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     

Collusion and group lending with adverse selection

In an environment with correlated returns, this paper characterizes optimal lending contracts when the bank faces adverse selection and borrowers have limited liability. Group lending contracts are shown to be dominated by revelation mechanisms which do not use the ex post observability of the partners' performances. However, when collusion between borrowers under complete information is allowed, group lending contracts are optimal in the class of simple revelation mechanisms (which elicit only the borrower's own private information) and remain useful with extended revelation mechanisms.     

Homotopy methods to compute equilibria in game theory

This paper presents a survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present the relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke–Howson algorithm and the van den Elzen–Talman algorithm to compute Nash equilibria in 2-person games, and the Herings–van den Elzen, Herings–Peeters, and McKelvey–Palfrey algorithms to compute Nash equilibria in general n-person games. We explain how the main ideas can be extended to compute equilibria in extensive form and dynamic games, and how homotopies can be used to compute all Nash equilibria.