ON A CLASS OF GAMES
Abstract
This paper describes qualitatively the nature of optimal strategies for a payoff kernel K(x,y) defined on the unit square satisfying K sub y...y (x, y) > or = 0 with n partial derivatives taken with respect to y. A complete analysis for n < or = 4 is presented. However, the method employed easily extends and enables, by enumerating cases, the situation for general n to be solved. Specifically, it is shown that for n = 3 and n = 4 the maximizing player has optimal strategies involving respectively at most 3 points and at most 4 points of increase. For general n, it can be shown that the maximizing player has optimal strategies using at most n points. For the minimizing player the statement of the nature of an optimal strategy is more precise. There always exist for the general case optimal solutions using at most n/2 points, with the understanding that the end points 0 or 1 when used are each counted only half. For example when n is odd, then n/2 is a half integer and hence must use a single end point if a full optimal strategy exists employing n/2 points. This counting procedure applies only to the minimizing strategies.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 16, 1951
- Accession Number
- AD0603995
Entities
People
- Samuel Karlin
Organizations
- RAND Corporation