01453146

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01453146 Heifetz, A. Eliminating redundancies in partition spaces. Bacharach, M. O. L. (ed.) et al., Epistemic logic and the theory of games and decisions. Based on the conference, Marseille, France, January 1994. Dordrecht: Kluwer Academic Publishers. Theory Decis. Libr., Ser. C. 20, 95-103 (1997). 1997 Dordrecht: Kluwer Academic Publishers EN models of uncertainty partition spaces uncertainty of agents redundancy non-well-founded sets Zbl 0379.62003 From the introduction: Redundancy in models of uncertainty is a non-trivial concept. We address this issue in the framework of partition spaces, introduced to game theory by {\it R. J. Aumann} [Ann. Stat. 4, 1236-1239 (1976; Zbl 0379.62003)]. In these models the uncertainty of agents is about a space of points, which are called states of the world. Each such state stands for a combination of physical, environmental parameters -- the state of nature, and possible mutual uncertainties of the agents over these parameters. These uncertainties are expressed by a partition each agent has over the states of the world: every partition member contains all those states the agent conceives as possible when one of them prevails. When is such a space redundant, with different states which stand for essentially the same natural parameters and uncertainties? At first sight it might seem that if any two states are separated either by nature or by the partition of some agent then the space is non-redundant. However, the union of two copies of such a space, which is clearly redundant, still has this property. It turns out that a more subtle definition is needed: one has to unfold step by step the mutual beliefs of the agents in each point, and check whether any two points differ in some step. This process may be transfinite. For instance, two points may have the same state of nature, the same beliefs of each agent about nature, the same beliefs of each agent about the other agents' belief's about nature, and so on; but still it may be the case that an agent considers as possible in one point an infinite hierarchy of beliefs of another agent which he excludes in the second point. In what follows, we suggest a new tool to attack the issue of redundancy -- non-well-founded sets. These constitute an enlargement of the classical universe of sets, by allowing sets to contain themselves as members, to contain members that contain them, etc. The basic idea of the approach we suggest is simple: instead of describing the agents' partitions after the space of states of the world is established, each state of the world is initially required to be composed of its nature state and the relevant members of the agents' partitions as coordinates. Each state is thus non-well-founded -- it has as coordinates subsets of states. The main theme of the current work is that the non-well-founded version of the model has another merit: all the states which are epistemically equivalent in the classical form of the model ``collapse'' at once to one point of its non-well-founded version. This provides an equivalent characterization of non-redundancy which does not appeal to any infinite process.