  Commercial entries, such as hotels, are ranked according to score by a search
engine or recommendation system, and the score of each can be improved upon by
making a targeted investment, e.g., advertising. We study the problem of how a
principal, who owns or supports a set of entries, can optimally allocate a
budget to maximize their ranking. Representing the set of ranked scores as a
probability distribution over scores, we treat this question as a game between
distributions.
  We show that, in the general case, the best ranking is achieved by equalizing
the scores of several disjoint score ranges. We show that there is a unique
optimal reinforcement strategy, and provide an efficient algorithm implementing
it.
