A book review: A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature
Author: Tom Siegfried, Science editor of Dallas Morning News
Publisher: The Joseph Henry Press of the National Academy of Sciences
Many of us saw the fascinating movie A Beautiful Mind, directed by Ron Howard. It is about the tragic life of John Nash, the mathematician who was awarded the Nobel Prize for Economics in 1994. The sciences which owe much to John Nash and his mathematical achievements are many. Before him, the giant to be mentioned in the area of mathematical game theory is John von Neumann, who introduced the theory of zero-sum games and so-called mixed strategies, working jointly with his colleague, Oscar Morgenstern. The book by Tom Siegfried honours those creative scientists and their disciples. Tom Siegfried has been working as a science editor at Dallas Morning News. He covers a lot of ground in this book. It is also fascinating to find out how the seeds of important mathematical ideas can be traced to real life and the arts. Just to mention a few, a card game of the 1700s can be seen as capturing the minimax strategy; Edgar Allan Poe described the prisoner's dilemma, and something analogous to the ultimatum game - now understood to be so important - was played in the movie The Maltese Falcon with Humphrey Bogart.
Game Theory was developed as a mathematical tool for analysing economic behaviour. Today, its value in analysing almost any interaction in life is superior. In the 1970s, evolutionary biologists began making use of it in analyses of the survival of animals and plants. Researchers in political science, psychology, anthropology, sociology and neural sciences use it extensively. Many other well-known game theorists have been rewarded with a Nobel Prize, including Reinhard Selten, John Harsányi, Thomas Shelling and Robert Aumann. Game theory is nothing less than the universal language of behavioural sciences.
What about chemistry and physics? It is known that Game Theory and statistical mechanics have a formal connection. Molecules in chemical reactions do not appear to play or jostle for survival. However, they are seeking a stable state, characterised by minimal energy. The proposition examined in the book by Siegfried is that there is in fact a formal link, a mathematical similarity exploited by John Nash himself. Indeed, he studied chemistry. Psychologists and social scientists often emphasize the freedom of decision-making in humans, who are subject to the straitjacket created by their genetic inheritance. Just moves by molecules? I am an economist and do not want to get involved. If it's true, to the horror of humanities there exists a universal mathematical language to investigate the processes of both organic and non-organic nature!
The mathematics developed by John Nash has produced a useful way to approach and study the regularity of our world. Where Robinson Crusoe had to optimise, i.e. choose his best action, against non-optimising nature, an interaction between two decision-makers (or similar) is a question of choosing the right strategy. That is exactly what Game Theory does. By using Game Theory, one is able to characterise the equilibrium between competing strategies.
To be sure, no living organism could have time in the jungle or in the street to undertake game-theory calculations. Brains, however, have a "common currency" to assess the quality of different courses of action. And this takes place extremely rapidly. Neural science and Game Theory have consequently met each other. This is but part of the story. Such previously vague ideas as "fairness", "morality" and "trust" can now be precisely defined in terms of Game Theory. As evolutionary psychology tells us, such feelings developed within us far back in our past, at the time when our ancestors used to live in small societies, making their living as hunter-gatherers. From that perspective, modern man is nothing more than a hunter-gatherer with clothes.
Game Theory is not just a descriptive tool. Its utility extends to making predictions of behaviour, since it captures constant patterns. It helps us to judge whether nuclear war will break out; it helps us to understand the information role of gossiping; it makes it possible to understand why ducks co-operate, and so forth. And Game Theory helps us to understand why we observe so many forms of life on our planet. Indeed, evolution by natural selection is a game played by life, and all living nature is part of that game in the struggle for fitness to survive.
Game Theory helps us to understand why institutions have been developed to replace less efficient ones. It helps us to understand why some individuals are aggressive hawks, while others are doves. It is illuminating on why there is so much violence in society. And it tells us why the modern state with laws evolved.
Lack of trust destroys relationships, and the outcome is a bad equilibrium, with everyone losing out. The collective and private benefits do not match. Why then do we observe that co-operation is so common among humans, not forgetting it exists among animals too? Why is trust capital so strong in the economy? It is Game Theory which is the tool to understanding the laws of life. It makes it feasible to understand both competition and co-operation.
The desire by Siegfried to unify the regularity is perhaps extended a little too far. He searches for the link between the thinking of such giants as Isaac Newton, Adam Smith and Charles Darwin. As a critical remark, it is not always clear whether he is talking about the Nash equilibrium or the Nash bargaining solution. The former formalises the interaction when there is a lack of trust between the agents. In the latter, the players are able to enjoy the benefits of co-operation. Life is a struggle for a place in the sun, but also subject of mutually-beneficial transactions, social exchange. Altruism can be profitable. Though the book has links to evolutionary psychology, it pays no attention to meme theory, which has become important.
To summarise, the book by Siegfried is not a textbook. Instead, it represents an exciting story of the human mind in developing a universal mathematical theory to be helpful in understanding human behaviour within societies, interactions between animals, evolutionary mechanisms and (perhaps) also the laws of physics.
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