An Introduction to Simplicial Algorithms on the Simplotope

Abstract

Several ways have been introduced to solve the fixed point problem on the simplotope S. For example one can adapt the set S or one can generalize the algorithms described on Sⁿ. First of all, the set S = $\prod\nolimits_{j\, = \,1}^N {S^{n_j } }$ can be viewed as a subset of the (n+N−1)-dimensional set S(N), with S(N) given by

$$S(N)\,\{ x\, \in \,R_ + ^{n + N} |\,\sum\nolimits_i {x_i \, = \,N\} ,}$$

and $n\, = \,\sum\nolimits_{j\, = \,1}^N {n_j }$. A function f on S can easily be extended to a function on S(N) with only fixed points in S. However, the dimension of S is n while S(N) is N−1 dimensions higher. The algorithms discussed in Chapters 4–6 can be applied directly on S(N). Clearly, the disadvantage of doing so is that the dimension of the problem is increased by N−1. Garcia, Lemke and Lüthi [1973] developed an algorithm on S(N) which is very close to Kuhn’s variable dimension algorithm on the unit simplex. One of the applications of their algorithm was the computation of a Nash equilibrium in a noncooperative N-person game.