  Due to the lack of coordination, it is unlikely that the selfish players of a
strategic game reach a socially good state. A possible way to cope with
selfishness is to compute a desired outcome (if it is tractable) and impose it.
However this answer is often inappropriate because compelling an agent can be
costly, unpopular or just hard to implement. Since both situations (no
coordination and full coordination) show opposite advantages and drawbacks, it
is natural to study possible tradeoffs. In this paper we study a strategic game
where the nodes of a simple graph G are independent agents who try to form
pairs: e.g. jobs and applicants, tennis players for a match, etc. In many
instances of the game, a Nash equilibrium significantly deviates from a social
optimum. We analyze a scenario where we fix the strategy of some players; the
other players are free to make their choice. The goal is to compel a minimum
number of players and guarantee that any possible equilibrium of the modified
game is a social optimum, i.e. created pairs must form a maximum matching of G.
We mainly show that this intriguing problem is NP-hard and propose an
approximation algorithm with a constant ratio.
