Game Theory: Fundamentals and Applications

By Fouad Sabry

What Is Game Theory

The study of mathematical models of strategic interactions among rational actors is what's known as "game theory." It also has applications in the sciences of logic, systems science, and computer science, in addition to the various areas of social science. The ideas contained within game theory find widespread application in the field of economics. The conventional approaches to game theory dealt with two-player games known as zero-sum games, in which the gains and losses of each participant are precisely balanced by those of the other players. The more advanced game theories can be applied to a greater variety of behavioral relations in the 21st century; the phrase "game theory" is now used as an umbrella word for the science of logical decision making in humans, animals, and computers.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Game Theory

Chapter 2: Strategy (Game Theory)

Chapter 3: Nash Equilibrium

Chapter 4: Evolutionarily Stable Strategy

Chapter 5: Non-cooperative Game Theory

Chapter 6: Backward Induction

Chapter 7: Correlated Equilibrium

Chapter 8: Subgame Perfect Equilibrium

Chapter 9: Cooperative Bargaining

Chapter 10: Mertens-stable Equilibrium

(II) Answering the public top questions about game theory.

(III) Real world examples for the usage of game theory in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of game theory' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of game theory.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 27, 2023

Book preview

Chapter 1: Game theory

The study of mathematical models of strategic interactions among rational actors is what's known as game theory. The conventional approaches to game theory dealt with two-player games known as zero-sum games, in which the profits and losses of each participant are precisely balanced by those of the other players. It is currently an umbrella word encompassing the study of rational decision making in people, animals, as well as computers. In the 21st century, the improved game theories apply to a larger variety of behavioral connections.

The concept of mixed-strategy equilibria in a two-person zero-sum game and John von Neumann's demonstration of this concept are considered to be the origins of modern game theory. The original proof that Von Neumann developed made use of the Brouwer fixed-point theorem on continuous mappings into compact convex sets. This theorem is now considered to be a basic approach in game theory and mathematical economics. His study was followed by the 1944 book Theory of Games and Economic Behavior, which he co-wrote with Oskar Morgenstern and discussed competitive and cooperative games involving several participants. Following the announcement that game theorists Paul Milgrom and Robert B. Wilson would share the Nobel Memorial Prize in Economic Sciences, a total of fifteen game theorists have been awarded the Nobel Prize in Economics. For his contributions to the field of evolutionary game theory, John Maynard Smith was recognized with the Crafoord Prize.

Long before the development of contemporary mathematical game theory, discussions on the mathematics of games were already well under way.

Cardano's contributions to the book Liber de Ludo Aleae, in which he discusses games of chance (Book on Games of Chance), This was composed around the year 1564 but wasn't published until after the author's death in 1663, developed some of the fundamental concepts that underpin the subject.

Around the year 1650, Pascal and Huygens are credited with developing the idea of anticipation based on reasoning about the structure of games of chance, and Huygens published his gambling calculus in De ratiociniis in ludo aleæ (On Reasoning in Games of Chance) in 1657.

In 1713, A game known as le Her was dissected in a letter that was ascribed to Charles Waldegrave.

He was a prominent member of the Jacobite movement and the uncle of James Waldegrave, a senior official in the British government.

In the following letter:, A minimax mixed strategy answer that Waldegrave supplied for a two-player version of the card game le was given by him. Her, The issue is now often referred to as the Waldegrave dilemma.

In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot was thinking about a duopoly when he came up with a solution that turned out to be the Nash equilibrium of the game.

In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess), which demonstrated that the best chess strategy can be predicted with absolute certainty.

This opened the door for the development of more broad theorems.

In the book Applications aux Jeux de Hasard that he published in 1938 as well as previous remarks, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game).

Borel hypothesized that in two-player finite games with zero-sum outcomes, there are no mixed-strategy equilibria that can arise, a hypothesis that did not hold up after being tested by von Neumann.

Before John von Neumann's 1928 publication of the article On the Theory of Games of Strategy, there was no such thing as the distinct discipline known as game theory..

The first mathematical treatment of the prisoner's dilemma was published in 1950, and prominent mathematicians Merrill M. Flood and Melvin Dresher conducted an experiment in the same year as part of the RAND Corporation's study into game theory. RAND decided to carry out the study because of the potential relevance to international nuclear policy. John Nash, around the same time, created a criteria for reciprocal consistency of player tactics that he called the Nash equilibrium. This criterion is applicable to a larger range of games than the one suggested by von Neumann and Morgenstern. It was shown by Nash that any finite n-player, non-zero-sum, non-cooperative game (not only two-player zero-sum games), had what is now known as a Nash equilibrium in mixed tactics. This was a breakthrough in game theory.

In the 1950s, there was a burst of work in the field of game theory, which resulted in the development of a number of important ideas, including the core, the extended form game, fictional play, repeated games, and the Shapley value. The decade of the 1950s was also notable for the introduction of game theory to the fields of philosophy and political science.

Reinhard Selten presented his solution notion of subgame perfect equilibria in the year 1965. This further enhanced the Nash equilibrium. Later on, he would also establish the concept of trembling hand perfection. In recognition of their significant contributions to economic game theory, Nash, Selten, and Harsanyi were awarded the Nobel Prize in Economics in 1994.

The field of biology made substantial use of game theory in the 1970s, partly because to the contributions made by John Maynard Smith and his evolutionarily stable strategy. This trend continued well into the 1980s. Additionally, an examination and presentation of the ideas of linked balance, trembling hand perfection, and common knowledge were included in this study.

Because of his significant contributions to game theory, John Nash was presented with the Nobel Memorial Prize in Economic Sciences in the year 1994. The notion of the Nash equilibrium, which is a solution concept for non-cooperative games, is perhaps the most well-known contribution that Nash has made to the field of game theory. A Nash equilibrium is a collection of strategies, one for each player, such that no player can increase their reward by unilaterally altering their strategy. An equilibrium is said to be in a state of Nash if it is impossible for any player to do so.

Thomas Schelling and Robert Aumann, both game theorists, were awarded the Nobel Prize in 2005, following in the footsteps of John Nash, David Selten, and John Harsanyi. Schelling contributed to the development of dynamic models, which served as early instances of evolutionary game theory. Aumann made further contributions to the equilibrium school by establishing a comprehensive formal study of the assumption of common knowledge and of its repercussions. He also contributed to the equilibrium school by introducing equilibrium coarsening and linked equilibria.

The economists Leonid Hurwicz, Eric Maskin, and Roger Myerson were recognized for having developed the foundations of mechanism design theory and won the Nobel Prize in Economics in 2007. Myerson's contributions include the idea of correct equilibrium as well as a significant graduate work called Game Theory, Analysis of Conflict. Both of these are his ideas. The idea of incentive compatibility was first presented by Hurwicz, who also gave it a formal name.

The theory of stable allocations and the practice of market design earned Alvin E. Roth and Lloyd S. Shapley the Nobel Prize in Economics in 2012. The award was given to them for the practice of market design. Jean Tirole, a game theorist, was awarded the Nobel Prize in 2014.

A game is considered to be cooperative if the participants are able to create commitments that are externally enforced and legally enforceable (e.g. through contract law). A game is said to be non-cooperative if the players are unable to establish alliances with one another or if every agreement must be self-enforcing (e.g. through credible threats).

The cooperative game theory provides a high-level approach because it only describes the structure, strategies, and payoffs of coalitions, whereas the non-cooperative game theory also examines how bargaining procedures will affect the distribution of payoffs within each coalition. Cooperative game theory provides a high-level approach because it describes only the structure of coalitions. Because non-cooperative game theory is more general than cooperative game theory, cooperative games can be analyzed using the approach of non-cooperative game theory (the converse does not hold), provided that sufficient assumptions are made to encompass all of the possible strategies available to players due to the possibility of external enforcement of cooperation. The analysis of cooperative games using the approach of non-cooperative game theory is not possible. The application of a single theory may be desirable; however, in many cases, there is insufficient information available to accurately model the formal procedures available during the process of strategic bargaining. Alternatively, the model that would be produced as a result would be too complex to provide a useful tool in the real world. In situations like these, cooperative game theory offers a streamlined method that enables analysis of the whole game without the need to presuppose anything on the parties' respective negotiating positions.

A game is said to be symmetric