Game theory

Game theory is “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.” Originally, it addressed zero-sum games, in which one person’s gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann’s original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, eleven game-theorists have now won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

A game is composed of:

• Players: usually rational players.
• Rules: in stochastic games can change.
• Actions or strategies: the possible actions of each player.
• Payoffs: The payoffs of each player regarding the actions taken by each player and the existing rules.

A game could be represented as:

• Extended form: decision tree.
• Normal form: payoff matrix.

The mainly descriptive properties of games are:

• Cooperative / Non-cooperative:
• Symmetric / Asymmetric:
• Zero-sum / Non-zero-sum:
• Simultaneous / Sequential:
• Complete information / incomplete information:
• Perfect information / imperfect information:
• Combinatorial games:
• Finite-time games / Infinitely long games:
• Discrete games / continuous games:
• Differential games:
• Many-player and population games:
• Stochastic outcomes:
• Metagames:
• Pooling Games:

Game theory is a interdisciplinary cross-field of science in which interacts among others with:

• Description and modeling: giving explanations to phenomena of multi-agent systems in different fields of science.
• Economics and business: best decision making in a market system.
• Social science: understand human patterns of behavior.
• Political science: social choice, how to decide collective actions, or how states interact in order to understand negotiations and international diplomatics.
• Biology: understanding some cell phenomena.
• Ecology: understanding patterns and phenomena in animal interaction.
• Computer science: understanding and implementing better algorithms with algorithmic game theory.
• Philosophy: Game theory has also challenged philosophers to think in terms of interactive epistemology and develop ethics around some games studied in game theory.

See also

Game Theory

Material

• https://en.wikipedia.org/wiki/Glossary_of_game_theory
• Game Theory Coursera’s course. Coursera.
• Christopher Griffin (2010) Game Theory: Penn State Math 486 Lecture Notes
• Adam Kalai: Game Theory and Computer Science - Lecture notes on Game Theory and Computer Science.
• Marek M. Kaminski: Game Theory and Politics - Syllabuses and lecture notes for game theory and political science.

Papers

• Nash, John (1950), Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America 36 (1): 48-49,
• Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095-1100.
• von Neumann, John (1928), “Zur Theorie der Gesellschaftsspiele”, Mathematische Annalen 100 (1): 295-320

Books

• von Neumann, John; Morgenstern, Oskar (1944), Theory of Games and Economic Behavior, Princeton.
• Osborne, M.J. and Rubinstein, A. (1994) A Course in Game Theory, MIT Press
• Ken Binmore (2007). Playing for real: a text on game theory. Oxford University Press US.
• Shoham, Yoav; Leyton-Brown, Kevin (2009), Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, New York: Cambridge University
• Leyton-Brown, Kevin; Shoham, Yoav (2008), Essentials of Game Theory: A Concise, Multidisciplinary Introduction, San Rafael, CA: Morgan & Claypool Publishers
• Rasmusen, Eric (2006), Games and Information: An Introduction to Game Theory (4th ed.), Wiley-Blackwell
• Maynard Smith, John (1982), Evolution and the theory of games. Cambridge University Press
• Webb, James N. (2007), Game theory: decisions, interaction and evolution, Springer undergraduate mathematics series, Springer