From the other side, whenever a member of my team has come to me asking to match an offer I've always refused and wished them well in their new role.
If someone is unhappy enough to start job hunting rather than talking to me then it's a clear sign that they'll leave soon regardless of any pay rise. Psychologically, every pay increase makes someone happy for a short time, and then they get used to it and they start to wonder if they should have asked for more, or they focus on other reasons why they don't like the role they're in and wonder if the other one would have been more fulfilling. If you match an offer, or even beat it, the person is likely to have left in a year anyway.
If someone is unhappy enough to start job hunting rather than talking to me then it's a clear sign that they'll leave soon regardless of any pay rise. Psychologically, every pay increase makes someone happy for a short time, and then they get used to it and they start to wonder if they should have asked for more, or they focus on other reasons why they don't like the role they're in and wonder if the other one would have been more fulfilling. If you match an offer, or even beat it, the person is likely to have left in a year anyway.