  Learning the optimal ordering of content is an important challenge in website
design. The learning to rank (LTR) framework models this problem as a
sequential problem of selecting lists of content and observing where users
decide to click. Most previous work on LTR assumes that the user considers each
item in the list in isolation, and makes binary choices to click or not on
each. We introduce a multinomial logit (MNL) choice model to the LTR framework,
which captures the behaviour of users who consider the ordered list of items as
a whole and make a single choice among all the items and a no-click option.
Under the MNL model, the user favours items which are either inherently more
attractive, or placed in a preferable position within the list. We propose
upper confidence bound (UCB) algorithms to minimise regret in two settings -
where the position dependent parameters are known, and unknown. We present
theoretical analysis leading to an $\Omega(\sqrt{JT})$ lower bound for the
problem, an $\tilde{O}(\sqrt{JT})$ upper bound on regret of the UCB algorithm
in the known-parameter setting, and an $\tilde{O}(K^2\sqrt{JT})$ upper bound on
regret, the first, in the more challenging unknown-position-parameter setting.
Our analyses are based on tight new concentration results for Geometric random
variables, and novel functional inequalities for maximum likelihood estimators
computed on discrete data.
