Calling all statisticians, card players (new card game)

Factual Questions
1

So I’m playing Knights of the Old Republic on the X-Box. It’s a fantasy role playing game, and has a blackjackish mini-game inside called Pazaak. I’m trying to come up with a decent strategy, but nothing is coming to me.

Here’s a rundown of the game.

It is a two player game, no dealer hand. The deck is comprised of 40 cards, 1 through 10, four of each. The object of the game is to get to 20 without busting. Busting is an automatic loss. You are always Player 1 (you draw first each time). Best of 5 hands. Deck is shuffled between hands.

However, each player has a mini-hand, which you assemble. Cards you are allowed to add are +1 through +5, -1 through -5 and +/-1 through +/-5. You pick 10 cards at the beginning, and the computer randomly chooses 4 cards.

So play begins with each player with 4 cards in his mini-hand (you cannot see the opponent’s mini-hand). You are given a card immediately from the deck (1-10). You have three options: (1) play a card from your mini-deck (each card can only be played once in the best of 5 round); (2) hold (player 2 will have a turn, then you will get another card from the deck); or (3) stand.

Strategy in this game is two-fold: (1) which cards are the best to accumulate and add to your possible mini-deck and (2) what is a good number to hold on.

Lemme know if you need any more information.

2

Another thing I should add is regarding the mini-deck. You can add any number of those cards (+1 through +5, -1 through -5 and +/-1 through +/-5). Assume I have 10 of each of those. (Currently, I don’t - you assemble your deck throughout the game).

3

Can you see your opponent’s draw cards?

4

Yes, his draw cards, not his mini-deck.

5

Some questions :

How does one ‘assemble’ a mini-deck? Do you get 4 cards per hand?

If so, are they picked from the same 10 or a different group of your choice?

What’s the difference between ‘stand’ and ‘hold’? If you stand, do you not get any more cards and hope that Player 2 busts?

Is the bust immediate or can you play a mini-deck card to prevent it?

Do you get the choice of using a +/- card, or is it random? If you choose, it’s pretty clear any of those 5 are the best to get.

My initial feeling is that keeping your points down and collecting larger value cards is better, if busting is immediate. Any time you’re above 10 you have a risk of losing. With the large + mini-deck card, you can win with a large draw and your mini-deck card. A large - will keep your totals down if they get too high.

Playing conservatively like this, I’d hold if ahead in the game (forcing the opponent to use up mini-deck cards) or if I didn’t have any useful mini-deck cards and I’m over 14.

6

Actually, the more I think about this, the less sense I can make of it. Why do you put 10 cards in your “mini-hand” but only have four available to you during play? Do you get to replenish the mini-hand up to four from your stock of ten after each hand?

Superficially, I’d say stock your mini hand with +3 cards. Draw two cards normally and:

Hit if you have 12 or less
Improve from mini-hand if you have 13-17
Stand on 18+

Logic is as follows:
You’ll get 10 or less on two cards 45% of the time. Draw, and you have no chance of busting.
You’ll get exactly 11 10% of the time. Draw and you have 10% chance of busting.
Youll get exactly 12 9% of the time. Draw and you have 20% chance of busting.

So far, you only bust (0.1 * 0.1) + (0.09 * 0.2) = 2.8% of the time.

Keep drawing until you bust or have more than 12.

If you get 13, draw 2 of your +3 cards, giving you 19 and a good shot at winning.
If you get 14, draw 2 of your +3 cards and stand on 20.
If you get 15-17, draw one +3 for 18-20.
If you get 18+, stand.

I figure you have a small chance of busting and every other outcome gives you at least 18. You could replace one of your +3 with a +6, if you like, meaning you’ll draw that card instead of two +3 cards on a 13 or 14, but the mechanics of the mini-hand remain unclear to me.

7

“How does one ‘assemble’ a mini-deck? Do you get 4 cards per hand?”

You select 10 cards to create a stack from which your 4 cards will selected for your mini-deck. You get 4 and only 4 for the entire best of 5 round. Use a card, you have 3 remaining for the rest of your hands, etc.

“What’s the difference between ‘stand’ and ‘hold’? If you stand, do you not get any more cards and hope that Player 2 busts?”

“Hold” means you want a card in the next turn. “Stand” means you’re done, and hope the other player busts.

“Is the bust immediate or can you play a mini-deck card to prevent it?”

You can play a mini-deck card. But only 1.

“Do you get the choice of using a +/- card, or is it random?”

You can choose. Obviously, these are your best bet. But do you mix them, go with higher cards, or lower cards?

“Why do you put 10 cards in your “mini-hand” but only have four available to you during play?”

It’s probably just to add more randomness to it. You most likely want a variety of +/- cards, and put 2 of each in. But you still have the possibility of getting 2 +/-5 and 2 +/-4 cards, with no 1, 2 or 3 cards.

8

Well, heck, if the computer was going to randomly choose which four (of ten) cards I’d have in my mini-hand, I’d go with all +3 cards (I’d misread the rules when I mentioned selecting a +6 card above). You only need the negative cards if you’re close to busting, and the strategy I described above offers only a small chance of that, so I wouldn’t bother.

The only modification I’d make is to keep saying “hold” until you opponent stops drawing cards. There’s no point wasting two +3 cards from your-mini hand if you have 14 and he’s showing 20. You may as well draw and hope for the best.

Keeping only +3 cards gives you a good chance of winning each hand, and since you only need 3 out of 5, there’s no point making it more complicated than it has to be.

9

I know, it’s extremely unclear, but every time you say “hold” you get a card the start of the next turn. “Hold” equals “hit me at the end of Player 2’s decision (unless he busts)”. For instance:

Player 1: draws a 5 (5) – hold
Player 2: draws a 9 (9) – hold

Player 1: draws a 3 (8) – hold
Player 2: draws a 3 (12) – hold

Player 1: draws a 6 (14) – hold
Player 2: draws an 8 (20) – wins

10

The average value of a single card in this game is 5.5 (with ranks 1-10).

Thus, after two cards, the average hand will be 11.

After three cards, 16.5. (Half the time the hand will be 16, the other half 17… on average).

And after four cards, 22 (bust!).

So, your best mini-hand cards are -2 and [3 or 4]. Of course, +/-2, +/-3, and +/-4 are not only just as good, but better because of their versatility. My guess here is that any +/- card is better than even the -2, 3, and 4, because of the versatility.

Of course, you have to play strategically depending on what shows in the other player’s hand. If they have a 17 and you have 14, then you hit (‘hold’ in game terms) because the average card of 5.5 will put you in the lead and will bust your opponent.

If you have a -3 mini card, then you’re playing up to 23, and so you can hit on a 17 (17 + 5.5 = 22.5) if your opponent shows 18. If your opponent shows 16 in this case, then stand.

Of course, your opponent also has mini-hand cards. However, if your opponent (the computer) chooses these randomly and they are distributed between +5 and -5 evenly, then you can assume that the overall average value of the computer’s mini-hand is zero, and therefore, not statistically significant, overall (although, how they’re played each hand is).

Hope that helps.

Peace.

Got any threes?

11

moriah, thanks a bunch. That makes a lot of sense. I’m sure Bryan will update his suggestion if he gets a chance.

Regarding the computer’s mini-hand, I assume that it gets better as the game moves forward, or as my mini-hand improves. In any case, I’d wager that it curves with my hand.

12

You know, if you want the win the damn game, you can just use the force.

[sub]yeah, yeah, dark side… lighten up, Yoda[/sub]

13

You know, even for the chosing the dark side path, that’s not an option!