´´ Value Investing And The Art of K- Level Thinking

Sunday, September 27, 2015

Value Investing And The Art of K- Level Thinking

Recently, I stumbled over a New York Times puzzle that is followed by an article. It asked readers to pick a number between 0 and 100 that is two-thirds of the average guess of the participants of the game (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 on the stock market is not to pick the best company. It is to pick the company that others think will be the best performing company on 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 situations in business and politics can be well approximated by outguessing games. The Nash equilibrium can make pretty good predictions about the behavior of the participants of such a game. But they are only accurate when participants have enough experience with analogous games. Or such games are repeated so that players can develop enough experience to learn to predict other people's responses.

This NYT game cannot be repeated and it appears to be a novel strategic situation to many readers. Thus, there is no stable Nash Equilibrium.

Very likely many people entering the game start reasoning that 50 is roughly the average answer if everyone picked a random number. Hence, it is stupid to guess more than 50. It is more rational to calculate two- third of fifty, which gets you to 33. Nevertheless, unless you think the fellow players are stupid, 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 a number close to the Nash Equilibrium is by no means a winning strategy in the NYT game. So far the reader’s average guess has been 28 and the winning number 19.

I find this game intriguing, because it tells so much about the limited knowledge and ignorance of human beings. Apparently, many NYT readers just jump into the game without even contemplating doing the most important thing in order to win it. Namely, doing nothing and think.

So it should come as no surprise that those kind of games are standard games for many psychologists, game theorists 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 some games the empirically significant Level- k type (LK) predictions coincide with equilibrium. But in this NYT game Level- k type predictions deviate systematically from equilibrium. Game theorists call such a game a structural, non-equilibrium game of initial responses based on "level-k thinking".

In order to come closer to winning this game the participants have to use level-k thinking models. They have to predict the deviations of the best guesses of the crowd. Hence, the best guess must rather come from thinking than learning, because predictions are less reliable for initial responses.

For the average person k is a small number. It is hard and unnatural for humans to think multiple steps ahead. The average of participants of the NYT game thinks about one and a half step ahead of random guessers. The average of all guesses is 28. This is roughly half way between the first order answer of 33 and the second order answer of 22.

In this NYT game, someone who guessed randomly is a 0-step thinker. Lk (with k>0), on the other hand, anchors its beliefs with a naive, non- strategic prior L0, and adjusts them via thought-experiments with iterated best responses:

● 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 also 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.

Reading The Mind of The NYT Readers

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 well aware of how the game works. But interestingly, he is slightly underestimating the competence of his fellow players.

By far the most popular number was 33. That is fascinating! Because it is one of the most ignorant, arrogant and cynical number one can choose. It is L1's best answer over L0 players. Those “33” pickers must think that they are the only smart guys in the room, and all others are just blockheads. Those guys exhibiting the Dunning- Kruger Syndrome marvellously. They are below average performers that underestimate the competence of the fellow players.

The pickers of numbers around 0 are very interesting too. They also clearly fall into the Dunning- Kruger Syndrome category. 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, and do not place any wager on similar games. Unfortunately, very likely they are surprised about their poor performance.

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 what 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)

One does not go too far assuming that Graham's quote about Mr. Market is his most famous. Concerning the interpretation of this quote, the majority tend to see it only as an advice to keep investors from becoming too emotionally involved to stock market and portfolio volatility. But this quote goes much further. Because Benjamin Graham is also suggesting a best response to Mr. Market, who is just a representative (or a 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 my have actually 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 and make sure not overachieving. Because being a super smart market participant can just be as bad as being an overconfident Idiot.

Thus, the moral of this little post is that the reasonably intelligent investor should find the market where the >50 pickers are dominant. The market of the clueless, overconfident, ignorant and unknowledgeable counter parties. On such a market the investor with average thinking skills has the best chance to win the stock market outperformance game.


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