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Papers by Robert J. Aumann

2012 27th Annual IEEE Symposium on Logic in Computer Science, 2012

One of the areas of Game Theory that are of most interest to Computer Scientists, and in which Fo... more One of the areas of Game Theory that are of most interest to Computer Scientists, and in which Formal Logic is most heavily used, is that of Games of Perfect Information (like Chess or Checkers). Of central interest in this connection is the Backward Induction algorithm, which has generated a good deal of controversy, and with it, a large literature. We will review some of this literature, culminating with an as yet unpublished result of Itai Arieli and the speaker.

Economists have long expressed dissatisfaction with the complex models of strict rationality that... more Economists have long expressed dissatisfaction with the complex models of strict rationality that are so pervasive in economic theory. There are several objections to such models. First, casual empiricism or even just simple introspection leads to the conclusion that even in quite simple

Subjectivity and correlation, though formally related, are conceptually distinct and independent ... more Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a

Advances in Game Theory. (AM-52), 1964

Values of Non-Atomic Games, 2015

W ars and other conflicts are among the main sources of human misery.'' Thus begins the Advanced ... more W ars and other conflicts are among the main sources of human misery.'' Thus begins the Advanced Information announcement of the 2005 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, awarded for Game Theory Analysis of Conflict and Cooperation. So, it is appropriate to devote this lecture to one of the most pressing and profound issues that confront humanity: that of war and peace. I would like to suggest that we should perhaps change direction in our efforts to bring about world peace. Up to now, all the effort has been put into resolving specific conf licts: India-Pakistan, North-South Ireland, various African wars, Balkan wars, Russia-Chechnya, Israel-Arab, etc., etc. I'd like to suggest that we should shift emphasis and study war in general.

Letters from Robert Aumann and Ken Binmore on knowledge and belief, from Sudha Shenoy on mathemat... more Letters from Robert Aumann and Ken Binmore on knowledge and belief, from Sudha Shenoy on mathematical economics, and from Casey Mulligan on the Ph.D. circle in academic economics.

The paper presents an approach to the philosophy of Game Theory; it is not a survey. The examples... more The paper presents an approach to the philosophy of Game Theory; it is not a survey. The examples and references are an eclectic, largely haphazard selection-strongly skewed towards applications-from a truly enormous literature. The idea was merely to illustrate some of the points in the text; the references are neither representative nor systematically chosen, even within the particular subjects actually discussed.

Jerusalem. We are greatly indebted to Kenneth Arrow for an extremely helpful conversation on the ... more Jerusalem. We are greatly indebted to Kenneth Arrow for an extremely helpful conversation on the subject of this paper; the interpretation of d t in terms of ''fear of ruin'' (Section 6) is an outcome of that conversation. 2. We use the term ''money'' where often economists use the term ''income.'' We wish to keep the ideas behind these terms distinct: in our terminology, the commodity ''money'' provides the units in which wealth or income are measured. Thus we can talk of ''income,'' ''gross income,'' and ''net income'' as specific quantities of money received by individuals (or groups) in various circumstances. Coalitional Games: Economic and Political Applications 258 3. No adverse value judgment is intended. Pressure groups are what democracy is all about-they are as essential to healthy politics as competition is to a healthy economy. Power and Taxes 259 9. sup x;t u t ðxÞ < y. 10. inf t u t ð1Þ > 0. This implies that inf t u t ðxÞ > 0 for all x > 0.

Contributions to Probability Theory, 1967

We introduce and study a unified reasoning process that allows to model the beliefs of a fully ra... more We introduce and study a unified reasoning process that allows to model the beliefs of a fully rational agent and of an unaware one. This reasoning process provides natural properties to introspection and unawareness. The corresponding model for the rational or boundedly rational agents is both easy to describe and to work with, and the agent's full system of beliefs has natural descriptions using a reduced number of parameters.

World Scientific Series in Economic Theory, 2022

Nature Human Behaviour, 2019

International Journal of Game Theory, 2019

Bulletin of the American Mathematical Society, 1960

By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subs... more By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subsets, called measurable sets, whose union is the whole space. The structure of a space will be the <r-ring of its measurable subsets. A measurable transformation from one space to another is a mapping such that the inverse image of every measurable set is measurable. Let X and F be spaces, F a set of measurable transformations from X into'F, and cj> F : FXX-^Y the natural mapping defined by $F(/> #)=ƒ(#)• A structure R on F will be called admissible if Fl considered as a mapping from the product space (F, R)XX into F, is a measurable transformation. 2 It may not be possible to define an admissible structure on F; if it is, F itself will also be called admissible. We are concerned with the problem of characterizing, for given X and F, the admissible sets F and the admissible structures R on the admissible sets. The following three theorems may be established fairly easily: THEOREM A. A set consisting of a single measurable transformation is admissible. THEOREM B. A subset of an admissible set is admissible. Indeed, if G(ZF, Ris an admissible structure on F, and Ro is the sub space structure on G induced* by R, then RG is admissible on G. THEOREM C. The union of denumerably many admissible sets is admissible. Indeed, if F = \J^Lx F{ and R\, R 21 • • • are admissible structures on Fi, F 2l • • • respectively y then the structure R on F generated by the members of all the Ri is admissible on G.

Journal of the Society for Industrial and Applied Mathematics, 1961

Naval Research Logistics Quarterly, 1958

2012 27th Annual IEEE Symposium on Logic in Computer Science, 2012

One of the areas of Game Theory that are of most interest to Computer Scientists, and in which Fo... more One of the areas of Game Theory that are of most interest to Computer Scientists, and in which Formal Logic is most heavily used, is that of Games of Perfect Information (like Chess or Checkers). Of central interest in this connection is the Backward Induction algorithm, which has generated a good deal of controversy, and with it, a large literature. We will review some of this literature, culminating with an as yet unpublished result of Itai Arieli and the speaker.

Economists have long expressed dissatisfaction with the complex models of strict rationality that... more Economists have long expressed dissatisfaction with the complex models of strict rationality that are so pervasive in economic theory. There are several objections to such models. First, casual empiricism or even just simple introspection leads to the conclusion that even in quite simple

Subjectivity and correlation, though formally related, are conceptually distinct and independent ... more Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a

Advances in Game Theory. (AM-52), 1964

Values of Non-Atomic Games, 2015

W ars and other conflicts are among the main sources of human misery.'' Thus begins the Advanced ... more W ars and other conflicts are among the main sources of human misery.'' Thus begins the Advanced Information announcement of the 2005 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, awarded for Game Theory Analysis of Conflict and Cooperation. So, it is appropriate to devote this lecture to one of the most pressing and profound issues that confront humanity: that of war and peace. I would like to suggest that we should perhaps change direction in our efforts to bring about world peace. Up to now, all the effort has been put into resolving specific conf licts: India-Pakistan, North-South Ireland, various African wars, Balkan wars, Russia-Chechnya, Israel-Arab, etc., etc. I'd like to suggest that we should shift emphasis and study war in general.

Letters from Robert Aumann and Ken Binmore on knowledge and belief, from Sudha Shenoy on mathemat... more Letters from Robert Aumann and Ken Binmore on knowledge and belief, from Sudha Shenoy on mathematical economics, and from Casey Mulligan on the Ph.D. circle in academic economics.

The paper presents an approach to the philosophy of Game Theory; it is not a survey. The examples... more The paper presents an approach to the philosophy of Game Theory; it is not a survey. The examples and references are an eclectic, largely haphazard selection-strongly skewed towards applications-from a truly enormous literature. The idea was merely to illustrate some of the points in the text; the references are neither representative nor systematically chosen, even within the particular subjects actually discussed.

Jerusalem. We are greatly indebted to Kenneth Arrow for an extremely helpful conversation on the ... more Jerusalem. We are greatly indebted to Kenneth Arrow for an extremely helpful conversation on the subject of this paper; the interpretation of d t in terms of ''fear of ruin'' (Section 6) is an outcome of that conversation. 2. We use the term ''money'' where often economists use the term ''income.'' We wish to keep the ideas behind these terms distinct: in our terminology, the commodity ''money'' provides the units in which wealth or income are measured. Thus we can talk of ''income,'' ''gross income,'' and ''net income'' as specific quantities of money received by individuals (or groups) in various circumstances. Coalitional Games: Economic and Political Applications 258 3. No adverse value judgment is intended. Pressure groups are what democracy is all about-they are as essential to healthy politics as competition is to a healthy economy. Power and Taxes 259 9. sup x;t u t ðxÞ < y. 10. inf t u t ð1Þ > 0. This implies that inf t u t ðxÞ > 0 for all x > 0.

Contributions to Probability Theory, 1967

We introduce and study a unified reasoning process that allows to model the beliefs of a fully ra... more We introduce and study a unified reasoning process that allows to model the beliefs of a fully rational agent and of an unaware one. This reasoning process provides natural properties to introspection and unawareness. The corresponding model for the rational or boundedly rational agents is both easy to describe and to work with, and the agent's full system of beliefs has natural descriptions using a reduced number of parameters.

World Scientific Series in Economic Theory, 2022

Nature Human Behaviour, 2019

International Journal of Game Theory, 2019

Bulletin of the American Mathematical Society, 1960

By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subs... more By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subsets, called measurable sets, whose union is the whole space. The structure of a space will be the <r-ring of its measurable subsets. A measurable transformation from one space to another is a mapping such that the inverse image of every measurable set is measurable. Let X and F be spaces, F a set of measurable transformations from X into'F, and cj> F : FXX-^Y the natural mapping defined by $F(/> #)=ƒ(#)• A structure R on F will be called admissible if Fl considered as a mapping from the product space (F, R)XX into F, is a measurable transformation. 2 It may not be possible to define an admissible structure on F; if it is, F itself will also be called admissible. We are concerned with the problem of characterizing, for given X and F, the admissible sets F and the admissible structures R on the admissible sets. The following three theorems may be established fairly easily: THEOREM A. A set consisting of a single measurable transformation is admissible. THEOREM B. A subset of an admissible set is admissible. Indeed, if G(ZF, Ris an admissible structure on F, and Ro is the sub space structure on G induced* by R, then RG is admissible on G. THEOREM C. The union of denumerably many admissible sets is admissible. Indeed, if F = \J^Lx F{ and R\, R 21 • • • are admissible structures on Fi, F 2l • • • respectively y then the structure R on F generated by the members of all the Ri is admissible on G.

Journal of the Society for Industrial and Applied Mathematics, 1961

Naval Research Logistics Quarterly, 1958