There is a game! I played it some time ago. I just can't remember the name... There I would go through the city as a father with my daughter, looking for my wife maybe? It's like I was banged in the head and have amnesia right now. I'll try to remember the game

Edit: I Am Alive! That's how it is called. I remember that I may have a gun, yet without bullets. And I could bluff on some bandits that I shall shoot 'em. Or I could shoot in the air, or someone in the leg. And sometimes they would beg for mercy.

FO3 (may it come here) enemies can run in fear.

Team Fortress 2.

When the enemy team loses, they lose their weapons and try to flee. Some of them start to dance, thinking that way the winning team members will not kill them, as in "Hey look I am a funny guy, don't kill me mmmmkay?". I'll kill them anyway, after all they were shooting at me just seconds before, and now I am supposed to forgive that?!?

As for single-player games, in Far Cry 2 wounded enemies at least moan and crawl, not sure if they beg for mercy. Probably not as they sometimes still try to take some potshots at you.

In Metro Last Light Redux there are occasions in that someone surrenders but they seem scripted.

I believe in the Jedi Knight series you can force pull weapons from the stormtroopers, and then they will surrender... until you turn around, they rush to their weapons, and keep shooting at you.

There's something like that in Bloodrayne .

The arcade game N.A.R.C. ('Busted')

Pretty sure this happens in the original Red Faction.

No, I'm pretty sure you're thinking of I Am Alive, not The Last of Us...

If you're violent, npcs may beg for their lives in Postal 2. They'll also cower and try to run.

I stand corrected. Yes, it's I Am Alive. Thanks

In SWAT3 the bad guys can be made to drop their weapons and surrender if you overwhelm them sufficiently.

I do. The hostages! :P