There were many other ideas being suggested in regards the horses. All of them have rights to exist and probably they are better than mine. I just would like to share my point of view. It is alpha, the best time for bold test, especially with the current state of the server.
The horses should be tamable.
Parameters of a domesticated horse:
Essence
Substance
Vitality
Stamina %
Speed %
Hide quality
Meat quality
Quality of Milk
Quantity of Milk
Breeding
The domesticated horse should have additional parameters: Hunger, Water, Life. When one of the parameters will not be filled properly, the horse will not be able to gallop.
With the default stamina parameter the horse will be able to run for 10 minutes on a basic speed (with full Hunger and Water bars). After this the horse stops and drops the rider out of its back. You have to feed the horse with 4 units of any type of plants and with 1 l of water, until the Hunger and Water bars will be full again. After the feeding it will take 15 minutes to be able to ride the horse. But you could leash it and then your horse will follow you, however the parameters of Hunger and Water will stop restoring during the movement.
Stamina. During the breeding process the parameter for the newborn will vary from -1, 0, 1 (28.6%) or +2 (14.3% chance).
Speed. During the breeding process the parameter for the newborn will vary from -1, 0, 1 (28.6%) or +2 (14.3% chance). The max increase might be +1, which is a matter for a discussion.
For bonuses for stamina and speed you will have to equip the horse with a saddle.
A bit of analytics.
Getting 2% increase in speed or stamina is not so easy. To get a chance of 1 to 7 , you need to have 7 mares and 2 weeks of gestation and growth. And given that it would be desirable to receive increase of 2% Stamina and 2% run speed , the chance of getting such an increase is quite low . Only large village can afford maintaining big numbers of mares to keep efficient breeding. Another reason to group up in bigger villages. There might be horse races between villages as well.
I do not know how exactly it would affect chasing someone, but those parameters are logical and represent the only motivation for horse selection.
Kumis. The new type of drink, similar to wine. You will need to milk the horse and pour the milk into a container and wait until it will turn to kumis.
A character might perform any actions while riding a horse, such as PvP or foraging. However the character cannot break objects or enter a house.
You could also tie one or a few domesticated horse to a tree.
Mares move with 5% speed and stamina penalty.
You cannot ride a horse until it will reach maturity.
You cannot agro a horse if someone is riding it or when it is tied to an object.
Customization.
Someone has suggested the idea of donkeys with container bags. The idea is quite good and could be done as combination of a saddle and saddle bags (crafted separately, freely attached or unattached). A horse carrying bags will have the default speed and will consume 5% more stamina.
There might be situations when you might need to differentiate your own horse from other’s. There is a wide field for customization, starting from different colored saddle bags, cloths, battle paintings (another use for sketchbooks ), hair cuts and so on.