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door-in-the-face technique

n. (context psychology English) A compliance tactic that involves getting a person to agree to a lesser request by first having them reject a larger one.

Wikipedia
Door-in-the-face technique

The door-in-the-face (DITF) technique is a compliance method commonly studied in social psychology. The persuader attempts to convince the respondent to comply by making a large request that the respondent will most likely turn down, much like a metaphorical slamming of a door in the persuader's face. The respondent is then more likely to agree to a second, more reasonable request, compared to the same reasonable request made in isolation. The DITF technique can be contrasted with the foot-in-the-door (FITD) technique, in which a persuader begins with a small request and gradually increases the demands of each request. While the FITD technique differs from DITF, it is also a persuasion technique that increases the likelihood a respondent will agree to the second request.