Robert Aumann

Robert Aumann ישראל אומן
Born	Robert John Aumann 8 6, 1930
Birthplace	Frankfurt am Main, Prussia, Germany
Nationality	Israeli, American
Occupation	Mathematician, economist
Employer	Hebrew University of Jerusalem, Stony Brook University
Known for	Aumann's agreement theorem, correlated equilibrium, folk theorem, Aumann–Shapley value
Education	Massachusetts Institute of Technology (MS, PhD)
Awards	Nobel Memorial Prize in Economic Sciences (2005), Israel Prize (1994), Harvey Prize (1983)

Robert John Aumann (Hebrew: ישראל אומן, Yisrael Aumann; born June 8, 1930) is an Israeli-American mathematician and economist whose pioneering work in game theory has shaped the modern understanding of conflict, cooperation, and strategic interaction. Born in Frankfurt am Main, Germany, Aumann fled with his family to the United States in 1938, just two weeks before the Kristallnacht pogrom that devastated Jewish communities across Nazi Germany. He went on to build a distinguished academic career spanning more than six decades, primarily at the Hebrew University of Jerusalem, where he serves as a professor at the Center for the Study of Rationality. In 2005, Aumann was awarded the Nobel Memorial Prize in Economic Sciences, shared with Thomas Schelling, "for having enhanced our understanding of conflict and cooperation through game-theory analysis." His contributions include foundational concepts such as correlated equilibrium, Aumann's agreement theorem, and the folk theorem, as well as the Aumann–Shapley value developed in collaboration with Lloyd Shapley. A member of the United States National Academy of Sciences, Aumann is also one of the founding members of the Stony Brook Center for Game Theory at Stony Brook University.

Early Life

Robert John Aumann was born on June 8, 1930, in Frankfurt am Main, in the Free State of Prussia, Germany. He was born into a Jewish family during a period of escalating antisemitism in Germany. As the political situation deteriorated under the Nazi regime, the Aumann family made the decision to leave the country. In 1938, when Robert was eight years old, the family fled Germany to the United States, arriving just two weeks before the Kristallnacht pogrom of November 9–10, 1938, during which Nazi paramilitary forces and civilians carried out coordinated attacks against Jewish people and property across Germany and Austria.

After settling in the United States, the young Aumann grew up in New York City. He attended the Rabbi Jacob Joseph School, a yeshiva high school in New York City, where he received both secular and religious education. This formative period in a yeshiva environment instilled in Aumann a deep engagement with Talmudic scholarship, an interest he would carry throughout his life and eventually connect to his academic work in mathematics and game theory. Aumann has written scholarly papers exploring game-theoretic and economic concepts within the Talmud, including analyses of risk aversion in Talmudic texts.

The experience of fleeing Nazi Germany as a child profoundly shaped Aumann's worldview and his later intellectual focus on conflict and cooperation between parties. His personal history of displacement and survival informed the urgency he brought to understanding the strategic dynamics that underlie both war and peace, themes that would become central to his life's work in game theory.

Education

Aumann pursued his undergraduate studies at the City College of New York (CCNY), where he earned a Bachelor of Science degree in mathematics in 1950. CCNY was known during this era for its rigorous academic environment and as a pathway for talented students from immigrant families in New York City.

Following his undergraduate education, Aumann enrolled at the Massachusetts Institute of Technology (MIT) for graduate studies in mathematics. He received his Master of Science degree from MIT in 1952 and completed his Doctor of Philosophy degree in mathematics in 1955. His doctoral dissertation, titled Asphericity of Alternating Linkages, was a contribution to knot theory, a branch of topology in pure mathematics. His doctoral advisor was the distinguished topologist George Whitehead, Jr., who was a prominent figure in algebraic topology at MIT.

Although Aumann's doctoral work was in pure mathematics rather than economics or game theory, his rigorous mathematical training at MIT provided the analytical foundation upon which he would build his subsequent contributions to mathematical economics and the theory of games. The transition from pure mathematics to game theory came in the years following his doctorate, as Aumann recognized the deep mathematical structures underlying strategic interaction.

Career

Early Career and Move to Israel

After completing his doctorate at MIT in 1955, Aumann began his professional career. In 1956, he joined the mathematics faculty of the Hebrew University of Jerusalem, a move that represented both a professional appointment and a personal commitment to the newly established State of Israel. The Hebrew University would become his primary academic home for the rest of his career.

At the Hebrew University, Aumann began shifting his research focus from pure mathematics toward game theory and mathematical economics. This transition was facilitated by the intellectual environment in Israel, where the intersection of mathematics, economics, and strategic thinking held particular relevance given the young nation's geopolitical circumstances. Aumann became affiliated with the Einstein Institute of Mathematics at the Hebrew University, where he would develop many of his most significant contributions to the field.

Contributions to Game Theory

Aumann's scholarly output over the following decades fundamentally advanced the mathematical foundations of game theory. His contributions span several distinct but interrelated areas of the discipline.

One of Aumann's most notable contributions was the introduction of the concept of correlated equilibrium in game theory. Correlated equilibrium represents a type of equilibrium in non-cooperative games that is more flexible than the classical Nash equilibrium introduced by John Nash. In a correlated equilibrium, players may coordinate their strategies through signals from an external mediator, even in non-cooperative settings. This concept expanded the toolkit available to game theorists and provided a more realistic model for understanding how players in strategic situations might achieve coordination without binding agreements.

Aumann also introduced the first purely formal account of the notion of common knowledge in game theory. Common knowledge refers to information that all players know, all players know that all players know, and so on ad infinitum. This seemingly simple concept had profound implications for understanding strategic interaction, as it clarified the informational conditions under which certain equilibrium outcomes could be sustained.

Closely related to his work on common knowledge is Aumann's agreement theorem, published in his 1976 paper "Agreeing to Disagree" in The Annals of Statistics. In this influential paper, Aumann demonstrated that two Bayesian rationalists who share common prior beliefs cannot "agree to disagree"—that is, if their posterior beliefs about an event are common knowledge, those posteriors must be equal. This result had far-reaching implications for economics, epistemology, and the study of information aggregation, challenging the intuition that rational agents with access to different information could persistently hold different beliefs about the same proposition once those beliefs were mutually known.

Aumann made significant contributions to the theory of repeated games, including his work on the folk theorem of game theory. The folk theorem establishes that in infinitely repeated games, a wide range of outcomes can be sustained as equilibria, provided players are sufficiently patient. This result helped explain how cooperation can emerge and be sustained in long-term relationships, even among self-interested agents, by providing a formal framework for understanding the role of reputation and punishment strategies in maintaining cooperative behavior.

In collaboration with Lloyd Shapley, Aumann developed the Aumann–Shapley value, an extension of the Shapley value to games with a continuum of players. This work was part of a broader research program on large games—games with many players—that Aumann pursued extensively. The Aumann–Shapley value provided a method for fairly allocating costs or benefits among an infinite number of participants, with applications in economics, cost allocation, and public finance.

Aumann also made contributions to the study of decision-making under uncertainty. The Anscombe-Aumann framework, developed with Frank J. Anscombe, provided a foundational approach to subjective expected utility theory that complemented the earlier work of Leonard Savage. This framework offered a mathematically elegant way to derive subjective probabilities and utilities from preferences over acts, and it remains a standard tool in decision theory.

Additionally, Aumann made mathematical contributions through his work on the integral of a correspondence (also known as the Aumann integral), which provided a way to integrate set-valued functions. This technical contribution had applications beyond game theory, in areas such as mathematical economics and control theory.

The Nobel Prize

In 2005, the Royal Swedish Academy of Sciences awarded the Nobel Memorial Prize in Economic Sciences jointly to Robert Aumann and Thomas Schelling "for having enhanced our understanding of conflict and cooperation through game-theory analysis." The prize committee's Advanced Information announcement opened with the observation that "Wars and other conflicts are among the main sources of human misery," framing the laureates' work as fundamental to understanding how such conflicts arise and how they might be prevented or resolved.

The Nobel committee recognized Aumann's work for demonstrating, through rigorous mathematical analysis, how long-term relationships and repeated interactions can sustain cooperation even among parties with conflicting short-term interests. His analysis of infinitely repeated games showed that threats of future punishment could deter defection from cooperative arrangements, providing a formal basis for understanding phenomena such as price wars, arms races, and trade disputes. The committee noted that Aumann's theoretical framework shed light on why some communities succeed in managing common resources while others fail, and why some organizations achieve cooperation while others are plagued by conflict.

While Schelling's contributions were recognized for their intuitive and applied character, Aumann's work was cited for its mathematical precision and generality. Together, the two laureates represented complementary approaches to the game-theoretic analysis of conflict and cooperation.

Visiting Positions and Institutional Roles

Throughout his career at the Hebrew University, Aumann held numerous visiting positions at leading academic institutions around the world. He served as a visiting professor at the University of California, Berkeley in 1971 and again in 1985–1986. He held visiting professorships at Stanford University in 1975–1976 and 1980–1981, and at the Université catholique de Louvain in 1972, 1978, and 1984.

Since 1989, Aumann has held a visiting position at Stony Brook University in New York, where he became one of the founding members of the Stony Brook Center for Game Theory. This center became a significant hub for game theory research and annual conferences, attracting scholars from around the world.

At the Hebrew University, Aumann was instrumental in establishing the Center for the Study of Rationality, an interdisciplinary research center that brings together scholars from mathematics, economics, computer science, philosophy, psychology, biology, and law to study rational decision-making and strategic interaction.

Doctoral Students

Aumann supervised several doctoral students who went on to make significant contributions to game theory and mathematical economics. His doctoral students include David Schmeidler, who made important contributions to decision theory and cooperative game theory; Sergiu Hart, who became a leading figure in game theory at the Hebrew University; Abraham Neyman, known for his work on repeated games and stochastic games; and Yair Tauman, who contributed to the theory of cost allocation and mechanism design.

Public Commentary

In addition to his academic work, Aumann has been a public figure in Israel, occasionally applying game-theoretic reasoning to contemporary political and strategic issues. In interviews following the outbreak of the Israel-Hamas war in 2023, Aumann expressed views on the hostage negotiations and the broader conflict. He argued that the public campaign by families of hostages held in Gaza served to increase the "price" demanded by captors, a position he framed in terms of strategic bargaining theory. He also stated that Israel should occupy Gaza, applying his understanding of conflict dynamics to advocate for a particular strategic approach.

Aumann's public statements on political matters have placed him in the broader discussion of how game-theoretic reasoning is applied to real-world geopolitical situations, a topic examined in both scholarly and journalistic contexts.

Personal Life

Aumann immigrated to Israel in 1956, the same year he joined the Hebrew University faculty, and has lived in the country since that time. He is known within academic circles by both his English name, Robert Aumann, and his Hebrew name, Yisrael Aumann (ישראל אומן).

Aumann has maintained a lifelong engagement with Jewish religious scholarship alongside his academic career in mathematics and economics. His published work includes papers examining game-theoretic and economic concepts within Talmudic texts, reflecting the intersection of his religious and intellectual life. His paper on the "Man with Three Wives" explores a Talmudic problem of estate division using cooperative game theory.

In December 2021, Aumann delivered a keynote address at the O'Malley School of Business Research Seminar at Manhattan College (now Manhattan University), demonstrating his continued engagement with academic and public audiences well into his ninth decade.

Recognition

Aumann has received numerous awards and honors over the course of his career, reflecting the significance of his contributions to mathematics, economics, and game theory.

His most prominent recognition was the 2005 Nobel Memorial Prize in Economic Sciences, shared with Thomas Schelling, for their work on understanding conflict and cooperation through game-theory analysis.

Prior to the Nobel Prize, Aumann received the Israel Prize in 1994, one of the highest civilian honors awarded by the State of Israel. He was also a recipient of the Harvey Prize from the Technion – Israel Institute of Technology.

Aumann received the Erwin Plein Nemmers Prize in Economics from Northwestern University, an award recognizing outstanding contributions to economics.

He was elected to membership in the United States National Academy of Sciences and the American Academy of Arts and Sciences. He also received the Yakir Yerushalayim (Worthy Citizen of Jerusalem) award from the city of Jerusalem.

Legacy

Robert Aumann's contributions to game theory have had a lasting impact on multiple disciplines, including economics, political science, computer science, evolutionary biology, and philosophy. His mathematical formalization of concepts such as correlated equilibrium, common knowledge, and the theory of repeated games provided the rigorous foundations upon which subsequent generations of researchers have built.

The concept of correlated equilibrium, which Aumann introduced, expanded the notion of what constitutes rational strategic behavior beyond the Nash equilibrium framework. Correlated equilibrium has found applications in algorithmic game theory and mechanism design, areas of growing importance in the age of digital platforms and automated decision-making systems.

Aumann's agreement theorem continues to be a reference point in epistemology, information economics, and the study of financial markets. The theorem's implication—that rational agents with common priors who share their posterior beliefs cannot persistently disagree—has generated extensive scholarly debate about the nature of disagreement, speculation, and information processing in both academic and practical contexts.

His work on repeated games provided a formal explanation for the emergence and sustainability of cooperation in long-term relationships, a finding with implications for understanding international diplomacy, business partnerships, regulatory compliance, and social norms. The folk theorem and related results demonstrated that the "shadow of the future" could serve as a powerful mechanism for enforcing cooperative behavior without external enforcement.

Through his role in founding the Center for the Study of Rationality at the Hebrew University and the Stony Brook Center for Game Theory, Aumann helped establish institutional frameworks that continue to support interdisciplinary research at the intersection of mathematics, economics, and the social sciences. His doctoral students and academic collaborators have carried forward his research program, ensuring that his methodological approaches and theoretical insights continue to influence the field.

Aumann's unusual combination of deep mathematical rigor with attention to practical applications of strategic reasoning—from Talmudic estate division to international conflict—has made him a distinctive figure in the intellectual landscape of the twentieth and twenty-first centuries.