Agreeing to disagree in infinite information structures

Several authors have recently studied the game theory aspects of generalized information structures, that is, information structures that are not partitions. Such structures are needed when we wish to impose some restrictions on the concept of knowledge or bound rationality (see [3] and [4]). Some theories that were developed for partitions do not hold for generalized information structures or at least require changes and adjustments (e.g., correlated equilibria in [2] and the 'no-trade' theorem in [5]). Other results continue to hold for some families of generalized information structures. Thus, for example, it has been shown in [4] that the impossibility of 'agreeing to disagree', which was proved in [1] for partitions, also holds for more general information structures. We want to draw attention to some non-trivial differences between finite and infinite structures where generalized information structures are concerned. For partitions, there is almost no interesting distinction between finite structures and infinite ones. Theorems for the finite case can be repeated and proved almost verbatim for the infinite, countable case. We show here that this is no longer true for generalized structures. More specifically, we show that the impossibility of 'agreeing to disagree', which holds invariably for finite and infinite partitions, holds generally for refl.exive-transitive structures only in the finite case. For infinite structures one has to impose some restrictions on the infiniteness of the structure. Thus in [4], in order to prove the 'agreeing to disagree' theorem an assumption is made that agents' knowledge is finitely generated. This means that for each state and each agent there is a finite number of facts from which everything he knows is implied. Here we show, by constructing a counter-example, that with no restriction on the infinite structure the 'agreeing to disagree' theorem fails. Infinite information structures are by no means a mathematical luxury. They are indispensable in the theory of information and knowledge in many cases. For example, in developing models in which the description of a state of the world includes the state of knowledge of various agents (as in [1], [4] and many others) we can hardly avoid using infinite information structures. This is also the case for models of the 'coordinated attack' type in which communication is iterated with no bound on the number of iterations (see [6]). Finally, we should note that models with infinite information structures are commonplace in the literature of economic