1. Picking up a girl (in some contexts guy, see 1a.) with the intent of using her for sex. Or in other words, using your game to run (on) a girl. It can also mean cheating on someone.
2. In a general context, running game means cheating or otherwise playing someone for a fool.
1a. 50 cent (Ya Don't Love Me) - Tryna run game on me you punk bitch, I know you don't love me.
1b. Jhene (Dog) - Tryin run game on me you're a dog - thinkin you could cheat on me
1c. Ciara (Pick Up the Phone) - run game on a rookie cuz I'm too sharp for that why, don't you pick up the phone?
2. Wu-tang (Shame on a nigga) - Shame on a nigga, who try to run game on a nigga...To the young youth, ya wanna get gun? Shoot!...Punk! You playing me, chump, you get DUMPED
to run game on a shorty is to straight spit the ill flow in the dime pieces ear soo you can get up in that ass
to actively attempt to entice a lady...
or to hit on a girl yo?
mafucka: yo should i run game on that beotchhh
mafucka2: hell nah yo thats a dude you fuckin gay bitch