This paper studies how society should be risk-averse accordingly to how individuals are risk-averse. We consider two axioms, one that if everybody prefers one risky prospect over another so should the society, the other that if everybody is more risk-averse so should the society. We show that the two axioms imply dictatorship, in the sense that the social ranking over risky prospects has to be always identical with one individual's risk preference in each equivalence class of risk preferences which yield the same profile of ordinal preferences over deterministic outcomes.

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This paper provides a rigorous and gap-free proof of the index theorem used in the theory of regular economy. In the index theorem that is the subject of this paper, the assumptions for the excess demand function are only several usual assumptions and continuous differentiability around any equilibrium price, and thus it has a form that is applicable to many economies. However, the textbooks on this theme contain only abbreviated proofs, and no known monograph contains a rigorous proof of this theorem. Hence, the purpose of this paper is to make this theorem available to more economists by constructing a readable proof.

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Cournot's economics was deeply rooted in French mathematics in the nineteenth century. The main stream was Laplace-Lagrange's physico-mathematics. But in examining what drove Cournot to economics, we must not dismiss another stream, that is the tradition of social mathematics. It was probability theory which supported social mathematics as a principal tool of analysis. However, the mathematical machinery which Cournot used in economics was not probability theory but mathematical analysis while he succeeded the tradition of social mathematics. The transformation of mathematical methodology of economics due to Cournot was made possible by his peculiar grasp of the law of large numbers.

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In contrast to Mandler's (1999a; Theorem 6.1) impossibility result about the Sraffian indeterminacy of the steady-state equilibrium, we have proposed a simple example of overlapping generation economy in which generic indeterminacy occurs in the Sraffian steady-state equilibrium.

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Poincaré-Miranda's theorem is an n-dimensional zero-point theorem, which was conjectured by Poincaré [6] and proved by Miranda [5]. Also, Miranda [5] proved its equivalence to Brouwer's fixed point theorem. The author [3] presented an n-dimensional intermediate value theorem by using Poincaré-Miranda's theorem. He applied it to two-person and three-person strategic games with two pure strategies, and showed a range of the payoffs that can be achieved by mixed strategies. In this paper, we extend the result of [3] to an n-person strategic game.

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