"Odds in card-based systems" Topic

It's been decades since my college prob-stat course, which (oddly) I don't recall ever dealing with card game theory anyway (it probably did, but, you know, it was a requirements course, there were girls around, and I was an English major, so *fsssst* on memory retention of that).

Still, I'm okay with basic card game concepts: odds of card of a type, suit, or a unique card given a deck of whatever size and number of draws, that I can do. But what about specific combinations of cards in various sizes of draws?

To set this up, I recently bought a copy of the Han Solo Card Game (my advice is treat it like the Solo movie— unless you're an obsessed fan, don't waste your money). The rules for this game (which suck) are based on a deck of 62 cards consisting of cards ranging from 1 to 10 in two suits, three cards of each number per suit, and two 0 cards of no suit. The players start with two cards each, and then draw and/or discard once per round over three rounds in an attempt to create a hand that totals 0 (one suit is negative, one positive), or as close as possible, with positive hands beating negative hands if 0 is not reached. There are a few other rules, but the interesting aspect is that a player does not have to draw or discard, and can do either without also doing the other. So after three rounds, you could have situation where one player holds two cards, one holds four, one three, and one five. So I'm befuddled as to how the odds work on all this, especially as the scoring system is based both on nearness to zero AND number of cards in the hand AND (if that's still a tie) the absolute value of all cards in a hand. So, for example, +2,-3,+4,-1,-2 is a five card hand totaling 0, with an absolute value of 12, while +5,+3,+1,-5,-4 is a five card hand totaling 0 with an absolute value of 18. The latter wins the tie.
BUT a three card hand of +10,-10, 0 is always the winning hand (the designers seem to have not realized that +10, +10, -10, -10, 0 is also a possibility, and one that can happen simultaneous to their "best hand," given the deck has three +10, three -10, and two 0s. Like I said, the rules suck.)
What's got me befuddled is calculating the odds given the three rounds with varying hand sizes AND varying absolute value totals.

So given the above, how do I calculate the odds, based on different player numbers, for:
Odds of getting a hand of value 0 with 2 cards, 3 cards, 4 cards, 5 cards of any value combination.
Odds for various absolute value calculations, again based on hand sizes?
(So, for example, a 0 value two card hand could be 0, 2, 4, 6, 8,…20. But don't forget hands that don't reach 0, but reach either a positive or negative number. And then you have 3 card hands, etc.)
Odds of hands with a 0 value which also include one or both 0s, in addition to positive and negative cards which total 0.

I may be overthinking this, and it still just comes down to rather basic card math.

In any case, I thought the above question might also apply to games that use card activation systems, especially where the cards are drawn and held in a hand (like C&C), and the hands can potentially wind up uneven.

(PS: Yes, I've been working on a better, more interactive set of rules for this game. I think I've come up with some that don't suck, but I'm going by "feel," not actual odds calculation. Did I say, "English major?")

— Parzival, English Major="You Do the Math."

If you really want the true odds, this is a bit of a challenge.
See: link

Consider Zero first
There are 2/62*1/61 ways to be dealt two zero cards (0.053%)
The odds of being dealt Zero is the number of combo say -1 and +1 is 3/62*2/61 (~0.16%)
But also paired 2s..10s at 0.16% each so total the %s
So: 0,0; -1,+1; … -10,+10 that is 11 possible ways to be dealt a hand of two cards that equal zero at 1.64% total.

But no one is playing by themselves, so that already confounds the odds. Say two players: Player A 62 possible card 1, Player B 61 possible card 1, Player A 60 possible card 2, player B 59 possible card 2:
And each draw depletes the deck of a value.
So limit your case to just zero:
It is actually better to calculate the chance that the case will not happen. One has to draw out a branching system of all possible outcomes and calculate the odds for each branch and sum the odds:
A 60/62 the first card is not zero (~97%)
And 2/62 the first card is a zero (~3%)
Now the second card to player B, if A no zero:
B 59/61 not a zero
If A was a zero then 60/61 B not a zero

Already more then can be covered here. You may be able to image the branching if a play can draw and discard.

If all cards are used for each draw, then the fractional/ percentage chance remains the same.
If there is a discard pile then it will change each draw. Total cards minus thos in the discard pile. So the positive draws are the numerator and the total draw, minus discards, is the denumenator.

This is not a situation covered by basic probability rules because there is an element of player choice involved.

If you were to just consider the likelihood of being dealt two (or more) cards that totalled zero (or any other score) it would use basic probability.

To build up 3 or 4 card hands a players has to decide a course of action and this human intervention means that all probabilities are modified by those player decisions.

The best you can do is consider each possible draw and estimate the best action for that draw (for the player) and assume that is what happens. That will however only give you an estimate of results and that after a lot of arithmetic.

This is a problem in game theory, not probability.

As GildasFacit says, this is a game theory (specifically a Markov chain) problem, but there are some insights you can get from probability and some simple "good enough" game theory things you can learn without going through the many (yatto is the 10^24 prefix, and the number is well beyond that) explicit combinations.

The average of all the cards is … ta-da … zero, but the standard deviation is a little more than 6. This is a fairly fat distribution, so with four players (I am assuming) there are unlikely to be ties.

Also with four players, you only go through 20 cards, fewer than a third, so in most cases the draws don't affect the probability distribution much, at least from the standpoint of decision making.

Most importantly, you are really only making two decisions in the game (Do I discard a card after the first draw? Do I discard a card after the second draw?) unless there is betting and raising. The last discard is pretty much a closed form answer (Which discard (including the null discard) gets me closer to zero with the highest magnitude total?)

Without betting and raising, this makes for a pretty sucky (as you say) game. It also makes it similar to a number of other card games that you can play with regular cheap cards, so buying a special Han Solo deck for it is an aesthetic choice.

What it would be good for is a theme-based role playing interlude in a larger game to resolve some "random", "luck", "hutzpah" situation.

So, looking at your two actual decisions.

Do I discard something after the first draw?

Well, you have seen between three and six cards, depending on turn order and others' discard decisions. Your info doesn't affect what you know about the odds that much, unless you have a rare extreme case. F'r'ex, if you are holding three high cards (6+) and you see three low cards (-6 or lower) discarded.

But even in that extreme case, the decision is still pretty simple. Discard what gives you the lowest total, giving you the greatest odds of getting nearer to zero.

If you have a low total (+/-six or fewer points), again, the answer is discard if that can get you closer to zero.

With only 12 (max) of 62 cards draw, and you having seen a max of half of those, you don't have enough info to go for a finesse approach.

The same applies for the decision about the second discard, except there are slightly better, though still slim, odds of an extreme case that provides you better info than just the basic distribution of cards and the basic "closest to zero" criterion.

Compare to counting cards for blackjack. The principle is you watch what cards have already been played, and when you get into the last half of the deck, you know whether the odds of the remaining cards favor the player or the house. With four players, you never get to 1/3 of the deck gone.

If you had eight to twelve players, you could start doing a similar "counting cards" for the odds. Like blackjack, I would recommend dividing cards into groups, so something like:

Low 1-3
Mid 4-6
High 7-10

Track this for positive and negative.

So you know, based on what you are currently holding whether low, mid, or high cards are going to help you get closer to zero, and you have a decent look at how many cards in that range have already been burnt.

You could get by with high pos (6-10), low pos (1-5) high neg (-6--10), and low neg (-1--5) as the categories.

Also, ignore the zeroes. They are a low probability distractor, and don't really impact the total Markov chain that much.

This would be the same if you played three rounds from the same deck with no reshuffle with four players.

Now, if there's betting involved, it's a different game. Now you are playing the players as much or more than you are playing the game …

My head hurts….

So does mine, says the history major.

Jim

Here's how the basic structure of the game works:

Cards are shuffled and two are dealt to each player. The top card of the deck is revealed and forms the discard pile.
On his turn a player may draw from either the deck of the discard (or not draw at all) and may also discard a card (or not).
This happens three times, with no reshuffle.
Cards are revealed and compared for the winning hand (described above).
Yes, this sucks as a game.
There are dice involved, in an attempt to "liven" things up a bit (I guess), and to match the dangling pair of gold dice on the Millennium Falcon (visible in the original film). 2d6 are rolled after each round of drawing/discarding; on doubles, all hands are discarded and players are dealt replacement hands (of the same number of cards as each discarded) from the deck. This, of course, sucks too, as it blows any strategy out the water, especially on the third round.

And that (aside from some gussied up credit bars, which players get to choose from each hand, based on order of winning, trying to claim the Falcon) is the entire game.
As I said, it sucks.

What I've done is to add an actual betting structure, plus an option to "lock" cards to protect them from the dice results (at the cost of revealing cards to other players), add some unique hand results that are automatically better than others, and include a "Treasury" aside from the betting pot which can also be won, plus a few unique concepts to make the dice rolls a bit more interesting. Some of these ideas aren't original to me, but in the end I think I've made a better game (with actually more Star Wars flavor than the original). But I'm neither a gambler or a poker player, so I'm not the best judge of how it works as a gambler's card game.

(Yes, I am aware of the West End Games game-within-a-game of Sabacc— that's where my "lock" and "Treasury" pot ideas come from.)

If anyone wants to try it out, shoot me a PM and I'll send over my rules. (I call ‘em "Smuggler's Run," which I came up with while completely unaware of the name of the new Disneyland ride! Darn it. Oh well, ain't changin' it now.) You'll need a deck of 62 cards in two suits 1-10 each, three of each number and suit, and two jokers as your 0 cards. (Best approach is to take two identically-backed decks, use two suits of diamonds, one suit of hearts, two suits of spades and one suit of clubs, no face cards, and treat all red cards as negatives and all black cards as positive, ignoring the actual suits. And, of course, add two jokers. You'll also need two d6s, something to act as "credit bars" (poker chips are fine, just ignore mentions of "face values" in the rules), and two special tokens to represent "The Smuggler" and "The Governor" (concepts original to me).

You can also e-mail me using my TMP member name as the address with AOL as the domain. (I'm assuming all here know where to put the @ and the .com in that mix.)

Thanks to all for the probability answers! I think I've got a better grasp of it, and also think I don't really want to go much further than that!

So, I know what this game is good for …

BARFIGHT!

Put your five or six minis (one per player) around a table in the middle of a bar scene. Play the card game (pretty short). Now write the magnitude scores (minuses become plusses) in order lowest to highest. Next to them write a column of players in reverse victory order.

So you get this

0 – Yoda
0 – Darth Maul
2 – Lando
3 – Chewie
4 – Han

Yoda lost worst with 4, then Darth Maul with 3. Lando did OK with 2, and Han and Chewie tied for first with 0 points, but Han won with one of the tie breakers.

Now go down the list and each character gets to make the number of move or non-combat actions (flip over furniture, charge up a blaster, turn invisible, etc.). Now start the fight with characters taking turns in list order.

So, the losers have to preposition first, and get the fewest number of actions. The winners get to see the others' preposition actions and get the most pre actions.

Of course, it wouldn't be a bsrfight without random civilians. Civvies get some random, low-level, non-heroic stats.

* – After a turn, a player rolls a d3 (you figure this out on your own; it's easy). Move that many civvies a "standard move" (by your system) directly away from your character.

* – If a civilian runs into a character, it attacks the character.

* – If a civilian touches a wall, it leaves the bar. Dramatically, if possible.

So with strategic thinking, you can "move" civilians to thwart your opponents and set up your own awesome moves.

Heroes can also run away by leaving through an exit. Not very heroic. Then again he who fights and runs away …

Victory Points – Total damage you caused to characters (not civilians) minus total damage taken by you. For each civilian you KOd/killed, roll a d3; on a 3, take a 25% (of your original score) penalty to your score.

If there's a tie (unlikely), put the tied characters in the center of the board and have them trade punches (close combat attacks) until last man (woman, icthioid, etc.) standing.

So In your system, Han shoots first?

Actually, what you've written sounds more like Mal & Co in an Alliance bar on Unification Day.
"Them's not kosherized rules!"

Yeah, with the full version of the scenario, including stuff like representative stats, bar layouts, and special conditions and civilians (like the cops arriving, or the bartender pulling out a double barrel shotgun at some point), it have a much more Western feel like Firefly.

So, yeah, any genre where you get the seedy bar trope should be able to accommodate this type of bar fight.

If it fits your concept, I also created a "character" system where players roll a character type (a simple d6 chart); the character type (as "gambler," "pilot," "bounty hunter," etc.) is seeking to win specific items such as spacecraft, helmets, weapons, etc. (the "faces" of the credit bars show these), and gets extra points for doing so. Not sure it works (haven't tested this at all), but it might fit your scenario, so that IF Han has won his favorite blaster, he can always shoot first, but if he doesn't have it, he never goes first.

I'm not sure I like that gambling aspect of this game, given the high luck and undecidability. I feel, if you are gambling/risking, it's better to have reasonably (not necessarily precisely) known odds, with significant spreads complimented by rising potential gains. If you're just laying money down on a set of (relatively) equally likely outcomes, it doesn't interest me.

So, as you may have guessed, I don't "gamble". For money, that is.

The idea of competing for bonuses, however, is always welcome. I think an aspect like specific bonuses wagered by players would work better for a Monte Banque type game. Faro is good and Old-Westy, though a round is a bit long as a precursor for a short skirmish.

I think something like that might work best in the "Epic Barfight" realm, where the fight spills out into the surrounding streets and most of the bonuses only work in certain areas. So you would have to spend effort to manipulate the fight into the area for your bonus, if you wanted to use it.