Lower bounds on the probability that one random variable is greater than a set of others

Question

Let $X_1, \cdots, X_n$ be $n$ random variables (not necessarily independent) such that $E[X_i] > E[X_j]$ whenever $i < j$. I am interested in obtaining lower bounds on the following probability: $P[ (X_1 > X_2) \wedge (X_1 > X_3) \wedge \cdots \wedge (X_1 > X_n) ]$.

Answer 1

The lower bound is $0$.

For example let $X_i = -(i^{k-1})$ with probability $\frac{i^2-1}{i^k-1}$ for some $k>2$, and $X_i=-\frac{1}{i}$ otherwise. Then $E[X_i]=-i$, satisfying the condition.

If $i <j$ then both possible values of $X_i$ are between the two possible values of $X_j$. In particular, $X_1 = -1$ with probability 1. Meanwhile $\frac{i^2-1}{i^k-1}$ is a decreasing function of $k$, so as $k$ increases the more negative value of $X_i$ becomes less likely.

So $\Pr(X_1 > X_2)$ and the longer expression can be made arbitrarily small by increasing $k$.