**Dr. Ziv Hellman, a lecturer in the Department of Economics at Bar-Ilan University tells about his acquaintance with the mathematician, winner of the Nobel Prize in Economics John Nash and his wife who were killed this week in a car accident and also about the Nash equilibrium, and wonders what would have happened if the mental illness had not attacked the Nash**

by Dr. Ziv Hellman

On Sunday I received a message from a friend. I was informed that we suddenly had 40 minutes left at a large conference on game theory that will be held in Jerusalem at the end of June. The message directed me to a news site with an explanation for the gap created in the conference schedule: Nobel laureate John Nash and his wife were killed in a car accident in the state of New Jersey, while they were traveling in a taxi from the airport to their home.

It took me a few minutes to digest the news. John Nash will not speak at the conference, neither in Israel nor anywhere else. We will not see him standing on the lecture stage again, radiating his shy smile. After all the difficult struggles he went through, his life was cut short in a second by a taxi driver's lack of control over the steering wheel of his vehicle. John Nash is gone.

The life story of John Nash is well known all over the world, perhaps more than the life story of any other mathematician in recent years. This is thanks to "Wonders of Reason", a movie that received many Oscar awards. The movie was produced 14 years ago, but to this day when I teach the concept of Nash equilibrium to university students, it is enough to mention that Nash is the hero of that movie so that everyone knows exactly who it is.

The film has rightly received a lot of interest, not least because the story at its center contains the elements of almost any successful story: a gifted and talented story hero, an attractive man with a bright future in front of him, forced to face a severe crisis (in this case hallucinations of mental illness) that causes him to lose all dear to him Despite the decline, our hero manages to rise on his own, subdue the inner demons in his soul, return to sanity and even win the highest recognition given to a scientist, the Nobel Prize.

In the case of John Nash, the plot is not a figment of the imagination of a Hollywood screenwriter, but he told the truth. After completing his doctorate with honors at Princeton University in 1950, Nash joined the academic staff of the mathematics department at the prestigious MIT University. There he began to develop an impressive career, with a number of amazing achievements in a short time, until the schizophrenia he suffered from was discovered. In 1959, a long period of deterioration began. In the seventies of the last century, students in Princeton's mathematics department gave him the nickname "the ghost of the department" because he used to walk around the corridors of the campus buildings at night and write indecipherable formulas on classroom blackboards.

Slowly, Nash managed to return to sanity and even return to the ranks of the academy as a researcher. In 1994, after convincing the members of the Nobel Prize Committee in Economics that his mental state was stable enough, he was awarded the prize for the equivalence theorem he proved in his doctoral thesis in 1950, aka the equivalence that bears his name.

Every day, in every department of economics around the world, the words "Nash equilibrium" are heard countless times. Roger Myerson, a professor of economics at the University of Chicago and himself a Nobel laureate, compared the importance of Nash equilibrium in the study of economics to the importance of understanding the structure of DNA in biology. It is simply impossible to imagine research in entire fields in today's economy without the cornerstone of the Nash equilibrium. Moreover, the concept is used as an important to central element in a variety of fields, from evolutionary biology, computer science and artificial intelligence to psychology and political science.

The main idea behind SHM Nash is relatively simple. In the game, as we know, each player chooses a strategy. The totality of all the strategies of the players determines the outcome of the game from which the payments to each player are derived. Each player strives to choose a strategy that will guarantee him a maximum payout, but cannot ignore the fact that his payout also depends on the strategies of the other players.

Suppose the players have chosen their strategies, but just before the game is played, one of the players is revealed the list of the other players' strategies. Will this lucky player change his strategy in the XNUMXth minute following the information he received? It is not difficult to imagine many situations in which exactly this will happen - after all, all intelligence and espionage institutions exist for this purpose.

A Nash equilibrium is a special situation where this does not happen. That is, a Nash equilibrium is a set of strategies that maintains the property that even if we reveal everything to the players and there is common knowledge about the strategies being played, there will be no incentive for any player to change his choice unilaterally because he can only lose if he takes such a step. A Nash equilibrium is similar to a self-enforcing contract. By its very existence, each player, out of a completely vertical incentive, will do what is expected of him.

According to this description, SH Nash is depicted as a special and rare condition. Indeed, it is easy to think of games in which "pure strategies" do not exist. A good example is the game "rock, paper and scissors". If, for example, I intend to play "paper" but at the last minute I learn that my opponent will play "scissors", of course I will have an incentive to change my strategy to choose "stone" instead of "paper". It is not difficult to see that for each pair of strategies in this game, it is always worthwhile for one of the players to change his choice.

But suppose we allow the players to play "mixed strategies", that is, each player can randomly choose the action to play, according to a certain probability. So we can easily find a Nash value in this game: if I choose "stone" with a probability of one third, "paper" with a probability of one third, and "scissors" with a probability of one third, and my opponent uses the same mixed strategy, it is clear that no one will have a one-sided deviation advantage from this pattern . Therefore this solution fulfills the Nash condition.

Nash's great achievement was to prove that if mixed strategies are allowed then in every game (with a finite number of possible actions) there is always, without exception, at least one Nash s.m. This result is far from easy and immediate. In his doctoral thesis, Nash applied heavy theorems from the study of "Sabbath points" in order to arrive at a proof that gave him fame for generations.

There is no doubt that John Nash will be remembered mainly for the equilibrium named after him, but Nash had many other contributions, to game theory in particular and mathematics in general. Some of them show an even more impressive depth than the brilliance underlying his famous equilibrium theorem. Among his contributions are solution concepts for bargaining games and cooperative games, theorems about canvases in algebraic geometry, groundbreaking results in the study of differential equations and a particularly deep housing theorem in geometry.

It is too short to describe all these results. We can only touch the tip of the fork in the housing law. Calculate a sphere, meaning the surface area of a sphere, similar to an orange peel or the Earth's crust. Locally, a small area around each point on the sphere can be described as a two-dimensional plane - after all, this is exactly what we do when we draw a map of a city or country on a flat sheet of paper. But even though the sphere locally resembles a two-dimensional plane, we know that it resides within a three-dimensional space.

These concepts can be generalized to higher dimensions. An n-dimensional "sheet" is a geometric structure that is locally similar to an n-dimensional Euclidean space. There are situations where an n-dimensional sheet is naturally nested within a larger dimensional space. But is this true for every sheet? In his housing theorem, Nash showed that under certain conditions the answer is yes, every sheet can be housed within a Euclidean space. Nash even calculated an upper limit on the dimension needed to house an n-dimensional sheet, for any natural number n.

John Nash had all these achievements to his credit before he turned 30, before mental illness took years of his life. Although over the years Nash returned to academic activity and research, the truth is that he was unable to reproduce the brilliant successes of his youth. Large audiences filled halls to hear his lectures, out of curiosity and reverence, but the quality of the results he presented was poor compared to the huge expectations that arose from his reputation. One can only regret the loss to mathematics and the world and imagine what would have been enough to do if he had not suffered from his illness.

Acquaintances of the young John Nash described him as quick-thinking, ambitious and even arrogant in his personality. I, who only knew him in recent years, remember a man very different from this description. I remember a quiet, shy John Nash, maybe a little embarrassed by the amount of attention that the movie about his life focused around him. I discovered an approachable person, who was willing to talk about his mathematical research, from marriages in gaming theory to housing law, with anyone who asked for a few minutes of his time. He and his wife, Alicia, were always warm and generous to me, inviting me to share transportation with them to lecture halls or to join them for a cup of coffee and a conversation about Ha Veda at conferences. They will be missed.

Dr. Ziv Hellman is a lecturer in the Department of Economics at Bar-Ilan University

## 2 תגובות

Most of the research and major breakthroughs in science are done at a relatively young age.

The age of 40 is even considered by mathematicians as an age after which not much is renewed. Even if it is an inaccurate "magic number", the principle is quite correct.

It's a bit sad that this is so, especially since the life expectancy in the West is about 80, so that most of the scientific output is done in the second quarter of life, after which we mostly enjoy the glory of the past.

So it was not really possible to expect John Nash to create much in his sixties and seventies, in particular that it came after a "scientific and cognitive coma" of about 30 years in which he suffered from schizophrenia.

The fact that he managed to overcome the disease, and return to a creative life is a great achievement.

With the existing high life expectancy, and the expectation of a further increase in it, one of the important challenges of neuroscience is to enable the preservation of a high cognitive level in old age in general, and the ability to learn and create new things in particular.

With all due respect, why underestimate the taxi driver? Where did he know that he is guilty and even if so there is no medical reason for his guilt?