Why can we write any lottery as a convex combination of the degenerate lotteries?

Question

I know that a degenerate lottery is a lottery that yields outcome $n$ with probability $1$ and I also know the definition of convex combination: given $x_{1},x_{2}, \cdots ,x_{n} \in \mathbb{R}$, a convex combination of these points is a point of the form $\alpha_{1}x_{1}+...+\alpha_{n}x_{n}$ such that $\alpha_1+...+\alpha_n=1$.

But I am quiet confused about why we can write any $L=(p_1,...,p_n)$ as a convex combination of the degenerate lotteries $(L^1,...,L^n)$. Can someone please explain this for me? Many thanks!

Answer 1

Pick any set of non-negative $p_1,\dots,p_n$ such that $p_1+\cdots+p_n=1$. The convex combination of degenerate lotteries $L^1,\dots,L^n$ with the $p_i$'s can be written as \begin{align} p_1L^1+\cdots+p_nL^n&=p_1(1,0,\dots,0)+\cdots+p_n(0,\dots,0,1)\\ &=\bigl(p_1(1)+p_2(0)+\cdots+p_n(0),\;\dots,\;p_1(0)+\cdots+p_{n-1}(0)+p_n(1)\bigr)\\ &=(p_1,\dots,p_n). \end{align} This is just a lottery $L=(p_1,\dots,p_n)$.