Dov Samet

Before publication

Belief consistency and trade consistency (Last version: June, 2012)
(with E. Lehrer)

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agents' beliefs are derived is of the first kind. The nonexistence of an agreeable bet, that is a contingent zero-sum trade which is always favorable to all agents is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinstein's email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.

The table below summarizes the three equivalence theorems. The conditions on the left column are economic and on the right, epistemic. Equivalence holds in each row. The two notions in red were proved in previous works to be equivalent in finite spaces. In this case all the implications in the columns are equivalences.

Three equivalence theorems

weak trade consistency ⇔ weak belief consistency

no agreeable bet^* uniformly existence of common

bounded away from 0 ε-priors^** for all ε > 0

⇑ ⇑

trade consistency ⇔ belief consistency

no agreeable bet^* the common ε-priors^** of some type

vanish infinitely more slowly than ε

⇑ ⇑

strong trade consistency ⇔ strong belief consistency

for some state and agent, the gains existence of a common prior

in ε-agreeable bets^*** vanish with ε

A glossary

^* A bet is a contingent zero-sum trade.

A bet is agreeable if the expected gains of all agents are always positive.

^** An ε-prior of an agent is a probability distribution ε-close to a prior of the agent.

A common ε-prior is a probability distribution which is an ε-prior of all agents.

^*** An ε-agreeable bet is a bet such the expected loss of each agents is at most ε.

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.

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).

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

Publications available in e-journals

weak trade consistency	⇔	weak belief consistency
no agreeable bet^* uniformly		existence of common
bounded away from 0		ε-priors^** for all ε > 0
⇑		⇑
trade consistency	⇔	belief consistency
no agreeable bet^*		the common ε-priors^** of some type
		vanish infinitely more slowly than ε
⇑		⇑
strong trade consistency	⇔	strong belief consistency
for some state and agent, the gains		existence of a common prior
in ε-agreeable bets^*** vanish with ε

^*	A bet is a contingent zero-sum trade.
	A bet is agreeable if the expected gains of all agents are always positive.
^**	An ε-prior of an agent is a probability distribution ε-close to a prior of the agent.
	A common ε-prior is a probability distribution which is an ε-prior of all agents.
^***	An ε-agreeable bet is a bet such the expected loss of each agents is at most ε.

Before publication

Belief consistency and trade consistency (Last version: June, 2012) (with E. Lehrer)

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)

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

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)

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 (Last version: June, 2012)
(with E. Lehrer)

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)

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