Matching mechanisms rely on agents submitting complete preference relations however, in various matching frameworks (e.g. National Residency Matching Program, New York High School Matching) the length of reported relations is either exogenously constrained, or agents myopically shorten the list. To what extent the shortening of preferences hurts agents? Is it possible to improve social welfare using the complete preference profiles instead of shortened versions? To answer these questions it is necessary to have a legitimate procedure to complete agents’ preferences. The paper introduces a procedure that allows us to extend reported incomplete preferences under the assumption that preferences of agents are strongly correlated. The procedure can be applied to any assignment problem; however, we focus our attention on many-to-one matching problem (school choice/college admission) as one with the largest number of applications. The procedure does not require any additional information beyond the reported preferences. We use computational experiments to estimate (i) the welfare loss due to incompleteness of preferences and (ii) the potential efficiency gains from the completion mechanism. We investigate how welfare losses and efficiency gains depend on size (thickness) of the market, length of reported relations and the amount of information that agents possess on their potential on the matching market. Information defines the accuracy of agents’ perception of their own relative rank and narrows the agents’ shortened profile around his potential match. We provide an estimate of welfare losses (and potential gains from the completion) for a given length of reported preference relation in a market of a given size. Thus, we can parametrically characterize (in terms of sizes of markets and lengths of reported preference profiles) the class of problems, which can be significantly affected by the shortening of preference profiles. Moreover, we show for which of them the problem can be resolved by completion procedure we propose. Finally, information on relative rankings of agents is often available to the designer and can be disseminated to agents at zero cost. The paper also provides the estimate on what share of inefficiency can be eliminated by spreading this information as the first step to improve the possible outcomes.