A symmetric distribution of probabilites among N objectives

Question

Suppose there are $n$ objectives each have a given distinct feature measured by $x_i$ which is a positive real number. Now I intend to see whether there is a way to distribute the probability mass of $n-1$ (summation of probs equals $n-1$) among these objectives such that the relative difference in probabilities equal the relative difference in expectation of x's (their probability times their x)? Precisely,I mean $(p_i-p_j)/(p_j-p_k)=(p_ix_i-p_jx_j)/(p_jx_j-p_kx_k)$ for all $i, j, k$. Are there conditions such a probability distribution is unique?