It's random... ish. Every turn your merchants monopolise a resource there is a percentage chance that they will gain a level. By the way, IIRC you are only monopolising a resource if there are two (or more) of the same resource type in the same province, and you have merchants sitting on both of them.
Edit : correction, there seems to be no random element involved - it's 20 turns to get to monopolist level (see below).
Here's a repost of something i wrote ages ago:
More on merchant skill levels
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Natural Skill
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On creation of a merchant, you have:
1/2 chance of skill level 1
1/3 chance of skill level 2
1/6 chance of skill level 3
This is completelty independent of any buildings etc you may have
Merchant's Guild effects
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If the player has a master merchant's guild, or a guild HQ anywhere in their empire, then all created merchants get skill bonuses:
Master Guild : Merchants Guild Trained Trait : +1 Skill
Guild HQ : Merchants Guild Member Trait : +2 Skill
Legal/Shady Dealer Traits
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On creation of a merchant, if the settlement has a town hall or better, there is a chance of the merchant having the legal dealer trait:
1/2 chance of no trait
1/3 chance of skill +1
1/6 chance of skill +2
The inverse of this is, if the settlement does NOT have at least a town hall in it, the merchant instead has a chance of aquiring the shady dealer trait:
1/2 chance of no trait
1/3 chance of skill -1
1/6 chance of skill -2
Religious Merchant Trait
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On creation of a merchant, if a cathedral or better is present at the settlement, then there is a chance of the merchant aquiring this trait:
1/2 chance of no trait
1/3 chance of skill -1
1/6 chance of skill -2
Monopolist Trait
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Every turn that your merchant monopolises trade in a region then the value of this trait is increased.
Value 5 to 9 -> Skill +1
Value 10 to 19 -> Skill +2
Value 20 or more -> Skill +3
Aquisition Skills (good merchant trait)
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For every successful aquisition your merchant makes, this value goes up by 1
Value 1 -> Skill +1
Value 2 to 3 -> Skill +2
Value 4 to 7 -> Skill +3
Value 8 or more -> Skill +4
Failed Aquisitions (bad merchant trait)
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For every unsuccessful aquisition attempt by your merchant, there is a 1/3 chance that the value will go down by 1 (this is tied to the good merchant trait above).
Value -1 to -2 -> Skill -1
Value -3 to -5 -> Skill -2
Value -6 or less -> Skill -3
Dont worry too much about the bad merchant trait - if youve failed enough aquisitions in a row to get a significant value to this trait, chances are your merchant will have been eliminated in the process. In order to get skill -3 your merchant will have had to fail 18 (on average) consecutive attempts, and yet not have been eliminated - pretty unlikely.
Making Enemies
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Every time your merchant performs a successful aquisition mission, they have 25% chance of increasing the value of this trait.
Value 1 or 2 -> Personal Security -1
Value 3 to 5 -> Personal Security -2
Value 6 or more -> Personal Security -4
In theory this makes more highly skilled merchants easier to assassinate - in practice this will probably just cancel out some of the effects of the skill level rise itself.
Surviving Aquisitions (secure merchant trait)
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Every time your merchant survives an aquisition attempt by another merchant, they have a 50% chance of gaining an increase in the value of this trait.
Value 1 or 2 -> Skill +1
Value 3 or more -> Skill +2
Surviving Assassinations
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Every time a merchant survives assassination, they have a 50% chance of increasing the value of this trait
Value 1 or 2 -> Personal Security +1
Value 3 or more -> Personal Security +2, Finance -1
Note the -1 to finance. If you repeatedly attempt assassinations on enemy merchants they will eventually lose 1 finance. However, i'm not sure it's really worth the effort.
Worldly Merchant Trait
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Trading a resource far away from your capital (distance >= 40) has a 20% chance to increase the value for this trait each turn.
Value 1 or 2 -> Skill +1
Value 3 or more -> Skill +2
Self Serving Merchant
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If your merchant is very far away from your capital (distance >= 60) then he has a 3% chance every turn of increasing the value of this trait.
Value 1 or 2 -> Skill -1
Value 3 or more -> Skill -2
Moral : dont let your merchants get too far from your capital, unless its really worth it for the money.
Ancillaries
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There are 3 merchant ancillaries, each of while can only be obtained at initial creation of the merchant, and each of which gives +1 to finance skill.
Counterfeiter - 15% chance to obtain this if the settlement where the merchant is trained has an Alchemists Lab
Merchant Clerk - 50% chance to obtain this if the settlement where the merchant is trained has an Alchemy School or greater
Trick Abacus - 5% chance to obtain this if the settlement where the merchant is trained has a Town Hall or greater (note - this trigger occurs twice in v1.1 - so this probably increases the chances to around 10%)
Notes
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w.r.t. the Legal Dealer Trait mentioned by Foz above -
this trait seems to be bugged in that it seems to be impossible to get the second level of the trait (trigger adds 1 to make a total of 2, but the threshold is 3)
There is a trait series called 'Exotic Tastes' mentioned in the game textfiles, which gives penalties to finance skill and personal security. However, i havent found any triggers listed for this trait.
Caveat: all of the above stuff is (i think) from vanilla 1.0 of the game. Some stuff may have been tweaked in the patches.
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