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I know there is a similar question, but that one is related to D&D.

I'm planning a scene for a Vampire: The Dark Ages campaign where the players will need to participate in a gambling game - specifically, the Hazard game as explained here - but I don't see any Ability that can be used to perform such a scene. Should I just use Intelligence, Dexterity, or a random dice?

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Games of Skill and Not

First, estimate whether you want the scene to involve a game of skill, or a game of pure chance, or something in-between. Among the games colloquially known as gambling, some have a huge skill component, while others only have a 'non-stupid' strategy and you can't get much better than that.

If the game leans towards the skill end of the spectrum, regular skill rolls are appropriate. If the chance end, then at most make a single, relatively easy, intelligence-based skill roll, to figure out the 'non-bad' strategy, and then, if it succeeded, proceed to only roll dice with no dependency on the character sheet; if it failed, just rule that the PC made terrible decisions and lost almost as much money as was willing to bet.

Judging by what's written in your link, Hazard seems to lean towards needing to be aware of some very static mathematical truths, but there seems to be opportunity for more dynamic decisions based on who's betting what at the table, and even for some social manipulation of the caster into different kinds of choices.

Skill to Use

While the skill description doesn't make it explicit, I'd say that Larceny seems the most associated with the kind of 'shady' knowledge that fits both with the proficiency in fair gambling, and the definitely-non-fair cheating that may accompany it.

(In rarer cases such as betting on horse races or knightly tournaments, you may want to use a different skill - one relevant to knowing the participants and predicting which ones of them are better, like Ride or Mêlée. In case of Hazard, Science might be of use if none of the people at the table are familiar with the basic mathematical truths and all the participants are trying to figure them out on their own.)

Attribute to Use

If it's a quick cerebral game, like some card games, you probably want to ask for using Wits. If it's a cerebral one with very long opportunities to ponder the next strategy, only then should you call for Intelligence rolls (but that seems more characteristic of games of chess; gambling tends to want a bit more fast-paced adrenaline!). Again, in your case, Intelligence may be relevant if everyone's unfamiliar with the maths of the game and is slowly figuring it out over the course of a long evening.

For a game with a strong social component (like the predecessors of modern poker), you may want to use Manipulation (and/or perhaps Perception), but if you go that way, maybe you're actually looking for involving even more skills and attributes (below).

Additional Considerations

If you decide you want to make the social component really detailed, you may actually want to make separate opposed rolls of Manipulation+Acting vs. Perception+Empathy instead, and have the results of those contests provide a bonus to one of the two sides in the 'main' roll. But that's a lot of rolling, so only break out additions if you're sure it'll be fun for everyone. (IME, chances are higher that players will enjoy that if they have characters who can capitalize on social competence and don't get bored by too many dice.)

Finally, for cheating, you will definitely want Dexterity+Larceny used against Perception + either Alertness or Larceny, with a cheater's victory providing a big bonus (such as difficulty reduction), and a cheater's defeat resulting in revealing the cheating.

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It depends entirely on how you and your players choose to interact with the situation, emphasis on your players.

Assuming they just want to play the game itself, with pure random chance, a simple dice roll should work fine. If you want to work out even more rules, you could make the dice rolling more complex for the sake of making it more interesting.

On the other hand, if they start describing their participation in the game in terms of trickery, out thinking their opponents or trying to use fancy hand movements to effect the outcome of the game, then I would allow them to use Intelligence, Dexterity, sleight of hand, or quite literally anything they could justify in the situation you think is legitimate.

This is something I've done in my own games (To a broader extent). It allows players a much greater sense of personal autonomy, and encourages engagement with the scene. Often times certain players due to their character design or personality will check out until they become relevant or their goals highlighted. Instead, they get to brainstorm their own way of being important rather than relying on you to do so as a DM or simply being irrelevant to the scene.

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