At three, we started off with cooperative board games, which means we won or lost collectively. I can't tell you how glad I am to start with 'low emotional risk' activity.:)
Now Kiddo's five and still, we let him choose what he wants to play. Sometimes, it is the cooperative games, and sometimes not. I would say that on days which are feeling 'hard' for him, he usually self-governs and chooses the cooperative game. (This is very appropriate for five; one of the developmental traits of this age is not dealing well with competition, esp. losing, so I do keep this in mind.) Some days, I might not play as strategically as I usually do; sometimes its unavoidable (bingo and other games of chance) that I win. My husband does let our son win-- but usually it's a lightsaber 'duel' and not a game. I also modify some competitive games which are more difficult and make them cooperative so he can learn some of the strategies/thinking about a game before we make it competitive.
If you are looking for some great cooperative games which employ strategy and are more than just counting spaces, check out the Family Pastimes website. I've used these games with my son and preschoolers-- they are rather beloved, esp. Max and Granny's House.
http://www.familypastimes.ca/
For what it's worth, I am not against winning while playing with my son, however, I'm not in a hurry for him to learn competition, which is why the noncompetitive games are available. He's going to have plenty of it in life, has experienced it... I want to respect that learning losing/winning behaviors IS hard for kids this age, and respect his own personal development in this. As he chooses more competitive games, I know he's feeling more ready to deal with the emotions that come with losing.:)