Dov Samet

Belief consistency and trade consistency (November, 2011)
(with E. Lehrer)
A twelve minute talk about the highlights of the paper

Interpersonal consistency can be defined in epistemic terms as consistency of beliefs, and in economic terms as the impossibility of certain trades. More specifically, beliefs are consistent when they have a common prior, that is, when they are derived by Bayesian updating from a prior belief common to all agents. Trade consistency requires that there is no contingent zero-sum trade which is always commonly known to be favorable to all agents. It is well established that in finite, and more generally in compact type spaces, the two notions of consistency are equivalent. However, for countable type spaces trade consistency may hold even when there is no common prior. We map the relations between various notions of belief and economic consistency in countable type spaces. Our main result is an equivalence theorem for finite and infinite countable type spaces between trade consistency and a new notion of belief consistency. This equivalence is a powerful tool that enables us to fully analyze the consistency of type spaces based on the knowledge structure of Rubinstein's email game. It also helps to justify the requirement of boundedness of trade in countable type spaces by showing that in a large class of spaces there exists an agreeable bet which is possibly unbounded even when a common prior exists.

Conditional belief types (September, 2011)
(with A. Di Tillio and J. Halpern)

Decision making requires that agents have beliefs about what happens given events that are believed or known not to happen. Such beliefs can be modeled by conditional probability functions which allow conditioning on unconditionally null events. Players with such beliefs must have conditional beliefs about conditional beliefs. We model this using a type space where a player's type at a state is a conditional probability on the space. We axiomatize type spaces using conditional belief operators, and examine additional three axioms of increasing strength: introspection, that requires that the agent is unconditionally certain of her beliefs; echo, according to which the unconditional beliefs that are implied by the condition must be held given the condition; anddetermination, which says that the conditional beliefs are the unconditional beliefs that are conditionally certain. The echo axiom implies that conditioning events must be unconditionally certain. Thus, conditioning on an event is conditioning on the agent being certain of the event. This formalizes the meaning frequently given to conditioning in probability theory. The echo axiom also implies that the probability given an event is a prior of the unconditional probability. Type spaces are closely related to the sphere models of counterfactual conditionals and to models of hypothetical knowledge.

On the dispensable role of time in games of perfect information (July, 2011)

Time present and time past
Are both perhaps present in time future,
And time future contained in time past.

T. S. Eliot, Burnt Norton—the Four Quartets

In two papers on rationality in games with perfect information, Aumann (1995) and Aumann (1998), time is assumed implicitly in the description of games of perfect information, and it is part of the epistemic distinction between ex-ante and ex-post knowledge. We show that ex-post knowledge can be expressed by ex-ante knowledge and therefore epistemically, time is irrelevant to the analysis. Furthermore, we show that material rationality by weak dominance and by expectation can be expressed in the timeless language of the strategic form of the game.

Common belief of rationality in games of perfect information (July, 2011)

Aumann (1995) showed that for games with perfect information common knowledge of substantive rationality implies backward induction. Substantive rationality is defined in epistemic terms, that is in terms of knowledge. We show that when substantive rationality is defined in doxastic terms, that is, in terms of belief, then common belief of substantive rationality implies backward induction. Aumann (1998) showed that material rationality implies backward induction in the centipede game. This result does not hold when rationality is defined doxastically. However, if beliefs are interpersonally consistent then common belief of material rationality in the centipede game implies common belief of backward induction.

What if Achilles and the tortoise were to bargain?
An argument against interim agreements? (January, 2010. Last version, April, 2010)

Engaging in a dynamic process of interim agreements guarantees that agreement will never be reached. Arguments of Zeno, Aristotle, von Neumann, Nash, Raiffa, and C. Northcote Parkinson lead to this grim conclusion. Is the everlasting Israeli-Palestinian peace process a case in point?

(In the picture, Achilles and the tortoise bargain on the splitting of a drachma).

Generalized Raiffa solutions (January, 2010)
(with A. Diskin and M. Koppel) , forthcoming GEB

Between the discrete and continuous Raiffa solutions to bargaining problems lie a continuum of discrete solutions: the generalized Raiffa solutions. We propose a simple, "classical" axiomatization of this family in terms of a "non-classical" solution concept: the stepwise solution which consists of a step function and a solution function.

Matching of like rank and the size of the core in the marriage problem (last version, March, 2010)

When men and women are objectively ranked in a marriage problem, say by beauty, then pairing individuals of equal rank is the only stable matching. We generalize this observation by providing bounds on the size of the rank gap between mates in a stable matchings in terms of the size of the ranking sets. Using a metric on the set of matchings, we provide bounds on the diameter of the core---the set of stable matchings---in terms of the size of the ranking sets and in terms of the size of the rank gap. We conclude that when the set of rankings is small, so are the core and the rank gap in stable matchings.

The sure thing principle and independence of irrelevant knowledge (2008)

Savage (1954) introduced the sure thing principle in terms of the dependence of decisions on knowledge, but gave up on formalizing it in epistemic terms for lack of a formal definition of knowledge. Using simple models of knowledge, we examine the sure thing principle, presenting two ways to capture it. One is in terms of the union of future events, for which we reserve the original name — the sure thing principle; the other is in terms of the intersection of kens — bodies of agents' knowledge — which we call independence of irrelevant knowledge. We show that the two principles are equivalent and that the only property of knowledge required for this equivalence is the axiom of truth--the requirement that whatever is known is true. We present a symmetric version of the independence of irrelevant knowledge which is equivalent to the impossibility of agreeing to disagree on the decision made by agents, namely the impossibility of agents making different decisions being common knowledge

A commitment folk theorem (2007) forthcoming GEB
(with Adam Tauman Kalai, Ehud Kalai, and Ehud Lehrer)

Real world players often increase their payoffs by voluntarily committing to play a fixed strategy, prior to the start of a strategic game. In fact, the players may further benefit from commitments that are conditional on the commitments of others. This paper proposes a model of conditional commitments that unifies earlier models while avoiding circularities that often arise in such models. A commitment folk theorem shows that the potential of voluntary conditional commitments is essentially unlimited. All feasible and individually-rational payoffs of a two-person strategic game can be attained at the equilibria of one (universal) commitment game that uses simple commitment devices. The commitments are voluntary in the sense that each player maintains the option of playing the game without commitment, as originally defined.

Before publication

Belief consistency and trade consistency (November, 2011) (with E. Lehrer) A twelve minute talk about the highlights of the paper

Conditional belief types (September, 2011) (with A. Di Tillio and J. Halpern)

On the dispensable role of time in games of perfect information (July, 2011)

Common belief of rationality in games of perfect information (July, 2011)

What if Achilles and the tortoise were to bargain? An argument against interim agreements? (January, 2010. Last version, April, 2010)

Generalized Raiffa solutions (January, 2010) (with A. Diskin and M. Koppel) , forthcoming GEB

Matching of like rank and the size of the core in the marriage problem (last version, March, 2010)

The sure thing principle and independence of irrelevant knowledge (2008)

A commitment folk theorem (2007) forthcoming GEB (with Adam Tauman Kalai, Ehud Kalai, and Ehud Lehrer)

PowerPoint Presentations

A double feature: One observation behind two envelope puzzles + Agreeing to agree Between Liberalism and Democracy

Publications available in e-journals

Belief consistency and trade consistency (November, 2011)
(with E. Lehrer)
A twelve minute talk about the highlights of the paper

Conditional belief types (September, 2011)
(with A. Di Tillio and J. Halpern)

What if Achilles and the tortoise were to bargain?
An argument against interim agreements? (January, 2010. Last version, April, 2010)

Generalized Raiffa solutions (January, 2010)
(with A. Diskin and M. Koppel) , forthcoming GEB

A commitment folk theorem (2007) forthcoming GEB
(with Adam Tauman Kalai, Ehud Kalai, and Ehud Lehrer)

A double feature:
One observation behind two envelope puzzles + Agreeing to agree
Between Liberalism and Democracy