Game theory

Math helps us study how we make choices.

It looks at how people play games. It also looks at animals and computers.

Some games are about winning and losing. In those games, one person wins what another loses.

Other games let everyone win together. This is called a cooperative game. People can make plans with each other. This math helps us see how groups work. It can even help us understand nature. Math is a great tool for many things.

Imagine you are playing a game. Your next move depends on what your friend does. This is what math can study. We call this study game theory. It is a way to look at how people, animals, and computers make choices.

Some games are zero-sum. This means if you win, someone else must lose. A game of poker is like this. One player gets the money that others lose.

Other games are not zero-sum. In these games, players might both win or both lose. John Nash was a famous thinker in this field. He found a way to look at choices called the Nash equilibrium. This helps us see when players are making the best moves for themselves.

Game theory helps us in many ways. It helps us study how businesses compete. It even helps us understand how animals live and grow in nature. Many smart people have won Nobel Prizes for this work. They used these math tools to understand the world better.

Have you ever wondered why people make certain choices? Sometimes our decisions depend on what others do. This is the core idea of game theory. It is a mathematical way to study strategic interactions.

This science helps us understand humans, animals, and even computers. It is used in many areas like economics and computer science. It helps us see how different players act to reach a goal.

There are different ways to look at these games. Some are called zero-sum games. In these, one person's gain is exactly balanced by another's loss. A game of poker is a good example of this.

Other games are non-zero-sum. In these, players might both win or both lose together. Some games are simultaneous, where everyone moves at the same time. Other games are sequential, where one person moves after another.

People have studied the math of games for a long time. Around 1564, Cardano wrote about games of chance. In 1657, Huygens published work on reasoning about games. Later, in 1713, Charles Waldegrave analyzed a card game called "le her." In 1838, Antoine Augustin Cournot created a model for business competition. By 1913, Ernst Zermelo showed how math applies to chess strategies. These early thinkers helped build the foundation for modern math.

Modern game theory really began in the 20th century. John von Neumann published important work in 1928. He and Oskar Morgenstern wrote a famous book in 1944. In 1950, John Nash developed a new idea. He found a way to see when players' choices are consistent. This is called a Nash equilibrium.

This discovery was very important for many different types of games. It helped move the science forward into new areas.

Today, game theory is a huge tool for many scientists. In the 1970s, it was used to study how animals evolve. John Maynard Smith won the Crafoord Prize in 1999 for this work. Many people have even won the Nobel Prize in economics. This includes John Nash in 1994 and Paul Milgrom in 2020. Others like Robert B. Wilson and Lloyd S. Shapley also won. These thinkers use math to solve real-world puzzles about how we live.

Game theory is the mathematical study of strategic interactions. It examines how different players make decisions when their success depends on the choices of others. This science acts as an umbrella term for the study of rational decision-making. It applies to humans, animals, and computers alike. Today, it is a vital tool in social sciences, economics, logic, systems science, and computer science.

To understand how these models work, one must look at the relationship between players. In a zero-sum game, any gain for one participant is exactly balanced by a loss for another. This means the total resources remain constant. For example, in poker, one player's winnings are equal to the opponents' losses. In contrast, non-zero-sum games allow for outcomes where the net result is not zero. In these scenarios, players might both gain or both lose depending on their strategies.

Games are also categorized by how players interact and the rules they follow. Cooperative games involve players who can form binding commitments, often enforced by external rules like contract law. Non-cooperative games occur when players cannot form alliances or must rely on self-enforcing agreements. Furthermore, games can be simultaneous or sequential. In simultaneous games, players move at the same time or without knowing previous actions. In sequential games, players take turns, and earlier decisions affect later ones.

Another way to classify games is through symmetry. A symmetric game is one where each player earns the same payoff for making the same choice. The identity of the player does not change the game's structure. Common examples include the prisoner's dilemma or the game of chicken. Asymmetric games involve different strategy sets for different players. An example is the ultimatum game, where players face different options.

History shows that mathematical interest in games predates modern theory. Around 1564, Cardano wrote about games of chance in his book *Liber de ludo aleae*. In 1657, Huygens published work on the concept of expectation. In 1713, Charles Waldegrave analyzed the card game "le her," providing a minimax mixed strategy solution. Later, in 1838, Antoine Augustin Cournot modeled competition in oligopolies. By 1913, Ernst Zermelo proved that optimal chess strategy is strictly determined.

Modern game theory emerged in the 20th century through the work of John von Neumann. In 1928, he published a paper on the theory of games of strategy. He used the Brouwer fixed-point theorem to prove results for two-person zero-sum games. In 1944, von Neumann and Oskar Morgenstern published *Theory of Games and Economic Behavior*. This work introduced cooperative games and an axiomatic theory of expected utility. This allowed economists to model decision-making under uncertainty.

In 1950, John Nash expanded the field significantly. He developed the Nash equilibrium, a concept for non-cooperative games. A Nash equilibrium occurs when no player can improve their payoff by changing their strategy alone. Nash proved this exists for every finite n-player, non-zero-sum, non-cooperative game. This was a major leap beyond von Neumann's earlier work. His contributions were so impactful that he, Reinhard Selten, and John Harsanyi were awarded the Nobel Prize in 1994.

The impact of game theory is visible in many prestigious awards. As of 2020, fifteen game theorists have won the Nobel Prize in economics. This includes recent winners Paul Milgrom and Robert B. Wilson. In the 1970s, the field moved into biology through evolutionary game theory. John Maynard Smith applied these ideas to evolution and won the Crafoord Prize in 1999. From designing markets to understanding nuclear strategy at the RAND Corporation, game theory remains essential.