Connect Four for Three

This is an odd line of design thinking: how can you turn a two player game into a three player game?

Say, hypothetically, you had two cousins playing a game like Connect Four and a third one becomes interested? Could you find a variation for them to play?

Rules

1. Players are NOT the colors. They play for themselves, not as the black tokens or as the red tokens.
2. The goal is to be the player to place the fourth token in a row going vertical, horizontal, or diagonal. That player is the winner.
3. On the first turn, the first player places a red token which slides down as normal (unless you’re not subject to Earth’s gravity or in space in which case play a different game).
4. The next player clockwise places a black token.
5. The next player clockwises places a red token.
6. Continue steps 4 and 5, alternating players and token colors until a player reaches the goal of step 2.

There you go, a two player game into a three player one.

Any other games that can have a treatment like this? Chess? Tic-Tac-Toe? Mancala?

As with other three player games, it’s best if the design doesn’t encourage or support kingmaking… Many years of playing with my dad and brother has taught me that ain’t a good move.