This dissertation has two chapters. Chapter 1 analyzes the convex hull of the parametric curve $\left(t, t.

2, \cdots, t.

N\right)$, where $t$ is in the closed interval $\left[t_{min}, t_{max}\right]$. It finds that every point in the convex hull is representable as a convex combination of $\frac{N+1}{2}$ points on the curve if $N$ is odd, or of $\frac{N+2}{2}$ points on the curve if $N$ is even. Furthermore, if $N$ is even, then one of these $\frac{N+2}{2}$ points can be required to be the point whose $t$ is $t_{min}$. This is a representation without any redundant information, because it gives an $N$-parameter description of every point in an $N$-dimensional object. It is also found that the evaluation of the convex combination is a homeomorphism from the convex combinations of $\frac{N+1}{2}$ (or $\frac{N+2}{2}$ with one of them at $t = t_{min}$ if $N$ is even) points on the curve to the convex hull of the curve, as long as the points are listed in increasing order, and as long as two representations that are reachable from each other by removing terms with coefficient zero, combining terms with the same point, the inverses of these operations, or a sequence of these operations in any order, are considered to be equivalent.

Chapter 2 analyzes the structure of mixed-strategy equilibria for Colonel Blotto games, where the outcome on each battlefield is a polynomial function of the difference between the two players' allocations. This chapter severely reduces the set of strategies that needs to be searched to find a Nash equilibrium. It finds that there exists a Nash equilibrium where both players' mixed strategies are discrete distributions, and it places an upper bound on the number of points in the supports of these discrete distributions. An extension uses the theorem from the first chapter to reduce the number of parameters that describe the space in which a Nash equilibrium strategy is to be found.