Game Theorists

"# or relational mathematics, including non-metrical fields such as network and graph theory."

"# Factor analysis, i.e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields"

"# General system theory in the narrower sense (G.S.T.), trying to derive from a general definition of “system” as complex of interacting components, concepts characteristic of organized wholes such as interaction, sum, mechanization, centralization, competition, finality, etc., and to apply them to concrete phenomena."

"Game theory is the study of individual decision-making in the face of competing boundedly rational actors."

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory."

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution."

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises."

"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 point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a in infinitely repeated games with sufficiently little discounting."

"By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war."

"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."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. 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 , if it is also true for a finite number of agents of a finite choice space."

"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."

"Chapter 2 describes the most demanding rational choice theory of all, game theory, which was developed by a genius and assumes that other people are geniuses."

"Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."

"Game theory, however, deals only with the way in which ultrasmart, all knowing people should behave in competitive situations, and has little to say to Mr. X as he confronts the morass of his problem."

"At present game theory has, in my opinion, two important uses, neither of them related to games nor to conflict directly. First, game theory stimulates us to think about conflict in a novel way. Second, game theory leads to some genuine impasses, that is, to situations where its axiomatic base is shown to be insufficient for dealing even theoretically with certain types of conflict situations... Thus, the impact is made on our thinking process themselves, rather than on the actual content of our knowledge."

"(Game theory is) essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not."

"[G]ame theory has already established itself as an essential tool in the , where it is widely regarded as a unifying language for investigating human behavior. Game theory's prominence in evolutionary biology builds a natural bridge between the life sciences and the behavioral sciences. And connections have been established between game theory and the two most prominent pillars of physics: and quantum theory. ...[M]any physicists, neuroscientists, and social scientists... are... pursuing the dream of a quantitative science of human behavior. Game theory is showing signs of... an increasing important role in that endeavor. It's a story of exploration along the shoreline separating the continent of knowledge from an ocean of ignorance... a story worth telling."

"The last decade has seen a steady increase in the application of concepts from the theory of games to the study of evolution. Fields as diverse as sex ratio theory, animal distribution, contest behaviour and have contributed to what is now emerging as a universal way of thinking about phenotypic evolution... Paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behavior for which it was originally designed"

"The theory of games was first formalised by Von Neumann & Morgenstern (1953) in reference to human economic behaviour. Since that time, the theory has undergone extensive development... Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest. Such an assumption would clearly be out of place in an evolutionary context. Instead, the criterion of rationality is replaced by that of and stability, and the criterion of self-interest by Darwinian fitness."

"Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find s... a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells in favour of the analogy of mathematics to analysis of games played by other... agents, and against the analogy of mathematics to the expert play of the game itself."

"One should remember that mathematical logic itself or the study of mathematics as a formal system can be considered a branch of combinatorial analysis. Metamathematics introduces a class of games—"solitaires"—to be played with symbols according to formal rules. One sense of Gödel's theorem is that some properties of these games can be ascertained only by playing them."

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence:"

"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."

"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."

"Then what explains war among states? Rousseau's answer is really that war occurs because there is nothing to prevent it."

"“Some have even said that Iran with nuclear weapons would stabilize the Middle East, I think people who say this have set a new standard for human stupidity.”"

"We differed on the last point, and sometimes on the key, recurring question of American foreign policy: When is military intervention justified, by which he meant, when is it in the national interest? Waltz had no patience for "liberal intervention," or what we might now call humanitarian intervention, because we could never be sure that we would succeed in making things better over the long-term. And he had little patience for supposed national security arguments that could not identify a threat to vital interests — we are not, and therefore should not act like, an empire. Waltz was not an isolationist, but he was definitely a minimalist when it came to the use of force."

"Waltz is to the study of international relations what Darwin is to the study of biology. I make this claim in terms of the sheer intellectual significance of his theoretical contribution. One cannot make sense of the biological world apart from Darwin’s theory of evolution: equally, Waltz’s structural framework for understanding how states interact under anarchy, with an uneven distribution of power and a desire to survive, offers a powerful theory for making sense of the international system. Neither theory explains everything in their domain — one always needs to know more about particularities — but both provide compelling big-picture explanations of their domains."

"Like any truly great thinker, Ken Waltz defied stereotypes. The preeminent hard-headed theorist of power politics, he savaged the hopes of those who believe that moral energy, liberal principles, or democratic crusades can end war. But he belied the common assumption that realists are callous hawks who relish the use of force. He opposed the Vietnam War, the Reagan military buildup, and the 2003 invasion of Iraq as overreactions to threats that were exaggerated and could be handled by calm containment and deterrence."

"Differences in the incidence of destruction and "death" do not account for the reluctance to refer to international politics as a harmonious realm, while competitive economies are often so described. Instead, one may say that the standards of performance now applied to international political systems are higher, or at least widely different. As John Maynard Keynes once remarked, those who believe that unhampered processes of natural selection lead to progress do not "count the cost of the struggle" (1926, p. 37). In international politics, we often count nothing but the costs of the struggle."

"I am not saying that such a theory cannot be constructed, but only that I cannot see how to do it in any way that might be useful. The decisive point, anyway, is that a macrotheory of international politics would lack the practical implications of macroeconomic theory. National governments can manipulate system-wide economic variables. No agencies with comparable capabilities exist internationally. Who would act on the possibilities of adjustment that a macrotheory of international politics might reveal?"

"If we gather more and more data and establish more and more associations, however, we will not finally find that we know something. We will simply end up having more and more data and larger sets of correlations."

"Each state pursues its own interest's, however defined, in ways it judges best. Force is a means of achieving the external ends of states because there exists no consistent, reliable process of reconciling the conflicts of interest that inevitably arise among similar units in a condition of anarchy."

"It is not true that were the Soviet Union to disappear the remaining states could easily live in peace."

"War may achieve a redistribution of resources, but labor, not war, creates wealth."

"No system of balance functions automatically."

"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."

"In a zero-sum game, the problem is entirely one of distribution, not at all one of production."

"The best critical consideration of the inherent weakness of a federation of states in which the law of the federation has to be enforced on the states who are its members is contained in the Federalist Papers."

"States in the world are like individuals in the state of nature. They are neither perfectly good nor are they controlled by law."

"In anarchy there is no automatic harmony."

"With many sovereign states, with no system of law enforceable among them, with each state judging its grievances and ambitions according to the dictates of its own reason or desire - conflict, sometimes leading to war, is bound to occur."

"Once socialism replaces capitalism, reason will determine the policies of states."

"External pressure seems to produce internal unity."

"Is it capitalism or states that must be destroyed in order to get peace, or must both be abolished?"

"To build a theory of international relations on accidents of geography and history is dangerous."

"The transitory interests of royal houses may be advanced in war; the real interests of all people are furthered by the peace."

"War most often promotes the internal unity of each state involved. The state plagued by internal strife may then, instead of waiting for the accidental attack, seek the war that will bring internal peace."

"If we are to have peace, we must learn loyalty to a larger group. And before we can learn loyalty, the thing to which we are to be loyal must be created."