Non-linearity

A Chessboard Full of Rice

According to myth, the Emperor’s wise adviser once did him a great favor. So grateful was the Emperor that he begged his wise advisor to take any gift she might like from the vast treasures of gold or jewels, any lands or gardens, any of the Emperor’s many male children to be her companion. However, the advisor answered as follows: “Thank you for your generosity, oh mighty Emperor. I have no need of great material wealth. My needs and wants are simple. I do get hungry and thirsty, of course, as do we all, and sometimes my household runs short of rice. You see this fine chessboard?”

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“Oh, yes, my wise counselor, it is indeed finely made of gold and silver and I would gladly give you twenty such!” 

“Thank you again for your generosity, but I only wish for a some grains of rice. Give me one grain on this space and tomorrow, two grains on this space and the next day, four grains on this space. Each day for 64 days, double the number of grains of rice you gave me the day before. At the end of the 64 days, I will ask for no more.” 

The Emperor looked puzzled. “Surely, you must have something more valuable than rice! Name it!” 

“No, Sire, that is all I desire. Just the doubled rice will do quite nicely.” 

“Well, it shall be so!” And thus the Emperor told his staff that they were to provide a grain of rice for the first day, two grains of rice for the next day and to double the amount each day until all 64 days had passed. At first, it seemed such a pathetic gift for such a great favor. 

Even after 8 days, the wise counselor only received 128 grains of rice – not even a bowlful. 

Readers familiar with exponential growth realize that on the 64th day, the Emperor has promised to deliver 2**63 grains of rice. This is not only more rice than the Emperor had at his disposal. It is more grains of rice than exist in all the kingdoms of earth. To be exact, the last payment is meant to be 9,223,372,036,854,775,808 grains of rice while the total is one less than 2**64. To put the matter scientifically — it’s a lot of rice! Much more than exists in the world. 

How would you like the story to end? A wise Emperor, to my mind, would thank the counselor after a couple weeks and say, “I see, oh wise Counselor, that you used my gift to give me another gift to enhance my wisdom. For I now understand that what seemed at first an easy thing to do is actually quite hard. Doubling soon undoes even the richest king. I will keep this in mind when I think about interest rates and population growth.” 

A crummy Emperor, on the other hand, might say, “I offer you a gift and you see fit to embarrass me by making me agree to an impossible task? Boil her in oil!”

The Lily Pad Pond Puzzle. 

Beside my house is a pond. In this pond, a lily pad began to grow. Every day, it doubled in size. On day 20, it completely covered the surface of the pond. The surface of the pond is 400 square feet. How many days did it take to cover half of the pond? 

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At first glance, you might think this problem is insoluble because you don’t know how big the lily pad was initially. In fact, you don’t even need to know how large the pond is. It will cover half the pond on day 19.  

The Ping Pong Table Ping Pong Player Population

When I began at IBM Research in 1973, I soon discovered that a fair number of researchers were avid table tennis players. At lunch time, somewhere between six and twenty researchers would show up to play. There were two tables and some small amount of room for spectators to stand on the edges of the two ping-pong rooms and watch. Our rule was that if a person won, they would stay at the table and a new challenger would play. However, if you won three times in a row, you had to sit down regardlessly. I didn’t go over every lunch time, but I went over quite a few times over the course of my first ten years there and there was invariably someone to play with. Sometimes, I had a longer wait time than others, but it was never too long a wait. 

Then, because management wanted to use one of the two ping-pong rooms for other purposes, they repurposed one of the rooms. Now, there was only one ping pong table. In the two ping-pong table case, remember, I never had to wait too long nor did I ever go there and have no-one to play. As I said, the number of players varied between somewhere around six to twenty. What is your prediction about how many players showed up when there was only one ping pong table? 

Here’s what happened. The first day after this change happened, I went over and about fifteen people showed up. I, like everyone else, waited a long time for a game. Our “official” lunch hour was actually 42 minutes and the building was a five minute walk away. So, if you had to wait a half hour for your chance to play, it really wasn’t that much fun. In addition, there were some more subtle effects. All the players were good, but there some substantial differences in skill level. People tried to arrange it so that they played someone at about the same level. WIth only one table, this was trickier. In addition, when a relatively large number of people showed up, it was too crowded for everyone to see the match without interfering with play. It happened that I was too busy to go for a few days. The next time I showed up, no-one was there. Some of us talked about trying to “organize” the ping pong to insure that enough people showed up but everyone was busy and no-one wanted to take this on. Scheduling researchers is harder than you might think. It was hard for people to make a commitment to show up at noon because a meeting might run over, their manager might give them extra work, etc. The number of people showing up swung wildly for about two weeks and then stabilized. 

At zero. 

What had been a vibrant community with two ping pong tables did not stay the same size, or shrink to half when we were limited to one table. It went to zero. 

Warring Positive Feedback Loops. 

We’ve already talked about “positive feedback loops” which are also known as “vicious circles.” Sometimes, there are actually (at least) two positive feedback loops hiding beneath what appears to be a stable system. In the Case of the Missing Ping Pong Table described above, one positive feedback loop was simply that when you went there and had a good time through some combination of watching good matches or playing yourself, you were more likely to go there again. There was also a positive feedback loop that was more of a social nature. The more people who were there, the more likely it was you would find a good or interesting match. It was also more likely to be able to find someone you wanted to have a conversation with although the venue prevented this from being a big part of the adventure. Another way that having more people there increased the chances that more people would be there the next day was that it was kind of exciting to have a larger audience watching, cheering, throwing the ball back when the ball crept under the radiator after pin-balling around for awhile after a decent slam. 

At the same time, there were other feedback loops, sometimes of the same factors but in a different range. For instance, beyond the point of having the periphery of the playing field covered one or two deep, additional spectators added only a little excitement and they were more likely to infringe on the needed space around the table. In addition, while the first ring of spectators felt very much a part of the action, the experience for the second ring of spectators was far less engaging. While I mentioned above that more players meant a better change of finding a good match, it also meant that one had to wait longer between matches. The worst case scenario, of course, is that you are the only one who shows up. 

Behind Every Abstraction are a Host of Personal Stories. 

Yes, you can practice against the wall, and I did this a few times, but it is significantly less fun than a real match. I love to serve, for instance. I have a raft of difficult serves. Just to give you one example, with most set-ups, I can hit the right side of the ball so thinly that I put enough side-spin for the ball to appear as though it isn’t even going to hit the table on the second side, but it does; it curves radically back around the left. Sometimes people are so surprised that they miss it entirely. Even if they get there, the sidespin often makes them hit it off the table or the curve causes them to mis-hit the ball on their thumb or finger. I can also add a fair amount of top-spin or under-spin as well. Anyway, I didn’t get to do any of that just hitting the ball against the wall. The wall was not perfectly smooth either. So I might hit three of four shots and then the ball would hit a little imperfection in the plaster and careen off to scribble scrabble along the floor and then crawl under the radiator. It’s the kind of annoyance that everyone has experienced. And if someone else is there, you can kind of glance at your friend who nods nearly imperceptibly as you get down on your hands and knees and stretch your fingers into the territory of God-knows what spiders or broken glass and feel around through the grit and dust until you retrieved the ball. And that little glance and that little nod actually make quite a difference. If you’re on your own, it’s not any fun at all. It’s just an annoyance. The only reason I even bother to hit against the wall is to learn to keep focus for extended periods of time. For this, it is good practice and a good challenge. But, if I’m interrupting this to go fish my hand into a pile of dust every couple minutes, it isn’t so likely I’ll come back. 

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These various factors were all in a dynamic balance so long as there were two tables. When the tables went from two to one, however, what had been a stable equilibrium became a very unstable one. Eventually, of course, it did find a new equilibrium point and that was zero. To crawl out of that, one person might show up. But most of the time, they were the only one. So, they would be less likely to come again. Even if two showed up, since no-one could play every day, you might still find yourself wondering whether someone would be there the next time. 

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You might have read this whole story and wondered why the hell this building full of Ph.D.’s couldn’t get their act together and arrange some matches. It’s an interesting question and here is my personal opinion. When it came to these brilliant scientists and engineers, they came from every part of the globe and they came in all shapes and sizes. Some were vastly overweight and others were ultra marathoners. But the ones who liked to play table tennis were, by and large, athletic and “hyper” – an impatient lot. What all of us really loved was working to find out the truth. And, these truths that we sought were ones the company that we worked for wanted us to seek. True enough, but by the same token, that meant the truth found and utilized would make people’s lives better in some way in the not too distant future. But working in a corporation also meant doing a bunch of administrivia. So, the ping pong set in particular, wanted to get up from their intense sedentary mental and administrative work and play hard at something completely physical and different. The last thing any of us wanted to do was add more administriva to our lives. 

The Takeaway

 It’s easy and common to assume implicitly that the systems you deal with are linear.

They often aren’t. 

Things can go out of control extremely quickly (into a dominant positive feedback loop) once the dynamic equilibrium is disturbed. 

Would the invention of the iPhone have kept the ping pong community going? 

Another takeaway: there are two quite distinct ways of analyzing that are going on in the essay above: a fairly abstract one (even if it uses concrete examples like rice and lily pads) and a very concrete and experiential one. In my experience, both of these modes are useful and valid and if taken together give a fuller picture of what’s going on. My experience in this was mainly in human computer interaction but I think it is equally true for many in law, medicine, management and many other fields. What’s your experience? 

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