I absolutely agree with the statement that the most important thing is not winning but taking part. 

Firstly, if all players in a game just want themselves to be the winner, they may cheating or do all the things include bad things to get the "first prize". Players won't executive fair play rules. The play will be boring then because the cheated game will be easily "see through" by experienced cheaters. Unless the players play in a fair way, they won't enjoy the unpredictability of the game.  

Secondly, if all players in a game just want the win, then no one respect each other. This may produce bad emotions to the game's atmosphere. 

Last but not least, if all players in a game just want themselves to get the "golden trophy" but unfortunately can't afford that, they will get bad emotions, shedding tears, etc. This is an absolutely bad image, and the viewers who watch the game may feel bad themselves and can't enjoy their day.

Finally, I think that if people just taking part in a game then enjoy themselves and their own outcomes, they will feel great, happy and get good memories of the game they has participated in.