Dov Samet

Agreeing to Agree, (with E. Lehrer), (2003)

In a seminal paper, Aumann (1976) demonstrated the impossibility of agreeing to disagree. That is, if the agents have a common prior they cannot have common knowledge of their posteriors for event E if these posteriors do not coincide. We ask here under what which conditions is agreeing to agree possible: Given an event E, can the agents have posteriors with a common prior such that it is common knowledge that the posteriors for E do coincide? We show that a necessary and sufficient condition for this is the existence of a nonempty finite event F with the following two properties. First, it is common knowledge at F that the agents cannot tell whether or not E occurred. Second, this still holds true at F, when F itself becomes common knowledge.

S5 knowledge without partitions, (2006) (Last version, January 23, 2008)

We study set algebras with an operator (SAO) that satisfies the axioms of S5 knowledge. A necessary and sufficient condition is given for such SAOs that the knowledge operator is defined by a partition of the state space. SAOs are constructed for which the condition fails to hold. We conclude that no logic singles out the partitional SAOs among all SAOs.

Agreeing to disagree: The non-probabilistic case, (last version, October 20, 2006)

A non-probabilistic generalization of Aumann's (1976) agreement theorem is proved. Early attempts to formulate such a theorem (Cave (1983), Bacharach (1985)) were based on a version of the sure thing principle which assumes intrapersonal-interstate comparison of knowledge. These theorems were conceptually flawed since such comparisons are impossible in partition knowledge-structures. Therfefore, it was impossible to formulate the sure-thing pricnciple in terms of the partitions or the knowledge operators they define. Later attempts to solve this problem (Moses and Nachum (1990), Aumann and Hart (2006)), by restricting the domain of the decision functions, required unwarranted assumptions and did not generalize Aumann's agreement theorem. The theorem proved here is based on a new version of the sure thing principle that makes interpersonal-intrastate comparison of knowledge.

A commitment folk theorem (2007)
(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.

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

Matching of like rank and the size of the core in the marriage problem (last version, December 4, 2008)

The core of a marriage problem in which men and women are objectively ranked, say by beauty, is a singleton: pairing individuals of equal rank is the only stable matching. We generalize this observation by defining metrics on the sets of rankings and on the set of stable matchings, as well as a measure of the gap in rank between mates in a matching. We explore the relation between these three magnitudes and conclude that when the set of rankings is small so are the core and the rank gap in stable matchings. We also show that when the rank gap in stable matchings is small, then the core is small.

How common are common priors? (May, 2009)
(with Ziv Hellman)

To answer the question in the title we vary agents' beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent's knowledge results in an increase in common knowledge. We show that for partition profiles which are tight the set of consistent type profiles is topologically large, while for partition profiles which are not tight this set is topologically small.

Before publication

Agreeing to Agree, (with E. Lehrer), (2003)

S5 knowledge without partitions, (2006) (Last version, January 23, 2008)

Agreeing to disagree: The non-probabilistic case, (last version, October 20, 2006)

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

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

Matching of like rank and the size of the core in the marriage problem (last version, December 4, 2008)

How common are common priors? (May, 2009) (with Ziv Hellman)

PowerPoint Presentations

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

Publications available in e-journals

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

How common are common priors? (May, 2009)
(with Ziv Hellman)

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