Manipulation with(out) Money in Matching Markets

In a stable matching market, a central agency might try to steer the outcome by reordering or swapping entries in preference lists, removing agents or acceptable pairs, or paying passive agents to join. I present work by Sushmita Gupta and Pallavi Jain on which goals such an agency can reach within a budget: getting a designated agent matched, forcing a pair into or out of the stable matchings, or making a given matching the unique stable one. Most of these problems are NP-hard, one stays W[1]-hard even for small budgets, a few are polynomial-time, and questions remain open for preference lists of length three.