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CHAPTER 3

Spatial Games— Chessboard of Life

The chess-board is the world, the pieces are the phenomena of the universe, the rules of the game are what we call the laws of Nature. The player on the other side is hidden from us.

—Thomas Huxley

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

—John von Neumann

Where there’s life, there are lumps, clumps, and colonies. Bacteria grow in films. Slime molds aggregate in three-dimensional shapes, similar to Mexican hats. Bisons gather in herds. Ants work in colonies. Apes form troops. There are sleuths of bears, murders of crows, pods of whales, and gaggles of geese. And, of course, populations of people form structures too. We are organized in villages, towns, and cities. We gather in the workplace, at schools, and in theaters and pubs. We have mobs and crowds and posses and throngs.

When I started to work on ways reciprocity can solve the Prisoner’s Dilemma, I wondered if population structure could offer another way to solve the Dilemma. Remember that the calculations of the past two chapters were based on a simple assumption: the players in the Dilemma were in a well-mixed population, where every player has an equal chance of meeting every other player. In these uniform populations we found that defectors always outcompete cooperators. But, of course, all real populations have some structure. What difference does this make? Could population structure affect the outcome of the simple Prisoner’s Dilemma? Could there be structures that make cooperators triumph over defectors?

The very fact that people decide to live in the same patch, rather than dotted around at random, has to do with cooperation. Why? Some believe, for example, that the first communities evolved as a result of success in agriculture: surplus food from farming enabled people to settle down and specialize, from butcher to baker to candlestick maker. Others link it to ancient belief systems and religions. At Göbekli Tepe in Turkey (“Hill with a potbelly”), for example, is a sanctuary with limestone pillars that were carved and erected by hunter-gatherers more than eleven thousand years ago. That remarkable discovery would suggest that temples planted the seeds of cities.

Perhaps cities were born of the struggle for existence: in his book Whole Earth Discipline, Stewart Brand suggests that the very first urban invention was the defendable wall, followed by rectangular buildings that could pack people efficiently inside that wall. The Cambridge archaeologist Colin Renfrew argues that the first communities came with the birth of the modern mind. That is when the effects of new intellectual software kicked in, allowing our ancestors to work together in a more settled way. However, I would discover from my computer simulations that you do not need any brainpower at all to benefit from forming a huddle.

THE GOD GAME

One can trace the efforts to link life and geography—in the guise of what are called spatial automata—to a study by the great John von Neumann, who believed that biological organisms could be thought of as information-processing systems. The fact that it is now possible to write and synthesize the genetic code of a living thing in the laboratory, like a glorified computer program, shows how right he was. He puzzled over the difference between the trivial kind of reproduction that enables crystals to grow in a test tube, for example, and the clever kind that enables creatures to breed. To explore the difference, von Neumann wanted to design a machine that was complex enough to reproduce itself.

To this end, he devised a “self-reproducing automaton,” a robot that was afloat on a sea of its own components, just as living things on Earth thrive and abound among the chemical building blocks of life. He built on the work of Alan Turing, the English mathematician who had laid the logical foundations of computing with the idea of the “universal Turing machine,” which offered a splendid abstract device to explore the theoretical limits of mathematics.

Von Neumann showed that there also exists a universal automaton, an abstract simulation of a physical universal assembler. Logical errors within the automaton could be viewed as mutations, allowing the possibility of more complex varieties of automaton to emerge; in an environment with finite resources, selection pressure—survival of the fittest—would lead to Darwinian evolution. But there was no mathematically rigorous way to analyze his creation, let alone the means to build one.

The mathematician Stanislaw Ulam gave von Neumann advice on how to make his self-reproducing machine simpler. He put forward a way that Neumann’s “machines” could be built of pure logic. Ulam suggested replacing the floating automaton with what he christened “tessellation robots.” This quaint term refers to the growth of crystals, which occurs by the buildup of unit blocks, or “tessera.” Today Ulam’s tessellation robot is called a “cellular automaton” and consists of an abstract array of cells programmed to execute rules en masse. This collection of cells—like squares on a chessboard—carries out computations in unison and can be viewed as a kind of organism, running on pure logic, though the cells in question have little to do with the real thing.

Each cell in the array is in a particular “state” at a given time. A state might be a certain color, say red, green, or blue, a numeric value, or simply on or off. Time is not continuous in the automaton, but discrete, so that it advances with each tick of a clock. With each tick, simple rules determine how any cell changes state from one instant to the next. These rules depend not just on the state of that particular cell but also on the states of its neighbors—a predetermined “look-up” table based on rules decides what the cell has to do next. So, in a black-and-white automaton, it could be that if all neighboring squares are black, the central square is made white. Despite the apparent simplicity of this setup, it turned out that anything achieved in von Neumann’s original automaton could be aped by Ulam’s cellular system.

After von Neumann’s death, the torch for cellular automata passed to others, notably to John Holland, professor of Psychology and professor of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor. In 1960, Holland outlined an “iterative circuit computer” related to cellular automata that could mimic genetic processes. Despite the dry title, this research captured the public imagination. As one newspaper put it, “He’s the man who taught computers how to have sex.” This logical view of life now began to take hold and multiply in the minds of researchers in other laboratories. One of the best known is the British mathematician John Conway.

For several months in the late 1960s, the mathematics department at Cambridge University was taken over by Conway’s efforts to find a hypothetical machine that could build copies of itself. From a small table an assortment of poker chips, foreign coins, cowrie shells, and whatever else came to hand was used to mark out the “living” squares in patterns that spread across the floor of the common room as the rules were enacted during each tea break. Even then, Conway used a computer to study particularly long-lived populations.

In 1970 he unveiled his Game of Life. Now Conway had selected the rules of his automata with great care to strike a delicate balance between two extremes: the many patterns that grow quickly without limit and those that fade away rapidly. The evocative name reflected Conway’s fascination with how his combination of a few rules could produce global patterns that would expand, morph shape, or die out unpredictably.

Some people call it the “God Game” in recognition of how players have a toy universe at their beck and call. Taking part is easy, requiring no miracles, religious followings, or holy books. Imagine a checkerboard with counters in a few of the squares. Then follow these simple rules. If an empty square has three occupied neighbors (this includes the diagonal as well as adjacent sites), it comes “alive,” nurtured by its neighbors. If a square has two occupied neighbors, then it remains unchanged. Finally, if an occupied square has any other number of occupied neighbors, then it loses its counter—to anthropomorphize, the cell dies pining for neighborly love and the chance to cooperate.

Conway conjectured that no initially finite population could grow in number without limit and offered fifty dollars for the first proof or disproof. The prize was won in November 1970 by a group in the Artificial Intelligence Project at the Massachusetts Institute of Technology, which discovered a “glider gun,” a pattern that every thirty moves ejects a glider, a moving pattern consisting of five counters. Since each glider added five more counters, the population could grow without limit. Intersecting gliders were found to produce fantastic results, giving birth to strange patterns that in turn spawned still more gliders. Sometimes the collisions expanded to digest all guns. In other cases, the collision mass destroyed guns by “shooting back.”

The patterns produced can be complex; indeed the cellular automaton can be shown to be equivalent to a universal Turing machine, so the Game of Life is theoretically as powerful as any computer. It does not take much imagination to see that cellular automata provide a powerful tool to study the patterns of nature. Stephen Wolfram—who wrote his first paper at the age of sixteen and went up to study at the University of Oxford at the age of seventeen, and who devised a highly successful software application called Mathematica as well as a new kind of search engine (Wolfram Alpha)—turns this idea on its head in his book A New Kind of Science