Thirty-nine scientists

English engineer. He built one of the first British petrol motor cars and founded the Lanchester Engine Company, and his two-volume Aerial Flight developed the circulation theory of lift. In Aircraft in Warfare (1916) he proposed the differential equations of attrition now named after him and derived the $N$-square law, the quantitative case for concentration. M. Osipov published a closely related analysis independently in a Russian military journal in 1915.

French mathematician, trained at the École Normale Supérieure and later professor at the Faculty of Sciences in Paris, where he held the chair of probability and mathematical physics; he initiated the Institut Henri Poincaré and became its first director in 1928. He contributed to measure theory and probability, studied divergent series and their summation, and proved that almost all real numbers are normal. His notes of 1921–1927 on games of skill developed mixed strategies and asked whether a minimax solution exists in general; von Neumann proved the theorem for finite two-person zero-sum games in 1928.

Born in Kolozsvár (Cluj); doctorate in mathematics in Vienna, where he worked on geometry and then, at the Austrian Institute for Business Cycle Research, on economic models, including early existence results for specified models of a competitive economy. He moved to the United States in 1938 and joined Columbia University, where he founded sequential analysis and built statistical decision theory: a statistical procedure is a decision function, judged by its risk, and the minimax decision function is the one that guards against the worst case. He died in an air accident in India in 1950.

Born in Budapest, he studied mathematics there and chemical engineering in Zurich, taught in Berlin and Hamburg, and moved to Princeton in 1930; in 1933 he joined the first faculty of the Institute for Advanced Study. His work spans set theory, operator algebras, the mathematical foundations of quantum mechanics, ergodic theory, numerical analysis and the logical design of stored-program computers. In 1928 he proved the minimax theorem for finite two-person zero-sum games; with the economist Oskar Morgenstern he later wrote Theory of Games and Economic Behavior.

Mathematician at the Steklov Institute in Moscow, blind from the age of fourteen. He first worked in topology — duality theorems, topological groups, characteristic classes — and from the 1950s turned to control, leading the group whose maximum principle became the foundation of optimal control theory; he later developed linear differential games of pursuit and evasion, pursuing the subject independently of Isaacs' line.

Mathematician, educated at Lehigh University (bachelor's degree, 1933) and Yale (doctorate, 1937); by the late 1940s he was working on game theory at the RAND Corporation. In January 1950 he and Merrill Flood planned and ran the experiment with the game later known as the prisoner's dilemma. His book presents two-person games together with their computational methods, and his 1962 memorandum analyses the recursive inspection game with which this chapter begins.

Norwegian economist. He took his degree in economics at the University of Oslo in 1933 and worked as an assistant at Ragnar Frisch's Institute of Economics. In the United States from 1939 to 1947 he wrote The Probability Approach in Econometrics, his doctoral thesis, published as a supplement to Econometrica in 1944, which argued that economic theories should be formulated in probabilistic terms and estimated and tested by the methods of mathematical statistics; his 1943 article on simultaneous equations showed that estimating one relation in isolation can be highly misleading when relations are interdependent. He was professor of economics and statistics at Oslo from 1948 to 1979, and received the 1989 Nobel Memorial Prize in Economic Sciences for clarifying the probability foundations of econometrics and for his analyses of simultaneous economic structures.

Educated at Leningrad University, he became a professor there in 1934, moved to Novosibirsk in 1960 and to Moscow in 1971. He worked on ordered vector spaces, approximation theory and numerical methods. A problem of allocating machine capacity in plywood production, brought to him in 1938, led to his 1939 work on what is now called linear programming, with its resolving multipliers; in 1942 he formulated the mass-transport problem as an optimisation problem. He shared the 1975 Nobel Memorial Prize in Economic Sciences with Tjalling Koopmans for contributions to the theory of optimum allocation of resources.

Statistician at the United States Weather Bureau in Washington, where for many years he headed meteorological statistics (in 1958 he was chief of the Statistics Section in its Office of Meteorological Research); later he was associated with Colorado State University. He worked on the verification and evaluation of forecasts. His 1950 note introduced the quadratic score for probability forecasts that now carries his name; it is summed over all forecast categories, so that the binary score in this chapter is half of his.

Born in Portland, Oregon, he earned his Berkeley doctorate under Jerzy Neyman in 1946. As mathematical adviser to the US Air Force comptroller he formulated linear programming and, in 1947, the simplex method. He then worked at the RAND Corporation and at the University of California, Berkeley, and from 1966 at Stanford, where in 1974 he and Richard Cottle founded the Systems Optimization Laboratory. His later work includes decomposition methods for large structured programs and stochastic programming.

Mathematician who received his doctorate from Columbia University. At the RAND Corporation in the early 1950s he founded the theory of differential games and developed it systematically in a series of research memoranda — pursuit and evasion, the Main Equation now read as the Hamilton–Jacobi–Isaacs equation, dominance regions, and worked examples such as the homicidal chauffeur and the guarding of a territory — published a decade later as a book. He later worked in industry and defence analysis and taught at Johns Hopkins University.

Trained in political science at the University of Chicago (doctorate, 1943), he began with studies of municipal administration that he later classed as operations research, taught at the Illinois Institute of Technology from 1942, and from 1949 helped to develop the Graduate School of Industrial Administration at the Carnegie Institute of Technology, now Carnegie Mellon University. In Administrative Behavior and later works he described decision-makers in organisations whose capacity for rational action is limited and who settle for a satisfactory alternative; “bounded rationality” and “satisficing” are his terms for this. With Charles Holt, Franco Modigliani and John Muth he developed linear decision rules for production and inventory planning; with Allen Newell, whom he met at RAND in 1952, he wrote early computer programs that simulated human problem solving. He shared the 1975 Turing Award with Newell and received the 1978 Nobel Memorial Prize in Economic Sciences for research on decision-making in economic organisations.

Statistician whose The Foundations of Statistics built the modern theory of subjective probability and personal decision on axioms of coherent preference. He taught at the University of Chicago (1946–60), the University of Michigan (1960–64) and Yale (1964–71). His paper on elicitation, published posthumously in December 1971, characterised the scoring rules under which announcing one's true probability is optimal — the propriety theory this chapter instantiates.

Mathematician at the Computing Centre of the USSR Academy of Sciences and at Moscow State University, where in 1970 he founded and first headed the Department of Operations Research of the Faculty of Computational Mathematics and Cybernetics. He used guaranteed-result criteria in operations research and in games with non-opposed interests, and studied what a leader can secure by announcing a choice or a response rule when the follower's interests are not fully known. With N. N. Moiseev he also studied hierarchical control systems.

Born in Centralia, Illinois; doctorate at the University of Illinois in 1941 under Joseph Doob. He taught at Howard University, worked with the RAND Corporation on duels and games of timing, and joined the University of California, Berkeley, in 1954 as a visiting professor, becoming Professor of Statistics in 1955. His work spans the Rao–Blackwell theorem, renewal theory, the comparison of statistical experiments, approachability in repeated games with vector payoffs, dynamic programming, and the determinacy of infinite games.

Mathematician. She received her doctorate at Berkeley in 1948 under Alfred Tarski, remained associated with Berkeley and became professor there in 1976. Her 1951 paper proved that in every finite zero-sum game the minimax bounds computed from the fictitious-play frequencies converge to the value — the convergence used in this chapter; the actions themselves need not settle — and her work on Hilbert's tenth problem, with Martin Davis and Hilary Putnam, laid the ground for its negative solution by Yuri Matiyasevich in 1970. She was the first woman elected president of the American Mathematical Society (1983–84).

Mathematician at the RAND Corporation (1947–66) and then at the University of Southern California. He worked on branching processes and on interacting particle systems, and Harris recurrence of Markov chains is named after him. With F. S. Ross he wrote the RAND report on the capacity of a rail network, whose object of interest was a minimum cut.

Economist born in Budapest, who left Hungary in 1950, took a master's degree in Sydney (1953) while working in factories, and after a Stanford doctorate (1959), working closely with Kenneth Arrow, taught at the University of California, Berkeley, from 1964. He showed how games with incomplete information can be analysed by giving each player a type that summarises its beliefs, under a common prior — the construction this book uses from Chapter 0 on; the universal construction of belief hierarchies came later, from Mertens and Zamir — and shared the 1994 Nobel Memorial Prize in Economic Sciences with John Nash and Reinhard Selten.

Economist who received his doctorate from Harvard and taught at Yale, at Harvard and at the University of Maryland; he spent a year at the RAND Corporation. He studied bargaining, commitment and credibility — how a player can gain by limiting its own options or information — and later how individual choices aggregate into collective patterns. He shared the 2005 Nobel Memorial Prize in Economic Sciences with Robert Aumann for having enhanced the understanding of conflict and cooperation through game-theory analysis.

Born in New York City, he studied at the City College of New York (B.S. in social science with a mathematics major, 1940) and at Columbia University (M.A. in mathematics, 1941), and served as a weather officer in the US Army Air Corps from 1942 to 1946. After research at the Cowles Commission in Chicago he joined Stanford in 1949, moved to Harvard in 1968, and returned to Stanford in 1979. His Columbia doctoral dissertation was Social Choice and Individual Values, with the impossibility theorem; with Gérard Debreu he proved the existence of competitive equilibrium; and he studied information as an economic variable, in medical care, insurance and learning by doing. He shared the 1972 Nobel Memorial Prize in Economic Sciences with John Hicks.

Mathematician and game theorist. He received his doctorate from Princeton in 1953 under Albert W. Tucker, worked at the RAND Corporation from 1954 to 1981 and then at the University of California, Los Angeles. He introduced stochastic games and the value iteration for their discounted and stopping versions, the value for cooperative games that bears his name, and, with David Gale, the stable-matching algorithm; his 1964 example showed that fictitious play need not converge outside the zero-sum world. He shared the 2012 Nobel Memorial Prize in Economic Sciences with Alvin Roth for the theory of stable allocations and the practice of market design — not for any of the results used in this chapter.

Mathematician in Sverdlovsk, now Yekaterinburg, at the Ural Polytechnic Institute, the Ural State University and the Institute of Mathematics and Mechanics of the Academy of Sciences' Ural branch, which he directed from 1970 to 1977 and which now bears his name. He worked first on the stability of motion, extending Lyapunov's methods, and then built the theory of positional differential games: strategies as feedback rules, the extremal-shift aiming rule, and the alternative theorem on stable bridges, with A. I. Subbotin.

Mathematician and biophysicist, a Moscow State University graduate of 1952, who worked in Moscow, from 1958 in Israel Gelfand's seminar on the boundary of mathematics and physiology. His 1961 paper pioneered the study of finite automata in random media — how trivially simple devices can adapt to unknown environments — and the games of collectives of such automata, including the Goore game proposed in his circle in 1963 and published by others in 1965, showed that collectives can concentrate on an optimum of a function none of them observes; the line was continued by Varshavsky's work on the collective behaviour of automata. His papers were collected posthumously (Nauka, 1969; English edition 1973).

Mathematician at the RAND Corporation from 1951 and, from 1971, at Cornell University. He worked on network flows, combinatorial optimisation and the theory of blocking and anti-blocking polyhedra; with George Dantzig and Selmer Johnson he solved a 49-city travelling-salesman problem by the cutting-plane method. A prize in discrete mathematics bears his name.

Mathematician who worked on operations research at the Operations Evaluation Group from 1954, then at the Office of Naval Research from 1959 and, in the 1960s, as a scientific analyst at the Arms Control and Disarmament Agency; his 1968 book set out mathematical models of arms control and disarmament. He taught at the Wharton School from 1969 and at the University of Pittsburgh from 1979, and created the analytic hierarchy process, a method that decomposes a decision problem, compares its elements and synthesises the results into priorities.

Mathematician at the Hebrew University of Jerusalem. In cooperative game theory he introduced, with Robert Aumann, the bargaining set and, with Morton Davis, the kernel. With Aumann, in collaboration with Richard Stearns, he developed repeated games with incomplete information in a research project reported to the US Arms Control and Disarmament Agency in 1965–1968. His papers on the inspector's non-constant-sum game studied inspector leadership — an inspection policy announced in advance — including how the result depends on the way the inspectee resolves indifference.

Mathematician who worked at the RAND Corporation in the 1950s. With D. R. Fulkerson he proved the max-flow min-cut theorem and gave the augmenting-path algorithm for maximum flows; his name is also attached to the Bellman–Ford shortest-path algorithm and, with S. M. Johnson, to the Ford–Johnson merge-insertion sorting method.

Mathematician who studied at the Carnegie Institute of Technology and received his doctorate from Princeton in 1950 with a thesis on non-cooperative games. He taught at the Massachusetts Institute of Technology and later worked at Princeton. He introduced the equilibrium concept that bears his name and an axiomatic solution of the bargaining problem, and proved fundamental results in differential geometry and partial differential equations. He shared the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten, and the 2015 Abel Prize with Louis Nirenberg.

Physicist and mathematician, a graduate of the Faculty of Physics of Moscow State University (1953) and from 1969 a professor there. He built the theory of conditional Markov processes — the nonlinear filtering of a hidden process from noisy observations — introduced the symmetric form of the stochastic integral that now carries his name, and developed a theory of the value of information, in which an amount of Shannon information is priced by the largest reduction in expected loss it can buy.

Engineer and mathematician, born in Frankfurt am Main and educated at the Université Libre de Bruxelles, in electrical engineering and in mathematical physics, and at the Massachusetts Institute of Technology, where he received his doctorate in electrical engineering in 1966 with a thesis on the minimax control of uncertain systems. He then joined Bell Laboratories. He worked on stochastic control with decentralised information — the intrinsic model of information structures and the counterexample that bears his name — and on information theory, including the common information of dependent random variables.

Mathematician who received his doctorate from the Massachusetts Institute of Technology in 1955 and has worked since 1956 at the Hebrew University of Jerusalem, where he was a founding member of the Center for the Study of Rationality. He developed the theory of repeated games, including games with incomplete information with Michael Maschler, introduced correlated equilibrium, and formalised common knowledge. He shared the 2005 Nobel Memorial Prize in Economic Sciences with Thomas Schelling.

Born in Breslau, he studied mathematics at the University of Frankfurt, where he took his master's degree in 1957 and his doctorate in mathematics in 1961, and worked for about a decade as assistant to the economist Heinz Sauermann, with whom he ran early laboratory experiments on oligopoly. After a visiting professorship at Berkeley (1967–68) he was professor at the Free University of Berlin (1969–72), at Bielefeld (1972–84) and at Bonn from 1984. In a 1965 paper on an oligopoly model with demand inertia he defined subgame perfectness, and in 1975 the refinement now called trembling-hand perfectness; with John Harsanyi he wrote a general theory of equilibrium selection. He shared the 1994 Nobel Memorial Prize in Economic Sciences with Harsanyi and John Nash.

Mathematician, a Moscow State University graduate of 1957, who worked in Novosibirsk (1959–69) and from 1969 at the Computing Centre of the Soviet and then Russian Academy of Sciences; corresponding member of the Academy from 1984 and full member from 1992; professor at Moscow State University, where he held the chair of mathematical methods of forecasting from 1997 to 2022; working in discrete mathematics and pattern recognition. His algebraic approach to recognition treated whole families of imperfect classification algorithms as objects on which operations are defined, proving that suitable closures of such families contain correct algorithms — an algebraic perspective, developed in the late 1970s, that invites comparison with modern ensemble methods.

Trained in mathematics at Moscow State University, where he took a candidate's degree in psychology in 1971; from 1974 he worked at the University of California, Irvine. He developed mathematical models of reflexive processes — decision-makers holding explicit models of each other's models — introduced the concept of reflexive control, steering an opponent by shaping the model it holds, and later built a formal model of ethical systems and conscience.

Mathematician at Leningrad, later Saint Petersburg, State University, where he long led the Faculty of Applied Mathematics and Control Processes and its department of game theory and statistical decisions. He worked on differential games of pursuit and on dynamic cooperative games, and introduced the requirement that a cooperative solution be time-consistent, together with the imputation distribution procedure that implements it.

Economist who studied at Princeton, Oxford and Harvard, where he received his doctorate. He developed the theory of market signalling — how an informed party can credibly convey its type by an action that is costlier for other types — and taught at Harvard and Stanford, where he was dean of the Graduate School of Business. He shared the 2001 Nobel Memorial Prize in Economic Sciences with George Akerlof and Joseph Stiglitz for analyses of markets with asymmetric information.