A running martingale stops the horse from evading by throwing its head up. The rider has more leverage because the reins pass through two rings and help the rider to pull the head down.
It is a much debated device - the purists say they are not necessary if the horse has been trained properly to ride "on the bit".
They are not just used in show jumping - they can help a rider who is going out with a crowd where the horse might get excited. Rather than use a harsher bit then stay with the normal bit and add the standing martingale to obtain greater control.
There is a related but fiercer device - the fixed martingale which is not attached with the reins through the rings but runs directly up from the chest to the nose band. Not nice really - used by weaker riders on a sparky or badly schooled horse. It restricts the bend in the horse's neck
And can pull the head down into a false ramener position. If used it should be adjusted very carefully.
If your horse doesn't throw its head up -
If you have adequate strength in your arms to hold the horse back when it gets excited
Or if you use as standard procedure a harsh bit - then think twice and obtain advice before you buy and fit one.
Professional showjumpers select horses which can jump naturally - regardless of any other abilities they are concerned with the horse's agility and jumping ability. They keep the horse fit and they feed it up with energy foods. The horse then performs for a few minutes over 3 - 4 rounds. The professional show jumping horse can be quite a handful to control. So many riders look for any device which gives them better control.
A personal opinion - I won't use one - the rings interfere with my feel of the bit in the horses mouth. I prefer straight contact through to the mouth. Anyway my mare's mouth is very soft and in case of emergency I have strong arms.
I use one on my horse (running martingale) - If I don't her head goes everywhere... and there's nothing I can do about it with my weak little arms.
I don't see what the big deal is with it if it doesn't hurt the horse. Saddles make it easier to ride as do stirrups and bits... soo.. why can't we use martingales to make it easier to ride as well?
I use martingales on a couple of our horses.
Galen has one. Now, Galen, is a complete angel. . But she gets veeeeery strong sometimes. Her owner has one, but it's usually not used. But if we're going in the field or on trails, we're just like, "hey, why not." and we put it on.
Rupert has one, but I don't think it helps & it really doesn't fit. I have ridden him with & without it and it does not make a difference in any way.