Simple multiplayer card game

There is a shuffled deck of cards and 2 (or more) players. Each person draws from the deck, revealing the card each time they draw (drawn cards are not returned to the deck). They may draw whenever they like but the deck must be played successively.

For a player to maximize their chances of drawing an Ace of Hearts, for example, when should they enter play and how many cards should they draw? Obviously, the more cards they draw the better their chances, but it also increases the chances of the player after them drawing an Ace of Hearts. They also cannot wait too late into the game or risk another player getting the card first. Ultimately the goal for any player is to draw the Ace of Hearts, but they also want to minimize the number of cards drawn or time spent playing.

Hope I have made sense. This may be more of a game theory question than math but maybe you can give me some ideas

What does "the deck must be played successively" mean? I assume that means no one can draw two cards in a row, as otherwise the optimal strategy is to draw all 52 cards immediately.

Since there is no limit, the object is to draw as many cards as possible, as the "winning card" has an equal chance of being in any position in the deck. The strategy is to go first, and then draw whenever you have the opportunity.

When I say successively, I mean that players cannot pick a card from the bottom and then the middle during their turn, for example. They must draw in original deck order, one after another. Not that it matters, since the deck is shuffled, but that covers nefarious possibilities such as a clumped shuffle or edge sorting.

Yes I expected someone would say "draw all cards immediately" but that's why I added that players must also minimize the number of cards played, i.e. each card drawn gives a penalty. If Player A draws 5 cards and gets an Ace of Hearts but Player B only draws 2 cards and gets an Ace of Hearts, Player B has the better score.

Why would you want to go first? If you draw the first card and it isn't an Ace of Hearts, the person after you has a higher chance of drawing an Ace of Hearts, not to mention that you have a penalty. I'm still trying to wrap my head around this but am thinking it should be classified an optimal starting/stopping problem. I mean, if you know that all the Clubs and Spades have been drawn already but no Hearts, then you know that your chances are much better. But yet you don't want to jump in and play too many cards and then have your opponent draw the Ace of Hearts.

I guess in my mind I have a parabola of where the peak probability of drawing an Ace of Hearts would be for the player, given the previous cards drawn.

Quote: lmaths

When I say successively, I mean that players cannot pick a card from the bottom and then the middle during their turn, for example. They must draw in original deck order, one after another. Not that it matters, since the deck is shuffled, but that covers nefarious possibilities such as a clumped shuffle or edge sorting.

Yes I expected someone would say "draw all cards immediately" but that's why I added that players must also minimize the number of cards played, i.e. each card drawn gives a penalty. If Player A draws 5 cards and gets an Ace of Hearts but Player B only draws 2 cards and gets an Ace of Hearts, Player B has the better score.

Why would you want to go first? If you draw the first card and it isn't an Ace of Hearts, the person after you has a higher chance of drawing an Ace of Hearts, not to mention that you have a penalty. I'm still trying to wrap my head around this but am thinking it should be classified an optimal starting/stopping problem. I mean, if you know that all the Clubs and Spades have been drawn already but no Hearts, then you know that your chances are much better. But yet you don't want to jump in and play too many cards and then have your opponent draw the Ace of Hearts.

dude, how could "Player A draws 5 cards and gets an Ace of Hearts but Player B only draws 2 cards and gets an Ace of Hearts" if they are drawing from the same deck.

Quote: lmaths

There is a shuffled deck of cards and 2 (or more) players. Each person draws from the deck, revealing the card each time they draw (drawn cards are not returned to the deck). They may draw whenever they like but the deck must be played successively.

For a player to maximize their chances of drawing an Ace of Hearts, for example, when should they enter play and how many cards should they draw? Obviously, the more cards they draw the better their chances, but it also increases the chances of the player after them drawing an Ace of Hearts. They also cannot wait too late into the game or risk another player getting the card first. Ultimately the goal for any player is to draw the Ace of Hearts, but they also want to minimize the number of cards drawn or time spent playing.

Hope I have made sense. This may be more of a game theory question than math but maybe you can give me some ideas

you are giving 2 constraints:

1.maximize their chances of drawing an Ace of Hearts
2. minimize the number of cards drawn

and in my opinion these 2 constraints are contradictory.

Quote: andysif

dude, how could "Player A draws 5 cards and gets an Ace of Hearts but Player B only draws 2 cards and gets an Ace of Hearts" if they are drawing from the same deck.

Pretend it's 2 decks then, actual "winning card" is irrelevant as there is more than one but limited quantity. Sorry didn't specify earlier

I don't think it's contradictory at all. You want to maximize your chances but at the same time minimize the number of cards you draw. It's no different than maximizing utility with a limited shopping budget. You can do this with a hypergeometric distribution, but for our case the question is optimal time and quantity. It's only contradictory if you draw the entire deck. Like I said, if your opponent has drawn 30 cards with no Ace of Hearts, then you know right away that your chances are better because they have eliminated 30 non-winning cards. But the question is, should you jump in now or wait until they draw more?

Quote: lmaths

Pretend it's 2 decks then, the actual "winning card" is irrelevant as there is more than one but a limited quantity. Sorry I didn't specify earlier

I don't think it's contradictory at all. You want to maximize your chances but at the same time minimize the number of cards you draw. It's no different than maximizing utility with a limited shopping budget. You can do this with a hypergeometric distribution, but for our case the question is optimal time and quantity. It's only contradictory if you draw the entire deck.

you are totally screwed. please think clearly about what you are asking

if they are drawing from 2 decks, then these 2 conditions in your original question:

"when should they enter play and how many cards should they draw?"
"the more cards they draw the better their chances, but it also increases the chances of the player after them drawing an Ace of Hearts"

is not valid.

Quote: andysif

you are totally screwed. please think clearly about what you are asking

if they are drawing from 2 decks, then these 2 conditions in your original question:

"when should they enter play and how many cards should they draw?"
"the more cards they draw the better their chances, but it also increases the chances of the player after them drawing an Ace of Hearts"

is not valid.

No, I think your reading comprehension is off. Two decks combined together would not invalidate my statement at all.

If you draw X cards from finite deck(s) you have Y probability of drawing Z card.

However, this would NOT be the case for the player AFTER YOU because YOU HAVE REMOVED those cards from FINITE deck(s). They would have a HIGHER probability of drawing the Ace. This of course wouldn't be true if you could just draw all the cards before them. But, you can't. So the question is and has been, WHEN should you draw cards and HOW MANY should you draw? Given the constraint mentioned above, that the more cards you draw, the worse your penalty. This applies to the other player as well.

I suppose I am not being clear enough and because of that I will likely end up answering my own question :-)

Quote: lmaths

Yes I expected someone would say "draw all cards immediately" but that's why I added that players must also minimize the number of cards played, i.e. each card drawn gives a penalty. If Player A draws 5 cards and gets an Ace of Hearts but Player B only draws 2 cards and gets an Ace of Hearts, Player B has the better score.

I didn't see any mention of a penalty anywhere - and how can both players draw the Ace of Hearts, since you say, "Drawn cards are not returned to the deck"?

Quote: lmaths

Why would you want to go first? If you draw the first card and it isn't an Ace of Hearts, the person after you has a higher chance of drawing an Ace of Hearts, not to mention that you have a penalty.

Like I said - you didn't mention any penalty until after I replied. I was working under the assumption that the only real rule was, "Whoever draws the highest card wins."

You need to specify all of the rules, in detail (e.g. don't say "there's a penalty" - tell us what the penalty is) - then we can determine the strategy.

There's no real penalty, think of it as a constraint. It is imaginary because it is only an individual utility function. In fact, this is more of a multiobjective optimization problem. Maximize one variable while minimizing another.

For the sake of the scenario, say each card drawn gives a score of -1 but each Ace of Hearts (or really, any rare arbitrary card) gives a score of 10. Now you would probably calculated the expected values and on and on but that is not what I am looking for, besides it is partially of issue due to the finite deck(s).

I know how to calculate the probability using a hypergeometric distribution. If X cards have already been drawn without the winner, then I know the probabilities of the next cards being a winner. But how do I get to the point of those X cards? You can't just say OK when the probability is 10% I will play. Because when you stop, the other player's chances are higher. Yet if you don't stop you will have significant penalty. If you start too early you also get higher penalty. And if you wait too long to begin play then the opponent will get the winning card first. There has to be a parabola, I am looking for the peak optimal point. Do I enter play when 15 cards have been played with no Ace, and do I play 5, for example? Do I just draw as many cards as my opponent? Should I wait until an event occurs where all but the last card have been drawn, so I KNOW the last card is an Ace of Hearts? This would guarantee the constraint, but that's the extreme of it.

The best analogy I can give is Blackjack card counting with each bet diminishing in utility (not same as EV). You wouldn't want to play the whole game; optimally, you would wait until the deck was rich with 10s. At the same time you don't want to bet too much even when the deck is rich or you will go broke. So you would optimize your time to enter the game as well as optimize how much you bet so as to avoid loss. I am trying to apply this more generally, and let me stress that this is finite. Unlike Blackjack where you can just keep coming back to the deck again and the cards will always be the same. So pretend you can only play this specific game for a finite number of trials in a discrete space (which is why hypergeometric comes in handy).

If I'm confusing you all, that's OK, try not to take this thought exercise too seriously, just something I have been trying to get around.

It depends on how many other people are playing. It probably depends on their strategy as well.

If I draw one card and my opponent draws the other 51, I win - I have the higher score (either -1 or 10, vs their -51 or -40).

I'll let you do your own homework from there.

I still think the question is flawed somehow

If the optimal strategy is to draw x cards or wait or whatever, then BOTH player will want to do that. Since there is no rule on who got to draw first or what happen if neither wants to draw therefore there will be no game.

Quote: Dieter

It depends on how many other people are playing. It probably depends on their strategy as well.

If I draw one card and my opponent draws the other 51, I win - I have the higher score (either -1 or 10, vs their -51 or -40).

I'll let you do your own homework from there.

Yeah, am thinking this is game theory-ish. The value 10 for Ace of Hearts was just example, in reality it would be high enough that players wouldn't just stop playing in the event that even winning the Ace couldn't put them ahead. So the goal is survival but also highest score. It is conflicting but at the same time I feel like there is some point where it would be optimal to play or not, and how much to play, given the cards drawn and the cards remaining.

As for other players, some would just play randomly without any knowledge of the prior cards drawn. That's the difficulty of this I think. Not everyone would use the same strategy; maybe they believe the last half of the deck is better "just because," for example.