No trade theorem in an S-4 logic model (Algebraic Semigroups, Formal Languages and Computation)
Kazuki Hirase · Kyoto University Research Information Repository (Kyoto University) · 2001
This paper introduces an application of the S-4 logic.There are two aims in this paper.Aim 1is to check the relation between our model and the S-4 logic.We 11 see the soundness and completeness of the S-4 logic with respect to the model by using the concept of structure.Aim 2is to prove avariation of no trade theorem in the model.1IntroductionThe word "knowledge" and especialy "common knowledge" plays avery important role in game theory.Intuitively, an event is common knowledge if everyone knows it, everyone knows that everyone knows it, everyone knows that everyone knows that everyone knows it and so on.Then how can we treat (common) knowledge fomffiy?Aumann (1976) tried to solve this problem.He introduced the formal notion of common knowledge using set based and partitioned information structure and showed, so caUd, agreeing to disagree $\mathrm{t}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}^{1}$ After Aumann, many papers have studied knowledge.Milgrom (1981), and Monderer and Samet (1989) treated knowledge by different approaches.Milgrom (1981) applied axiomatic approach 2 to modelng knowledge.Monderer and Samet (1989) used probabilty appr0ach3.They managed to approximate knowledge with belief.We note that these approaches also use partiotional information structure.Samet (1990) have studied non-partitional information structure.He showed agreeing to disagree theorem based on non-partitional information structure.This paper also studies non-partitional information structure like Samet.We would h.ke to prove kind of no trade theorem which is introduced by Milgrom and Stokey(1982) *Mita2-15-45, Mnatoh, $\mathrm{T}\mathrm{o}\infty$ , $108^{\mathrm{q}}-\vee \mathfrak{N}5$ , Japan $\uparrow \mathrm{B}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{l}$ $\mathrm{d}\mathrm{d}\mathrm{r}\infty:\mathrm{h}\mathrm{h}\subset \mathrm{M}$) $\mathrm{p}\mathrm{c}$ highmy ne jp lIhk $\mathrm{t}\mathrm{h}\infty \mathrm{m}$ insists that if players' posteriors for agiven event are common knowledge, then these must be equal, even though they are based on different information.$2\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$ is the approach which defines the set of all states in which aplayer knows agiven event.After M 釉。 $\mathrm{m}'\epsilon$ paper mny papers have boen 一一吋 this approach.'Prob 市 ih.ty approach defines the event in which player $n$ believes $E$ with prolx 山 ilty at least $p$ .