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Subject: Game Design: Ranking Poker Hands rss

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Robert Seater
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It has always bothered me that many games with poker hands rank the hands based on a random draw of 5 cards (as in traditional poker). However, the relative frequencies of different hands can be quite different when you introduce mechanics such as partial redraws. To one extreme folly, I recall the Deadlands CCG whcih was customizable yet still ranked a straight as easier than a flush! When working with Rob Herman to refine Montana, I was determined to get this problem right.

To determine the ranks of hands, I first built a lattice of all the legal hands, indicating which hands were subsets of others (and thus strictly simpler). In the diagram, things get harder as you radiate out from 'single'.



In Montana, you keep unused cards between turns, and thus you have partial control over what partial hands you retain. To model this, I supposed that players were always looking at a hand of 8 cards and picking a hand from those. So, I dealt out a huge number of 8-card hands, and ticked off (on the lattice) all the hands that were present. The relative tallies provided a very good first cut at which hands are easier to build iteratively than others.

Now, that only captured the rarity, which is only part of the ranking. Since you only slowly draw new cards after playing a hand, bigger hands are more costly than smaller hands independent of rarity. I initially ranked the hands entirely by probability, and then went back and made a few swaps.



A flush pair is actually pretty rare, but (a) it's cheap to play, and (b) there is no skill in planning for one. So I put it lower than its rarity warranted. I gave 5-Garbage a little boost, simply because 5 cards is painful to play. However, I consider it an important quirk of the game that there are 3-card hands that can beat some 5-card hands, so I only adjusted the order where playtesting suggested that it was way too frustrating ranked by pure likelihood.

Note the oddity wherein a 3-straight is harder than a 3-flush and yet a 4-straight is easier than a 4-flush. It's funny how these things work out.

Also note that the deck has 5 suits, numbered 1 through 12, with 2 copies of each distinct card. This makes straights relatively easier compared to flushes, but doesn't actually change the ranking of hands all that much.
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Robert Seater
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3-straight-flush and 5-straight are also technically out of rarity order, However, it turned out that having two 5-card hands that are worse than the best 3-card had was too frustrating.

The garbage hand was, in any case, added specifically to ensure that (almost) any set of 8 cards hand would have at least a mid-value hand within it. As such, it's ok if it is a situational or costly hand to play. It's also a great bluffing surprise, since it looks like you are just dumping unwanted cards, when suddenly you have a decent hand (since, in this game, cards are revealed/played incrementally).
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Rob Herman
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Early versions of the game used an ordinary double poker deck with four suits. With the combination of: a) many cards of each suit, b) a large hand to work from and c) keeping partial hands, flushes were incredibly common. My initial solution was just to warn players not to expect a straight to win a showdown! The change to five suits was important and keeps the relative hand rankings feeling more natural.
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Scott Nelson
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I had an idea called: Dig! that was even more frustrating (theme was Surfing):

Draw cards till you have at least 5 cards (you may have more for other reasons), but then set aside 5 cards as you hand.
1 card each from the 5 cards is then chosen by each player upside down. Flip them up simutaneously. First card the winner is high card. That player takes a +1 VP token. Each hand has a 1,1,1,1,X tokens to earn in order.
Next, the players takes 2 cards from their 5 card hand set aside (now 4 card only), and flip them up simutaneously. Check for the best poker hand of those 2 cards; that player takes a 1 VP token.
Then the 3 cards; 1 VP token gained from best hand all 3 cards.

Then the next 2 cards ; 1 vp token on next 2 cards.
Then the final X token gain from all 5 cards. the x is a multiplier from all the vps tokens gained in this hand with min of 1 vp.

Trying to figure out the best poker hand in the various number of cards 2,3,4 "poker hand" size was a headache! I never even added a few of them like pair flush, but that would even add more headache. After looking at your chart, I'm thinking it would be less headache, more Nightmare!

edit: in 6 suits as well. Ack!
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Robert Seater
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ropearoni4 wrote:
Trying to figure out the best poker hand in the various number of cards 2,3,4 "poker hand" size was a headache!

I use to be in the field of combinatorics, but I still found it too hard to set up the equations right. Worse, small mistakes in the formula would produce wildly different result, so it was hard to know if I was modeling the domain accurately. A modeling mistake is just as bad as a math mistake (and computers/calculators won't help with the modeling part!)

I felt pretty confused until I set up the lattice and resolved to do straight empirical tests. It then got a whole lot easier. The lattice itself is straight-forward to make -- since it just captures strict subsets. I included more hands than were going to be needed in the game, since I could cherry pick the ones I actually wanted later.

Then I drew a bunch of cards and ticked off which hands I saw. That way, I actually got several pieces of data with each draw, which gave me a lot more data. I did that a few hundred times (while watching a really bad Bela Lugosi movie) and got a pretty clear rank order of likelihood.

I highly recommend a quick-to-execute empirical method over a straight computation, if only because I find that I have more confidence that my results are mistake free. It's easy to get caught up in determining the exactly probabilities, but remember that you only need a relative ranking! As such, empirical methods do the job nicely!
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Scott Nelson
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rseater wrote:
ropearoni4 wrote:
Trying to figure out the best poker hand in the various number of cards 2,3,4 "poker hand" size was a headache!

I use to be in the field of combinatorics, but I still found it too hard to set up the equations right. Worse, small mistakes in the formula would produce wildly different result, so it was hard to know if I was modeling the domain accurately. A modeling mistake is just as bad as a math mistake (and computers/calculators won't help with the modeling part!)

I felt pretty confused until I set up the lattice and resolved to do straight empirical tests. It then got a whole lot easier. The lattice itself is straight-forward to make -- since it just captures strict subsets. I included more hands than were going to be needed in the game, since I could cherry pick the ones I actually wanted later.

Then I drew a bunch of cards and ticked off which hands I saw. That way, I actually got several pieces of data with each draw, which gave me a lot more data. I did that a few hundred times (while watching a really bad Bela Lugosi movie) and got a pretty clear rank order of likelihood.

I highly recommend a quick-to-execute empirical method over a straight computation, if only because I find that I have more confidence that my results are mistake free. It's easy to get caught up in determining the exactly probabilities, but remember that you only need a relative ranking! As such, empirical methods do the job nicely!


Next time, I'll waste a good 2 hours of a bad movie and do something profitable It does sound like a real world example would be better, anyways.
 
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Jeremiah Lee
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Wow.

I can't believe they let me into this super exclusive club. I totally have no idea what they're talking about, or how I'm supposed to dress.
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Scott Nelson
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Jeremiah_Lee wrote:
Wow.

I can't believe they let me into this super exclusive club. I totally have no idea what they're talking about, or how I'm supposed to dress.


Just add a bunch of big words to your sentences, and you'll feel right at home with Rob and his thread. Dressing is optional.
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Robert Seater
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ropearoni4 wrote:
Jeremiah_Lee wrote:
Wow.

I can't believe they let me into this super exclusive club. I totally have no idea what they're talking about, or how I'm supposed to dress.


Just add a bunch of big words to your sentences, and you'll feel right at home with Rob and his thread. Dressing is optional.

Only the pants are optional.
 
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