A four player card game

Inspired by the paper by Euler, I've come up with a four-player card game. I hereby dub this game "Euler's Coincidence", in honor of the guy who analyzed it. North and East are partners on a team, and likewise South and West (so not like bridge). North gets the clubs, South the spades, East the Hearts, West the Diamonds. On each turn each player places a card on the table and they all simultaneously flip. The game is over if cards of like color match in number. If black cards match, North/East wins 1 point. If red cards match, South/West wins 1 point. On a match, everybody gets their 13 cards back and the game begins anew. Otherwise, the four cards are discarded and play continues. (Note that if (for example) a black 6 and a red 6 appear, then it's not a match.) So therefore, North and West are seeking a match with the player across from them, and South and East are trying to avoid matching. Of course, the optimal play is to play your cards in random order. The game might need to be tweaked still to make people avoid that. Stay tuned for future tweaks.