Processes of bargaining are studied in which the players reach interim agreements that serve as status quo points for further bargaining. This is modeled in Nash’s setup of bargaining problems, where the solution is a time parameterized path of interim agreements rather than a single point. We characterize path solutions for linear problems that satisfy the axioms of restarting and covariance, and show that if a Pareto efficient agreement is not reached immediately, then it is never reached in finite time. Adding the axioms of individual rationality, relevance, and monotonicity, we characterize the family of continuous Raiffa solutions and show that these solutions converge to a Pareto efficient agreement but never reach it in finite time. Finally, if a deadline is added to the bargaining problem, and the speed of bargaining is proportionally inverse to the deadline, then a Pareto efficient agreement is reached exactly at the deadline.

We show that knowledge satisfies interpersonal independence, meaning that a non-trivial sentence describing one agent’s knowledge cannot be equivalent to a sentence describing another agent’s knowledge. The same property of interpersonal independence holds, mutatis mutandis, for belief. In the case of knowledge, interpersonal independence is implied by the fact that there are no non-trivial sentences that are common knowledge in every model of knowledge. In the case of belief, interpersonal independence follows from a strong interpersonal independence that knowledge does not have. Specifically, there is no sentence describing the beliefs of one person that implies a sentence describing the beliefs of another person.

We propose a model of an agent’s probability and utility that is a compromise between Savage (The foundations of statistics, Wiley, 1954) and Jeffrey (The Logic of Decision, McGraw Hill, 1965). In Savage’s model the probability–utility pair is associated with preferences over acts which are assignments of consequences to states. The probability is defined on the state space, and the utility function on consequences. Jeffrey’s model has no consequences, and both probability and utility are defined on the same set of propositions. The probability–utility pair is associated with a desirability relation on propositions. Like Savage we assume a set of consequences and a state space. However, we assume that states are comprehensive , that is, each state describes a consequence, as in Aumann (Econometrica 55:1–18, 1987). Like Jeffrey, we assume that the agent has a preference relation, which we call desirability , over events, which by definition involves uncertainty about consequences. For a given probability and utility of consequences, the desirability relation is presented by conditional expected utility, given an event. We axiomatically characterize desirability relations that are represented by a probability–utility pair . We characterize the family of all the probability–utility pairs that represent a given desirability relation.

The impossibility of agreeing to disagree in the non-probabilistic setup means that agents cannot commonly know their decisions unless they are all the same. We study the relation of this property to the sure thing principle when it is expressed in epistemic terms. We show that it can be presented in two equivalent ways: one is in terms of knowledge operators, which we call the principle of follow the knowledgeable, the other is in terms of kens—bodies of agents' knowledge—which we call independence of irrelevant knowledge. The latter can be easily extended to a property which is equivalent to the impossibility of agreeing to disagree.

We study new non-Bayesian solutions of games in strategic form, based on four notions of dominance: weak or strict domination by either a pure or a mixed strategy. For each of these types of dominance, d, we define a family of sets of strategy profiles, called d-correlated equilibria. We study the structure and properties of these families. A player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows about the play of the other players. A set of profiles is a d-correlated equilibrium if and only if it is the set of profiles played in a model where d-dominance rationality is commonly known. When d denotes strict domination by a mixed strategy, a set of profiles is a d-correlated equilibrium if and only if it is the set of profiles played in a model where Bayesian rationality is commonly known.

The main purpose of this paper is to provide a simple criterion enabling to conclude that two agents do not share a common prior. The criterion is simple, as it does not require information about the agents' knowledge and beliefs, but rather only the record of a dialogue between the agents. In each stage of the dialogue, the agents tell each other the probability they ascribe to a fixed event and update their beliefs about the event. To characterize dialogues consistent with a common prior, we first study monologues, which are sequences of probabilities assigned by a single agent to a given event in an exogenous learning process. A dialogue is consistent with a common prior if and only if each selection sequence from the two monologues comprising the dialogue is itself a monologue.

The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem is a classic common-goods dilemma, it naturally lends itself to a game-theoretic analysis. Hence, we designed a model wherein physicians weigh whether antibiotics should be prescribed, given that antibiotic usage depletes its future effectiveness. The physicians' decisions rely on the probability of a bacterial infection before definitive laboratory results are available. We show that the physicians' equilibrium decision rule of antibiotic prescription is not socially optimal. However, we prove that discretizing the information provided to physicians can mitigate the gap between their equilibrium decisions and the social optimum of antibiotic prescription. Despite this problem's complexity, the effectiveness of the discretization solely depends on the type of information available to the physician to determine the nature of infection. This is demonstrated on theoretic distributions and a clinical dataset. Our results provide a game-theory based guide for optimal output of current and future decision support systems of antibiotic prescription.

Ceva's theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. The three-dimensional generalization of a triangle is a tetrahedron, and the n-dimensional generalization of these is an n-simplex. We extend Ceva's theorem to n-simplices and in doing so illustrate the considerations and choices that can be made in generalizing from plane geometry to high-dimensional geometries.

There are four types of dominance depending on whether domination is strict or weak and whether the dominating strategy is pure or mixed. Letting d vary over these four types of dominance, we say that a player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows. For weak dominance by a mixed strategy, Stalnaker (1994) introduced a process of iterative maximal elimination of certain profiles that we call here flaws. We define here, analogously, d-flaws for each type of dominance d, and show that for each d, iterative elimination of d-flaws is order independent. We then show that the characterization of common knowledge of d-dominance rationality is the same for each d. A strategy profile can be played when d-dominance rationality is commonly known if and only if it survives an iterative elimination of d-flaws.

Strict-dominance rationality is a non-Bayesian notion of rationality which means that players do not chose strategies they know to be strictly dominated. Similarly, weak -dominance rationality means that players do not choose strategies that they know to be weakly dominated. Iterative elimination of strictly dominated strategies can be intuitively and formally justified by players having common knowledge of strict-dominance rationality. In contrast, common knowledge of weak -dominance rationality fails to justify iterative elimination of weakly dominated strategies. Examining the reasons for this failure leads to a characterization of the strategy profiles played when weak-dominance rationality is commonly known. These are profiles that survive a process of iterative elimination of profiles called weak flaws that was introduced by Stalnaker (1994) to characterize certain Bayesian models of games. We define, analogously, strict flaws, and show that the iterative elimination of either weak or strict flaws is order independent. Our main result is that the case of weak dominance and strict dominance are completely analogous: Common knowledge of weak-dominance or strictdominance rationality is characterized by iterative elimination of weak or strict flaws correspondingly. Our results hold equally for domination by pure and mixed strategies, which distinguish them from characterizations in Bayesian models that hold only for mixed domination.

We consider a group of individuals, such that each coalition of them is endowed with a preference relation, which may be incomplete, over a given set of prospects, and such that the extended Pareto rule holds. We assume that each singleton coalition has complete vNM preferences. In this setup, Baucells and Shapley (2008) gave a sufficient condition for a coalition to have complete preferences, in terms of the completeness of preferences of certain pairs of individuals. The new property that we introduce of individual prospects requires each individual to have a pair of consequences between which only she is not indifferent. We show that with this property a weaker condition guarantees the completeness of preferences of a coalition: it suffices for a coalition to be a union of a connected family of coalitions with complete preferences.

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 a standard model of knowledge, the partition model, we examine the sure-thing principle, presenting two ways to capture it. One is in terms of knowledge operators, which we call the principle of follow the knowledgeable; the other is in terms of kens—bodies of agents’ knowledge—which we call independence of irrelevant knowledge. We show that the two principles are equivalent. We present a stronger version of the independence of irrelevant knowledge and show that it is equivalent to the impossibility of agreeing to disagree on the decision made by agents, namely the impossibility of different decisions made by agents being common knowledge.

We say that agreeing to agree is possible for an event E if there exist posterior beliefs of the agents with a common prior such that it is common knowledge that the agents' posteriors for E coincide. We propose a notion called Dutch book which is a profile of interim contracts between an outsider and the agents based on the occurrence of E, such that the outsider makes positive profit in all states. We show that in a finite state space, when the agents cannot tell whether E occurred or not, agreeing to agree is possible for E if and only if there is no Dutch book on E. This characterization also holds in countable state spaces with two agents. We weaken the notion of Dutch book to characterize agreeing to agree in a countable state space with multiple agents, when each set in each agent's information partition is finite.

The iterative elimination of strongly dominated strategies can be justified by common knowledge of rationality. Rationality in this context means that players do not play strategies which are strongly dominated in a game they know they play. Indeed, each iteration of elimination of strongly dominated strategies seems to correspond to an iteration of mutual knowledge, that is, an iteration of “all know that”. Thus, the first round of elimination is justified by rationality. The second is justified by mutual knowledge of rationality, the third, by mutual knowledge of mutual knowledge of rationality and so on. The example in Section 2 demonstrates that for games with an infinite number of strategies the said correspondence may fail to hold. For common knowledge, only the infinitely many finite iterations of “all know that” are required. In contrast, elimination of strongly dominated strategies may require transfinite iterations, that is, eliminations made after all the finite rounds of elimination. This result is puzzling. On one hand, it seems plausible that common knowledge of rationality cannot hold if the transfinite process of elimination is not completed. On the other hand, the finite iterations of “all know that” culminate in common knowledge of rationality before the task of elimination is exhausted. Common knowledge of rationality seems to sluggishly follow the process of elimination of strongly dominated strategies, which is carried on vigorously beyond all finite rounds. One is tempted to continue transfinitely the iteration of mutual knowledge. Indeed, Barwise (1988) showed that in a non-well-founded set theoretic model, common knowledge is reached after transfinite iterations of mutual knowledge. Also, Heifetz (1999) showed that in infinitary logic common belief requires a transfinite sequence of iterations of mutual belief. However, the puzzling phenomenon demonstrated here and in Lipman (1994) is presented in standard set theoretic terms and simple finitary language. In such a setup mutual knowledge of rationality of all finite order does define common knowledge of rationality, and no transfinite steps are required. We show that the failure to derive the transfinite process of iterative elimination of strongly dominated strategies by the finite iterations of “all know that” is due to the inadequacy of the proof and not to the notion of common knowledge. Common knowledge of rationality is perfectly captured by the finite iterations of mutual knowledge on one hand, and on the other hand it does imply the transfinite process of elimination of strongly dominated strategies.

In Aumann (Games Econ Behav 8(1):6–19, 1995 , Games Econ Behav 23(1):97–105, 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 in these papers 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 terms of the timeless strategic form of the game.

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 non-existence 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. (C) 2013 Elsevier Inc. All rights reserved.

What strategy profiles can be played when it is common knowledge that weakly dominated strategies are not played? A comparison to the case of strongly dominated strategy is in order. A common informal argument shows that if it is common knowledge that players do not play strongly dominated strategies then players can play only profiles that survive the iterative elimination of strongly dominated strategies. We formalize and prove this claim. However, the analogous claim for the case of weak dominance does not hold. We show that common knowledge that players do not play weakly dominated strategies implies that they must play profiles that survive an iterative elimination of profiles, called flaws of weakly dominated strategies, a process described by Stalnaker (1994). The iterative elimination of flaws of strongly dominated strategies results in the same set of profiles as the iterative elimination of strongly dominated strategies. Thus, the case of weak dominance and strong dominance are completely analogous: Common knowledge that players do not play weakly, or strongly dominated strategies implies iterative elimination of flaws of weakly, or strongly dominated strategies, correspondingly. These processes, for both weak and strong dominance, are independent of the order of elimination.

We study type spaces where a player's type at a state is a conditional probability on the space. We axiomatize these spaces using conditional belief operators, examining three additional axioms of increasing strength. First, introspection, which requires the agent to be unconditionally certain of her beliefs. Second, echo, according to which the unconditional beliefs implied by the condition must be held given the condition. Third, determination, which says that the conditional beliefs are the unconditional beliefs that are conditionally certain. Echo implies that conditioning on an event is the same as conditioning on the event being certain, which formalizes the standard informal interpretation of conditional probability. The game-theoretic application of our model, discussed within an example, sheds light on a number of issues in the analysis of extensive form games. Type spaces are closely related to the sphere models of counterfactual conditionals and to models of hypothetical knowledge.

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 matching 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. We construct examples showing that our bounds are essentially tight, and that certain natural variants of the bounds fail to hold. (C) 2014 Elsevier Inc. All rights reserved.

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.