📓On Literature: On Games (From Fiction to Science-Fiction)

"It was beyond him how someone could voluntarily let himself get involved in this game of dimension-shifting and mutant-battling. But it takes all sorts to make a continuum." (Robert Silverberg, "MUgwump 4" 1959)

"The exploration of alien worlds was just a monotonous and exhausting game." (Arkady Strugatsky & Boris Strugatsky, "Prisoners of Power", 1969)

"In a smoke-congested back room of the universe the God of my agnostic imagination oversees this crooked card game." (Robert Thurston, "Good-Bye, Shelley, Shirley, Charlotte, Charlene", 1972)

"The catastrophe story, whoever may tell it, represents a constructive and positive act by the imagination rather than a negative one, an attempt to confront the terrifying void of a patently meaningless universe by challenging it at its own game. [. . .] Each one of these fantasies represents an arraignment of the finite, an attempt to dismantle the formal structure of time and space which the universe wraps around us at the moment we first achieve consciousness." (James G Ballard, "Cataclysms and Dooms" 1977)

"Human brains back then had become such copious and irresponsible generators of suggestions as to what might be done with life, that they made acting for the benefit of future generations seem one of many arbitrary games which might be played by narrow enthusiasts - like poker or polo or the bond market, or the writing of science-fiction novels." (Kurt Vonnegut Jr, "Galapagos" 1985)

"All reality is a game. Physics at its most fundamental, the very fabric of our universe, results directly from the interaction of certain fairly simple rules, and chance; the same description may be applied to the best, most elegant and both intellectually and aesthetically satisfying games. By being unknowable, by resulting from events which, at the sub-atomic level, cannot be fully predicted, the future remains malleable, and retains the possibility of change, the hope of coming to prevail; victory, to use an unfashionable word. In this, the future is a game; time is one of its rules." (Iain Banks, "The Player of Games", 1988)

"The very first-rank games acknowledge the element of chance, even if they rightly restrict raw luck." (Iain Banks, "The Player of Games", 1988)

"To read fiction means to play a game by which we give sense to the immensity of things that happened, are happening, or will happen in the actual world. By reading narrative, we escape the anxiety that attacks us when we try to say something true about the world. This is the consoling function of narrative - the reason people tell stories, and have told stories from the beginning of time." (Umberto Eco, "Six Walks in the Fictional Woods", 1994)

"Programming is the Great Game. It consumes you, body and soul. When you're caught up in it, nothing else matters. When you emerge into daylight, you might well discover that you're a hundred pounds overweight, your underwear is older than the average first grader, and judging from the number of pizza boxes lying around, it must be spring already. But you don't care, because your program runs, and the code is fast and clever and tight. You won." (Orson Scott Card, "How Software Companies Die", Windows Sources: The Magazine for Windows Experts, 1995)

"The real training ground for leadership is in the game." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

Orson Scott Card - Collected Quotes

"Any decent person who knows what warfare is can never go into battle with a whole heart." (Orson Scott Card, "Ender’s Game", 1985)

"Fiction, because it is not about somebody who actually lived in the real world, always has the possibility of being about oneself." (Orson Scott Card, "Ender’s Game", 1985)

"I think that most of us, anyway, read these stories that we know are not 'true' because we're hungry for another kind of truth: the mythic truth about human nature in general, the particular truth about those life-communities that define our own identity, and the most specific truth of all: our own self-story. Fiction, because it is not about someone who lived in the real world, always has the possibility of being about oneself." (Orson Scott Card, "Ender’s Game", [introduction] 1985)

"No book, however good, can survive a hostile reading." (Orson Scott Card, "Ender’s Game", 1985)

"The story itself, the true story, is the one that the audience members create in their minds, guided and shaped by my text, but then transformed, elucidated, expanded, edited, and clarified by their own experience, their own desires, their own hopes and fears." (Orson Scott Card, "Ender’s Game", 1985)

"This is how humans are: We question all our beliefs, except for the ones we really believe, and those we never think to question." (Orson Scott Card, "Speaker for the Dead", 1986)

"Your dream is a good one. [. . .] The desire that is the very root of life itself: To grow until all the space you can see is part of you, under your control. It’s the desire for greatness." (Orson Scott Card, "Speaker for the Dead", 1986)

"All the universe is just a dream in God's mind, and as long as he's asleep, he believes in it, and things stay real." (Orson Scott Card, "The Tales of Alvin Maker: Seventh Son", 1987)

"Everything possible to be believed is an image of the truth." (Orson Scott Card, "The Tales of Alvin Maker: Seventh Son", 1987)

"[...] science fiction is about what could be but isn't; fantasy is about what couldn't be." (Orson Scott Card, "How to Write Science Fiction and Fantasy", 1990)

"The wise are not wise because they make no mistakes. They are wise because they correct their mistakes as soon as they recognize them." (Orson Scott Card, "Xenocide", 1991)

"These are the people who hold a community together, who lead. Unlike the sheep and the wolves, they perform a better role than the script given them by their inner fears and desires. They act out the script of decency, of selfsacrifice, of public honor - of civilization. And in the pretense, it becomes reality." (Orson Scott Card, "Xenocide", 1991)

"Coincidence is just the word we use when we have not yet discovered the cause." (Orson Scott Card, "The Call Of Earth", 1992)

"Here's the secret that every successful software company is based on: You can domesticate programmers the way beekeepers tame bees. You can't exactly communicate with them, but you can get them to swarm in one place and when they're not looking, you can carry off the honey." (Orson Scott Card, "How Software Companies Die", Windows Sources: The Magazine for Windows Experts, 1995)

"Programming is the Great Game. It consumes you, body and soul. When you're caught up in it, nothing else matters. When you emerge into daylight, you might well discover that you're a hundred pounds overweight, your underwear is older than the average first grader, and judging from the number of pizza boxes lying around, it must be spring already. But you don't care, because your program runs, and the code is fast and clever and tight. You won." (Orson Scott Card, "How Software Companies Die", Windows Sources: The Magazine for Windows Experts, 1995)

"The environment that nuture's creative programmers kills management and marketing types - and vice versa." (Orson Scott Card, "How Software Companies Die", Windows Sources: The Magazine for Windows Experts, 1995)

"Here is the truth. What human life is, what it’s for, what we do, is create communities." (Orson Scott Card, "Pastwatch: The Redemption of Christopher Columbus", 1996)

"History has no laws, and all patterns that we find there are useful illusions." (Orson Scott Card, "Children of the Mind", 1996)

"Knowledge is just opinion that you trust enough to act upon." (Orson Scott Card, "Children of the Mind", 1996)

"Time itself, as a phenomenon, is utterly linear and unidirectional." (Orson Scott Card, "PASTWATCH", 1996

"He had long since learned that when something unusual was going on, he would often find out more information faster by waiting than by asking." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"Human evolution is driven by community needs […]. How is that possible, since genetic information is passed only by and to individuals?" (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"Most human communities demand anti-survival behavior from large numbers of their members." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"'My preliminary assessment shows that when a civil society is mature and successful, the citizens become complacent and to satisfy various needs they reinvent tribes that eventually collapse the society from the inside.' 'So both failure and success lead to failure.' 'The only question is whether it’s inevitable.'" (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"People reach their peak ability as military commanders much earlier than we thought. Most of them in their late teens. The same age when poets do their most passionate and revolutionary work. And mathematicians. They peak, and then it falls off. They coast on what they learned back when they were still young enough to learn." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"That would be true […] and indeed is true in the main, except that there are only a few types of human communities that actually survive long enough to improve the chances of individual survival." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"The real training ground for leadership is in the game." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"There are community traits that promote survival of the individual, and individual traits that promote the survival of the community." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"We make each other think better […]. Together, we’re smarter." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"Patience, absolute self-control, learning the capabilities of those under you so you can make up for their deficits through training." (Orson Scott Card, "Ender in Exile", 2008)

"[...] you can’t stay committed to one pattern. It makes you too easy to predict." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"[…] that’s what science was - the sharing of information, the pooling of knowledge. (Orson Scott Card, "Ender in Exile", 2008)

"[…] the idea was for humans to colonize all their former worlds, so that humanity’s fate would not be tied to that of a single planet. (Orson Scott Card, "Ender in Exile", 2008)

"[...] the real barrier to interstellar flight wasn’t just getting up to lightspeed. It was overcoming the collision problem." (Orson Scott Card, "Ender in Exile", 2008)

"The absence of information is information.” (Orson Scott Card, "Ender in Exile", 2008)

"We arrive at an extremely high level of technology - but with nothing under it to hold it up. If we crash, we crash all the way down." (Orson Scott Card, "Ender in Exile", 2008)

"We humans can be destructive, but the universe’s thirst for creation goes on unslaked." (Orson Scott Card, "Ender in Exile", 2008)

"[...] science fiction is about what could be but isn't; fantasy is about what couldn't be." (Orson Scott Card, "How to Write Science Fiction and Fantasy", 1990)

David Robinson - Collected Quotes

"A game in strategic form is just a function with one input for each player (a strategy) and one output for each player (a payoff). More formally, a game in strategic form is a vector function and its do-main, the strategy space. The strategy space is just the set of all possible combinations of strategies, and therefore incorporates both the player and strategy sets." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"A game with no pure strategy equilibrium can be two minimal steps from a game with two equilibria. The significant observation is that for some games the equilibrium can be quite fragile. The topological approach provides a way to examine the robustness of payoffs and strategies by seeing how they vary for neighbouring games." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"A solution set is simply a subset of the possible outcomes that either predicts how a particular game will turn out or prescribes how it should turn out. A solution concept is a rationale for picking a solution based on the information specified in the form. No other information can be used." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"A symmetric game is often the convenient representative of a collection of related games, most of which are asymmetric. [...] The symmetric games are used so often, especially in introductions to game theory, that it is easy to forget they represent a very special case. For each strategy of the row player in a symmetric game, there must be an equivalent strategy for the column player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Any game in one of the classes can be converted into any other in the same region by some strictly monotonic transformation. Since a monotonic transform conserves order, all the games in an equivalence class are ordinally equivalent. These equivalence classes par-tition the 8-dimensional payoff space for the 2 × 2 games into 144 regions. An ordinal 2×2 game is a 2×2 game with a payoff function that maps from the strategy space to these equivalence classes." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Every game is related to every other in the sense that there is a transformation that converts the payoff structure for one into the payoff structure for the other." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Figures that plot the payoff vectors are common in the literature, as are figures that show the convex hull of the payoffs. The latter are used for discussing mixed strategies and bargaining games, and require real values. A complete representation of the strategic form requires that the strategic choices represented in the matrix be recoverable, and that is the reason for using different line styles to represent each player’s inducement correspondences." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Games of pure conflict, pure common-interest and mixed motives can be described in terms of the slopes of the inducement correspondences. In a game of pure conflict every inducement correspon-dence is negatively sloped. Any action that improves the outcome for one player must make the outcome worse for the other. In a game of pure common-interest, every inducement correspondence is positively sloped." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"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)

"How often even ordinal symmetry appears in the real world is an open question. A great many situations are approximately symmetric, and symmetric payoffs provide a useful starting point for analysis. On the other hand, symmetric games present a problem that human beings are equipped to evade. Symmetry theoretically erases distinctions between players, but real people are capable of exploiting very subtle distinctions." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Inducement correspondences provide a particularly easy way to explore the Nash equilibrium. Because payoffs are strictly ordered there will always be a single best response in a given inducement correspondence. The inducement correspondence can also be used to describe a number of other solution concepts, including maxi-min and solutions based on dominance." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"One important and standardized block of information in the formal descriptions used by game theorists is called a game form. A form specifies the payoffs associated with every possible combination of decisions. There are several widely used forms, including the strate-gic form, typically presented in a matrix, the extensive form, which is usually represented as a tree, and the characteristic function form, expressed as a function on subsets of players." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Order graphs allow us to describe games easily, and our indexing system lets us lay out the games in a systematic and revealing way. Symmetries in the order graph repre-sentation shed some light on the nature of symmetric and reflected games, and on the structure of the space of the 2 × 2 games. They are not directly useful for analysing behaviour." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Simplicity gives the 2 × 2 games their power: they provide re-markable diversity with the absolute minimum of machinery. The strategic situation involves only two players, each with only two al-ternatives. There are only four possible outcomes and each outcome is described by a single payoff for each player. A game is therefore fully described by just 8 numbers." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Since games are characterized by the payoff function, similar games must have similar payoff functions. To define meaningful neighbourhoods, we need to characterize the smallest significant change in the payoff function. Obviously a change affecting the payoffs of one player is smaller than a change affecting two players. The closest neighbouring games are therefore those games that differ only by a small change in the ordering of the outcomes for one player." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Starting with a torus with the graph of a layer embedded in it and a second torus with the the graph of a pipe, imagine gluing the two toruses together so that the four points of a shared tile coincide. Now imagine puncturing the two tiles that are glued together to make a door out of the pipe-torus and into the layer-torus. We now have a figure-eight, a two-holed torus." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The beauty of a topological approach to the 2×2 games is that every topological feature yields some surprising insight into the relationships among the games. Even ignoring features can be productive." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The discrete order topology of the ordinal games [25] is important to understanding the relationship among 2 × 2 games but is insufficient for describing and pre-dicting patterns of behaviour." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The graphical feature that ensures that none of these games have equilibria is that at every position, negatively sloped inducement correspondences lead away in the same direction, ensuring that if one player likes the position, the other has an improving move. The sequence of best response choices cycles around the order graph, and the games are sometimes called cycle games." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The matrix representation illustrates how a formal deep structure can be captured in an apparently simple surface structure. The fact that the rows and columns are at right angles to each other, for example, reflects the idea that the strategies available to the two players are independent. Independence means that it is possible to speak of changing Row’s strategy without changing Column’s strategy. An assumption about the nature of the world is displayed spatially in the matrix." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"The Nash equilibrium is often rationalized using a story about how people think and how their behaviour is related to their thoughts. Economists generally assume that, from a set of alternatives, a player will actively choose the one he likes best. This is the assumption of economic rationality, one of the core assumptions of standard game theory. Rationality alone will not predict behaviour in a game, but it leads us to single out the member of any inducement correspondence that yields the greatest payoff for the player that is making the choice. The resulting behaviour is sometimes described as 'myopic' because it fails to take into account how other players might respond to a given choice." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"To get beyond the typological approach requires a notion of what it means for games to be related. It turns out that preferences pro-vide the appropriate notion of closeness. The topology induced by preferences is beautiful and it makes the systematic treatment of the 2×2 games possible." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"When the graph is embedded within a surface it is called a map. The nature of the surface needed to embed the graph without cross-ing edges is a topological feature, and the topological structure of this payoff space is not only useful, but also beautiful." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"With four outcomes and two players, a 2 × 2 game is completely described by eight numbers. An array with eight numbers is just an address in an 8-dimensional Cartesian payoff space, and there are uncountably many 2 × 2 games, each fully described by an 8-number address." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

♟️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 (-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 Game Theory: 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 Game Theory: 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)

"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)

"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)

"The chaos game may seem like a special trick that can produce Sierpínski triangle but little else. However, the procedure of using randomness to make fractals turns out to be quite general. For example, if one plays the chaos game with four points arranged in a square, not surprisingly one obtains the Sierpínski carpet [...]. Slightly more complicated versions of the chaos game can yield many other fractal shapes. In fact, it turns out that almost any shape - fractal or nonfractal - can be generated by a version of the chaos game. A generalized version of the chaos game turns out to be remarkably powerful and flexible."  (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"This generalized chaos game is constructed as follows. Thus far each move in the game takes a single point and moves it to another location. Which location it gets moved to is determined by a random rule. We could also imagine a chaos game that uses slightly more complicated rules. It is easiest to think of these rules in terms of their effects on shapes rather than single points. To that end, consider operations that not only move a shape but also transform it - stretch, shrink, shear, and/or rotate it. The technical term for these sorts of transformation is affine. An affine transformation is any geometric transformation that keeps parallel lines parallel." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 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)

"A game in strategic form is just a function with one input for each player (a strategy) and one output for each player (a payoff). More formally, a game in strategic form is a vector function and its do-main, the strategy space. The strategy space is just the set of all possible combinations of strategies, and therefore incorporates both the player and strategy sets." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"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)

"The beauty of a topological approach to the 2×2 games is that every topological feature yields some surprising insight into the relationships among the games. Even ignoring features can be productive." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"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 Game Theory: 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)

"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)

"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)

"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)

"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)

"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)

"Science observes the game of Nature, played out on the vast board of the universe, and tries to deduce the rules. Reductionism works whenever the part that is being observed resembles a dream game, where simple rules give rise to simple strategies. It breaks down when it runs into a patch of nightmare game, with simple rules but complex strategies." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"The analogy with threading a maze runs deeper than games and puzzles. It illuminates the whole of mathematics. Indeed, one way to think about mathematics is as an exercise in threading an elaborate, infinitely large maze. A logical maze. A maze of ideas, whose pathways represent 'lines of thought' from one idea to another. A maze which, despite its apparent complexity, has a definite 'geography', to which mathematicians are unusually attuned." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"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 Game Theory: On Games (2000-2009)

"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)

"[...] 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)

"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)

"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)

"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)

"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)

"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)

"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 Game Theory: 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)

"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)

"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)

"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)

"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 Game Theory: 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)

"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)

"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)

"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) 

"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)