$\pi$ consist of the elements $\pi_j$ that are larger in value than all elements $\pi_i$ lying to the left. The [**inversions**]{} of $\pi$ consist of pairs $\pi_i > \pi_j$ where $i <
j$.

Now, let $c: {\mathfrak{S}}_N \rightarrow {\mathbb{N}}$ be a statistic on the symmetric group of permutations of size $N$. Then we can weight the permutation $\pi \in {\mathfrak{S}}_N$ by ${\theta}^{c(\pi)}$ where ${\theta}$ is a positive real number to obtain a discrete probability distribution on ${\mathfrak{S}}_N$. When ${\theta}= 1$, we recover the uniform model. Distributions of this form were introduced by Mallows where $c(\pi)$ represented some measure of distance from a fixed permutation, typically the identity. More recently, these distributions have been used by researchers in combinatorics; see [@elizalde; @bouvel; @log-book] for example. In our first result, Theorem \[t:main\], we show that for a large class of statistics the optimal strategy in the weighted game of best choice has the same form as that for the classical model: to reject an initial set of candidates and accept the next left-to-right maximum thereafter. This seems mildly surprising given that the full history of relative rankings are available at each step to guide a strategy; evidently, however, the positions of the 