Critical Strikes and Bashes

Both, critical strikes and bashes stack with diminishing returns, as according to the formula:

1 - ( 1 - A ) * ( 1 - B ) * ( 1 - C ) * ... * ( 1 - X ) = Chance to get at least one bash / critical strike

where A, B, C and so forth are the seperate chances for each item or skill that can bash / critical strike.
Note that all critical strikes and bashes in the game are rounded to the nearest 5% of chance before calculating them in this formula.
So a 36% chance to crit (such as Juggernaut currently has) is actually 35%. However, this does not mean that the percentage you get out of this formula will be rounded to the closest 5%.


Critical strikes

When calculating damage from critical strikes, if two occur at once, only the last critical strike skill or item you acquired will have effect.
Note that a skill counts as acquired when level 1 of it is learned.
An item counts as acquired every time its picked up.


Bashes

When calculating bashes, the result depends on whether you're a melee hero or a ranged hero.
It is true for both however, that if you bash an already stunned unit, the greater of the remaining stun time and the bash time of your bash will be used. That means that a bash can never shorten the amount of time a unit is stunned.

Melee heroes' bashes stack fully, so if two or more bashes occur at once, the longest stun time from all the bashes will be used (bashes never shorten stun time) and all the different bash damages will be added .
The damage is reduced by spell resistance, and eliminated by magic immunity.
The damage is a separate source and therefore does not count to lifesteal and does not interact with critical s trikes in any way.
Additionally, melee units' bashes have a clear advantage:
The stun part of the bash is NOT eliminated by magic immunity and therefore you can even stun magic immune units, though they will get no additional damage from the bash.
As melee heroes can proc more bashes on one attack, the formula above only gives the chance of having at least one bash ( and for that a stun ) per attack. This also means that on melee heroes, the damage output from the bashers are not affected by diminishing returns.
Two bashers means you have 27.75% chance to land at least one bash on the opponent, but 25.5% chance to get one bash and 2.25% chance to get both bashes. Since the damage stacks, this gives as much average damage as 30% chance to bash once.

As for ranged heroes, it's the other way round. The bash damage is added directly to the normal attack damage and therefore it behaves just like any normal damage increase (it's reduced by armour value and armour type, it grants lifesteal, etc.). The stun part is blockable by magic immunity. However, a ranged hero can not land multiple bashes in one hit - a proc of a later acquired bash will always completely overwrite an earlier achieved one.


Bash and crit interactions

On melee heroes crits and bashes will have no interaction with each other (they are independent of each other). This means the critical multiplier will never multiply the bash damage.

On ranged heroes every crit achieved after a bash or another crit will completely negate the earlier achieved crits and bashes on its procing. In essence, this means that if you proc multiple crits and bashes on a ranged hero, the last acquired crit that triggered will count as the only critical, and if any of the bashes that procced were acquired after the critical was acquired (not procced), the last of those bashes would add damage (which isn't multiplied) as well as the stun.