An extremely simple proof of the K-K-M-S Theorem

Summary.  An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer’s fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show -balanced versions of the K-K-M-S Theorem, with -balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to by Billera can be even weakened. Received: January 22, 1996; revised version June 9, 1996     

A simple proof of Nataf's theorem on consistent aggregation

In this paper we present a simple and short proof of Nataf's theorem on consistent aggregation for the case that all functions involved have non-zero first derivatives.     

A simple proof of Blackwell's “comparison of experiments” theorem

This note presents a simple proof of Blackwell's well-known theorem on the equivalence of two orderings of experiments, one based on the statistical concept of sufficiency, and the other on the economic concept of the value to the decision makers.     

A short proof of Harsanyi''s purification theorem

A short proof of more general version of Harsanyi's purification theorem is provided through an application of a powerful, yet intuitive, result from algebraic topology.     

On Kakutani's fixed point theorem,the K-K-M-S theorem and the core of a balanced game

Summary We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S thoerem, based on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.The results presented here were first reported in Shapley (1987) and Vohra (1987). A version of our proof of Theorem 1 has also been presented in a recent book by C.D. Aliprantis, D.J. Brown and O. Burkinshaw,Existence and Optimality of Competitive Equilibria (1989) Springer-Verlag. We are grateful to Ky Fan, Wanda Gorgol, Tatsuro Ichiishi and Ali Khan for comments on earlier drafts. Vohra's research has been supported in part by NSF grant SES-8605630.     

A geometric proof of Gibbard's random dictatorship theorem

Summary Gibbard has shown that a social choice function is strategy-proof if and only if it is a convex combination of dictatorships and pair-wise social choice functions. I use geometric techniques to prove the corollary that every strategy-proof and sovereign social choice function is a random dictatorship.I thank Kim Border for several useful discussions and many insightful comments.     

A constructive proof of the ordinal efficiency welfare theorem

We provide a non-geometric constructive short proof of the ordinal efficiency welfare theorem.     

A one-shot proof of Arrow’s impossibility theorem

We offer a new proof of the well-known Arrow??s impossibility theorem. The proof is simple, very short and it follows from the assumptions in a transparent way.     

A simple proof of the latent root sensitivity formula

This note provides a simple proof of the formula for the partial derivatives of the latent roots of a matrix with respect to its coefficients.     

A simple proof of the necessity of the transversality condition

Summary. This note provides a simple proof of the necessity of the transversality condition for the differentiable reduced-form model. The proof uses only an elementary perturbation argument without relying on dynamic programming. The proof makes it clear that, contrary to common belief, the necessity of the transversality condition can be shown in a straightforward way. Received: January 22, 2001; revised version: April 2, 2001     

Pivotal voters: A new proof of arrow's theorem

This paper presents a new proof of Arrow's ‘General Possibility Theorem’, focusing on the way how ‘social’ preferences change in response to changes in the preferences of individuals, under given social welfare functions.     

A simple proof of the Sato proposition on non-homothetic CES functions

Summary. Based on some elementary results on duality, the paper proposes a much simpler way of deriving the class of non-homothetic CES production functions which was derived as a solution to a partial differential equation that defines the elasticity of substitution. Received: February 11, 1998; revised version: April 28, 1998     

A simple proof for the Kleinberg-Weiss representation of the shapley value

We obtain a simple direct proof of the Kleinberg-Weiss representation of the Shapley value.     

A simple axiomatization and constructive representation proof for choquet expected utility

>P>Summary. We provide a set of simple and intuitive set of axioms that allow for a direct and constructive proof of the Choquet Expected Utility representation for decision making under uncertainty. Received: October 29, 2002; revised version: November 13, 2002 RID="*" ID="*" We thank Matthew Ryan for very useful comments and suggestions on related work and for encouraging us to write this note. Correspondence to: S. Grant     

An alternative proof for the linear utility representation theorem

Summary The paper presents an alternative short proof for the linear utility representation theorem. In particular a generalization of the theorem of Blackwell and Girshick (1954) and a special case of the theorem of Herstein and Milnor (1953) are proved by exploiting the topological group structure of finite-dimensional Euclidean vector space.I thank two careful referees for their helpful remarks.     

A simple proof of the LeChatelier-Samuelson principle and the theory of cost and production

We provide a simple proof of a version of the so-called strong LeChatelier-Samuelson principle particularly suitable for the application to the theory of cost and production. We will illustrate its usefulness and fundamental nature in the case of the classical theory of cost and production.     

A simple proof of the monotonicity of the state trajectories in autonomous control problems

This paper deals with the question whether in optimal control problems with one state variable cyclical solutions and other nonmonotonicities can be optimal. It is shown that this can only occur if exogeneous shocks are brought into the model while for autonomous problems the state trajectory must always be monotonic. The proof also applies for nondifferentiable control problems as it does not use any regularity assumptions for the model functions.