Game theory

branch of mathematics focused on strategic decision making

Game theory is the study of mathematical models of strategic interactions among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, 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.

A game of imperfect information (the dotted line represents ignorance on the part of player 2, formally called information set)
CONTENT : A - F , G - L , M - R , S - Z , See also , External links



Quotes are arranged alphabetically per author

A - F

  • "Interactive Decision Theory" would perhaps be a more descriptive name for the discipline usually called Game Theory
  • There are quite a number of novel developments intended to meet the needs of a general theory of systems. We may enumerate them in brief survey:
    1. Cybernetics, based upon the principle of feedback or circular causal trains providing mechanisms for goal-seeking and self-controlling behavior.
    2. Information theory, introducing the concept of information as a quantity measurable by an expression isomorphic to negative entropy in physics, and developing the principles of its transmission.
    3. Game theory, analyzing in a novel mathematical framework, rational competition between two or more antagonists for maximum gain and minimum loss.
    4. Decision theory, similarly analyzing rational choices, within human organizations, based upon examination of a given situation and its possible outcomes.
    5. Topology or relational mathematics, including non-metrical fields such as network and graph theory.
    6. Factor analysis, i.e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields
    7. General system theory in the narrower sense (G.S.T.), trying to derive from a general definition of “system” as complex of interacting components, concepts characteristic of organized wholes such as interaction, sum, mechanization, centralization, competition, finality, etc., and to apply them to concrete phenomena.
While systems theory in the broad sense has the character of a basic science, it has its correlate in applied science, sometimes subsumed under the general name of Systems Science.
  • An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory.
    • Ivar Ekeland (2006) The Best of All Possible Worlds. Chapter 7, May The Best One Win, p. 141.
  • A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution.
    • Richard Arnold Epstein (1977) ''The Theory of Gambling and Statistical Logic (Revised Edition) Chapter Two, Mathematical Preliminaries, p. 36.
  • Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises.
    • Thomas Flanagan (1998) Game Theory and Canadian Politics Chapter 10, What Have We Learned?, p. 164.
  • That strategic rivalry in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other’s actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated "Folk Theorem." An outcome that Pareto dominates the minimax point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a Nash equilibrium in infinitely repeated games with sufficiently little discounting.
  • By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war.
    • M. Fortun, and S.S. Schweber (1993) "Scientists and the Legacy of World War II: The Case of Operations Research (OR)." Social Studies of Science Vol 23, p. 604.

G - L

  • Game theory is about how people cooperate as much as how they compete... Game theory is about the emergence, transformation, diffusion and stabilization of forms of behavior.
    • Herbert Gintis, Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction (2000) Preface, pp. xxiv-xiv.
  • Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of quantum states. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a Banach space, if it is also true for a finite number of agents of a finite choice space.
    • Herbert Gintis, Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction (2009) 2nd edition, Preface, pp. xv-xvi.
  • Direct application of the theory of games to the solution of real problems has been rare, and its chief uses have been to offer some insight and understanding into the problems of competition (without actually solving them), and to provide mathematicians with new fields to conquer. Many important real problems involve more than two opponents, are not zero-sum, and exceed the bounds of the most developed versions of game theory.
    • Lindsey, G. R. (1979) "Looking back over the Development and Progress of Operational Research." In: K. B. Haley (ed.) Operational Research ‘78. Amsterdam: North-Holland Publishing Company 1979, p. 13–31.
  • Chapter 2 describes the most demanding rational choice theory of all, game theory, which was developed by a genius and assumes that other people are geniuses.
    • Harford, Tim (2008). The Logic Of Life. Random House.

M - R

  • Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.
  • Game theory, however, deals only with the way in which ultrasmart, all knowing people should behave in competitive situations, and has little to say to Mr. X as he confronts the morass of his problem.
    • Howard Raiffa (1982) The Art and Science of Negotiation Prologue, p. 2.
  • At present game theory has, in my opinion, two important uses, neither of them related to games nor to conflict directly. First, game theory stimulates us to think about conflict in a novel way. Second, game theory leads to some genuine impasses, that is, to situations where its axiomatic base is shown to be insufficient for dealing even theoretically with certain types of conflict situations... Thus, the impact is made on our thinking process themselves, rather than on the actual content of our knowledge.
  • (Game theory is) essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not.
    • Anatol Rapoport Prisoner's dilemma: A study in conflict and cooperation. co-authored by Albert S. Chammah. Ann Arbor: The University of Michigan Press, 1965. p. 196.

S - Z

  • [G]ame theory has already established itself as an essential tool in the behavioral sciences, where it is widely regarded as a unifying language for investigating human behavior. Game theory's prominence in evolutionary biology builds a natural bridge between the life sciences and the behavioral sciences. And connections have been established between game theory and the two most prominent pillars of physics: statistical mechanics and quantum theory. ...[M]any physicists, neuroscientists, and social scientists... are... pursuing the dream of a quantitative science of human behavior. Game theory is showing signs of... an increasing important role in that endeavor. It's a story of exploration along the shoreline separating the continent of knowledge from an ocean of ignorance... a story worth telling.
    • Tom Siegfried, A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006) Preface
  • The last decade has seen a steady increase in the application of concepts from the theory of games to the study of evolution. Fields as diverse as sex ratio theory, animal distribution, contest behaviour and reciprocal altruism have contributed to what is now emerging as a universal way of thinking about phenotypic evolution... Paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behavior for which it was originally designed
  • The theory of games was first formalised by Von Neumann & Morgenstern (1953) in reference to human economic behaviour. Since that time, the theory has undergone extensive development... Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest. Such an assumption would clearly be out of place in an evolutionary context. Instead, the criterion of rationality is replaced by that of population dynamics and stability, and the criterion of self-interest by Darwinian fitness.
  • Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find prime numbers... a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells in favour of the analogy of mathematics to analysis of games played by other... agents, and against the analogy of mathematics to the expert play of the game itself.
    • Robert Spencer David Thomas, "Mathematics is Not a Game But..." (January, 2009) The Mathematical Intelligencer Vol. 31, No. 1, pp. 4-8. Also published in The Best Writing on Mathematics 2010 (2011) pp. 79-88.
  • One should remember that mathematical logic itself or the study of mathematics as a formal system can be considered a branch of combinatorial analysis. Metamathematics introduces a class of games—"solitaires"—to be played with symbols according to formal rules. One sense of Gödel's theorem is that some properties of these games can be ascertained only by playing them.
    • Stanislaw Ulam: "Random processes and transformations." In Proc. Intern. Math. Congress, Cambridge, Mass., vol. 2, pp. 264–275. 1950. (quote from p. 266)
  • The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence:
    • the use of mathematical logic to understand the operation of the nervous system;
    • the invention of information processing machines (as digital computers), thus laying the basis for artificial intelligence;
    • the establishment of the metadiscipline of system theory, which has had an imprint in many branches of science, such as engineering (systems analysis, control theory), biology (regulatory physiology, ecology), social sciences (family therapy, structural anthropology, management, urban studies), and economics (game theory);
    • information theory as a statistical theory of signal and communication channels;
    • the first examples of self-organizing systems.
This list is impressive: we tend to consider many of these notions and tools an integrative part of our life...
  • Game theory brings to the chaos-theory table the idea that generally, societies are not designed, and that most situations don’t come with a rulebook. Instead, people have their own plans and designs on how things should fit together. They want to determine how the game is played, and they see societal designers as myopic busybodies who would imprison them with their theories.
    • L.K. Samuels , In Defense of Chaos: The Chaology of Politics, Economics and Human Action, Cobden Press (2013) p. 372.
  • The implication of game theory, which is also the implication of the third image, is, however, that the freedom of choice of any one state is limited by the actions of the others.
    • Kenneth Waltz (1954) Man, the State, and War Chapter VII, Some Implications Of The Third Image, p. 204.

See also

Wikipedia has an article about:

(by country)

AbelAnaxagorasArchimedesAristarchus of SamosAverroesArnoldBanachCantorCartanChernCohenDescartesDiophantusErdősEuclidEulerFourierGaussGödelGrassmannGrothendieckHamiltonHilbertHypatiaLagrangeLaplaceLeibnizMilnorNewtonvon NeumannNoetherPenrosePerelmanPoincaréPólyaPythagorasRiemannRussellSchwartzSerreTaoTarskiThalesTuringWeilWeylWilesWitten


123360eπFibonacci numbersIrrational numberNegative numberNumberPrime numberQuaternion


AbstractionAlgorithmsAxiomatic systemCompletenessDeductive reasoningDifferential equationDimensionEllipseElliptic curveExponential growthInfinityIntegrationGeodesicInductionProofPartial differential equationPrinciple of least actionPrisoner's dilemmaProbabilityRandomnessTheoremTopological spaceWave equation


Euler's identityFermat's Last Theorem

Pure math

Abstract algebraAlgebraAnalysisAlgebraic geometry (Sheaf theory) • Algebraic topologyArithmeticCalculusCategory theoryCombinatoricsCommutative algebraComplex analysisDifferential calculusDifferential geometryDifferential topologyErgodic theoryFoundations of mathematicsFunctional analysisGame theoryGeometryGlobal analysisGraph theoryGroup theoryHarmonic analysisHomological algebraInvariant theoryLogicNon-Euclidean geometryNonstandard analysisNumber theoryNumerical analysisOperations researchRepresentation theoryRing theorySet theorySheaf theoryStatisticsSymplectic geometryTopology

Applied math

Computational fluid dynamicsEconometricsFluid mechanicsMathematical physicsScience

History of math

Ancient Greek mathematicsEuclid's ElementsHistory of algebraHistory of calculusHistory of logarithmsIndian mathematicsPrincipia Mathematica


Mathematics and mysticismMathematics educationMathematics, from the points of view of the Mathematician and of the PhysicistPhilosophy of mathematicsUnification in science and mathematics