Type space on a purely measurable parameter space

Mikl�s Pint�r
<span title="">2005</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ctdypx5b5vc7dgs6m6avzta2e4" style="color: black;">Economic Theory</a> </i> &nbsp;
Several game theoretical topics require the analysis of hierarchical beliefs, particularly in incomplete information situations. For the problem of incomplete information, Harsányi suggested the concept of the type space. Later Mertens & Zamir gave a construction of such a type space under topological assumptions imposed on the parameter space. The topological assumptions were weakened by Heifetz, and by Brandenburger & Dekel. In this paper we show that at very natural assumptions upon the
more &raquo; ... ture of the beliefs, the universal type space does exist. We construct a universal type space, which employs purely a measurable parameter space structure. * The author wishes to thank Péter Tallos, and Tamás Solymosi for their suggestions and comments. This work was supported by OTKA grant T046194. 1 assumed a probability measure, defined on the product of the parameter space and the type spaces. This probability measure induces hierarchies of beliefs, so we can consider this probability measure as a "summary of hierarchies of beliefs". However, the opposite question remains: how can we build a type space from hierarchies of beliefs? A very important step in this direction was made by Mertens & Zamir[10] who built a universal type space based on a compact parameter space. Later, Heifetz[4] relaxed the compactness, but other topological assumptions were retained. Almost parallel Brandenburger & Dekel[2] proved the existence of a universal type space in presence of a complete, separable metric (Polish) parameter space. More recently, Mertens & Sorin & Zamir[9] gave an elegant proof for the existence of a universal type space in cases of parameter spaces with various structures. Ultimately, all of the above proofs are based on the Kolmogorov's Existence Theorem and its generalizations. In 1998 Heifetz & Samet[5] proved the existence of a universal type space, which possesses a purely measurable structure. In contrast to our paper, the authors make a distinction between universal type space, and space of coherent hierarchies of beliefs. They also gave an illuminating discussion on the problem of type spaces, beliefs spaces. The same authors gave a counterexample showing that in general circumstances, coherent beliefs are not always types (see Heifetz & Samet[6]). Quite recently, Meier[8] investigated the problem of the existence of a universal type spaces, his model is based on finitely additive measures. By regarding the opinions as finitely additive measures, the problem of existence of σ-additive measures on type spaces can be eliminated. On the other hand, the author discusses how "rich" the structure of a universal type space can be. This work brings to the surface that, the problem of existence of σ-additive measures on type spaces is not only the problem of σ-additivity. Mertens & Zamir[10], Heifetz[4], Brandenburger & Dekel[2], and Mertens & Sorin & Zamir[9] use the concept of projective limit for proving the existence of a universal type space. In all four papers the structure of beliefs is inherited from the topology of lower ranked beliefs spaces or the parameter space, moreover beliefs are modeled by compact regular probability measures. Our main goal is to build a universal type space, that is apparently "purely measurable", and in which every coherent hierarchy of beliefs is a type. The structure on the beliefs is naturally generated by the Baire sets of the pointwise convergence topology. For metric spaces Baire sets and Borel sets coincide. However, in non-metrizable cases (for instance when the cardinality of the players is greater than countable), our approach results in a weaker then Borel structure, but this structure allows the players to distinguish between any pair of beliefs (i.e. regular probability measures) yet. An other new idea in this paper is that we cut the parameter space off the beliefs space. This truncated space has a sufficiently good topological structure (i.e. a projective system of completely regular topological spaces), so the measure projective limit exists. After this, we re-fit the parameter space to the measure projective limit, and we construct the universal type space. It is clear that the existence of a measure projective limit crucially depends on topological assumptions. However, if we remove finitely many elements of the projective system of measure spaces, it does not influence the existence of the measure projective limit.
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