The Art of K- Level Thinking

Why For The Average Stock Market Operator Deep Value Investing Is The Best Strategy

A few years ago, I stumbled over a New York Times puzzle (Link) that is followed by an article. It asked readers to pick a number between 0 and 100 that is two-thirds the average guess of the other players (please play before reading further!).

Game theorists call such a game an outguessing game.

Are You Smarter Than the Crowd? 

The best example for an outguessing game is probably trading on the stock market. The goal of traders is not to pick the best company. It is to pick the company that others think will be the best performing company in the short run.  In fact, it is to pick the company that others think, that others think that others think ... is the best performing company.

John Maynard Keynes used a “beauty contest” to describe this idea. He noted that: 

“(...) the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole (...)" 

Many strategic issues in business and politics can be approximated by outguessing games. The Nash equilibrium can make pretty good predictions about the behavior of the participants.  

But they are only accurate when players are familiar with similar games. Or such games are repeated, so participants develop a skill in predicting other people's responses.  

What makes the NYT game interesting is that it cannot be repeated. And apparently, for many players it is a novel strategic situation.

Reading The Mind of The Players

Probably, players that enter the game start reasoning that 50 will be roughly the average number of everyone else’s random pick. Hence, it is illogical to guess more than 50. It is more rational to calculate two- thirds of fifty, which gets you to 33. Nevertheless, unless you think that fellow players are behaving illogically, it is “stupid” to guess 33. Because if everyone picked 33, then 22 would be the winning number. 

To put it differently. If you do not underestimate your competitor's intelligence, and rather think that all the participants of the game can think as far ahead as you, then 15 is about two-thirds of 22, and 10 is two-thirds of 15 … and eventually, we will get to zero.

Zero would be what is known as the Nash equilibrium of this game. But choosing the Nash Equilibrium, or a number closely above, is by no means a winning strategy in the NYT game. The reader’s average guess has been 28 and thus, the winning number is 19. 

What makes the game so intriguing is that it reveals a lot about limited knowledge and ignorance of human beings. Apparently, many New York Times readers just jump into the game without even contemplating doing the most important thing to win it. Namely, doing nothing and think. 

Unsurprisingly, outguessing games are used by many psychologists and behavioral economists to study human behavior. It brings forth the trouble most humans have in thinking more than one or two steps ahead ("Level-K Thinking"). 

Level- K Type Thinking 

In the NYT article three game theorists noted that: "Guessers are fatigued, clueless, overwhelmed, uncooperative, or simply more willing to make a random guess in the first period of a game and learn from subsequent experience than to think hard before learning.” 

As mentioned above, in most games the empirically significant Level- K Type (LK) predictions tally with the Nash equilibrium. But in the NYT game Level- K Type predictions deviate systematically from it. Game theorists call such a game a structural, non-equilibrium game of initial responses based on "Level-K Thinking". 

The winning strategy involves Level-K Thinking Models. Participants must grasp the deviations of the crowd’s best guesses. Hence, the best guess must rather come from thinking than learning, because predictions are less reliable for initial responses. 

For the average human k is a small number, as they have difficulties to think multiple steps ahead. The average participant of the New York Times game thinks roughly one and a half step ahead of random guessers. The average of all guesses is 28, which is roughly halfway between the first order answer (33) and the second order answer (22). 

In this NYT game, someone who guesses randomly is a 0-step thinker.
 
● L0  is most often taken to be uniform random over the set of possible decisions
 
● L1 is the best response to L0. Thus, the player has a perfect model of the game but a naive model, or is ignorant, about the competence of the competitors 
 
● L2 (or L3) is the best response to L1 (or L2). Thus, the player has a perfect model of the game and a less naive model, or is less ignorant, about the competence of the competitors L1 (or L2). 
 
Hence, the player Lk, with k > 0, is rational in the sense that he understands the structure of the game and has the best response to beliefs about other player's decisions. 

The Cocky 33 Pickers

As shown above the average outcome of 28 should come as little surprise. It is roughly L1 and L2 players best response to L0 and L1 players. The average participant of the game is aware of how the game works. But interestingly, he is slightly underestimating the competence of fellow players. 

By far the most popular number chosen is 33, which is L1's best answer over L0 players. That is fascinating! Because it is the most ignorant and arrogant number. The cocky “33” pickers so to speak of.

Picking 33 implies a self- perception being the only smart guy in the room, and all others are just blockheads. Nevertheless, they are below average performers that underestimate the competence of fellow players. 

The pickers of numbers around 0 are very interesting too. They underestimate their performance in this game and overestimate those of the competitors. Basically, they are overachievers. 

Than we have the players that picked a number greater than 66. They expected a number close to 100 would be the average pick. Hopefully, those guys know about their very limited knowledge and incompetence. But I highly doubt it.  

Benjamin Graham the K- Level Thinker 

"(…) Imagine that in some private business you own a small share that cost you $1000. One of your partners, named Mr. Market, is very obliging indeed. Every day, he tells you what he thinks your interest is worth and furthermore offers either to buy you out or to sell you an additional interest on that basis. Sometimes his idea of value appears plausible and justified by business developments and prospects as you know them. Often (...) Mr. Market lets his enthusiasm, or his fears, run away with him, and the value he proposes seems to you a little short of silly (...)" (p.206-207) 

Benjamin Graham’s Mr. Market analogy is one of the most quoted in the investment community. The majority interpret it as advice to detach emotionally from market volatility.  But the quote goes much further. Because Benjamin Graham is also suggesting an investor’s best response to Mr. Market, i.e. the simplified model of other investors. 

Mr. Market is Graham's 0-step thinker (L0).  Mr. Market is random, but very likely not uniform. Hence, when dealing with Mr. Market Benjamin Graham is also advocating to the investors being a L1 thinker. But by publishing this quote, Graham himself may have already entered L2 thinking. 

Be a Knowledgeable Idiot 

If you ever find yourself in a situation where you need to outguess great minds like Benjamin Graham, you better be closer to the L3 thinking level. But make sure you are not overachieving. Because being too smart can just be as bad as being an overconfident Idiot. 

The rest of us is best advised to find cheap stocks in the market where the >50 pickers are dominant. The market of the clueless, overconfident, ignorant and unknowledgeable counter parties. Only here, being a knowledgable idiot is enough to stay ahead of the pack and win the outperformance game over the long haul.
 

Source/ Reference: 
 
Benjamin Graham; The Intelligent Investor; Harper Business Essentials; 2003 
 
Chris DeMuth Jr. - Poison Both Cups 
 
Puzzle: Are You Smarter Than 60,660 Other New York Times Readers? 
 
Outguessing and Deception in Novel Strategic Situations; Vincent P. Crawford, University of California, San Diego MEDS, Northwestern University, 4 October 2005