You asked

What games should I play with my four month old, and why?

By the time your child is three months old, he or she will probably be a lot more aware of the world around him or her. Your child is also learning to use his or her hands, albeit slowly, and there are plenty of development boosting games you can play together.
One of the most important things is repetition. Games you try might not work the first time around, so keep at it. Eventually your baby will get the hang of them. Playing with bubble is great at this age, particularly if your baby is sitting already. Watch out for bubble solution getting in eyes though! Your baby will probably have great fun trying to catch the bubbles you blow.
‘Coming to get you’ is another game small babies enjoy at around this age. Call your baby, and tell him or her that you’re coming to get them, and what you’ll do when you get there. Then creep closer, and kiss or hug your baby as you promised you would! The anticipation usually results in floods of giggles.
Games that simulate flying, where you lift your baby into the air, are another popular game at this age. Last, but not least, there’s tummy time. It’s important to your baby’s neck development, and you can use mirrors or other toys to make spending time on his or her tummy more fun for your baby.

More questions

By the time your baby is three to six months old, they’re ready for more interactive developmental games.
Taking your child to their first movie can be a thrilling experience – if you follow a few simple tips.
Sandpits are fun, as long as your child is supervised, but they’re not without their risks, so exercise a little caution.
While it may seem that young infants can swim, this is actually an involuntary reaction.  Your child actually does not know how to swim.
Babies of nine months to one year old are on the brink of toddler hood. Play games with your toddler to be to encourage development.
Six to nine month olds are on the brink of learning to cruise or walk, and they’re likely to be very mobile, and gaining dexterity. There are plenty of games that encourage those skills.
Play is how your child learns about the world, and it’s one of the most important things he or she will do throughout early childhood.
Reading to your child is possibly the most important thing you can do for his or her linguistic development.
Since music affects your child’s mood, what you play, and when you play it, can have an impact.
There has, as yet, not been any concrete proof made for this question, but the positive effects of music have been proven in other areas.