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)