Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally developed by John von Neumann and Oskar Morgenstern, it has significantly broadened to include behavioral economics, evolutionary models, and other tools.

Game theory attempts to mathematically capture behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others. While used in economics, game theory can be applied to understand any situation with strategic interactions between parties: auctions, bargaining, duopolies, fair division, oligopolies, social network formation, war, voting systems, and more.

## Basic Concepts

Game theory analyzes mathematical models of conflict and cooperation between intelligent rational decision-makers. An approach to achieve a definite objective from a mix of cooperation and conflict. It is about interdependent decision makers whose decisions affect each other.

### Players

The individuals or groups making strategic decisions are called players. A player can be an individual, firm, nation, or animal.

### Strategy

A strategy is a complete plan of action a player will take given the set of circumstances that might arise within a game. It describes what action a player will select.

### Payoff

Payoffs represent the motivations of players in a game. It reflects the outcome of the game for a player and is expressed numerically. Players aim to maximize their payoffs.

### Information Set

An information set is the collection of possible game states that a player can distinguish given the information available to them when making a move. It contains all the possible moves a player could have observed before making their move.

### Nash Equilibrium

A Nash equilibrium is a stable state of a game involving two or more players in which each player holds the correct expectation about the other players’ behavior and acts rationally, therefore no player has an incentive to change only his own strategy unilaterally.

### Backward Induction

Backward induction is a technique to solve finite sequential games. It first considers the last time a decision might be made and chooses what to do in any situation at that time. Using this strategy, it then considers the second-to-last time a decision might be made, and so on.

## Types of Games

There are different categories into which games can be classified based on certain parameters. The most common classifications are:

### Cooperative vs Non-cooperative Games

- Cooperative games – Players can communicate, make binding agreements and coordinate their strategies.
- Non-cooperative games – Players make decisions independently without collaboration with any other player.

### Zero-sum vs Non-zero-sum Games

- Zero-sum games – If the total gains of all players sum to zero. In zero-sum games the interests of the players are strictly opposed. One person’s gain strictly equals the other player’s loss.
- Non-zero-sum games – If the total gains of the players are positive or negative. Players’ interests are neither completely opposed nor completely coincident. There are some cooperative elements among players.

### Symmetric vs Asymmetric Games

- Symmetric games – All players have the same strategies available to them and have identical payoffs for any given combination of strategies.
- Asymmetric games – Players have different strategies available to them or different payoffs.

### Simultaneous vs Sequential Games

- Simultaneous games – Players choose strategies without knowing the choices of other players. Their decisions are made at the same time.
- Sequential games – Later players observe earlier actions by other players before choosing their strategies. Decisions are made in a sequential manner.

### Perfect vs Imperfect Information Games

- Perfect information games – All players know the strategies and payoffs available to other players. They have complete information about the game.
- Imperfect information games – At least one player has incomplete information about the game. Some private information is hidden from other players.

### One-shot vs Repeated Games

- One-shot games – Players make decisions simultaneously or sequentially, and these decisions determine the payoffs. The game is played only once.
- Repeated games – Players interact repeatedly and have some knowledge about earlier actions. Players can adopt strategies based on previous interactions.

## Game Theory Solutions and Analysis

Game theory provides tools to predict when and how players change strategies, the outcome of games if rational decisions are made, as well as equilibrium points where all players are content with their strategy. Some key game theory solutions include:

### Dominant Strategy Solution

A strictly dominant strategy provides the best payoff for a player, regardless of the strategies pursued by other players. It makes a certain strategy the clear choice for a rational player.

### Nash Equilibrium

A Nash equilibrium occurs when each player takes the best decision for themselves based on their predictions of what the other players will do. No player can benefit by changing their strategy as long as the other players’ strategies remain unchanged.

### Backward Induction

Backward induction can be used to solve sequential games. Starting from the end of the game tree, one first determines the optimal strategy in the last time step. Using this information, one can then determine the optimal strategy for the second-to-last time step, and so on.

### Subgame Perfection

A subgame perfect Nash equilibrium further refines Nash equilibria to eliminate non-credible threats in sequential games. A strategy is subgame perfect if players’ strategies represent a Nash equilibrium in every subgame of the original game.

### Bayesian Games

These games model situations of imperfect information. Players are uncertain about the characteristics of other players, such as their available strategies or payoffs. Bayesian equilibrium allows players to update their beliefs according to Bayes’ theorem.

### Correlated Equilibrium

In this more general concept, an outside party selects the strategy profiles for players. They provide a recommendation that, if followed, leads to better payoffs for players than if they chose independently.

## Formal Models

Formal models refer to mathematical models designed to shed light on complex real-world problems through theoretical analysis. They are simplified abstractions representing key variables and their interactions. Some examples of formal models used in game theory include:

### Normal Form Games

These games model simultaneous-move games. The game is represented in a matrix form, showing the players, strategies, and payoffs in a standard game matrix. This allows analysis to determine dominant strategies and Nash equilibria.

### Extensive Form Games

Extensive form games capture sequential moves using a game tree structure. They model the order of players’ decisions, what each player knows when making decisions, chance events, payoffs, and more. This visual representation enables backward induction.

### Evolutionary Game Theory

These games model populations where strategies evolve over time based on their relative payoff success. Strategies performing well are imitated and reproduced in future generations. Evolutionary stability is analogous to equilibrium.

### Repeated Games

Formal models of repeated games identify equilibria when players interact recurrently. Factors like discount rates and being able to punish/reward other players’ past actions alter dominant strategies relative to one-shot games.

### Coalitional Games

These games study cooperative games where players can form collaborative groups or coalitions. The coalition as a whole receives a certain payoff which can be divided amongst players according to the rules set by the coalition.

### Signaling Games

Signaling games are two-player games with asymmetric information and two stages. In the first stage, the informed sender chooses whether and how to signal the state. In the second stage, the uninformed receiver observes the signal and takes an action.

## Applications of Game Theory

Game theory has been applied to many fields due to its ability to model strategic interactions. Some examples include:

### Economics

Game theory is used to model competition and cooperation in markets, auctions, bargaining, negotiations, industrial organization, and incentivizing optimal performance. Concepts like Nash equilibrium inform our understanding of markets.

### Political Science

Voting systems, bargaining, auctions, and campaigns can be analyzed using game theory. It provides insight into political decision making, international relations, war strategy, and more.

### Biology

Evolutionary game theory is used to understand competition, conflict, and cooperation in ecology and biology. It provides insight into the evolution of biological systems and behaviors among competing life forms.

### Computer Science

Applications in computer science include algorithmic game theory, machine learning, cybersecurity, networking (e.g. traffic routing) and logic systems. Analysis of opponent algorithms and development of optimization algorithms rely on game theoretic concepts.

### Philosophy

Game theory informs key issues in philosophy like decision theory, ethics, rational choice theory, and logic. Philosophical paradigms like bounded rationality, Pareto optimality, risk dominance, and perfect rationality draw directly from game theory.

### Operations Research

Scheduling, traffic flow optimization, supply chain management, and resource allocation rely on game theoretic tools to model interactions between independent decision makers and identify efficient cooperative/non-cooperative solutions.

### Social Psychology

Game theory provides insight into human behaviors like altruism, reciprocity, bargaining, and attitudes towards risk and fairness. Psychological and experimental research relies on game theory’s mathematical models of strategic interaction.

### Law

Analysis of legal rules and regulations can be done based on how rational actors respond to incentives. This provides insights into policymaking. Game theory informs contracts, property rights, torts, criminal law, and constitutional law.

## Key Figures in Game Theory

Some foundational thinkers who helped develop and advance game theory include:

### John von Neumann

Known as the founder of game theory, von Neumann established the mathematical foundations of game theory and the field of study. With Oskar Morgenstern, he authored the groundbreaking 1944 book “Theory of Games and Economic Behavior”.

### John Nash

Nash made key contributions to non-cooperative games, Nash equilibrium, bargaining games, and more. For this foundational work, he was awarded the 1994 Nobel Memorial Prize in Economic Sciences.

### John Harsanyi

Harsanyi introduced the concept of incomplete information into game theory by developing the analysis of Bayesian games. This paved the way for the analysis of asymmetric information games. In 1994, he received the Nobel Memorial Prize in Economics along with Nash and Reinhard Selten.

### Reinhard Selten

Selten significantly advanced the analysis of non-cooperative games. He developed the concept of subgame perfect equilibria and bounded rationality models. He was awarded the 1994 Nobel Memorial Prize jointly with Nash and Harsanyi.

### Robert Aumann

Aumann’s work focused on repeated games and the analysis of long-run relationships. He helped develop the theory of correlated equilibrium and coauthored influential books like “Values of Non-Atomic Games”. He was awarded the 2005 Nobel Memorial Prize in Economic Sciences.

### John Maynard Smith

Maynard Smith applied game theory to biology, pioneering the field of evolutionary game theory. This adapts classical game theory to evolving populations changing their behavior by natural selection over time.

### Thomas Schelling

Schelling applied game theory to international relations, arms control, and tax compliance. He developed focal point theory and contributed to the formal analysis of bargaining. He received the 2005 Nobel Prize in Economics.

### Kenneth Arrow

Arrow contributed to general equilibrium theory, social choice theory, and made significant advances in game theory and decision theory. He analyzed issues like information asymmetry, risk aversion, and uncertainty. In 1972, he was awarded the Nobel Prize in Economics.

### Oskar Morgenstern

With John von Neumann, Morgenstern co-authored “Theory of Games and Economic Behavior” which established the foundations of game theory. He made key contributions to utility theory as the basis for decision-making under uncertainty.

## Key Concepts and Theorems

Some fundamental theorems, solutions, and concepts which form the foundations of game theory analysis include:

### Minimax Theorem

Provides conditions for the existence of saddle points in zero-sum games, establishing optimal mixed strategy solutions. Authored by John von Neumann in 1928.

### Nash Equilibrium

States that each player’s strategy is optimal against those chosen by other players. If players are in a Nash equilibrium, they have no incentive to deviate from their strategy. Proposed by John Nash in 1950.

### Arrow’s Impossibility Theorem

Shows difficulties in aggregating individual preferences into community preferences. Demonstrates difficulties of creating a satisfactory voting system. Proved by Kenneth Arrow in 1951.

### Folk Theorem

Describes outcomes that can emerge in repeated games. Shows under what circumstances cooperative outcomes are sustained over time in an equilibrium. Generalizes to many non-cooperative games.

### Harsanyi Transformation

Provides a method to convert a game with incomplete information into a game with imperfect information. Introduced by John Harsanyi in 1967.

### Pareto Optimality

A situation where no individual can be made better off without making another individual worse off. A Pareto optimal outcome provides an efficient allocation of resources.

### Subgame Perfection

Requires strategies to represent a Nash equilibrium in every subgame. Eliminates non-credible threats in sequential games. Refines Nash equilibria in extensive form games.

### Perfect Bayesian Equilibrium

Equilibrium refinement for extensive form games with incomplete information. Combines Nash equilibrium and Bayesian inference under consistency constraints. Allows updated beliefs.

### Trembling Hand Perfection

Refines subgame and Nash equilibria by considering deviations happen with small probability. Eliminates overly fragile equilibria reliant on non-credible threats.

### Correlated Equilibrium

An outside party selects strategy profiles for players and provides recommendations. Following recommendations leads to better payoffs than independent choices. Generalizes Nash equilibrium.

## Major Textbooks

Some widely used and influential graduate-level textbooks covering game theory’s key models and applications include:

- Game Theory (2013) by Erich Prisner
- Game Theory 101: The Complete Textbook (2014) by William Spaniel
- Game Theory: An Introduction (2013) by Steven Tadelis
- Game Theory (2015) by Michael Maschler, Eilon Solan, Shmuel Zamir
- Lectures on Game Theory (1989) by Eric Rasmusen
- Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations (2013) by Yoav Shoham and Kevin Leyton-Brown
- Games of Strategy (2015) by Avinash Dixit and Susan Skeath
- The Ideas Behind the Chess Openings (1989) by Reuben Fine
- Playing for Real: A Text on Game Theory (2007) by Ken Binmore
- An Introduction to Game Theory (2004) by Martin J. Osborne
- Game Theory 101: Bargaining (2014) by William Spaniel
- Game Theory (1991) by Drew Fudenberg and Jean Tirole

These books cover both classical and modern game theory. They provide comprehensive coverage of key models, applications, theorems and research directions through detailed technical exposition and worked examples. These texts are widely used for teaching at the graduate level.

## Major Journals

Some of the most prominent peer-reviewed academic journals that publish game theory research include:

- Games and Economic Behavior
- International Journal of Game Theory
- Journal of Economic Theory
- Economics Letters
- Mathematical Social Sciences
- Journal of Mathematical Economics
- Social Choice and Welfare
- Games
- Dynamic Games and Applications
- International Game Theory Review
- The BE Journal of Theoretical Economics

These journals highlight the interdisciplinary nature of modern game theory spanning economics, mathematics, operations research, and computer science. They publish research on all aspects of game theory including computational techniques, applications, and extensions into new problem domains.

## Major Conferences

Some of the major academic conferences that bring together game theory researchers include:

- World Congress of the Game Theory Society
- North American Annual Meetings of the Game Theory Society
- Conference on Web and Internet Economics
- International Conference on Autonomous Agents and Multiagent Systems
- International Joint Conference on Artificial Intelligence
- Conference on Economics and Computation
- International Conference on Algorithmic Game Theory
- Game Theory Festival
- Analytic Hierarchy Process International Symposium
- International Conference on Machine Learning

These conferences provide a platform for researchers across disciplines like computer science, mathematics, economics, operations research and optimization to share new developments in all aspects of game theory. They also publish proceedings highlighting cutting-edge research.

## Recent Research Directions

Some active modern research directions in game theory include:

### Algorithmic Game Theory

Developing computational methods and tools to analyze large and complex games. Using algorithms to compute solutions and equilibria. Bridging computer science and game theory.

### Behavioral Game Theory

Incorporating insights from psychology into game theoretic models to better capture bounded rationality, altruism, reciprocity and human biases. Explaining deviations from perfect rationality.

### Cryptographic Protocols

Using game theory to design information sharing protocols and analyze security against outside attacks and coalitions of dishonest players. Applications in blockchain, elections, auctions.

### Machine Learning

Using game theory to improve the stability and robustness of machine learning algorithms against manipulation and adversarial examples. Designing competitive environments and multi-agent training.

### Mechanism Design

Designing rules of interactions to achieve desired outcomes. Applications in auction design, public policy, regulations and incentives. Inverse game theory.

### Quantum Games

Extending game theory into the realm of quantum information and physics. Modeling the impact of superposition, entanglement, and interference on strategic interactions.

### Evolutionary Game Theory

Modeling evolving populations where agents replicate successful strategies. Applications in biology, cognition, language emergence, social norms, technology growth and economics.

### Network Games

Analyzing strategic interactions over networks, where agent payoffs depend on local network connections. Applications in social networks, telecommunication networks, cybersecurity.

## Conclusion

In summary, game theory is a powerful and versatile mathematical modeling framework to analyze optimal decision-making in situations of strategic interdependence. The formal models, key concepts, common games, and techniques discussed in this article form the foundation of modern game theory analysis across diverse disciplines. Ongoing research is expanding the boundaries of game theory into new domains and problems. The rich literature and active research community will continue developing novel applications and insights using game theory as a lens.

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