. She didn’t stop at a single iteration, either. She carried out four rounds, with a different value for *p* for different groups (1/2, 2/3, and 4/3). Of course, *p=2/3* is the value of the original “Guess 2/3 of the Average” game. I’m sure you can extrapolate from there where to insert the 1/2 and 4/3 values.

A few things about the results struck me. For one thing, there were some highly irrational actors, especially in the first round of each group. For instance, since the winning answer would be 2/3 of the average guess, there is a ceiling on possible winning answers. Even if everyone (including yourself) were to guess the maximum value of 100, 2/3 of that would be 66.6. So guessing anything above 67 is completely irrational (even if you assume everyone else will be guessing 100). And yet there were multiple people guessing above 67 (some even guessed 100).

However, counter to this irrational behavior was an overall rational group (and individual) movement towards the limit in each successive round. Regardless of the original guesses, the group’s behavior bent toward the limit of zero each and every time. The exact amount of this movement at each stage is interesting.

In most sessions and periods, at least 80 percent of the observations remain within the bounds of iteration step 0 and iteration step 3, with the modal frequency (30 percent or more) at iteration step 2 when the previous period’s mean is the reference point.

Translation: When the reference point is the previous round’s average, over 80% of people tend to act on analytical levels of thinking between the zeroth and third degree, with the largest clumping at the second degree. So, thinking back to the Keynesian Beauty Contest, this means that almost everyone creates strategies derived from random selection (0 degrees), thinking about what all of the contestants think other contestants are going to think (3 degrees), or somewhere in between, with a plurality of participants strategizing at the 2nd degree.

### Reason and Anti-reason

What are the practical implications of this knowledge? Well, we know that even when given scenarios with simple mathematical calculations, most people tend to stop at the second or third degree of analysis. Many even choose to stop at the zeroth or first degree. Again, this is all with discrete mathematical possibilities.

What would you expect to happen when the math is not so simple? Or if there’s no available math at all (returning to the beauty contest)? Are people likely to go through more or fewer degrees of analysis? Given the added complexity and opacity of information, I would wager fewer degrees. Let’s complicate it even further with cognitive dissonance, motivated reasoning, and other rational short circuits. Given all of these new variables, I would say it is entirely possible that someone who operates in the 4th degree of analysis in the simple math scenario could easily be operating at the 1st, 0th, or even negative degrees of thought in many real-life scenarios.

### Player X

If I broke your brain with the “negative” degrees of thought, bear with me. Let’s start with someone who tends to operate at the 4th degree in the simple math scenario. We’ll call this participant “Player X”. In a complex and opaque information environment Player X might only be able to function at the 2nd degree. I’m sure that makes sense.

However, throw in some extreme cognitive dissonance, and Player X is no longer simply putting in less effort, he’s actually motivated to put in *more* effort to *disprove* or *change* the reality he is facing. In other words, he will seek to bend reality to his perception or his ideal.

Depending on the environment and the size of the group, this can actually be accomplished to a limited extent. Think of the 2/3 guessing game again. If there are only 3 players, Player X can significantly impact the outcome if he changes his guess from 22 to 100. Assuming the other two participants guessed 22, the new average is 48, 2/3 of which is 32 (whereas if Player X had guessed 22 like the others, the final value would have rounded to 15).

However, even though Player X was able to significantly alter the final value in this 3-player game, he ends up losing to the others, whose guesses of 22 are closer to the final value of 32 than Player X’s guess of 100. What if we introduce direct communication between participants, and Player X gets Player Y to cooperate with him? Let’s say both of them changed their guesses to 100, while the remaining player stayed at 22. The new average would be 74, 2/3 of which rounds to 49. Player X and Player Y *both* lose in this case. They are fighting the inevitable.

Given a high enough ratio of participants willing to go along with Player X (e.g., nine out of ten participants), the group seeking to redefine reality can actually win. However, in that case, the lone participant still guessing 22 can simply adjust his guess upward a few points to regain victory the next round. Alternatively, one of Player X’s co-conspirators need only move his guess down a single point to claim total victory for himself alone, or choose to instead cooperate with the lone player–a significant temptation to defect from the group in either case, and which begs the question of how Player X would have convinced the group to go along with his plan in the first place.

At this point, I’m beginning to muddy the waters of the abstract problem with new variables that are turning a simple toy problem into a complex dilemma, so I’ll quit while I’m ahead.

### Lessons

Several points to ponder:

- Even given ideal mathematical conditions, most people only carry their calculations so far (generally no more than the 2nd or 3rd degrees).
- Over iterated play, individuals and groups both tend to migrate toward the Nash equilibrium, or mathematical limit, regardless of how “irrational” their initial guesses may have been.
- While a minority, there are still a surprisingly high number of irrational actors, particularly in the first iteration of a game. This affects the profitability of analyzing more than 2 or 3 degrees, even if you are capable of doing so.
- Fighting against the “gravity” of the Nash equilibrium is a fool’s errand.
- Nonetheless, individuals–and especially groups–can impact results by acting on “negative degree” strategies.