Bargaining Set Solution Concepts in Dynamic Cooperative Games

This paper is concerned with the question of defining the bargaining set, a cooperative game solution, when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face (finite or infinite) sequences of exogenously specified TU-games and receive sequences of imputations against those static cooperative games in each time period. Two alternative definitions of what a ‘sequence of coalitions’ means in such a context are considered, in respect to which the concept of a dynamic game bargaining set may be defined, and existence and non-existence results are studied. A solution concept we term ‘subgame-stable bargaining set sequences’ is also defined, and sufficient conditions are given for the non-emptiness of subgame-stable solutions in the case of a finite number of time periods.Cooperative game; Repeated game; Bargaining set

Common Knowledge and Disparate Priors: When it is O.K. to Agree to Disagree

Abandoning the oft-presumed common prior assumption, partitioned type spaces with disparate priors are studied. It is shown that in the two-player case, a unique fundamental pair of priors can be identified in each type space, from whose properties boundaries on the possible ranges of expected values under common knowledge can be derived. In the limit as the elements of this pair approach each other,a common prior is identified, and standard results stemming from the common prior assumption are recapitulated. It is further shown that this two-player fundamental pair of priors is a special case of the n-player situation, where a representative n-tuple of fundamentally associated priors can be selected, out of at most n-1 such n-tuples, to play an analogous role.common knowledge; heterogeneous prior beliefs; common prior assumption

Bayesian games with a continuum of states

We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowl- edge relation is smooth. Conversely, for any common knowledge rela- tion that is not smooth, there exists a type space that yields this common knowledge relation and payoffs such that the resulting Bayesian game will not have any Bayesian equilibrium. We show that our smoothness condition also rules out two paradoxes involving Bayesian games with a continuum of types: the impossibility of having a common prior on components when a common prior over the entire state space exists, and the possibility of interim betting/trade even when no such trade can be supported ex ante

Measurable selection for purely atomic games

A general selection theorem is presented constructing a measurable mapping from a state space to a parameter space under the assumption that the state space can be decomposed as a collection of countable equivalence classes under a smooth equivalence relation. It is then shown how this selection theorem can be used as a general purpose tool for proving the existence of measurable equilibria in broad classes of several branches of games when an appropriate smoothness condition holds, including Bayesian games with atomic knowledge spaces, stochastic games with countable orbits, and graphical games of countable degree—examples of a subclass of games with uncountable state spaces that we term purely atomic games. Applications to repeated games with symmetric incomplete information and acceptable bets are also presented

Bargaining Set Solution Concepts in Dynamic Cooperative Games

This paper is concerned with the question of defining the bargaining set, a cooperative game solution, when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face (finite or infinite) sequences of exogenously specified TU-games and receive sequences of imputations against those static cooperative games in each time period. Two alternative definitions of what a ‘sequence of coalitions’ means in such a context are considered, in respect to which the concept of a dynamic game bargaining set may be defined, and existence and non-existence results are studied. A solution concept we term ‘subgame-stable bargaining set sequences’ is also defined, and sufficient conditions are given for the non-emptiness of subgame-stable solutions in the case of a finite number of time periods

Iterated Expectations, Compact Spaces and Common Priors

Extending to infinite state spaces that are compact metric spaces a result previously attained by Dov Samet solely in the context of finite state spaces, a necessary and sufficient condition for the existence of a common prior for several players is given in terms of the players’ present beliefs only. A common prior exists if and only if for each random variable it is common knowledge that all its iterated expectations with respect to any permutation converge to the same value; this value is its expectation with respect to the common prior. It is further shown that the restriction to compact metric spaces is ‘natural’ when semantic type spaces are derived from syntactic models, and that compactness is a necessary condition. Many proofs are based on results from the theory of Markov chains

Common Knowledge and Disparate Priors: When it is O.K. to Agree to Disagree

Abandoning the oft-presumed common prior assumption, partitioned type spaces with disparate priors are studied. It is shown that in the two-player case, a unique fundamental pair of priors can be identified in each type space, from whose properties boundaries on the possible ranges of expected values under common knowledge can be derived. In the limit as the elements of this pair approach each other,a common prior is identified, and standard results stemming from the common prior assumption are recapitulated. It is further shown that this two-player fundamental pair of priors is a special case of the n-player situation, where a representative n-tuple of fundamentally associated priors can be selected, out of at most n-1 such n-tuples, to play an analogous role

Bargaining Set Solution Concepts in Dynamic Cooperative Games

This paper is concerned with the question of defining the bargaining set, a cooperative game solution, when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face (finite or infinite) sequences of exogenously specified TU-games and receive sequences of imputations against those static cooperative games in each time period. Two alternative definitions of what a ‘sequence of coalitions’ means in such a context are considered, in respect to which the concept of a dynamic game bargaining set may be defined, and existence and non-existence results are studied. A solution concept we term ‘subgame-stable bargaining set sequences’ is also defined, and sufficient conditions are given for the non-emptiness of subgame-stable solutions in the case of a finite number of time periods

Common Knowledge and Disparate Priors: When it is O.K. to Agree to Disagree

Abandoning the oft-presumed common prior assumption, partitioned type spaces with disparate priors are studied. It is shown that in the two-player case, a unique fundamental pair of priors can be identified in each type space, from whose properties boundaries on the possible ranges of expected values under common knowledge can be derived. In the limit as the elements of this pair approach each other,a common prior is identified, and standard results stemming from the common prior assumption are recapitulated. It is further shown that this two-player fundamental pair of priors is a special case of the n-player situation, where a representative n-tuple of fundamentally associated priors can be selected, out of at most n-1 such n-tuples, to play an analogous role