Quotable Mathematics: 2025

01 November 2025

On Game Theory (2000-)

"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, "Le meilleur des mondes possibles" ["The Best of All Possible Worlds"], 2000)

"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)

"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." (Herbert Gintis, "Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction", 2000)

"Game theory is a formal approach to analysing social situations employing highly stylized and parsimo-nious descriptions." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table". 2005)

"I think game theory creates ideas that are important in solving and approaching conflict in general." (Robert Aumann, 2005)

"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, "The Best of All Possible Worlds", 2006)

"Good decisions require that each decision-maker anticipate the decisions of the others. Game theory offers a systematic way of analysing strategic decision-making in interactive situations. [...] Game theory is not about 'playing' as usually understood. It is about conflict among rational but distrusting beings." (Geraldine Ryan & Seamus Coffey, "Games of Strategy", 2008)

"Game theory proposes a method called minimization-maximization (minimax) that determines the best possibility that is available to a player by following a decision tree that minimizes the opponent’s gain and maximizes the player’s own. This important algorithm is the basis for generating algorithms for chess programs." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Game theory postulates rational behavior for each participant. Each player is conscious of the rules and behaves in accordance with them, each player has sufficient knowledge of the situation in which he or she is involved to be able to evaluate what the best option is when it comes to taking action (a move), and each player takes into account the decisions that might be made by other participants and their repercussions with respect to his or her own decision. Game theory about zero-sum games with two participants is relevant for chess. In this type of situation, each action that is favorable to one participant" (player) is proportionally unfavorable for the opponent. Thus, the gain of one represents the loss of the other." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Game theory covers an incredibly broad spectrum of scenarios of cooperation and competition, but the field began with those resembling heads-up poker: two-person contests where one player’s gain is another player’s loss. Mathematicians analyzing these games seek to identify a so-called equilibrium: that is, a set of strategies that both players can follow such that neither player would want to change their own play, given the play of their opponent. It’s called an equilibrium because it’s stable - no amount of further reflection by either player will bring them to different choices. I’m content with my strategy, given yours, and you’re content with your strategy, given mine." (Brian Christian & Thomas L Griffiths, "Algorithms to Live By: The Computer Science of Human Decisions", 2016)

On Game Theory (1975-1999)

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution." (Richard A Epstein, "The Theory of Gambling and Statistical Logic" [Revised Edition], 1977)

"Game theory is a collection of mathematical models designed to study situations involving conflict and/or cooperation. It allows for a multiplicity of decision makers who may have different preferences and objectives. Such models involve a variety of different solution concepts concerned with strategic optimization, stability, bargaining, compromise, equity and coalition formation." (Notices of the American Mathematical Society Vol. 26 (1), 1979)

"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." (George R Lindsey, "Looking back over the Development and Progress of Operational Research, 1979)

"There are many difficulties in application of [the games] theory to the real world. [., ..] In general, competitors are not in complete opposition. As a matter of fact often they don't even have the same objectives. This difficulty can often be circumvented by using a different objective, 'games of survival'. Secondly, a decision is seldom made once. This motivated the study of multistage games [...]. Finally, decisions are not usually made simultaneously. Recognition of this fact leads to 'games of protocol' [...]. Games of protocol can also be used to handle processes involving three or more people." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"Cybernetics is concerned with scientific investigation of systemic processes of a highly varied nature, including such phenomena as regulation, information processing, information storage, adaptation, self-organization, self-reproduction, and strategic behavior. Within the general cybernetic approach, the following theoretical fields have developed: systems theory" (system), communication theory, game theory, and decision theory." (Fritz B Simon et al, "Language of Family Therapy: A Systemic Vocabulary and Source Book", 1985)

"But the answers provided by the theory of games are sometimes very puzzling and ambiguous. In many situations, no single course of action dominates all the others; instead, a whole set of possible solutions are all equally consistent with the postulates of rationality." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)

"Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." (Roger B Myerson, "Game Theory: Analysis of Conflict", 1991)

"Game theory is a theory of strategic interaction. That is to say, it is a theory of rational behavior in social situations in which each player has to choose his moves on the basis of what he thinks the other players' countermoves are likely to be." (John Harsanyi, "Games with Incomplete Information", 1997)

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises." (Thomas Flanagan, "Game Theory and Canadian Politics", 1998)

On Game Theory (-1974)

"While these games are not typical for major economic processes, they contain some universally important traits of all games and the results derived from them are the basis of the general theory of games." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"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, "Man, the State, and War", 1959)

"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." (Anatol Rapoport, Fights, games, and debates", 1960)

"It is the shortcomings of game theory (as originally formulated) which force the consideration of the role of ethics, of the dynamics of social structure, and of social structure and of individual psychology in situations of conflict." (Anatol Rapoport, "Fights, games, and debates", 1960)

"A thorough understanding of game theory, should dim these greedy hopes. Knowledge of game theory does not make one a better card player, businessman or military strategist." (Anatol Rapoport, "The Use and Misuse of Game Theory," 1962)

"Although the drama of games of strategy is strongly linked with the psychological aspects of the conflict, game theory is not concerned with these aspects. Game theory, so to speak, plays the board. It is concerned only with the logical aspects of strategy." (Anatol Rapoport, "The Use and Misuse of Game Theory", 1962)

"Game theory applies to a very different type of conflict, now technically called a game. The well-known games such as poker, chess, ticktacktoe and so forth are games in the strict technical Bark and counterbark sense. But what makes parlor games is not their entertainment value or detachment from real life." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"Whether game theory leads to clear-cut solutions, to vague solutions, or to impasses, it does achieve one thing. In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate." (Anatol Rapoport, "The Use and Misuse of Game Theory", Scientific American 207, 1962)

"The modern era has uncovered for combinatorics a wide range of fascinating new problems. These have arisen in abstract algebra, topology, the foundations of mathematics, graph theory, game theory, linear programming, and in many other areas. Combinatorics has always been diversified. During our day this diversification has increased manifold. Nor are its many and varied problems successfully attacked in terms of a unified theory. Much of what we have said up to now applies with equal force to the theory of numbers. In fact, combinatorics and number theory are sister disciplines. They share a certain intersection of common knowledge, and each genuinely enriches the other." (Herbert J Ryser, "Combinatorial Mathematics", 1963)

"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)

On Games (Unsourced)

"As in every discipline, so in astronomy, too, the conclusions that we teach the reader are seriously intended, and our discussion is no mere game." (Johannes Kepler)

"Chess is a game by its form, an art by its content and a science by the difficulty of gaining mastery in it. Chess can convey as much happiness as a good book or work of music can. However, it is necessary to learn to play well and only afterwards will one experience real delight." (Tigran Petrosian)

"How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play." (Tobias Dantzig)

"I love mathematics [...] principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game the encompassing of the infinite." (Rózsa Péter)

"If chess permits a virtually infinite variety of games, the rules of nature surely do. Science may be immortal after all." (John Horgan)

"It's a game of a million inferences. There are a lot of things to draw inferences from - cards played and not played. These inferences tell you something about the probabilities. It's got to be the best intellectual exercise out there. You're seeing through new situations every ten minutes. Bridge is about weighing gain/loss ratios. You're doing calculations all the time." (Warren Buffett)

"Mathematics should be learned through recreational games, the way the Egyptians do, through amusement and pleasure." (John A Comenius)

"Science is a game - but a game with reality, a game with sharpened knives." (Erwin Schrödinger)

"The chief weakness of the machine is that it will not learn by its mistakes. The only way to improve its play is by improving the program. Some thought has been given to designing a program that would develop its own improvements in strategy with increasing experience in play. Although it appears to be theoretically possible, the methods thought of so far do not seem to be very practical. One possibility is to devise a program that would change the terms and coefficients involved in the evaluation function on the basis of the results of games the machine had already played. Small variations might be introduced in these terms, and the values would be selected to give the greatest percentage of wins." (Claude E Shannon)

"We must regard classical mathematics as a combinatorial game played with symbols." (John von Neumann)

On Games (2010-2019)

"Game theory is designed to address situations in which the outcomes of a person’s decisions depend not just on how they choose among several options, but also on the choices made by the people with whom they interact." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"To understand the idea of Nash equilibrium, we should first ask why a pair of strategies that are not best responses to each other would not constitute an equilibrium. The answer is that the players cannot both believe that these strategies would actually be used in the game, since they know that at least one player would have an incentive to deviate to another strategy. So a Nash equilibrium can be thought of as an equilibrium in beliefs. If each player believes that the other player will actually play a strategy that is part of a Nash equilibrium, then she has an incentive to play her part of the Nash equilibrium." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Analyzing the behavior of a nonlinear system is like walking through a maze whose walls rearrange themselves with each step you take" (in other words, playing the game changes the game)." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Mathematicians are used to game-playing according to a set of rules they lay down in advance, despite the fact that nature always writes her own." (Philip W Anderson, "More and Different: Notes from a Thoughtful Curmudgeon", 2011)

"Natural science has discovered 'chaos'. Social science has encountered 'complexity'. But chaos and complexity are not characteristics of our new reality; they are features of our perceptions and understanding. We see the world as increasingly more complex and chaotic because we use inadequate concepts to explain it. When we understand something, we no longer see it as chaotic or complex. Maybe playing the new game requires learning a new language." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Operational thinking is about mapping relationships. It is about capturing interactions, interconnections, the sequence and flow of activities, and the rules of the game. It is about how systems do what they do, or the dynamic process of using elements of the structure to produce the desired functions. In a nutshell, it is about unlocking the black box that lies between system input and system output." (Jamshid Gharajedaghi, "Systems Thinking: Managing Chaos and Complexity A Platform for Designing Business Architecture" 3rd Ed., 2011)

"Mathematicians are used to game-playing according to a set of rules they lay down in advance, despite the fact that nature always writes her own. One acquires a great deal of humility by experiencing the real wiliness of nature." (Philip W Anderson, "More and Different: Notes from a Thoughtful Curmudgeon", 2011)

"The nice thing with Monte Carlo is that you play a game of let’s pretend, like this: first of all there are ten scenarios with different probabilities, so let’s first pick a probability. The dice in this case is a random number generator in the computer. You roll the dice and pick a scenario to work with. Then you roll the dice for a certain speed, and you roll the dice again to see what direction it took. The last thing is that it collided with the bottom at an unknown time so you roll dice for the unknown time. So now you have speed, direction, starting point, time. Given them all, I know precisely where it [could have] hit the bottom. You have the computer put a point there. Rolling dice, I come up with different factors for each scenario. If I had enough patience, I could do it with pencil and paper. We calculated ten thousand points. So you have ten thousand points on the bottom of the ocean that represent equally likely positions of the sub. Then you draw a grid, count the points in each cell of the grid, saying that 10% of the points fall in this cell, 1% in that cell, and those percentages are what you use for probabilities for the prior for the individual distributions." (Henry R Richardson) [in (Sharon B McGrayne, "The Theory That Would Not Die", 2011)]

"Chess is a perfect arena for just such an exerted exploration of the possible. Its chequered sea is very deep indeed. The mathematics behind the game’s complexity are staggering. […] For all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programmed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face." (Daniel Tammet, "Thinking in Numbers" , 2012)

"The barrier to an appreciation of mathematical beauty is not insurmountable, however. […] The beauty adored by mathematicians can be pursued through the everyday: through games, and music, and magic." (Daniel Tammet, "Thinking in Numbers" , 2012)

"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." (Lawrence K Samuels, "In Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Probability theory provides the best answer only when the rules of the game are certain, when all alternatives, consequences, and probabilities are known or can be calculated. [...] In the real game, probability theory is not enough. Good intuitions are needed, which can be more challenging than calculations. One way to reduce uncertainty is to rely on rules of thumb." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"The taming of chance created mathematical probability. [...] Probability is not one of a kind; it was born with three faces: frequency, physical design, and degrees of belief. [...] in the first of its identities, probability is about counting. [...] Second, probability is about constructing. For example, if a die is constructed to be perfectly symmetrical, then the probability of rolling a six is one in six. You don’t have to count. [...] Probabilities by design are called propensities. Historically, games of chance were the prototype for propensity. These risks are known because people crafted, not counted, them. [...] Third, probability is about degrees of belief. A degree of belief can be based on anything from experience to personal impression." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"According to the traditional distinction from economics, risk is measurable, whereas uncertainty is indefinite or incalculable. In truth, risk can never be measured precisely except in dice rolls and games of chance, called a priori probability. Risk can only be estimated from observations in the real world, but to do that, we need to take a sample, and estimate the underlying distribution. In a sense, our estimates of real-world volatility are themselves volatile. Failure to realize this fundamental untidiness of the real world is called the ludic fallacy from the Latin for games. […] However, when the term risk measurement is used as opposed to risk estimation, a degree of precision is suggested that is unrealistic, and the choice of language suggests that we know more than we do. Even the language 'risk management' implies we can do more than we can." (Paul Gibbons, "The Science of Successful Organizational Change", 2015)

"Chess, with its straightforward rules and tiny Cartesian playing field, is a game tailor-made for computers." (John Horgan, "The End of Science", 2015)

"In business, as in game theory and chess, all great strategies start with a vision of the future. In one sense, the recipe is simple: it should include a sense of where the organization should go, what customers are likely to pay for, and how the organization can offer a unique product or service that customers will buy. The devil, of course, lies in the details." (David B Yoffie & Michael A Cusumano, "Strategy Rules", 2015)

"Master strategists understand that day-to-day tactical decisions are just as important as big competitive moves. Strategy creates the playing field; tactics define how you play the game - and ultimately whether you win or survive to play another day." (David B Yoffie & Michael A Cusumano, "Strategy Rules", 2015)

"Mathematicians usually think not in terms of concrete realizations but in terms of rules that are given axiomatically. Mathematics is the art of arguing with some chosen logic and some chosen axioms. As such, it is simply one of the oldest games with symbols and words." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Game theory covers an incredibly broad spectrum of scenarios of cooperation and competition, but the field began with those resembling heads-up poker: two-person contests where one player’s gain is another player’s loss. Mathematicians analyzing these games seek to identify a so-called equilibrium: that is, a set of strategies that both players can follow such that neither player would want to change their own play, given the play of their opponent. It’s called an equilibrium because it’s stable - no amount of further reflection by either player will bring them to different choices. I’m content with my strategy, given yours, and you’re content with your strategy, given mine." (Brian Christian & Thomas L Griffiths, "Algorithms to Live By: The Computer Science of Human Decisions", 2016)

"So everyone has and uses mental representations. What sets expert performers apart from everyone else is the quality and quantity of their mental representations. Through years of practice, they develop highly complex and sophisticated representations of the various situations they are likely to encounter in their fields - such as the vast number of arrangements of chess pieces that can appear during games. These representations allow them to make faster, more accurate decisions and respond more quickly and effectively in a given situation. This, more than anything else, explains the difference in performance between novices and experts." (Anders Ericsson & Robert Pool, "Peak: Secrets from the New Science of Expertise", 2016)

"The foundations of a discipline are inseparable from the rules of its game, without which there is no discipline, just idle talk. The foundations of science reside in its epistemology, meaning that they lie in the mathematical formulation of knowledge, structured experimentation, and statistical characterization of validity. Rules impose limitations. These may be unpleasant, but they arise from the need to link ideas in the mind to natural phenomena. The mature scientist must overcome the desire for intuitive understanding and certainty, and must live with stringent limitations and radical uncertainty." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"The model (conceptual system) is a creation of the imagination, in accordance with the rules of the game. The manner of this creation is not part of the scientific theory. The classical manner is that the scientist combines an appreciation of the problem with reflections upon relevant phenomena and, based on mathematical knowledge, creates a model." (Edward R Dougherty, "The Evolution of Scientific Knowledge: From certainty to uncertainty", 2016)

"But chess is a limited game and every position will have patterns and markers our intuition can interpret. Each of the estimated tens of thousands of positions a strong master has imprinted in memory can also be broken down into component parts, rotated, twisted, and still be useful. Outside of the opening sequences that are indeed memorized, strong human players don’t rely on recall as much as on a super-fast analogy engine." (Garry Kasparov, "Deep Thinking", 2017)

"There is no such thing as randomness. No one who could detect every force operating on a pair of dice would ever play dice games, because there would never be any doubt about the outcome. The randomness, such as it is, applies to our ignorance of the possible outcomes. It doesn’t apply to the outcomes themselves. They are 100% determined and are not random in the slightest. Scientists have become so confused by this that they now imagine that things really do happen randomly, i.e. for no reason at all." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Games (2000-2009)

"[...] if 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", 2000)

"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the ‘ideal gases’ which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted." (Gerald Tenenbaum & Michael M France, "The Prime Numbers and Their Distribution", 2000)

"Strategy in complex systems must resemble strategy in board games. You develop a small and useful tree of options that is continuously revised based on the arrangement of pieces and the actions of your opponent. It is critical to keep the number of options open. It is important to develop a theory of what kinds of options you want to have open." (John H Holland, [presentation] 2000)

"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"Ordinary numbers have immediate connection to the world around us; they are used to count and measure every sort of thing. Adding, subtracting, multiplying and dividing all have simple interpretations in terms of the objects being counted and measured. When we pass to complex numbers, though, the arithmetic takes on a life of its own. Since -1 has no square root, we decided to create a new number game which supplies the missing piece. By adding in just this one new element √-1. we created a whole new world in which everything arithmetical, miraculously, works out just fine." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)

"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? [...] A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them." (Paul Cohen, "Skolem and pessimism about proof in mathematics", Philosophical Transactions of the Royal Society A 363 (1835), 2005)

"I think game theory creates ideas that are important in solving and approaching conflict in general. Robert Aumann, 2005)

"The players in a game are said to be in strategic equilibrium (or simply equilibrium) when their play is mutually optimal: when the actions and plans of each player are rational in the given strategic environment - i. e., when each knows the actions and plans of the others." (Robert Aumann, "War and Peace", 2005)

"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, "The Best of All Possible Worlds", 2006)

"Mathematics is like a game. It has rules, and to enjoy playing or watching it, you have to know and understand the rules. Mathematicians make up the rules as they go along." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"A tactician feels at home reacting to threats and seizing opportunities on the battlefield. When your opponent has blundered, a winning tactic can suddenly appear and serve as both means and end. […] Every time you make a move, you must consider your opponent’s response, your answer to that response, and so on. A tactic ignites an explosive chain reaction, a forceful sequence of moves that carries the players along on a wild ride. You analyze the position as deeply as you can, compute the dozens of variations, the hundreds of positions. If you don’t immediately exploit a tactical opportunity, the game will almost certainly turn against you; one slip and you are wiped out. But if you seize the opportunities that your strategy creates, you’ll play your game like a Grandmaster." (Garry Kasparov, "How Life Imitates Chess", 2007)

"The middle game requires alertness in general and alertness to patterns in particular. These are general ideas that anyone can learn with practice; the more you play, the better you become at recognizing the patterns and applying the solutions. That is, to find similarities to positions you have seen before and then to recall what worked" (or what didn’t work) in that situation. There is still potential for great creativity, if you are able to relate known patterns to new positions to find the unique solution: the best move." (Garry Kasparov, "How Life Imitates Chess", 2007)

"The worst enemy of the strategist is the clock. Time trouble, as we call it in chess, reduces us all to pure reflex and reaction, tactical play. Emotion and instinct cloud our strategic vision when there is no time for proper evaluation. A game of chess can suddenly seem a lot like a game of chance. Even the finest sense of intuition can’t flourish in the long term without accurate calculations." (Garry Kasparov, "How Life Imitates Chess", 2007)

"There is still a great deal of uncharted territory in the opening phase of the game. New ideas, new concepts, new plans in old and forgotten variations, there is still much to discover in the opening. The tactical patterns and strategic concepts of the middle game have been well mapped out by generations of Grandmasters, although there are occasional fresh twists. In the endgame, however, the plans and possibilities are open and known to all, an almost mathematical exercise. This isn’t to say that everything is predetermined. With flawless play from both sides, the endgame will advance toward a predictable conclusion. But since humans are flawed, damage can be inflicted or repaired. Even if one player is at a clear disadvantage, he may simply outplay his opponent." (Garry Kasparov, "How Life Imitates Chess", 2007)

"A game is a situation of strategic interdependence: the outcome of your choices (strategies) depends upon the choices of one or more other persons acting purposely. The decision makers involved in a game are called players, and their choices are called moves. The interests of the players in a game may be in strict conflict; one person’s gain is always another’s loss. Such games are called zero-sum. More typically, there are zones of commonality of interests as well as of conflict and so, there can be combinations of mutually gainful or mutually harmful strategies. Nevertheless, we usually refer to the other players in a game as one’s rivals." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"Chess reflects the real world in miniature. Endeavor, struggle, success, and defeat - they are part of each game ever played." (Bruce Pandolfini, "Pandolfini's Ultimate Guide to Chess", 2008)

"John Nash’s beautiful equilibrium was designed as a theoretical way to square just such circles of thinking about thinking about other people’s choices in games of strategy. The idea is to look for an outcome where each player in the game chooses the strategy that best serves his or her own interest, in response to the other’s strategy. If such a configuration of strategies arises, neither player has any reason to change his choice unilaterally. Therefore, this is a potentially stable outcome of a game where the players make individual and simultaneous choices of strategies." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"The essence of a game of strategy is the interdependence of the players’ decisions. These interactions arise in two ways. The first is sequential [...] The players make alternating moves. [...] The second kind of interaction is simultaneous, as in the prisoners’ dilemma [...] The players act at the same time, in ignorance of the others’ current actions. However, each must be aware that there are other active players, who in turn are similarly aware, and so on. Therefore each must figuratively put himself in the shoes of all and try to calculate the outcome. His own best action is an integral part of this overall calculation. When you find yourself playing a strategic game, you must determine whether the interaction is simultaneous or sequential." (Avinash K Dixit & Barry J Nalebuff, "The Art of Strategy: A Game Theorist's Guide to Success in Business and Life", 2008)

"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess, as a game of zero sum and total information is, theoretically, a game that can be solved. The problem is the immensity of the search tree: the total number of positions surpasses the number of atoms in our galaxy. When there are few pieces on the board, the search space is greatly reduced, and the problem becomes trivial for computers’ calculation capacity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Finally, chess has a science - like special attraction since it lets the player first propose hypotheses of different strategic plans that are based on the game rules and possible moves of the pieces and then refute those hypotheses after careful investigation of the different lines of play. This process is analogous to the everyday work of a scientist." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"From its mystical origins as a dialogue with the supernatural powers to a metaphor for war, chess passes through a period as a representation of order in the universe until it becomes the game-art-science that millions of people all over the world are passionate about and that has developed into a testing ground for the sciences of artificial intelligence and cognitive psychology." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Many terms that are used to comment on games are aesthetic allusions, indicating that among chess players it is hard to separate out the game’s creative and analytic aspects. Terms that are frequently used include subtlety, depth, beauty, surprise, vision, brilliance, elegance, harmony, and symmetry." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"On the surface, chess is a game that has a winner and a loser. However, a deeper look reveals that perhaps chess is not just a game but a line of communication between two brains. [...] chess is a communication device. As with any other act of communication, it is necessary to have someone who sends the message, a transmission medium, and someone who receives the message. Players are both the communicators and receivers; the board and the chess pieces are the transmission medium. In an exchange of messages, ideas, attitudes, and personal positions about the uncertainty of our world, however, where is the win, and where is the loss?" (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The problem of identifying the subset of good moves is much more complicated than simply counting the total number of possibilities and falls completely into the domain of strategy and tactics of chess as a game." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

On Games (1990-1999)

"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)

"Independence of thought is a most valuable quality in a chess-player, both at the board and when preparing for a game." (David Bronstein, "200 Open Games", 1991)

"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)

"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 […]" (Francisco Varela, "The Embodied Mind", 1991)

"An example, which, like tossing a coin, is intimately associated with games of chance, is the shuffling of a deck of cards. […] the process is not completely random, if by what happens next we mean the outcome of the next single riffle, since one riffle cannot change any given order of the cards in the deck to any other given order. In particular, a single riffle cannot completely reverse the order of the cards, although a sufficient number of successive riffles, of course, can produce any order." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Because chaos is deterministic, or nearly so, games of chance should not be expected to provide us with simple examples, but games that appear to involve chance ought to be able to take their place. Among the devices that can produce chaos, the one that is nearest of kin to the coin or the deck of cards may well be the pinball machine. It should be an old-fashioned one, with no flippers or flashing lights, and with nothing but simple pins to disturb the free roll of the ball until it scores or becomes dead." (Edward N Lorenz, "The Essence of Chaos", 1993)

"It is true that every aspect of the roll of dice may be suspect: the dice themselves, the form and texture of the surface, the person throwing them. If we push the analysis to its extreme, we may even wonder what chance has to do with it at all. Neither the course of the dice nor their rebounds rely on chance; they are governed by the strict determinism of rational mechanics. Billiards is based on the same principles, and it has never been considered a game of chance. So in the final analysis, chance lies in the clumsiness, the inexperience, or the naiveté of the thrower - or in the eye of the observer." (Ivar Ekeland, "The Broken Dice, and Other Mathematical Tales of Chance", 1993)

"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Chess players recognize and applaud good play, from a single smart move to a brilliant combination to an entire game which is considered a masterpiece. Mathematicians recognize and applaud good mathematics, from clever tricks to brilliant proofs, and from beautiful conceptions to grand and deep ideas which advance our understanding of mathematics as a whole. It takes imagination and insight to discover the best moves, at chess or mathematics, and the more difficult the position, the harder they are to find. Chess players learn by experience to recognize types of positions and situations and to know what kind of moves are likely to be successful; they exploit brilliant local tactics as well as deep stategical ideas. So do mathematicians. Neither games nor mathematics play themselves - they both need a player with understanding, good ideas, judgement and discrimination to play them. To develop these essential attributes, the player must explore the game by playing it, thinking about it and analysing it. For the chess player and the mathematician, this process is scientific: you test ideas, experiment with new possibilities, develop the ones that work and discard the ones that fail. This is how chess players and mathematicians develop their tactical and strategical understanding; it is how they give meaning to chess and mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"[…] a rule for choosing an action is termed a strategy. If the rule says to always take the same action, it's called a pure strategy; otherwise, the strategy is called mixed. A solution to a game is simply a strategy for each player that gives each of them the best possible payoff, in the sense of being a regret-free choice." (John L Casti, "Five Golden Rules", 1995)

"It is typical that there is more than one way of looking at a geometrical figure, just as there are many ways of looking at lines of algebra. Perception, 'seeing', is an essential feature of mathematics. This is obvious when we are looking for patterns - how can you possibly 'spot' a pattern if you cannot in some sense 'see' it? But it is just as true when the mathematician is looking for hidden connections, or studying a position in a mathematical game, searching for a tactical sequence, or trying to 'see' the possibilities clearly. Superficially, it might seem that it is only geometry (and related fields of mathematics) that depends on perception, but this is not so. Perception is everywhere in mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1995)

"No, nature is, in its own subtle way, simple. However, those simplicities do not present themselves to us directly. Instead, nature leaves clues for the mathematical detectives to puzzle over. It's a fascinating game, even to a spectator. And it's an absolutely irresistible one if you are a mathematical Sherlock Holmes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Playing the game of mathematics is much harder than investigating scientifically! To jot down some numbers, a few differences, and spot a pattern is child's play compared to playing the game of algebra." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)

"The Minimax Theorem applies to games in which there are just two players and for which the total payoff to both parties is zero, regardless of what actions the players choose. The advantage of these two properties is that with two players whose interests are directly opposed we have a game of pure competition, which allows us to define a clear-cut mathematical notion of rational behavior that leads to a single, unambiguous rule as to how each player should behave." (John L Casti, "Five Golden Rules", 1995)

"To talk about sensemaking is to talk about reality as an ongoing accomplishment that takes form when people make retrospective sense of the situations in which they find themselves and their creations. There is a strong reflexive quality to this process. People make sense of things by seeing a world on which they already imposed what they believe. In other words, people discover their own inventions. This is why sensemaking can be understood as invention and interpretations understood as discovery. These are complementary ideas. If sensemaking is viewed as an act of invention, then it is also possible to argue that the artifacts it produces include language games and texts." (Karl E Weick, "Sensemaking in Organizations", 1995)

"What's important about a saddle point is that it represents a decision by the two players that neither can improve upon by unilaterally departing from it. In short, either player can announce such a choice in advance to the other player and suffer no penalty by doing so. Consequently, the best choice for each player is at the saddle point, which is called a 'solution' to the game in pure strategies. This is because regardless of the number of times the game is played, the optimal choice for each player is to always take his or her saddle-point decision. […] the saddle point is at the same time the highest point on the payoff surface in one direction and the lowest in the other direction. Put in algebraic terms using the payoff matrix, the saddle point is where the largest of the row minima coincides with the smallest of the column maxima." (John L Casti, "Five Golden Rules", 1995)

"But real-life situations often require us to measure probability in precisely this fashion - from sample to universe. In only rare cases does life replicate games of chance, for which we can determine the probability of an outcome before an event even occurs - a priori […] . In most instances, we have to estimate probabilities from what happened after the fact - a posteriori. The very notion of a posteriori implies experimentation and changing degrees of belief." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"In a mathematical sense a zero-sum game is a loser's game when it is valued in terms of utility. The best decision for both is to refuse to play this game." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"Losing streaks and winning streaks occur frequently in games of chance, as they do in real life. Gamblers respond to these events in asymmetric fashion: they appeal to the law of averages to bring losing streaks to a speedy end. And they appeal to that same law of averages to suspend itself so that winning streaks will go on and on. The law of averages hears neither appeal. The last sequence of throws of the dice conveys absolutely no information about what the next throw will bring. Cards, coins, dice, and roulette wheels have no memory." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"The resolution of how to divide the stakes in an uncompleted game marked the beginning of a systematic analysis of probability - the measure of our confidence that something is going to happen. It brings us to the threshold of the quantification of risk." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"In principle, every social situation involves strategic interaction among the participants. Thus, one might argue that proper understanding of any social situation would require game-theoretic analysis. But in actual fact, classical economic theory did manage to sidestep the game-theoretic aspects of economic behavior by postulating perfect competition, i. e., by assuming that every buyer and every seller is very small as compared with the size of the relevant markets, so that nobody can significantly affect the existing market prices by his actions." (John Harsanyi, "Games with Incomplete Information" 1997)

"Mathematical logic deals not with the truth but only with the game of truth." (Gian-Carlo Rota,"Indiscrete Thoughts", 1997)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a 'state' of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises." (Thomas Flanagan, "Game Theory and Canadian Politics", 1998)

"A random walk is one in which future steps or directions cannot be predicted on the basis of past history. When the term is applied to the stock market, it means that short-run changes in stock prices are unpredictable. Investment advisory services, earnings forecasts, and chart patterns are useless. [...] What are often called 'persistent patterns' in the stock market occur no more frequently than the runs of luck in the fortunes of any gambler playing a game of chance. This is what economists mean when they say that stock prices behave very much like a random walk." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"From the moment we first roll a die in a children's board game, or pick a card (any card), we start to learn what probability is. But even as adults, it is not easy to tell what it is, in the general way." (David Stirzaker, "Probability and Random Variables: A Beginner's Guide", 1999)

"The first view of randomness is of clutter bred by complicated entanglements. Even though we know there are rules, the outcome is uncertain. Lotteries and card games are generally perceived to belong to this category. More troublesome is that nature's design itself is known imperfectly, and worse, the rules may be hidden from us, and therefore we cannot specify a cause or discern any pattern of order. When, for instance, an outcome takes place as the confluence of totally unrelated events, it may appear to be so surprising and bizarre that we say that it is due to blind chance." (Edward Beltrami. "What is Random?: Chance and Order in Mathematics and Life", 1999)

"Winning and losing is not simply a pastime; it is the model science uses to explore the universe. Flipping a coin or rolling a die is really asking a question: success or failure can be defined as getting a yes or no. So the distribution of probabilities in a game of chance is the same as that in any repeated test - even though the result of any one test is unpredictable." (John Haigh," Taking Chances: Winning With Probability", 1999)

On Games (1980-1989)

"The way the mathematics game is played, most variations lie outside the rules, while music can insist on perfect canon or tolerate a casual accompaniment." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the soul's quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"There are many difficulties in application of [the games] theory to the real world. [...] In general, competitors are not in complete opposition. As a matter of fact often they don't even have the same objectives. This difficulty can often be circumvented by using a different objective, 'games of survival'. Secondly, a decision is seldom made once. This motivated the study of multistage games [...]. Finally, decisions are not usually made simultaneously. Recognition of this fact leads to 'games of protocol' [...]. Games of protocol can also be used to handle processes involving three or more people." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"If doing mathematics or science is looked upon as a game, then one might say that in mathematics you compete against yourself or other mathematicians; in physics your adversary is nature and the stakes are higher." (Mark Kac, "Enigmas Of Chance", 1985)

"Since a point hypothesis is not to be expected in practice to be exactly true, but only approximate, a proper test of significance should almost always show significance for large enough samples. So the whole game of testing point hypotheses, power analysis notwithstanding, is but a mathematical game without empirical importance." (Louis Guttman, "The illogic of statistical inference for cumulative science", Applied Stochastic Models and Data Analysis, 1985)

"A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play." (James P Cars, "Finite and Infinite Games: A Vision of Life as Play and Possibility", 1986)

"Artificial intelligence is based on the assumption that the mind can be described as some kind of formal system manipulating symbols that stand for things in the world. Thus it doesn't matter what the brain is made of, or what it uses for tokens in the great game of thinking. Using an equivalent set of tokens and rules, we can do thinking with a digital computer, just as we can play chess using cups, salt and pepper shakers, knives, forks, and spoons. Using the right software, one system" (the mind) can be mapped onto the other (the computer)." (George Johnson, "Machinery of the Mind: Inside the New Science of Artificial Intelligence", 1986)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)

"Science is not a given set of answers but a system for obtaining answers. The method by which the search is conducted is more important than the nature of the solution. Questions need not be answered at all, or answers may be provided and then changed. It does not matter how often or how profoundly our view of the universe alters, as long as these changes take place in a way appropriate to science. For the practice of science, like the game of baseball, is covered by definite rules." (Robert Shapiro, "Origins: A Skeptic’s Guide to the Creation of Life on Earth", 1986)

"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." (Drew Fudenberg & Eric Maskin, "The folk theorem in repeated games with discounting or with incomplete information", Econometrica: Journal of the Econometric Society, 1986)

"The modern theory of decision making under risk emerged from a logical analysis of games of chance rather than from a psychological analysis of risk and value. The theory was conceived as a normative model of an idealized decision maker, not as a description of the behavior of real people." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59" (4), 1986)

"Linear relationships are easy to think about: the more the merrier. Linear equations are solvable, which makes them suitable for textbooks. Linear systems have an important modular virtue: you can take them apart and put them together again - the pieces add up. Nonlinear systems generally cannot be solved and cannot be added together. [...] Nonlinearity means that the act of playing the game has a way of changing the rules. [...] That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems." (James Gleick, "Chaos: Making a New Science", 1987)

On Games (1970-1979)

"To the average mathematician who merely wants to know that his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency." (Paul Cohen, "Axiomatic set theory, American Mathematical Society", 1971)

"When we propose to apply mathematics we are stepping outside our own realm, and such a venture is not without dangers. For having stepped out, we must be prepared to be judged by standards not of our own making and to play games whose rules have been laid down with little or no consultation with us. Of course, we do not have to play, but if we do we have to abide by the rules and above all not try to change them merely because we find them uncomfortable or restrictive." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)

"Evolutionary game theory is a way of thinking about evolution at the phenotypic level when the fitnesses of particular phenotypes depend on their frequencies in the population." (John M Smith, "Evolution and the Theory of Games", 1973)

"In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." (Martin Gardner, "Mathematical Games", Scientific American, 1973)

"General systems theory deals with the most fundamental concepts and aspects of systems. Many theories dealing with more specific types of systems (e. g., dynamical systems, automata, control systems, game-theoretic systems, among many others) have been under development for quite some time. General systems theory is concerned with the basic issues common to all these specialized treatments. Also, for truly complex phenomena, such as those found predominantly in the social and biological sciences, the specialized descriptions used in classical theories (which are based on special mathematical structures such as differential or difference equations, numerical or abstract algebras, etc.) do not adequately and properly represent the actual events. Either because of this inadequate match between the events and types of descriptions available or because of the pure lack of knowledge, for many truly complex problems one can give only the most general statements, which are qualitative and too often even only verbal. General systems theory is aimed at providing a description and explanation for such complex phenomena." (Mihajlo D. Mesarovic & Yasuhiko Takahare, "General Systems Theory: Mathematical foundations", 1975)

"Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub." (Richard J Trudeau, "Dots and Lines", 1976)

"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)

"There may be such a thing as habitual luck. People who are said to be lucky at cards probably have certain hidden talents for those games in which skill plays a role. It is like hidden parameters in physics, this ability that does not surface and that I like to call 'habitual luck'." (Stanislaw Ulam, "Adventures of a Mathematician", 1976)

"To be a pioneer in science has lost much of its attraction: significant scientific facts and, even more, fruitful scientific concepts pale into oblivion long before their potential value has been utilized. New facts, new concepts keep crowding in and are in turn, within a year or two, displaced by even newer ones. [...] Now, however, in our miserable scientific mass society, nearly all discoveries are born dead; papers are tokens in a power game, evanescent reflections on the screen of a spectator sport, news items that do not outlive the day on which they appeared." (Erwin Chargaff, "Heraclitean Fire: Sketches from a Life Before Nature", 1978)

"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." (George R Lindsey, "Looking back over the Development and Progress of Operational Research, 1979)

"In the long run, qualitative changes always outweigh quantitative ones. Quantitative predictions of economic and social trends are made obsolete by qualitative changes in the rules of the game. Quantitative predictions of technological progress are made obsolete by unpredictable new inventions. I am interested in the long run, the remote future, where quantitative predictions are meaningless. The only certainty in that remote future is that radically new things will be happening." (Freeman J Dyson, "Disturbing the Universe", 1979)

On Games (1960-1969)

"It is the shortcomings of game theory" (as originally formulated) which force the consideration of the role of ethics, of the dynamics of social structure, and of social structure and of individual psychology in situations of conflict." (Anatol Rapoport, "Fights, games, and debates", 1960)

"Engineering is the art of skillful approximation; the practice of gamesmanship in the highest form. In the end it is a method broad enough to tame the unknown, a means of combing disciplined judgment with intuition, courage with responsibility, and scientific competence within the practical aspects of time, of cost, and of talent. This is the exciting view of modern-day engineering that a vigorous profession can insist be the theme for education and training of its youth. It is an outlook that generates its strength and its grandeur not in the discovery of facts but in their application; not in receiving, but in giving. It is an outlook that requires many tools of science and the ability to manipulate them intelligently In the end, it is a welding of theory and practice to build an early, strong, and useful result. Except as a valuable discipline of the mind, a formal education in technology is sterile until it is applied." (Ronald B Smith, "Professional Responsibility of Engineering", Mechanical Engineering Vol. 86 (1), 1964)

"Imagine that [...] the world is something like a great chess game being played by the gods, and we are observers of the game. [...] If we watch long enough, we may eventually catch on to a few of the rules [...]. However, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited [...]. We must limit ourselves to the more basic question of the rules of the game. If we know the rules, we consider that we 'understand' the world." (Richard P. Feynman, "The Feynman Lectures on Physics", 1964)

"[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", 1965)

"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)

On Games (1950-1959)

"Historically, the original purpose of the theory of probability was to describe the exceedingly narrow domain of experience connected with games of chance, and the main effort was directed to the calculation of certain probabilities." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"People prefer theory to practice because it involves them in no more real responsibility than a game of checkers, while it permits them to feel they're doing something serious and important." (Leo Stein,"Journey into the Self", 1950)

"The classical theory of probability was devoted mainly to a study of the gamble's gain, which is again a random variable; in fact, every random variable can be interpreted as the gain of a real or imaginary gambler in a suitable game." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men." (Friedrich Waismann & Karl Menger, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1951)

"Rather than solve the two-person cooperative game by analyzing the bargaining process, one can attack the problem axiomatically by stating general properties that 'any reasonable solution' should possess. By specifying enough such properties one excludes all but one solution. " (John F Nash, "Two-Person Cooperative Games", 1953)

"The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation." (Norbert Wiener, "Ex-Prodigy: My Childhood and Youth", 1953)

"Electronic computers are normally used for the solution of numerical problems arising in science or industry. The fundamental design of these computers, however, is so flexible and so universal in conception that they maybe programmed to perform many operations which do not involve numbers at all - operations such as the translation of language, the analysis of a logical situation or the playing of games. The same orders which are used in constructing a numerical program maybe used to symbolize operations on abstract entities such as the words of a language or the positions in a chess game." (Claude E Shannon, "Game Playing Machines, 1955)

"Chess combines the beauty of mathematical structure with the recreational delights of a competitive game." (Martin Gardner, "Mathematics, Magic, and Mystery", 1956)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956)

"In no subject is there a rule, compliance with which will lead to new knowledge or better understanding. Skillful observations, ingenious ideas, cunning tricks, daring suggestions, laborious calculations, all these may be required to advance a subject. Occasionally the conventional approach in a subject has to be studiously followed; on other occasions it has to be ruthlessly disregarded. Which of these methods, or in what order they should be employed is generally unpredictable. Analogies drawn from the history of science are frequently claimed to be a guide; but, as with forecasting the next game of roulette, the existence of the best analogy to the present is no guide whatever to the future. The most valuable lesson to be learnt from the history of scientific progress is how misleading and strangling such analogies have been, and how success has come to those who ignored them." (Thomas Gold," Cosmology", 1956)

"Science is the search for truth. It is not a game in which one tries to beat his opponent, to do harm to others. We need to have the spirit of science in international affairs, to make the conduct of international affairs the effort to find the right solution, the just solution of international problems, not the effort by each nation to get the better of other nations, to do harm to them when it is possible." (Linus Pauling, "No More War!", 1958)

On Games (1900-1924)

"If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force - and every other representation remains indefinite and therefore useless - it follows that, in the great dice game of existence, it must pass through calculable number of combinations. In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combination conditions of the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum. This conception is not simply a mechanistic conception; for if it were that, it would not condition an infinite recurrence of identical cases, but a final state. Because the world has not reached this, mechanistic theory must be considered an imperfect and merely provisional hypothesis." (Friedrich Nietzsche, "The Will to Power", [notes written 1883-1888] 1901)

"So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the intimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality." (Henri Poincaré, "The Value of Science", 1905)

"The game of chess has always fascinated mathematicians, and there is reason to suppose that the possession of great powers of playing that game is in many features very much like the possession of great mathematical ability. There are the different pieces to learn, the pawns, the knights, the bishops, the castles, and the queen and king. The board possesses certain possible combinations of squares, as in rows, diagonals, etc. The pieces are subject to certain rules by which their motions are governed, and there are other rules governing the players. [...] One has only to increase the number of pieces, to enlarge the field of the board, and to produce new rules which are to govern either the pieces or the player, to have a pretty good idea of what mathematics consists." (James B Shaw, "What is Mathematics?", Bulletin American Mathematical Society Vol. 18, 1912)

"Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." (David Hilbert,"Natur und Mathematisches Erkennen", 1919–20)

"[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words." (Charles Nordmann, Einstein and the Universe", 1922)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; "Die logischen Grundlagen der Mathematik." Mathematische Annalen 88"(1), 1923)

"We have come to believe that a pupil in school should feel that he is living his own life naturally. with a minimum of restraint and without tasks that are unduly irksome; that he should find his way through arithmetic largely hoy his own spirit of curiosity; and that he should be directed in arithmetic as he would he directed in any other game, - not harshly driven, hardly even led, but proceeding with the feeling that he is being accompanied and that he is doing his share in finding the way." (David E Smith, "The Progress of Arithmetic", 1923)

On Games (-1899)

"I believe the calculation of the quantity of probability might be improved to a very useful and pleasant speculation, and applied to a great many events which are accidental, besides those of games; only these cases would be infinitely more confused, as depending on chances which the most part of men are ignorant of." (John Arbuthnot, "Of the Laws of Chance", 1692)

"But here it seems to me that we are at a loss, since one is at liberty to do this only just in very few cases, and indeed one may hardly succeed else where other than in games of chance, the first inventors of which, doing their best to bring about fairness, arranged things for themselves in such a way that the numbers of cases in which gain or loss ought to follow, might be definite and known, and that all these cases might happen with equal facility. For in most other situations depending either on the working of nature or on the judgement of men, this is by no means the case." (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713)

"The man of system, on the contrary, is apt to be very wise in his own conceit; and is often so enamoured with the supposed beauty of his own ideal plan of government, that he cannot suffer the smallest deviation from any part of it. He goes on to establish it completely and in all its parts, without any regard either to the great interests, or to the strong prejudices which may oppose it. He seems to imagine that he can arrange the different members of a great society with as much ease as the hand arranges the different pieces upon a chess-board. He does not consider that the pieces upon the chess-board have no other principle of motion besides that which the hand impresses upon them; but that, in the great chess-board of human society, every single piece has a principle of motion of its own, altogether different from that which the legislature might choose to impress upon it. If those two principles coincide and act in the same direction, the game of human society will go on easily and harmoniously, and is very likely to be happy and successful. If they are opposite or different, the game will go on miserably, and the society must be at all times in the highest degree of disorder." (Adam Smith, "The Theory of Moral Sentiments", 1759)

"In another sense, the term ‘rule’ is used for ‘means’: to recognize an underlying truth through a single obviously relevant feature enables us to derive a general law of action from this feature. Rules in games are like this, and so are the short cuts used in mathematics, and so on." (Carl von Clausewitz, "On War", 1832)

"In short, absolute, so-called mathematical factors never find a firm basis in military calculations. From the very start there is an interplay of possibilities, probabilities, good luck and bad that weaves its way throughout the length and breadth of the tapestry. In the whole range the human activities war most closely resembles a game of cards." (Carl von Clausewitz, "On War", 1832)

"[...] in the game of heads or tails, the arrival of heads results from the constitution of the tossed coin. It can be regarded as physically impossible that the chances of both outcomes are the same; however, if that constitution is unknown to us, and we did not yet try out the coin, the probability of heads is for us absolutely the same as that of tails. Actually, we have no reason to believe in one of these events rather than in the other one. This will not be the same after many tosses of the coin: the chance of each side does not change during the trials, but for someone who knows their results, the probability of the future occurrence of heads and tails varies in accord with the number of times they happened." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The chemists who uphold dualism are far from being agreed among themselves; nevertheless, all of them in maintaining their opinion, rely upon the phenomena of chemical reactions. For a long time the uncertainty of this method has been pointed out: it has been shown repeatedly, that the atoms put into movement during a reaction take at that time a new arrangement, and that it is impossible to deduce the old arrangement from the new one. It is as if, in the middle of a game of chess, after the disarrangement of all the pieces, one of the players should wish, from the inspection of the new place occupied by each piece, to determine that which it originally occupied." (Auguste Laurent, "Chemical Method", 1855)

"The chess board is the world, the pieces are the phenomena of the universe, the rules of the game are what we call the laws of Nature. The player on the other side is hidden from us. We know that his play is always fair, just, and patient. But also we know, to our cost, that he never overlooks a mistake, or makes the smallest allowance for ignorance." (Thomas H Huxley, "A Liberal Education", 1868)

"And as the number of combinations that can be made on the chess-board, is so great that probably no two games exactly alike were ever played; so no two games which the student plays with nature to wrest from her hidden truths, which were worth playing at all, ever made use of quite the same methods in quite the same way." (Alfred Marshall, "Principles of Economics", 1890)

"Observe, finally, that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science: the different moves of the game do not resemble one another." (Henri Poincaré, "On the Nature of Mathematical Reasoning", 1894)

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01 November 2025