Probabilities maximizing products

Question

Given is an expression of the form $P=P_1\times P_2\times\dots\times P_n$, where each $P_i$ is a sum of some distinct elements from $\{x_1,x_2,\dots,x_k\}$. (For example, $P=x_1(x_1+x_2)(x_1+x_3)$.) We want to maximize this expression subject to the constraints $x_i\geq 0$ for all $i$, and $\sum_{i=1}^k x_i=1$. Let $A$ be the value of $P_1$ at the maximum. Let $B$ be the value of $P_1$ at the maximum if we instead maximize the expression $P'=P_2\times P_3\times\dots\times P_n$, subject to the same constraints.

Is it true that $A\geq \frac{n-1}{n}B+\frac{1}{n}$?

Answer 1

For what it's worth, I tried some Lagrange multipliers: Let $X=(x_1,...,x_k)$ and $Y_i\in \{0,1\}^k$ be such that $P_i=Y_i\cdot X$. Putting $$L(X)=\sum_i \ln(Y_i\cdot X)-\lambda(\textbf 1\cdot X-1)$$ gives for each $j$, either $x_j=0$ or $(\sum_i \frac1 {P_i}Y_i)_j=\lambda=n$. A couple observations:

1. $P_i=Y_i\cdot X$, so the above can be rewritten as $(\sum_i X+Z_i)_j=n$ where $Z_i\cdot X=0$.
2. $(\sum_i \frac1 {P_i}Y_i)\cdot X=n$

Hopefully this leads to something useful...