There are a lot of reins available on the market; leather, rope, cotton, nylon, rubber, BioThane, split, buckled, short and long. Different riders prefer different styles and materials for different horses and different applications.
Mounted games is no exception and US riders tend to use a variety of reins, but there are certainly trends. Some riders prefer nylon rope reins which come in tons of colors and styles and some prefer traditional leather. But one of the most popular types are cotton gaming reins. They are inexpensive, soft and wide and come in a variety of colors. You can pick them up from Chicks Saddlery for $13 among many other places.
The knotted rein gives the advantage that the rein can be adjusted throughout the session from race to race making shorter and longer reins. It also leaves a tail on the reins for a relaxed rein for between races.
Some riders do not like a knot but do like a short rein. This is easy to accomplish with the flat cotton reins by cutting them to the desired length then fixing them back through the snap and securing them with electric tape, by sewing, duct tape, etc.
I have used both the knot and the short rein for mounted games depending on the pony I am riding. Last year I rode Simon in a knotted rein so I could adjust the length throughout the session which was handy when he was still learning. But I am currently moving to a shorter rein and this weekend I worked with two different lengths.
I decided to go with the shorter of the two lengths I tried, which are a little uncomfortable for riding on a loose rein between races and during warm up but are much easier for riding through the actual games.