To create the Dilemma, it is important to arrange the relative size of each of the payoffs for cooperation and defection in the matrix in the correct way. The Dilemma is defined by the exact ranking of the payoff values, where R is the reward for mutual cooperation; S is the sucker’s payoff for cooperating when your fellow player defects; T is the temptation to renege when your fellow player cooperates, and P is the punishment if both players defect. Let’s spell this out. When the players both cooperate, the payoff (R) is greater than the punishment (P) if they both defect. But when one cooperates and one defects, the person who is tempted to renege gets the highest payoff (T) while the hapless cooperator ends up with the lowest of all, the sucker’s payoff (S). Overall, we can create the Dilemma if T is greater than R which is greater than P which is greater than S. We can rank the payoffs in the basic game in other, different ways and still end up with cooperative dilemmas. But of all of them, the Prisoner’s Dilemma is by far the hardest to solve. You can think of it as the ultimate dilemma of cooperation.
We all encounter the Dilemma in one form or another all the time in everyday life. Do I want to help a competitor in the office—for instance, offer to do his work during his holiday—when this person is competing with me for a promotion? When two rival firms set prices, should they both go for as much as they can, colluding in some way, or should one company try to undercut its competitor? Arms races between superpowers, local rival nations, or even different species offer other examples of the Dilemma at work. Rival countries are better off when they cooperate to avoid an arms race. Yet the dominant strategy for each nation is to arm itself heavily. And so on and so forth.
INCARCERATION
On my first encounter with the Prisoner’s Dilemma in that Alpine hut, I was transfixed. By that time, Karl had actually become my prisoner. He didn’t have any transport and I offered him a ride back to Vienna. We discussed the Dilemma as we drove back the next day in the same VW that my father still uses today to putter around Austria. Even after I dropped Karl off, I kept him in my sights. Before long, I was doing a PhD with him at the Institute for Mathematics in Vienna. Students who had studied there before me include the great physicist Ludwig Boltzmann, the logician Kurt Gödel, and the father of genetics, Gregor Mendel.
As I pursued my doctorate, Karl and I would often meet in local coffeehouses, the genius loci of past glory. In these inspiring surroundings Gödel had announced his incompleteness theorem, Boltzmann had worked on entropy, and Wittgenstein had challenged the Vienna Circle, a group of intellectuals who would gather to discuss mathematics and philosophy. One day we sat in the Café Central, an imposing building with arched ceilings and marble columns, where Trotsky had planned the Russian revolution.
As we sipped thick, strong coffee and chatted about how to solve the Prisoner’s Dilemma, Karl and I rediscovered the subtleties of a problem that had transfixed bright minds for generations. Little did we realize that in the decades that followed, we would devise new mathematics to explore the Dilemma. We would create communities of agents in a computer, study how they evolved, and conduct analyses to reveal the mechanisms able to solve the Dilemma. I would establish teams at Oxford, Princeton, and Harvard as well as collaborations with mathematicians, biologists, chemists, doctors, and economists around the world to understand how these mechanisms worked and what their wider implications were.
Some scientists regard the Prisoner’s Dilemma as a remarkably revealing metaphor of biological behavior, evolution, and life. Others regard it as far too simple to take into account all the subtle forces at play in real societies and in biology. I agree with both camps. The Dilemma is not itself the key to understanding life. For the Dilemma to tell us something useful about the biological world, we need to place it in the context of evolution.
Evolution can only take place in populations of reproducing individuals. In these populations, mistakes in reproduction lead to mutation. The resulting mutants might reproduce at different rates, as one mutant does better in one environment than another. And reproduction at different rates leads to selection—the faster-reproducing individuals are selected and thrive. In this context we can think about the payouts of the Prisoner’s Dilemma in terms of what evolutionary scientists call “fitness” (think of it as the rate of reproduction). Now we can express what cooperation in the Prisoner’s Dilemma means when placed in an evolutionary context: if I help you then I lower my fitness and increase your fitness.
Here’s where the story gets fascinating. Now that we have put the Dilemma in an evolutionary form, we discover that there is a fundamental problem. Natural selection actually opposes cooperation in a basic Prisoner’s Dilemma. At its heart, natural selection undermines our ability to work together. Why is this? Because in what mathematicians call a well-mixed population, where any two individuals meet equally often, cooperators always have a lower fitness than defectors—they’re always less likely to survive. As they die off, natural selection will slowly increase the number of defectors until all the cooperators have been exterminated. This is striking because a population consisting entirely of cooperators has a higher average fitness than a population made entirely of defectors. Natural selection actually destroys what would be best for the entire population. Natural selection undermines the greater good.
To favor cooperation, natural selection needs help in the form of mechanisms for the evolution of cooperation. We know such mechanisms exist because all around us is abundant evidence that it does pay to cooperate, from the towering termite mound to the stadium rock concert to the surge of commuters in and out of a city during a working day. In reality, evolution has used these various mechanisms to overcome the limitations of natural selection. Over the millennia they have shaped genetic evolution, in cells or microbes or animals. Nature smiles on cooperation.
These mechanisms of cooperation shape cultural evolution too, the patterns of change in how we behave, the things we wear, what we say, the art we produce, and so on. This aspect of evolution is more familiar: when we learn from each other and alter the way we act accordingly. It also takes place over much shorter timescales. Think about a population of humans in which people learn different strategies to cope with the world around them, whether religion or boat building or hammering a nail into a piece of wood. The impact of cooperation on culture is huge and, for me, the central reason why life is so beguiling and beautiful.
QUEST FOR THE EVOLUTION
OF COOPERATION
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
—Bertrand Russell, Study of Mathematics
My overall approach to reveal and understand the mechanisms of cooperation is easy to explain, even if my detailed workings might appear mysterious. I like to take informal ideas, instincts, even impressions of life and render them into a mathematical form. Mathematics allows me to chisel down into messy, complicated issues and—with judgment and a little luck—reveal simplicity and grandeur beneath. At the heart of a successful mathematical model is a law of nature, an expression of truth that is capable of generating awe in the same way as Michelangelo’s extraordinary sculptures, whose power to amaze comes from the truth they capture about physical beauty.
Legend has it that when asked how he had created David, his masterpiece, Michelangelo explained that he simply took away everything from the block of marble that was not David. A mathematician, when confronted by the awesome complexity of nature, also has to hack away at a wealth of observations and ideas until the very essence of the problem becomes clear, along with a mathematical idea of unparalleled beauty. Just as Michelangelo wanted his figures to break free from the stone that imprisoned them, so I want mathematical models to take on a life beyond my expectations, and work in circumstances other than those in which they were conceived.
Michelangelo sought inspiration from the human form, notably the male nude, and also from ideas such as Neoplatonism, a philosophy that regards the body as a vessel for a soul that longs to return