Classic search engine results are presented as an ordered list of documents and the problem of presentation trivially reduces to ordering documents by their scores. This is because users scan a list presentation from top to bottom. This leads to natural list optimization measures such as the discounted cumulative gain (DCG) and the rank-biased precision (RBP).

Increasingly, search engines are using two-dimensional results presentations; image and shopping search results are long-standing examples. The simplistic heuristic used in practice is to place images by row-major order in the matrix presentation. However, a variety of evidence suggests that users' scan of pages is not in this matrix order. In this paper we (1) view users' scan of a results page as a Markov chain, which yields DCG and RBP as special cases for linear lists; (2) formulate, study, and develop solutions for the problem of inferring the Markov chain from click logs; (3) from these inferred Markov chains, empirically validate folklore phenomena (e.g., the "golden triangle" of user scans in two dimensions); and (4) develop and experimentally compare algorithms for optimizing user utility in matrix presentations. The theory and algorithms extend naturally beyond matrix presentations.